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MODELING AND CHARACTERIZATION OF ELECTRICAL RESISTIVITY
OF CARBON COMPOSITE LAMINATES
by
Hong Yu
A dissertation submitted to the Faculty of the University of Delaware in partial
fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering
Winter 2018
© 2018 Hong Yu
All Rights Reserved
MODELING AND CHARACTERIZATION OF ELECTRICAL RESISTIVITY
OF CARBON COMPOSITE LAMINATES
by
Hong Yu
Approved: __________________________________________________________
Ajay K Prasad, Ph.D. Chair of the Department of Mechanical Engineering
Approved: __________________________________________________________ Babatunde Ogunnaike, Ph.D. Dean of the College of Engineering
Approved: __________________________________________________________ Ann L. Ardis, Ph.D.
Senior Vice Provost for Graduate and Professional Education
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________
Suresh G. Advani, Ph.D. Professor in charge of dissertation
I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Dirk Heider, Ph.D.
Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________
Erik T. Thostenson, Ph.D. Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Michael Keefe, Ph.D. Member of dissertation committee
iv
I would like to express my special appreciation and thanks to my advisor Dr.
Suresh Advani and co-advisor Dr. Dirk Heider, for the continuous support, patience,
and enthusiasm they have provided during my Ph.D. journey. Dr. Advani’s guidance
helped me in all the time of research and writing of this thesis. I could not have
imagined having a better advisor and mentor for my Ph.D study. His advice on both
research as well as on my career have been priceless. Dr. Heider has been supportive
and I am grateful to his scientific advice and knowledge and many insightful
discussions and suggestions, especially on experimental designs. I hope that I could be
as lively, enthusiastic, and energetic as him.
Besides my advisors, I would like to thank the rest of my thesis committee: Prof.
Erik T. Thostenson, and Prof. Michael Keefe for serving as my committee members. I
am also thankful to my collaborators from industry: Dr. Henry Zhang, and Dr. Kyu-
Pyung (Gabriel) Hwang, for their insightful discussions.
I would like to thank many colleagues who worked with me during my time in
Delaware. I had the privilege to work with Hang Yu, Gaurav Pandey and Jiayin Wang.
I was lucky to have Jessica Sun for her help in the lab. Also, I am grateful for the
support and friendship from my office mates in CCM123.
I would also like to thank the administrative staff of the Mechanical Engineering
Department: Lisa Katzmire, Ann Connor and Letitia Toto and Center for Composite
Materials: Corinne Hamed, Robin Mack, Penny O’Donnell, Therese Stratton and
Megan Hancock.
ACKNOWLEDGMENTS
v
I especially thank my family for their huge support and motivation throughout
my entire study and life. My hard-working parents have sacrificed their lives for my
sisters and myself and provided unconditional love and care.
vi
LIST OF TABLES ......................................................................................................... xi LIST OF FIGURES .......................................................................................................xii
ABSTRACT..................................................................................................................xxi
Chapter
1 INTRODUCTION .............................................................................................. 1
1.1 Background ................................................................................................ 1
1.1.1 CFRP used in aircraft industry....................................................... 1 1.1.2 Lightning strike to CFRP used on aircraft structures..................... 2
1.1.3 Common practice for lightning strike protection ........................... 4
1.2 Research Motivation and Objectives ......................................................... 4
1.2.1 Research motivation....................................................................... 4
1.2.2 Research objectives........................................................................ 6
1.3 Structure of Dissertation ............................................................................ 7
2 3D MICROSTRUCTURE BASED RESISTOR NETWORK MODEL ............ 9
2.1 Introduction ................................................................................................ 9 2.2 Electrical Conduction Mechanisms of CFRP .......................................... 11
2.2.1 Review of existing models for electrical conduction of unidirectional CFRP..................................................................... 11
2.2.2 Electrical conduction mechanism in longitudinal direction......... 14 2.2.3 Electrical conduction mechanism in transverse direction............ 15 2.2.4 Overview of model formulation................................................... 16
2.2.5 Numerical implementation........................................................... 18 2.2.6 Parameterization of 3D fiber geometry........................................ 20
2.2.6.1 Generation of 2D fiber arrangement ............................. 21
2.2.6.2 Parameterization of fiber waviness ............................... 23
TABLE OF CONTENTS
vii
2.2.7 Estimation of contact resistance................................................... 26
2.2.7.1 Hertz contact theory ...................................................... 27 2.2.7.2 Electrical constriction resistance between two
conductors ..................................................................... 28 2.2.7.3 Estimation of contact force ........................................... 29
2.3 Model Convergence and Validation ........................................................ 31
2.3.1 Model convergence ...................................................................... 31 2.3.2 Comparison between simulations and literature data .................. 34
2.4 Sensitivity Analysis ................................................................................. 36
2.4.1 Resistivity as function of processing pressure ............................. 37 2.4.2 Sensitivity study for all relevant material and process parameters39
2.5 Summary and Conclusions ...................................................................... 41
3 EXPERIMENTAL INVESTIGATION OF THROUGH-THICKNESS
RESISTIVITY OF CARBON FIBER TOWS .................................................. 43
3.1 Introduction .............................................................................................. 43 3.2 Experimental Setup and Methodology..................................................... 44
3.2.1 Setup............................................................................................. 44 3.2.2 Specimen preparation and experimental procedure ..................... 46
3.2.3 Characterization of fiber volume fraction .................................... 48 3.2.4 Characterization of fiber waviness............................................... 49
3.3 3D Resistor Network Model Implementation .......................................... 52
3.4 Experimental Results and Model Comparison ........................................ 54
3.4.1 Effect of fiber volume fraction..................................................... 55
3.4.2 Effect of fiber sizing .................................................................... 57 3.4.3 Effect of debulking....................................................................... 61
3.5 Conclusions .............................................................................................. 64
4 MODELING ELECTRICAL CONDUCTION BEHAVIOR OF COMPOSITE LAMINATES CONSIDERING RESIN-RICH LAYER .......... 66
4.1 Introduction .............................................................................................. 66 4.2 Equivalent Fiber Bundle Model ............................................................... 68 4.3 Angle-ply Model ...................................................................................... 70
viii
4.4 Multi-ply Model with Resin Rich Layer.................................................. 74
4.4.1 Number of inter-ply connections ................................................. 74 4.4.2 Inter-ply contact resistance .......................................................... 77
4.4.2.1 Composition of contact resistance ................................ 77 4.4.2.2 Critical processing pressure .......................................... 81 4.4.2.3 Constriction resistance for fibers with direct contact.... 82
4.4.2.4 Tunneling resistance between fibers with small separation distance ........................................................ 82
4.4.3 Construction of resistor network.................................................. 85
4.5 Model Convergence Tests........................................................................ 88 4.6 Model Validation ..................................................................................... 91
4.6.1 Through-thickness resistivity compared with reported experimental data for CFRP......................................................... 91
4.6.2 Resistor network model compared with FEM, analytical and experimental results ..................................................................... 95
4.6.3 Parametric study of the impact of resin rich layer ..................... 102
4.6.4 Impact of inter-ply connectivity on resistivity in the three principal directions..................................................................... 102
4.7 Summary and Conclusions .................................................................... 104
5 MODELING HIGH ELECTRIC CURRENT IMPACT ................................ 107
5.1 Introduction ............................................................................................ 107
5.2 Current Concentration at Micro-Scale Level ......................................... 108
5.2.1 Current concentration within carbon fibers ............................... 108
5.2.2 Current concentration at contact spots ....................................... 113
5.3 Joule Heating Effect............................................................................... 116
5.3.1 Within carbon fibers................................................................... 116
5.3.2 At contact spots .......................................................................... 118
5.4 Temperature Dependent Electrical Resistivity ...................................... 120
5.5 Temperature and Electric Field Induced Material Degradation............. 123
5.5.1 Thermal breakdown ................................................................... 124 5.5.2 Electric breakdown .................................................................... 124
ix
5.6 Modeling Approach ............................................................................... 126
5.6.1 Model overview ......................................................................... 126 5.6.2 Thermal-electrical RC circuit..................................................... 127
5.7 Results and Discussions ......................................................................... 133
5.7.1 Variations among simulations using same modeling parameters133 5.7.2 Impact of resin-rich layer ........................................................... 135
5.7.3 Parametric studies on the impact of model parameters.............. 138
5.8 Summary and Conclusions .................................................................... 142
6 EXPERIMENTAL INVESTIGATION OF HIGH CURRENT IMPACT ..... 144
6.1 Electrical Characterization of Dry Fiber Tows ...................................... 144
6.1.1 Materials and preparation........................................................... 145
6.1.2 Experimental setup..................................................................... 145 6.1.3 Typical resistance response........................................................ 147
6.1.4 Influence of processing pressure................................................ 149 6.1.5 Resistivity change after repetitive current application............... 152
6.2 Electrical Characterization of Cured Composite Laminates under
Medium-High Currents .......................................................................... 157
6.2.1 Materials and preparations ......................................................... 157
6.2.2 Setup and specimen fixtures ...................................................... 158
6.2.2.1 Specimen fixture for resistance characterization in the in-plane direction ........................................................ 158
6.2.2.2 Specimen fixture for resistance characterization in the through-thickness direction......................................... 159
6.2.3 Current waveform ...................................................................... 161 6.2.4 Typical resistance response (first observations) ........................ 161 6.2.5 In-plane resistance compared with simulation results ............... 163
6.2.6 Through-thickness resistance compared with simulation results165 6.2.7 Impact of current duration.......................................................... 177
6.2.8 Residue resistivity change after repetitive current applications . 180
6.3 Resistance Response under Simulated Lightning Impulses................... 183
6.3.1 Descriptions of the experimental data........................................ 183
6.3.2 Comparisons between simulation results and experimental data186
x
6.3.3 Residue resistivity change after repetitive current applications . 188
6.4 Summary and Conclusions .................................................................... 191
7 CONCLUSIONS, CONTRIBUTIONS, AND FUTURE WORK .................. 193
7.1 Conclusions ............................................................................................ 193 7.2 Unique Contributions ............................................................................. 195 7.3 Future Work ........................................................................................... 196
REFERENCES ........................................................................................................... 200
Appendix
A COPYRIGHT PERMISSIONS....................................................................... 206
xi
LIST OF TABLES
Table 2-1 Parameters used in the 3D resistor network model ........................................ 18
Table 2-2 Properties of HTA-7 fiber............................................................................... 32
Table 2-3 Resistivity of UD CFRP reported by Abry [31] ............................................. 35
Table 3-1 Specimen parameters ...................................................................................... 47
Table 3-2 Fiber waviness and Gutowski fiber volume fraction terms for five fiber types............................................................................................................. 50
Table 4-1 Properties of IM7 and T700 carbon fiber ....................................................... 78
Table 4-2 Properties of HTA-7 fiber and model parameters .......................................... 89
Table 5-1 Analogy between thermal and electrical conduction.................................... 128
Table 5-2 Model parameters ......................................................................................... 133
Table 6-1 Properties of fiber groups for high current density tests .............................. 145
Table 6-2 Current waveforms used in the repetitive current application tests ............. 152
Table 6-3 Specimen layup............................................................................................. 165
Table 6-4 Current waveforms used in the repetitive current application tests. ............ 180
Table 6-5 Current durations in the 33 cycles. ............................................................... 182
Table 6-6 Parameter values used for modeling resistivity of [0/90]2s AS4 laminate. . 186
Table 6-7 Desired peak voltage in each cycle............................................................... 188
xii
Figure 1.1 More than 50% (by weight) of Boeing 787 is composed of carbon composites in various forms ........................................................................ 2
Figure 1.2 Typical lightning current waveforms as defined in the MIL-STD-464 standard. Reproduced with permission [8] .................................................. 3
Figure 2.1 Conduction mechanisms of CFRP laminates in three primary directions. a) X direction: intrinsic fiber resistivity and fiber volume fraction; b) random contacts between carbon fibers forming conductive paths; c) Z direction:
fiber contacts within one ply and limited connections between laminas due to resin rich interface. The blue plates represent electrodes in various
directions. ................................................................................................... 10
Figure 2.2 Schematic representation of conductive path created by fiber-to-fiber contacts. Reproduced with permission [34] ............................................... 15
Figure 2.3 Flow chart of model formulation: 3D microstructure is generated from 2D fiber arrangement and fiber waviness parameter. Reproduced with
permission [35] .......................................................................................... 17
Figure 2.4 Flow chart of the algorithm to formulate and solve 3D resistor network. Reproduced with permission [35] .............................................................. 20
Figure 2.5 Schematic representations of various fiber arrangements: (a) hexagonal packing; (b) square packing; (c) random packing; (d) from micrograph
using image processing techniques. Reproduced with permission [35] .... 22
Figure 2.6 Schematic representation of fiber waviness. Reproduced with permission [25] ............................................................................................................. 23
Figure 2.7 Schematic representation of resistor network model: fractional fiber lengths in between contacts and contact resistances form the resistor network model. 2D presentation is shown here for clarity while the model is 3D.
Reproduced with permission [35] .............................................................. 26
Figure 2.8 Schematic illustration of load sharing among carbon fibers. Applied force is
assumed to be shared evenly by the total number of contact points at each layer............................................................................................................ 30
LIST OF FIGURES
xiii
Figure 2.9 Model convergence as a function of packing arrangement. Resistivity
normalized with value obtained for 6000 nodes using the fiber positions obtained from the micrograph. Reproduced with permission [35]. ........... 33
Figure 2.10 Comparison of simulation results and reported experimental data. Reproduced with permission [35] .............................................................. 36
Figure 2.11 Transverse resistivity as a function of processing pressure. Reproduced with
permission [35] .......................................................................................... 38
Figure 2.12 Fiber resistance drops with increasing processing pressure as higher
processing pressure yields higher fiber volume fraction, reducing inter-fiber spacing and thus shortening the length of each fiber section. Reduction in contact resistance results from higher contact force due to
processing pressure. Reproduced with permission [35] ............................ 39
Figure 2.13 Sensitivity study of model parameters. Reproduced with permission [35] . 41
Figure 3.1 Experimental setup with its schematic for characterization of through-thickness resistivity of carbon fiber tows................................................... 46
Figure 3.2 Fiber volume fraction calculations using data from Instron and from image
processing................................................................................................... 49
Figure 3.3 Compaction data for all five fiber types ........................................................ 52
Figure 3.4 Typical dataset recorded during the compression process for Fiber A ......... 54
Figure 3.5 Through-thickness resistivity of unsized Fiber A and unsized Fiber B as function of fiber volume fraction. Model describes experimental data well
at volume fraction below 60%. Fiber waviness term beta is 620 for Fiber A, and 365 for Fiber B, as listed in Table 3-2............................................ 56
Figure 3.6 Comparison of contact resistance and fiber resistance for unsized Fiber A and Fiber B. Both fiber resistance and contact resistance of Fiber B are larger than Fiber A in the present study. .................................................... 57
Figure 3.7 Experimental resistivity and model results of sized fibers (Fiber C, D, and E). Fiber C and D have same amount sizing (1%) and demonstrate similar
resistivity, while Fiber E with less sizing (0.25%) demonstrates smaller resistivity. ................................................................................................... 59
Figure 3.8 Comparison of contact resistance and fiber resistance for sized fibers.
Significant drop in contact resistance Rc in Fiber E is observed, which may
xiv
due to the breakage of the thin sizing layer on Fiber E at higher
pressures..................................................................................................... 60
Figure 3.9 (a) Fiber A compaction data for multiple debulking cycles and (b) β for first
three compaction cycles for all five fiber types examined. For each fiber type, 5 specimens were fabricated and tested and the average β terms and their variations are plotted. ......................................................................... 62
Figure 3.10 Through-thickness resistivity during the first three debulking cycles for 5 fiber types. The figures in the top row represent unsized fibers, while
figures in the bottom row represent sized fibers: (a) Fiber A (unsized); (b) Fiber B (unsized); (c) Fiber C (sized); (d) Fiber D (sized); (e) Fiber E (sized)......................................................................................................... 64
Figure 4.1 Demonstration of resin rich layer; inter-lamina boundaries between plies are noticeable with carbon fibers separated by excessive resin. Reproduced
with permission [47] .................................................................................. 67
Figure 4.2 Schematic illustration of fiber bundle model. A fiber bundle can be represented with 3 resistors whose values can be calculated from the
resistor network model with current injected from three primary directions respectively. ............................................................................................... 69
Figure 4.3 UD lamina represented by fiber bundle model. Each line section represents a fiber bundle, instead of single fiber as in the previous resistor network model discussed in Chapter 2. ................................................................... 70
Figure 4.4 For an angle ply, there is an angle θ between the material coordinate (𝒖 − 𝒗 )
and the structure coordinate (𝒙 − 𝒚). ........................................................ 71
Figure 4.5 Schematic drawing of a minimum bounding rectangle (MBR) for a 45 ∘ ply............................................................................................................... 71
Figure 4.6 Schematic illustration of the workflow for constructing a 3D resistor network for an angle ply .......................................................................................... 73
Figure 4.7 Ply orientations in a multi-ply CFRP laminate. Carbon fiber tows are
schematically shown with black lines. ....................................................... 74
Figure 4.8 Reduction in number of contacts due to fiber undulation. (a) contacts between fibers in [0-90] layup assuming fibers are straight; (b) reduced
contacts between fibers in [0-90] layup considering fiber undulation; (c) contacts between fibers in [0-45] layup assuming fibers are straight; (d)
contacts between fibers in [0-45] layup considering fiber undulation. ..... 75
xv
Figure 4.9 Sizing thickness as a function of sizing weight fraction and fiber radius ..... 79
Figure 4.10 Schematic illustration of the parts of contact resistance. (a) tunneling resistance is dominant when a thin resin layer exists between carbon fibers;
(b) constriction resistance becomes dominant if direct contact between carbon fibers is formed. ............................................................................. 80
Figure 4.11 Critical processing pressure for fibers in direct contact. Below critical
pressure, thin sizing layer exists between carbon fibers, and the dominant conduction mechanism is tunneling conduction. Above critical pressure,
direct contact between carbon fiber becomes the dominant conduction mechanism. ................................................................................................ 82
Figure 4.12 Tunneling resistance as function of separation distance ............................. 84
Figure 4.13 Combined constriction resistance and tunneling resistance as function of processing pressure. Below the critical pressure (denoted with the red
dashed line), contact resistance is calculated with the tunneling resistance formula, while constriction resistance formula is used under pressure higher than the critical pressure. ................................................................ 85
Figure 4.14 Schematic illustration of the workflow for constructing a 3D resistor network for multi-ply CFRP laminate with resin rich layer. [0/0] layup is
presented for clarity; the model can also consider a random ply orientation..................................................................................................................... 86
Figure 4.15 Demonstrations of model for multi-ply laminate. For the sake of simplicity,
only one layer of resistors is plotted for each ply, while in real calculations, multiple layers of resistors are used for each ply. (a) [0/90] two ply
laminate with 60% inter-ply connectivity; (b) [0/45] two play laminate with 40% inter-ply connectivity................................................................. 86
Figure 4.16 Convergence tests for two cases: (a) connectivity = 0.1; (b) connectivity =
1.0. Large variations are observed for resistivity in Z direction, especially for laminate with resin-rich interface (inter- lamina connectivity = 0.1) ... 90
Figure 4.17 Schematic illustrating unconnected fiber at the edge of resistor network. . 91
Figure 4.18 Comparison between simulation results and reported experimental data from Abry [51] ........................................................................................... 94
Figure 4.19 Cross-section of the unidirectional specimens. (a) Vf=0.43; (b) Vf=0.59. Reproduced with permission [51] .............................................................. 95
xvi
Figure 4.20 Schematic illustration of specimen aspect ratio (λ). λ is defined as the length to width ratio of a laminate plate, where length direction is aligned
with the test direction (direction in which current/ voltage is applied). .... 96
Figure 4.21 Experimental and theoretical results as function of aspect ratio (λ) and the
fiber direction (θ) of the UD preform for thickness h = 0.18 mm. Reproduced with permission [43] .............................................................. 98
Figure 4.22 Results from the virtual tests. Solid represent results from FE model lines (denoted with “FEM” in legend), while dashed lines represent resistor
network model (denoted with “ResNet” in legend). The solid black line
denotes the “critical aspect ratio” 𝜆𝑐𝑟 , and the two dashed black lines
denote the rough boundary for 𝜆 ≪ 𝜆𝑐𝑟 and 𝜆 ≫ 𝜆𝑐𝑟 respectively. ....... 100
Figure 4.23 Current streamline plot from FE model for an angle ply with 45 ∘ fiber
orientation. (a) aspect ratio 𝜆 = 0.2, representing conduction in Region II;
(b) aspect ratio 𝜆 = 5, representing conduction in Region III. ................ 102
Figure 4.24 Impact of inter-lamina interface. Three levels of inter-lamina connectivity are demonstrated with the inserts. Resistivity in Z direction is sensitive to
changes in inter-lamina connectivity especially in lower connectivity range, while the influence of inter-ply connectivity term is negligible on
resistivity in the X and Y directions. ....................................................... 104
Figure 5.1 Schematic illustration of current concentration at intra-ply contact points. (a) current applied to top surface of CFRP laminate; (b) RVE containing two
contacting fiber sections; (c) current path through carbon fibers and contact points. Current is concentrated at the contact points due to small
contact area compared to carbon fiber cross section area. ....................... 111
Figure 5.2 𝐾𝑓𝑖𝑏𝑒𝑟 as function of fiber volume fraction 𝑣𝑓 and on fiber waviness term
𝛽. .............................................................................................................. 113
Figure 5.3 Kcontact as function of a )processing pressure (other parameters are fixed:
𝑉𝑓 = 0.55, 𝐸 = 273 𝐺𝑃𝑎, 𝛽 = 400,𝑅𝑓𝑖𝑏𝑒𝑟 = 3.5 𝜇𝑚); and b) fiber
waviness term β (other parameters are fixed: 𝑉𝑓 = 0.55,𝐸 =273 𝐺𝑃𝑎,𝑃 = 800,000 𝑃𝑎, 𝑅𝑓𝑖𝑏𝑒𝑟 = 3.5 𝜇𝑚). ..................................... 115
Figure 5.4 Typical current waveforms: a) constant current; and b) current ramp. ....... 117
Figure 5.5 Temperature at the contact spot according to Equation 5.17 plotted as a function of the voltage drop over the contact region for three different
ambient temperatures ............................................................................... 120
xvii
Figure 5.6 Arrhenius plot for IM7 and T700 carbon fiber. Activation energy can be back
calculated from the slope of the linear fit. ............................................... 122
Figure 5.7 Activation energy in three primary directions for typical carbon composite
laminates. Activation energy has the unit of micro electronvolt (meV), the amount of energy gained by the charge of a single electron moved across an electric potential difference of one volt, and is defined as the minimum
amount of energy required to trigger a temperature-accelerated failure mechanism. .............................................................................................. 123
Figure 5.8 ON-OFF model for resin breakdown .......................................................... 125
Figure 5.9 Workflow for implementing the 3D resistor capacitor network work with thermal-electrical coupling ...................................................................... 127
Figure 5.10 Coupled thermal electrical resistor capacitor network model. .................. 129
Figure 5.11 Flowchart showing the Coupling between electrical and thermal conduction
networks ................................................................................................... 131
Figure 5.12 Variations of simulated through-thickness resistivity using same model parameters ................................................................................................ 134
Figure 5.13 Resistivity change over time. Laminate with small inter-lamina connectivity (resin-rich interface) undergoes quicker and larger resistivity drop in
through-thickness direction. Sudden drop in resistivity around 10ms can be explained by the localized heating ........................................................... 136
Figure 5.14 Temperature profile at selected location: contact between carbon fibers, at
fiber-fiber contacts, and at inter-lamina connection points for two types of composites: a) with resin-rich interface and b) without resin-rich interface.
.................................................................................................................. 138
Figure 5.15 Parametric study on fiber waviness term and activation energy. .............. 140
Figure 5.16 Impact of inter-ply resistance. A large inter-ply resistance not only affects
the absolute resistivity value before current application, but also changes the resistivity reduction after current application. ................................... 141
Figure 6.1 Schematic illustration of electrical characterization apparatus. .................. 146
Figure 6.2 Typical resistance response for dry fiber tow under a voltage ramp. Voltage ramp and the corresponding current response are also plotted. ............... 148
xviii
Figure 6.3 Resistance response for unsized IM7 and sized T700SC fiber tows: a)
unsized IM7 fiber tows see less than 5% drop in resistance; b) sized T700SC fiber tows yields larger resistance drops (18%) at the end of the
current waveform. .................................................................................... 149
Figure 6.4 Resistivity response under various load amount for unsized and sized fibers. Resistivity is normalized with the first measured value. a) unsized IM7
fiber tows: no noticeable change in resistivity under high compressive force; a) sized T700SC fiber tows: drop in resistivity decreases with the
increase of compressive load. Drop in resistivity is still noticeable (~15%) even under high compressive force (1000 N) .......................................... 150
Figure 6.5 Through-thickness resistivity of (a)unsized IM7 and (b)sized T700SC carbon
fiber tows after repetitive current applications. After each test, resistivity is partially recovered. Smaller residue resistivity change can be observed for
unsized IM7 fibers, while large change (~90%) in residue resistivity can be observed for sized T700SC fibers. ........................................................... 154
Figure 6.6 Electrical response of sized T700SC in the first three 100ms current cycles.
Most significant difference is observed between the first and second cycle, while subsequent cycles demonstrate little difference in current and
resistance response. .................................................................................. 156
Figure 6.7 Polished specimen surface. Carbon fibers are exposed for better contact with the electrodes............................................................................................ 158
Figure 6.8 Specimen fixture for mounting composite specimens in the in-plane tests 159
Figure 6.9 Specimen fixture for through-thickness tests .............................................. 160
Figure 6.10 Electrical response for 4-ply T700 CFRP, in fiber length direction (X) and through-thickness direction (Z) respectively. .......................................... 162
Figure 6.11 Comparison between simulated and measured resistivity response under
high current density for [0]4 IM7-977/3 cured composites. The green vertical line and arrow in (b) denotes the range of current density used in
the tests..................................................................................................... 164
Figure 6.12 Microscopic image of the cross-section of specimen E. Thermoplastic powers were added between plies, creating resin-rich layers. ................. 166
Figure 6.13 comparison between simulation results and experimental data for IM7 specimens without thermoplastic powders as listed in Table 6-3. (a)
xix
Specimen A with [0]2 layup; (b) Specimen B with [0/45] layup; (c)
Specimen C with [0]8 layup. .................................................................... 168
Figure 6.14 Characterization of resistance response for (a) 02 Specimen D, and (b) 08 Specimen E. The 8-ply specimens show larger resistivity than the 2-ply specimens, indicate even worse inter-ply contact quality induced from the
increased difficulties to get the excess resin out of the thicker laminates 170
Figure 6.15 Simulation results considering only the effect of temperature dependent
material properties. Predicted resistivity drops are smaller than observed from experiments. .................................................................................... 171
Figure 6.16 Limitations of a model considering only temperature dependent resistivity,
but not considering resin breakthrough. Maximum resistivity drop is only about 30% even for a small inter-ply connectivity (5%). ........................ 173
Figure 6.17 Micrograph showing the crack found in Specimen E after high current application ................................................................................................ 174
Figure 6.18 (a) CFRP model without electrical conduction contributed by resin matrix;
(b) CFRP model considering both direction carbon to carbon contact, and tunneling conduction through thin resin layer. ........................................ 175
Figure 6.19 Parametric studies on the effect of resin breakthrough on resistivity. ...... 177
Figure 6.20 Current, voltage, resistance and load recordings during application of current waveform with three durations: (a) 100ms, (b) 1000ms, and (c)
2000ms. Blue, orange, grey and gold lines represent voltage [V], current [A], normalized resistivity and normalized load respectively. ................ 180
Figure 6.21 Accumulated resistivity response for two types of tests: (a) same current waveform with 100ms current duration applied in the two cycles; (b) current duration is 100ms in the first cycle, and 1000ms in the second
cycle. ........................................................................................................ 181
Figure 6.22 Residue resistivity change as the test cycle progresses. A 7% reduction in
residual resistivity after applying a high electric current for 100ms for the first time, further reducing to 91% after 16 cycles. Total reduction in resistivity is about 35% after the last current cycle, where excessive
heating is observed................................................................................... 183
Figure 6.23 Typical voltage (a), current (b), and resistivity (c) response of 8-ply 1’’ by
1’’ IM7/-773 composite specimen. It also represents typical voltage, current, and resistivity response for other carbon composites tested in this
xx
study. Most significant changes in resistivity normally happen in the first
cycle. ........................................................................................................ 186
Figure 6.24 Comparison between simulations results and experimental data for a 8-ply
AS4 composite laminate with layup of [0/90]2s, and size of 1 inch by 1 inch. Experimental voltage waveform is extracted and used as input in the model. Five simulations are run using the same model parameters. ....... 188
Figure 6.25 Accumulated resistivity response during repetitive current applications. Irreversible resistivity reduction (denoted by blue arrows) is significant in
the first cycle and decreases in the following cycles, while reversible resistivity change (denoted by red arrows) is similar in all cycles. ......... 190
Figure 7.1 Modeling broken fiber with resistor network .............................................. 198
Figure 7.2 Schematic illustration of Joule heating induced damage propagation in CFRP ........................................................................................................ 199
xxi
In the past few decades, composite materials especially carbon fiber reinforced
polymers (CFRP) have been widely used as structural materials for its high strength to
weight ratio, tailorable properties, and excellent corrosion properties. Applications that
require better understanding of the electrical properties of CFRP laminates include
carbon fiber assisted heating during composites manufacturing, self-sensing of damage
of composite structures, integrated electromagnetic shielding, and lightning strike
protection. Accurate predictive model describing the electrical conduction behavior of
CFRP laminates is the key for them to be used for such applications.
Different approaches have been explored to model the electrical conduction of
CFRP under various current conditions. A comprehensive literature review revealed
that most methods used to model electrical conduction of CFRP fail to capture the
impact of micro-structure of CFRP, especially the fiber-fiber contact, and resin-rich
layer between plies, which can drastically change the conduction pattern.
The aim of this dissertation work is to develop a model that capture key
electrical conduction mechanisms of CFRP, which address the impact of the micro-
structure and geometrical parameters. The model is constructed in a modular fashion by
validating the model with experimental validation after the addition of each key
mechanism module. First, the model constructs a resistor network framework for
describing electrical conduction behavior of UD laminas and fiber tows subjected to
ABSTRACT
xxii
low DC currents. The model is validated with reported experimental results, and by
characterization of resistivity of dry carbon fiber tows.
The next module investigates the specific features of a multi-ply laminate such
as: varying ply orientation, existence of resin-rich layer, and dependence on geometric
parameters that influence the local resistivity. A meso-scale fiber bundle model is
proposed to strike a balance between the level of details modeled and the computational
cost. Influence of the resin-rich layer is described with an inter-ply connectivity term.
Expressions for estimating contact resistance from multiple sources including direct
fiber-fiber contact and tunneling resistance across thin resin layer are introduced. The
refined model is compared against experimental results and finite element model. A
parametric study is conducted to investigate the impact of geometrical parameters.
Finally, the dissertation work investigates the impact of high current density
both numerically and experimentally. Simplified analytical model examining the impact
of localized Joule heating revealed that current concentrations due to microstructure
constraints can introduce excessive Joule heating at contact spots. Thus, it is vital not to
under-estimate the temperature rise at contact points, even at seemingly small overall
applied currents. Based on these analysis, the model is further refined with the
implementation of the module that introduces Joule heating. Both reversible change in
resistivity such as temperature dependent resistivity and irreversible change such as
thermal and electric degradation of resin matrix is considered.
Electrical characterization under high current density is carried out for dry fiber
tows and cured composites experimentally. The contributions of reversible and
irreversible resistivity change are identified with carefully designed repetitive current
tests. It is found that for dry fiber tows with sizing and for cured composites, thermal
xxiii
breakdown of the thin resin/sizing layer contributes significantly to the nonlinear
conduction behavior under high current density. The developed model captures
important characteristics of the electrical conduction behavior when compared with
experimental results. Possible explanations are offered for cases and regions where the
model shows discrepancies with experimental results. This model should prove useful
to address and design and fabricate composite components in which electric and thermal
conductivity play a key role in defining their functional properties.
1
INTRODUCTION
1.1 Background
1.1.1 CFRP used in aircraft industry
In the past few decades, composite materials especially carbon fiber reinforced
polymer (CFRP) have been widely used as structural materials for its high strength-
weight ratio, tailorable properties, and excellent corrosion properties. In particular,
composites draw special attention from aircraft industries, and the percentage of
composites on aircraft has steadily increased in the past 30 years. According to a recent
NASA report [1], applications of carbon composites on aircraft have shifted from non-
critical parts such as stabilizer to critical structural parts such as main frame.
The Boeing 787 has more than 50% carbon composites by weight, which
includes major load carrying structures. Almost the entire outer surface of Boeing 787
uses CFRP (as shown in Figure 1.1), and is subjected to direct exposure to lightning.
Chapter 1
2
Figure 1.1 More than 50% (by weight) of Boeing 787 is composed of carbon
composites in various forms [2]
1.1.2 Lightning strike to CFRP used on aircraft structures
Commercial aircraft as well as military aircraft are designed to safely fly under
various conditions including severe and extreme weather conditions such as lightning
strike. It is reported [3]-[5] that on average two lightning strikes happen to commercial
aircraft every year; for military aircraft, this rate is even higher as they fly under severe
conditions. Some cases of catastrophic aircraft accidents are directly attributed to
lightning strike [6], [7]. Lightning strike induced damages to aircraft and can be
classified into two categories: direct damage, which is the structural damage caused by
heating and high electric field from lightning strike; and indirect damage, which is the
electromagnetic damage caused to onboard electrical equipment even though there is no
visible structural damage.
To better understand lightning strike effects, the Federal Aviation
Administration (FAA) defined lightning strike current waveform as shown in Figure 1.2
[8]. Natural lightning current is characterized into four idealized components:
3
a. Waveform A, which is a pulse representing the first arc. It has the largest
amplitude of all of the elementary lightning waveforms (200 kA) and a duration of
about 500 µs;
b. Waveform B is the intermediate pulse current waveform making the
slow transition from the impulse waveform A around 2000 A to the constant C
waveform at a level between 200 and 800 A, on a time scale ranging from 500 µs to 500
ms;
c. Waveform C is the intermediate pulse current waveform transitioning
between the waveform B to waveform D, on a time scale ranging from .25s to 1s;
d. The waveform D is another impulse waveform representing the second
arc with a peak current equal to half of the peak of the A waveform on a time scale of
500 µs.
Figure 1.2 Typical lightning current waveforms as defined in the MIL-STD-464 standard. Reproduced with permission [8]
4
Carbon composites undergo more severe damage than their metal counterparts,
due to the inherent electrical conduction mechanisms. CFRP consists of two component
materials: carbon fibers as the reinforcement and polymer resin as the surrounding
matrix. While carbon fibers are electrically conductive (with electrical conductivity of
1×105 S/m), the polymer matrix is usually a poor conductor and hence is considered as
good insulator. Composite structures in these applications thus cannot readily conduct
away the extreme electric currents and electromagnetic forces generated by lightning
strikes. For that reason, lightning strike protection (LSP) has been a significant concern
since the first composites were used on aircraft more than 30 years ago.
1.1.3 Common practice for lightning strike protection
Lightning protection is needed for composite structural component on aircraft
that have the greatest likelihood of a direct lightning interaction. The most common
practice for lightning strike protection is to add conductive attachments to composite
structures to guide the electric current induced from lightning strike to flow through the
least resistive path. The conductive attachments come in various forms: wire bundles,
strips, foils and wire meshes.
1.2 Research Motivation and Objectives
1.2.1 Research motivation
Good understanding of the electrical properties of CFRP is the key to effective
lightning strike protection. Other applications directly utilizing the electrical properties
of CFRP laminates include carbon fiber assisted heating during composites
manufacturing [9], [10], self-sensing of damage of composite structures [11] and
5
integrated electromagnetic shielding [12], [13]. Accurate predictive models describing
the electrical conduction behavior of CFRP laminates are necessary to design the
properties for their success in such applications.
Due to its electrically conducting property, carbon fibers can be used as sensors
in composite structures to detect cracks and delamination. In this context, several
researchers have focused [14-16] on the modeling of the electromechanical behavior of
CFRP materials under tensile loading. They have found that mechanical deformation
and electrical resistance especially of CFRP in axial direction are coupled due to fiber
deformation and breakage during the loading process. Wang [17] proposed that the
waviness of carbon fibers contributes to the formation of continuous electrical
conduction path and is the primary factor influencing through-thickness electrical
conductivity. However, no quantitative work has been done to implement a numerical
or analytical model based on fiber waviness.
Park [14], [18] proposed the concept of “electrically ineffective length” which is
the typical length over which a broken fiber regains its current-carrying capability due
to electrical contacts between fibers. They used a Monte Carlo technique to randomly
generate distributed contact points. Representing fibers with electrical resistors, the
CFRP is then modeled by a DC network circuit and solved with Kirchhoff’s rules. But
they did not present an approach to connect this variable parameter to physical
measurements.
Modeling of electrical conduction of carbon composites under high current
density such as lightning strike is currently limited to macro-scale finite element (FE)
models. Ogasawara [19] implemented a multi-physics FE model considering thermal
nonlinearity to study electrical conduction in multilayer CFRP. CFRP is modeled as
6
anisotropic material with conductivity tensor in three directions as material input. There
is another FE model [20] that follows similar methodology.
In the above models, microstructure of composites is not addressed in detail.
Therefore, they are not able to model the statistical variations of composite properties
which is inherent in such materials. Microstructure based models are needed to
understand the intrinsic relation between CFRP properties and its microstructure.
However, a detailed microstructure based model can be computationally inefficient
considering the complicated geometry of CFRP. Certain simplifications and
assumptions are necessary for implementing micro-structure based models.
This research is focused on the development of a microstructure based model to
predict electrical conduction behavior of CFRP. High current impact such as Joule
heating is also considered with a coupled thermal-electrical resistor-capacitor network
model.
1.2.2 Research objectives
This research will focus on the model development for prediction of electrical
property of CFRP under various electric fields. Based on the proposed model, a
methodology can be developed for the design of electrically tunable CFRP. The
objectives of this proposed research are as follows:
I. Develop and implement a numerical model that correlates the resistivity
of UD lamina with its material and (micro-)structural properties and
captures the electrical behavior of UD lamina (a single ply) when
subjected to low direct current;
7
II. Formulate an efficient modeling methodology that incorporates the
influence of ply orientation and inter-ply resin rich layer for large
composite laminates;
III. Investigate the electrical conduction behavior of CFRP under high
current densities similar to the ones encountered during a lightning strike
and identify key conduction mechanisms.
IV. Experimentally characterize the electrical response of carbon composites
under various current conditions from low DC to high current densities
to validate the model, as well as to identify other conduction mechanisms
not captured in the model.
1.3 Structure of Dissertation
This dissertation is organized as follows. This chapter (Chapter 1) gave a brief
introduction to CFRP used on aircraft and the critical issues encountered when aircraft
with carbon composite parts are struck by lightning. The motivation and the scope of
current study was also discussed.
A micromechanics based model for electrical resistivity of dry fiber tows and
unidirectional (UD) laminates without resin rich layer is discussed in Chapter 2,
followed by experimental investigation of the link between microstructure and electrical
resistivity of dry carbon fiber tows in Chapter 3.
In Chapter 4, the model discussed in Chapter 2 is extended to provide an
efficient solution for calculating resistivity tensor of a multi-ply laminate. The impact of
ply orientation, and laminate dimensions (aspect ratio to be specific) can be modeled. In
addition, resin rich layer between plies can be described by a newly defined parameter.
8
After validating the model with reported experimental data, parametric studies are
carried out to investigate the impact of resin rich layer.
Chapter 5 presents a numerical investigation of the influence of high current
density on electrical conduction of carbon composites. Current concentration within
carbon fibers and at contact spots is addressed in detail for the first time. Analytical
models are developed to quantify localized Joule heating. Nonlinear conduction
mechanisms such as temperature dependent intrinsic resistivity of carbon fiber and
degradation of thin resin layer are considered in the model. Parametric studies were
conducted to show the impact of local Joule heating on the temperature profile and
resistivity response of carbon composites under high current density.
Chapter 6 presents experimental investigations of both dry carbon fiber tows and
cured composites under high current density up to lightning strike level and discusses
the mechanisms not captured by the model that can be addressed in future work.
9
3D MICROSTRUCTURE BASED RESISTOR NETWORK MODEL
2.1 Introduction
Electrical properties of the constituent materials (fibers and polymer matrix) of
the CFRP differ by several orders of magnitude. Carbon fiber itself is a good conductor
with electrical resistivity in the range of 1 ×10-5 Ωm[20], [21]. In contrast, the polymer
matrix can be regarded as a good insulator with electrical resistivity ranging from
1×1010 Ωm to 1×1020 Ωm. Thus, CFRP conductivity along the fibers is governed by the
continuous conduction mechanism along the fibers while electrical properties in the
other directions are influenced by the shortest conduction path of connected fibers in the
width or thickness direction. Under low DC current, resin behaves as an insulator and
doesn’t contribute to the electrical conduction process. Therefore, electrical conduction
behaviors of dry carbon fiber tows and a single ply, also known as unidirectional (UD)
lamina as all fibers are aligned in one direction without the presence of resin-rich
interface, under low DC current are similar.
Chapter 2
10
Figure 2.1 Conduction mechanisms of CFRP laminates in three primary directions. a) X direction: intrinsic fiber resistivity and fiber volume fraction; b) random contacts between carbon fibers forming conductive paths; c) Z direction: fiber contacts within
one ply and limited connections between laminas due to resin rich interface. The blue plates represent electrodes in various directions.
Microstructure based modeling and analysis has been used to study the
mechanical behavior [22], [23] and thermal conduction [24] of unidirectional fiber
reinforced composites. The common modeling approach assumes that the fibers are
straight parallel cylinders and create the fiber spacing information from the cross
section of a composite specimen to represent the microstructure. A more detailed study
of the microstructure was performed by Gutowski et al. [25] which relaxed the
assumption of straight fibers, and introduced a parameter that quantified multiple
contact points along its length with the neighboring fibers due to the fiber waviness.
The waviness resulted in electrical contacts between neighboring fibers creating a
continuous conductive path, which governs the electrical conduction in transverse
direction.
In this work 2D micromechanics models of electrical conductivity of composites
are reviewed. A two-step scheme for generating the 3D microstructure of CFRP is
introduced. The microstructure describing the relationship between neighboring fibers
including distances between contact points and their waviness is used to build an
equivalent electrical resistor network model. A contact resistance term is integrated into
11
the model to represent the resistance between fiber contact points. The model predicts
electrical resistivity as a function of fiber volume fraction, intrinsic carbon fiber
properties, fiber waviness and applied pressure during processing of such composites. A
sensitivity study is conducted to identify the key factors that influence the electrical
resistivity of CFRP.
2.2 Electrical Conduction Mechanisms of CFRP
2.2.1 Review of existing models for electrical conduction of unidirectional CFRP
The longitudinal electrical resistivity of single carbon fiber or carbon fiber tow
has been reported by carbon fiber manufacturers or previous researchers [26], [27] and
for most PAN fibers is in the range of 1×10-5 Ωm. To our knowledge, the transverse
electrical resistivity/conductivity of dry fiber tow hasn’t been reported. The
conventional rule-of-mixture (ROM) model is accurate in describing the composite
conductivity along the fiber direction but fails to describe the electrical properties in the
transverse direction.
Continuum models have been widely used to consider the microstructure of
composites in the study of thermal conduction and micromechanics of composite
material [28], [29]. It is common practice to use a 2-component continuum medium
model to study the transport of heat in composite materials. However, when it comes to
electrical conduction in carbon fiber composites, the 2-component model fails due to
the large orders of magnitude difference between the electrical conductivity of carbon
fibers and the polymer matrix. The polymer matrix in such a composite system can be
regarded as an insulator and doesn’t contribute to the electrical conduction in
composites.
12
Resistor network model has also been adopted and modified by previous
researchers to study the thermal and electrical conduction behavior of composites. Self-
sensing of damage of composites has been achieved [11], [30], [31] mainly based on the
change of longitudinal electrical resistance of composites after structural damage, while
the through-thickness resistivity was not the focus in these investigations.
Park et al. [14] proposed the concept of “electrical ineffective length” and used a
resistor network model incorporating mechanical loading to model the change of
longitudinal electrical resistance of unidirectional CFRP under loading in the fiber
length direction. Hexagonal packing order of carbon fibers was assumed in their model.
Since it is believed that longitudinal electrical resistance is more useful in self-sensing
of composite structural damage, this model doesn’t consider the through-thickness
electrical resistivity of CFRP, which may be important in other situations such as
composite panels subject to lightning strike and characterization of electromagnetic
shielding property of CFRP.
Xia et al. [32] adopted a similar approach to model transverse conduction
behavior of CFRP as well as longitudinal resistance. Hexagonal and square packing
arrangements were assumed in their model. Random fiber-fiber contacts were
introduced and compared with uniform fiber-fiber contact distributions. While
assumption of ideal periodic packing arrangement simplifies the calculations, it cannot
address the impact of random fiber distribution on the property. Also, in this model,
fiber-fiber contact resistance is not considered. Similar to Park’s model, the electrical
ineffective length is back calculated by fitting the model results with the experimental
data, making it less effective in predicting resistivity of composites.
13
Many of these resistor network models have been developed to relate electrical
resistance change to mechanical loading in composite structures. Evolving fiber
breakage under loading is believed to be the major factor contributing to electrical
resistance change. However, the underlying mechanisms of electrical behavior of
composites without mechanical loading where fiber breakage is not involved have
received much less attention.
Finite element method (FEM) has been adopted by researchers to study the
orthotropic electrical conduction behavior of composites, especially for extreme
conditions where Joule heating of composites needs to be considered. Todoroki et al.
[11] applied FEA to study the effect of measured orthotropic electric conductance on
delamination. Ogasawara et al. [19] studied the coupled thermal-electrical behavior of
CFRP exposed to simulated lightning current using FEM. In these FE models, bulk
properties of composites are used without considering the microstructure of composites.
Due to the complexity of the composite microstructure, it is impractical to model the
composite microstructure in full detail with FEM.
While conventional models can provide general information of electrical and
mechanical interaction of CFRP, they are unable to accurately predict the effective
electrical properties that are inherently dependent on the microstructure. It follows that
an accurate prediction of macroscopic electrical conduction behavior can only be
accomplished by capturing the microstructure of the material as a basis for the model.
Another advantage of a microstructure-based model is that multi-physics simulation
such as thermo-electric-mechanical interactions can be addressed at the local scale.
Modeling of electrical conductivity in through-thickness direction must take into
consideration the contact between fibers, the fiber waviness, and the intrinsic single
14
fiber property. Of these mechanisms, the contact between fibers is quite important
because it influences continuous electrical conduction path in the through-thickness
direction and thus dictates the overall electrical conductivity of CFRP in the through-
thickness direction. Clearly, other mechanisms such as fiber breakage and sizing of
fibers can also be important, although limited number of numerical studies [33] have
modeled the effects of mechanical breakage of single fibers within a tow and the surface
treatment of carbon fibers on the overall electrical conductivity of carbon fiber tow.
2.2.2 Electrical conduction mechanism in longitudinal direction
Since the electrical conductivity of typical polymer matrix systems are 10 to 20
orders smaller than that of carbon fibers, polymer matrices can be regarded as
insulators. The single layer UD CFRP is therefore comparable to a carbon fiber tow in
terms of electrical conduction. The highly anisotropic behavior of the electrical
conductivity of unidirectional CFRP is due to different conduction mechanisms in
transverse and along the fiber direction.
Along the fiber direction, the current flows through the fibers and the carbon
fiber tow can be regarded as resistors in parallel. The resistivity of the unidirectional
carbon fiber tow depends on the intrinsic resistivity of the fibers and on the fiber
volume fraction. The longitudinal electrical conductivity of CFRP with fiber volume
fraction 𝑉𝑓 can be calculated by the rule of mixture:
σL = 𝜎𝑓𝑖𝑏𝑒𝑟𝑉𝑓 (2.1)
Where σfiber is the intrinsic electrical resistivity of carbon fiber under investigation.
15
2.2.3 Electrical conduction mechanism in transverse direction
In carbon fiber reinforced polymer composites, undulating carbon fibers lead to
electrical contacts between fibers. It’s noted by Wang [34] that the random fiber-to-
fiber contacts contribute to transverse electrical conduction and explains the anisotropy
of electrical conductivity. Fiber-to-fiber contacts create the continuous electrical
conduction path, contributing to overall electrical conductivity, as illustrated in Figure
2.2. Applied pressure and elastic modulus of fibers can affect the fiber waviness during
the fabrication process and thus influence electrical resistivity of carbon fiber tow.
Although this mechanism has been mentioned by other researchers when commenting
on their data qualitatively, there is no quantitative model built based on this mechanism.
Figure 2.2 Schematic representation of conductive path created by fiber-to-fiber contacts. Reproduced with permission [34]
16
The loading applied in the through-thickness direction during processing
impacts the overall microstructure and contact point geometry. First, the through-
thickness loading influences electrical resistivity by changing the fiber volume fraction
and thus fiber-fiber spacing, reducing the waviness. Second, applied pressure can
change the area and number of the contact points and the contact resistance. Applied
load during processing in the through-thickness direction can therefore influence the
transverse electrical resistivity significantly.
A micro-structure based resistor network is developed in this chapter to relate
the compressibility and relaxation behavior of fiber reinforcements during composite
processing with the electrical property.
2.2.4 Overview of model formulation
A 3D microstructure based resistor network model is proposed based on the
mechanisms discussed in the previous section. Contact points between neighboring
fibers are distributed along the longitudinal direction of composites while the dielectric
properties of the resin effectively insulate the remaining areas of the parallel fibers. This
assumption makes it possible to represent a CFRP structure as a large resistor network.
Figure 2.3 shows the flow chart describing the resistor network model
generation based on an existing microstructure. The fiber arrangement at a cross section
normal to the fiber length direction is generated first. Then the 2D fiber network is
extended along the fiber direction using the fiber waviness information describing the
full 3D microstructure. There are various ways to get fiber arrangement and fiber
waviness parameters experimentally or numerically. 2D fiber arrangement can be
generated numerically assuming square, hexagonal or random packing order or from
real composite structure micrographs. Fiber waviness parameter can be obtained
17
experimentally from compression behavior model developed by Gutowski [25] or from
numerical simulations.
Figure 2.3 Flow chart of model formulation: 3D microstructure is generated from 2D fiber arrangement and fiber waviness parameter. Reproduced with permission [35]
Each carbon fiber is divided into small sections separated by neighboring
contact points, which are modeled as nodes in the resistor network model. Each section
of carbon fiber is modeled by a resistor whose resistance value is determined by the
length of the carbon fiber section, diameter of carbon fiber, and intrinsic resistivity of
the carbon fiber. A contact resistor is added at the contact point to represent contact
resistance between carbon fibers. The resistors representing carbon fibers and contact
resistances form a 3D resistor network and can be solved using Kirchhoff’s law. A
uniform potential is applied across the sample allowing modeling of bulk resistance in
18
this direction. The electrical resistivity of unidirectional CFRP or carbon fiber tows in
all three primary directions can be calculated using known geometric information.
Parameters used in the proposed model are listed in Table 2-1.
Table 2-1 Parameters used in the 3D resistor network model
2.2.5 Numerical implementation
The proposed 3D resistor network model is implemented using MATLAB. As
illustrated in Figure 2.4, the procedure to generate the 3D resistor network consists of
the following steps:
Parameters Description Typical Value (AS4
carbon fiber)
VF Fiber volume fraction 50%
PackOrder Fiber packing order Hexagonal, Square,
Random, FromImage
Spec_Len Specimen length 1×10-2 [m]
Spec_Wid Specimen width 1×10-3 [m]
Spec_Th Specimen thickness 1×10-3 [m]
Pressure Processing pressure 8 bar (autoclave)
Beta Fiber waviness term ~300
Fiber_diameter Average fiber diameter 7 μm
E_mod Elastic modulus of carbon fiber 231 GPa
Fiber_rho Electrical resistivity of carbon fiber 1.7×10-5 Ωm
19
1. Generate 2D cross section fiber arrangement based on selected packing
order, specimen dimension, fiber volume fraction, and fiber radius. The
output is a matrix that stores the coordinates of fiber center and radius
for each fiber;
2. Create fiber connectivity based on fiber-fiber spacing. The output is an
array that stores the indexes of neighboring fibers for each fiber;
3. Randomly generate initial contact point for each fiber and add the
remaining contact points on that fiber based on fiber-fiber spacing, β,
and fiber connectivity as determined in step 2. The output is an array that
stores the 3 coordinate values of every contact point on each fiber and
index of neighboring contact points;
4. For each fiber, calculate contact resistance and fiber section resistance
( 𝐿/𝐴); where A is the cross-sectional area.
5. Formulate resistor network in matrix form based on contact points,
connectivity
6. Solve for nodal voltage and current using Kirchhoff’s 1st law; Calculate
overall resistance and resistivity.
20
Figure 2.4 Flow chart of the algorithm to formulate and solve 3D resistor network. Reproduced with permission [35]
2.2.6 Parameterization of 3D fiber geometry
Generation of 3D microstructure of carbon fiber tow or unidirectional CFRP is
not trivial considering the complex geometry details of CFRP. In this study, a two-step
procedure is employed to numerically generate the 3D structure of unidirectional CFRP.
First, 2D fiber arrangement is generated from the cross-section of a composite panel
either numerically or from a micrograph of a cross section of a real composite structure.
In the second step, the 2D model is then extruded in the fiber length direction with a
parameter that describes the fiber waviness.
21
2.2.6.1 Generation of 2D fiber arrangement
Hexagonal, square and random packing orders of carbon fibers are numerically
generated and fiber location and radius can be obtained as described in the next section.
Equation 2 gives the relation between fiber volume fraction, fiber diameter and inter
fiber spacing for square and hexagonal packing orders. Inter fiber spacing can be
derived by solving Equation 2 with known fiber volume fraction and fiber diameter.
𝑉𝑓 =𝜋𝑑𝑓
2
4(𝑑𝑓 +𝑑𝑠)2𝜂 (2.2)
Where Vf is fiber volume fraction, df is fiber diameter, ds is inter fiber spacing,
𝜂 as the packing order related parameter with 𝜂 = 1 for square packing and 𝜂 =2√3
3 for
hexagonal packing. This information is used to define fiber location and radius of all
fibers in the modeled 2D cross-section for hexagonal and square packing arrangements.
Fiber arrangements in unidirectional composites are typically non-uniform and
non-periodic. Periodic fiber distribution assumption leads to incorrect predictions of
mechanical behavior of composites [36]. Although there is no study available to
describe the influence of non-uniform fiber arrangement on electrical properties of
composites, it’s desirable to address practical representation of fiber arrangements. A
numerical random microstructure generator is programmed in MATLAB with the
following input: the fiber radius distribution, required fiber volume fraction, and the
size of the domain. The routine is based on Random Sequential Addition (RSA)
algorithm, and provides fiber location and radii of the 2D cross-section.
To obtain the distribution from a real composite, a microscopy image of the
cross-section of a CFRP is first transformed to a binary image representing resin and
22
fibers only. An edge detection algorithm is utilized to find the location of the circular
fibers from the binary image and locations of fiber center and fiber radius are extracted.
This method provides the most realistic microstructures but can be time consuming, and
the number of available datasets is limited. Figure 2.5 gives a schematic representation
of the four approaches used to generate the 2D fiber arrangement. Coordinates of fiber
centers and fiber diameters are stored in a matrix for later use. Connections between
fibers are created based on fiber-fiber distance. Indexes of neighboring fibers are stored
in a list for each fiber.
Figure 2.5 Schematic representations of various fiber arrangements: (a) hexagonal packing; (b) square packing; (c) random packing; (d) from micrograph using image
processing techniques. Reproduced with permission [35]
23
2.2.6.2 Parameterization of fiber waviness
Undulation of fibers is naturally present as fibers are placed together in a fiber
tow due to their continuous length and small diameter which makes them less rigid and
stiff. When these fibers are compacted due to processing pressure during manufacturing
of composite structures, the undulating fibers will contact their neighbors at various
locations along the length. As shown in Figure 2.6, instead of being regarded as straight
rigid rods, fibers are modeled as slightly arched so that the contact points between
neighboring fibers carry the applied force. Fiber waviness can be characterized from
Equation 3 as the ratio between the arc height 𝑑𝑠 and the contact span 𝐿.
𝛽 =𝑑𝑠𝐿=ℎ − 𝑑
𝐿 (2.3)
where dS is the arch height (average distance between fibers), L is the distance between
two contact points.
Figure 2.6 Schematic representation of fiber waviness. Reproduced with permission [25]
24
Gutowski relates the deformation of the curved fiber stack to axial and
compressive loads. In the case of transverse compression, the functional relationship
between compression stress and fiber deformation data is given by Equation 2.4.
σz =3𝜋𝐸
(𝛽)4∗
1 −√𝑉𝑓𝑉0
(√𝑉𝑎𝑉𝑓− 1)
4(2.4)
At or below a certain initial fiber volume fraction, V0, the fibers carry no load.
As fiber volume fraction, Vf, is increased, the network can carry a rapidly increasing
load. Eventually, the fiber volume fraction of the network approaches a theoretical
maximum based on close packed geometry. In this region, Vf approaches a maximum
available fiber volume fraction, Va. When the fiber network is perfectly aligned, Va falls
between the limits for a square packing order, Va=0.785, and a hexagonal packing
order, Va=0.907. The fiber waviness term β can be obtained using a three dimensional
least-square optimization fitting Equation 2.4 to the experimental obtained relationship
between compression pressure σz and fiber volume fraction Vf.
After characterizing the fiber waviness term, the 2D microstructure model is
extended in fiber length direction by generating contacts points along the fibers. The
location of contact points on each fiber can be calculated by adding contact span L to
the previous contact point location, while the location of the first contact point is
randomly selected to reflect the random nature of contacts between fibers. Figure 2.7
shows schematically the fiber network and the equivalent electrical representation of the
25
fiber system when extended along the fiber and through-thickness direction. Each
contact point as well as the two ends of a fiber are represented by an electrical node. At
each contact point a contact resistance is introduced to consider any interface contact
resistance. Note that for clarity only a 2D model is demonstrated in the schematic
drawing while the model actually considers the 3D configuration of the carbon fiber
system.
26
Figure 2.7 Schematic representation of resistor network model: fractional fiber lengths
in between contacts and contact resistances form the resistor network model. 2D presentation is shown here for clarity while the model is 3D. Reproduced with permission [35]
2.2.7 Estimation of contact resistance
Contact resistance between neighboring carbon fibers needs to be estimated
since it will influence the overall conductivity of CFRP especially in the transverse
27
direction. Resistor network models have been employed by other researchers to study
the electrical behavior of composite materials; but in these models, contact resistance is
either neglected [37], [38] or back calculated from curve fitting with experimental data
[16], [39]. Preliminary experimental investigations show that contact resistances
between carbon fibers vary by a large range depending on carbon fiber surface
treatments. The contact resistance of carbon fibers with nonconductive sizing can be
orders of magnitude higher than carbon fibers without sizing, which makes the fiber-to-
fiber contact resistance term important. In this study, contact resistance is considered as
demonstrated by the resistors in blue in Figure 2.7. Contact resistance between single
fibers is hard to characterize experimentally due to handling issues and measurement
accuracy. In many cases, sizing of carbon fibers is proprietary commercial information
of carbon fiber manufacturers and thus is usually not disclosed. This makes it difficult
to characterize the contact resistance accurately and its impact on the overall
conductance. Therefore, in this work we will focus on unsized fibers. A simplified
analytical model to estimate contact resistance for non-sized fibers based on Hertz
contact theory and Holm’s formula for constriction resistance is presented.
2.2.7.1 Hertz contact theory
The complicated load transfer between neighboring fibers makes an exact
analytical solution for contact mechanics impossible. With some geometry
simplifications, the theory of contact between elastic bodies can be used to find contact
areas and indentation depths for contacting carbon fibers. Due to undulation of fibers,
two fibers contact through points rather than a line. The contacting part between two
fibers is taken as two spheres with the same radius of the carbon fibers. Hertz
[40]derived the analytical solution for contact area, indentation depth and contact stress
28
for the ideal case of two spheres with radius of R1 and R2 respectively. The contact area
is directly related to the electrical contact resistance as described in the next section.
The equivalent radius contact area is related to the applied load F by the equation
𝑎3 =3𝐹𝑅𝑒4𝐸
(2.5)
Here E is the elastic modulus of the fiber and the effective radius Re is defined as
1
𝑅𝑒=1
𝑅1+1
𝑅2 (2.6)
For two carbon fibers with the same radius df/2, the radius of the contact circle
is given by
𝑎 = √3𝐹𝑑𝑓16𝐸
3
(2.7)
2.2.7.2 Electrical constriction resistance between two conductors
Constriction resistance, Rc, exists between two conductors connecting through a
small area. Holm [41]studied the constriction resistance between conductors with
various configurations and found constriction resistance at the contact location of two
conductors depends on the resistivity of the two conductors and the area of the contact
spot as follows,
𝑅𝑐 =𝜌1 +𝜌24𝑎
(2.8)
29
Where ρ1 and ρ2 are resistivity of the two conductors and a is radius of the
contact spot as defined in Equation 2.7.
For the case of two carbon fibers with the same intrinsic resistivity ρ, the
constriction resistance is further simplified to
𝑅𝑐 =𝜌
√3𝐹𝑑𝑓2𝐸
3
(2.9)
2.2.7.3 Estimation of contact force
Force applied to CFRP in the through-thickness direction is modeled by the
following characteristics. At or below a certain initial fiber volume fraction, the fibers
carry no load; as fiber volume fraction is increased, the fiber network can carry rapidly
increasing load as transferred by fibers through contacts between fibers.
Assuming that each contact point carries similar amount of load, contact force at
each contact point can be estimated by dividing total applied force F by the total
number of contact points Nc at each layer, as demonstrated in Figure 2.8.
30
Figure 2.8 Schematic illustration of load sharing among carbon fibers. Applied force is
assumed to be shared evenly by the total number of contact points at each layer.
Total number of contact points Nc is counted by multiplying it by the number of
fibers Nf at each load-carrying layer with average number of contact points Nci on each
fiber. These parameters are related to composite geometry and processing pressure as
defined in Equation 2.10 -13.
𝑁𝑓 =𝑊
𝑑𝑓 +𝑑𝑠 (2.10)
𝑁𝑐𝑖 =𝐿
𝑑𝑠 ∗ 𝛽 (2.11)
𝑁𝑐 = 𝑁𝑓 ∗ 𝑁𝑐𝑖 (2.12)
𝐹𝑐 =𝐹
𝑁𝑐=𝑃𝑊𝐿
𝑁𝑐 (2.13)
31
Where df is fiber diameter; ds is average distance between fibers; L is specimen
length; W is specimen width; F is total load applied to specimen; P is the applied
processing pressure.
Combing Equation 2.10 – 2.13 gives
𝐹𝑐 = 𝑃(𝑑𝑓 + 𝑑𝑠)𝑑𝑠𝛽 (2.14)
Contact force depends on applied processing pressure, fiber diameter and inter-
fiber spacing. Solving Equation 2.9 and Equation 2.14 gives contact resistance as
function of processing pressure, fiber diameter, fiber waviness and inter-fiber spacing
(Equation 2.15).
𝑅𝑐 =𝜌
√3𝑃(𝑑𝑓 + 𝑑𝑠)𝑑𝑠𝛽𝑑𝑓2𝐸
3
(2.15)
2.3 Model Convergence and Validation
2.3.1 Model convergence
Determining the size of the computational model is an important part of the
microstructure-based modeling approach. In the proposed 3D resistor network model,
each contact point is regarded as a computation node. While more nodes can represent
more details of the composite structure, the computation time increases exponentially.
Total number of contact points/nodes is determined by number of fibers and
number of contact points on each fiber. For the cross section, the representative volume
32
element (RVE) consists of around 200 fibers (about 15 fibers in through-thickness
direction) and is considered as representative of a real composite structure [36]. The
number of nodes is increased by increasing the length of fibers and thus increasing the
number of contact points on each fiber. Figure 2.9 provides the convergence plot for 4
different configurations to generate the 2D fiber arrangement – hexagonal packing,
square packing, random packing and micrograph fiber arrangement. Predicted
resistivity is normalized by dividing the predicted values with the predicted resistivity
from micrograph fiber arrangement at 6000 nodes. The fiber properties used in these
simulations are listed in Table 2-2 and fiber volume fraction is taken as 0.6 for all
simulations.
Table 2-2 Properties of HTA-7 fiber
Filament diameter Fiber modulus
Fiber resistivity β(estimated)
7 μm 238 GPa 1.6 X 10-5 Ωm 350
33
Figure 2.9 Model convergence as a function of packing arrangement. Resistivity
normalized with value obtained for 6000 nodes using the fiber positions obtained from the micrograph. Reproduced with permission [35].
The hexagonal and square configuration converges faster than that of the
random packing and micrograph microstructure. The irregular arrangement of fibers in
real composite structure and in random packing order introduces more uncertainties in
the model and requires more nodes to converge. Model predictions were relatively
unchanged at 3000 nodes and above for square and hexagonal packing and 5000 nodes
and above for random packing. Hence all computational models used in this study
ensured that the number of nodes used ensured convergence of results.
Transverse resistivity drops with increasing specimen length as observed by
Abry et al. [31] from experiments. When the specimen length is smaller than a critical
34
length in the range of the fiber to fiber contact distance, the number of contact points is
drastically reduced and an increase in resistivity is observed. Above the critical length,
the number of contact points increases almost linearly with specimen length and the
resistivity attains a constant value.
The model using random packing order is within 10%-15% of the predicted
resistivity of the real microstructure, while the hexagonal and square packing order over
predicts the resistivity by about 30%-50%. In random packing order model, fiber-fiber
spacing is randomized. Instead of having uniform distribution of contact length as in
uniform packing order models, there exists local small fiber-fiber spacing in fiber
system with random packing order, resulting in larger number of contact point. With
square packing order, each fiber except those close to boundaries has 4 neighbor fibers,
while with hexagonal packing order, each fiber has 6 neighbors. More neighbor fibers
leads to easier formulation of connecting conduction network, which lowers resistivity.
This explains the larger resistivity given by square packing order model than hexagonal
packing order model. It can also be noted that standard deviation of predicted
resistivity generally drops with increasing number of nodes. Once the appropriate model
scale was established, the next step was to validate the model and understand the effect
of input parameters on resistivity.
2.3.2 Comparison between simulations and literature data
The simulation results are compared with reported resistivity values of
unidirectional CFRP to validate the proposed model. Abry [31] reported experimental
results of electrical resistivity of unidirectional multi-ply CFRP using HTA-7 carbon
fibers as shown in Table 2-3. It can be seen that the transverse resistivity of the CFRP
35
through the thickness is at least one order different compared to the in-plane transverse
resistivity at low fiber volume fraction. The laminate has resin-rich layers in between
the unidirectional plies creating an additional resistance term not considered in the
current model implementation. The transverse resistivity for both directions converges
at higher fiber volume fraction. The in-plane transverse direction microstructure
represents the model implementation in all cases and is used for comparison.
Table 2-3 Resistivity of UD CFRP reported by Abry [31]
Vf Longitudinal resistivity (Ωm)
Transverse resistivity (Ωm)
In-plane transverse
Through-
thickness
0.43 4.72 X 10-5 4.67 X 10-1 16 0.49 3.71 X 10-5 1.13 X 10-1 2.83 0.58 2.93 X 10-5 4.16 X 10-2 4.82 X 10-2
Although fiber waviness term is not reported for this specific fiber type,
Gutowski [25] noticed that this term is similar for various carbon fibers and is
characterized as approximately 350 from repeated experiments. In the following
simulations, the mechanical and electrical properties of HTA-7 as shown in Table 2-2
are used in all following simulations as baseline parameters.
Simulations results of in-plane transverse resistivity of unidirectional CFRP or
carbon fiber tows are compared with experimental values of unidirectional CFRP
resistivity. As shown from Figure 2.10, simulation results including the contact
resistance term agree with reported experimental data while the predicted values from
the model without the contact resistance underestimate the experimental data, especially
in the lower fiber volume fraction range. Good match between simulation results and
36
experimental values for in in-plane transverse direction indicates that the model
captures the correct conduction mechanisms for UD CFRP with no or little inter-ply
resin rich layers. The discrepancy in through-thickness direction reveals that conduction
behavior of interface layer between two single plies needs to be modeled for multi-ply
CFRP.
Figure 2.10 Comparison of simulation results and reported experimental data. Reproduced with permission [35]
2.4 Sensitivity Analysis
The proposed model can be used to evaluate the effect of the material and
process parameters on the resulting resistivity of the bulk composite. This paper
evaluates in more detail the effect of process pressure used during consolidation on
37
change of resistivity. This is followed by a sensitivity study where material and process
parameters are varied to provide insight in the relative importance of these parameters.
2.4.1 Resistivity as function of processing pressure
The influence of processing pressure is two-fold. First, pressure can influence
fiber volume fraction and fiber arrangement. Debulking behavior of carbon fiber bundle
under compression loading results in a fiber arrangement that will stabilize to a state
where minimal loading is required, while the fiber volume fraction remain unchanged
[42]. Second, as can be noted from Equation 2.14, contact resistance is a function of
contact force. Higher processing pressure leads to higher contact force and lower
contact resistance, reducing overall electrical resistivity of CFRP. Transverse resistivity
of unidirectional CFRP or carbon fiber tows is calculated with processing pressure.
Nonlinear change of resistivity with the increase of processing pressure is observed and
shown in Figure 2.11. At very low processing pressure, increasing pressure will reduce
transverse resistivity of the composite rapidly, while at higher pressure, increasing
pressure does not change significantly the resistivity of the composite as fiber volume
fraction changes are small and contact resistance does not change significantly.
38
Figure 2.11 Transverse resistivity as a function of processing pressure. Reproduced with permission [35]
Contact resistance and fiber resistance are compared in Figure 2.12. Fiber
resistance is taken as the average resistance of all fiber sections between two
neighboring contact points in our resistor network. Fiber resistance drops with
increasing processing pressure as higher processing pressure yields higher fiber volume
fraction, reducing inter-fiber spacing and thus shortening the length of each fiber
section. Reduction in contact resistance results from higher contact force due to
processing pressure. Fiber resistance is also plotted as a fraction of total resistance,
which is defined as the sum of fiber resistance and contact resistance. Fiber resistance
39
contributes approximately 66% to the overall bulk resistance at pressures above 1 bar,
while the contact resistance cannot be neglected for lower pressures.
Figure 2.12 Fiber resistance drops with increasing processing pressure as higher processing pressure yields higher fiber volume fraction, reducing inter-fiber spacing and
thus shortening the length of each fiber section. Reduction in contact resistance results from higher contact force due to processing pressure. Reproduced with permission [35]
2.4.2 Sensitivity study for all relevant material and process parameters
Material and process parameters are varied below and above the baseline
parameters from Table 2 under 100 kPa of processing pressure. Figure 2.13 gives the
40
sensitivity plot for 5 model parameters (processing pressure, fiber diameter, fiber
resistivity, fiber waviness, and fiber modulus). Predicted resistivity increases with
increasing fiber waviness and resistivity and decreases with the other 3 parameters. The
beta term is related to fiber waviness; a smaller beta term indicates more contact points
per fiber, thus a more undulated fiber. The influence of fiber waviness can be
considered by varying beta term. Sensitivity of fiber waviness term is studied by
varying it while keeping other parameters constant. Increasing fiber waviness term (beta
term) reduces number of contact points per unit length since fiber-fiber spacing doesn’t
change, making the effective conductive path longer. The resistivity shows a nonlinear
dependency on fiber waviness, fiber modulus and processing pressure while fiber
resistivity and diameter indicate a more linear contribution. Fiber diameter and fiber
modulus have relatively small influence on the predicted resistivity in the value range
considered. Fibers with smaller modulus are easier to deform under pressure and thus
have a larger contact area, reducing contact resistance.
41
Figure 2.13 Sensitivity study of model parameters. Reproduced with permission [35]
2.5 Summary and Conclusions
CFRP exhibit highly anisotropic behavior in electrical conductivity with very
high conductivity along the fiber direction and extremely low conductivity in the
transverse direction. The fundamental conduction physics changes significantly as along
the fiber direction a continuous conduction path within the fibers exists while in
transverse direction the conduction path has to transverse fibers at discrete contact
42
points to neighboring fibers increasing the effective conduction path by several orders
of magnitude.
A micromechanics model was developed to predict the electrical conductivity of
unidirectional (UD) ply based on this proposed conduction physics considering both
electrical material properties and microstructural parameters. A 2D cross-sectional
representation of the fiber network is extended along the fibers providing discrete
contact points along the fiber length. These assumptions follow the proposed
microstructure studied by Gutowski implying an inherent undulation of the fibers. The
microstructural 3D representation is converted into an equivalent resistor network
which accounts for fiber and contact resistance allowing prediction of resistivity for
unidirectional composites.
Estimation for contact resistance between neighboring fibers is derived using
Hertz contact theory and Holm’s constriction resistance model. Processing pressure not
only influences electrical resistivity of CFRP by changing fiber volume fraction and
fiber arrangement, but can also affect contact resistance between fibers, thus changing
the electrical conduction behavior of CFRP. Predictions are validated using
experimental data from literature and are in excellent agreement to the reported data.
A sensitivity study reveals that fiber resistivity and waviness are the key factors
determining the electrical resistivity of UD CFRP in transverse direction. Process
conditions influence the resistivity. Increasing process pressure increases fiber volume
fraction and reduces contact resistance leading to lower bulk resistivity. Currently, the
model is limited to laminates without significant resin rich layers where resistivity is
governed by this interface layer and a later study will consider this effect.
43
EXPERIMENTAL INVESTIGATION OF THROUGH-THICKNESS
RESISTIVITY OF CARBON FIBER TOWS
3.1 Introduction
In Chapter 2, a micromechanics based model for transverse electrical resistivity
of UD lamina and dry fiber bundles is discussed. Good match between the modeling
results and reported experimental values indicates that the model can describe the
electrical conduction behavior of UD composites. There exists limited literature on the
experimental characterization of through-thickness resistivity of dry carbon fiber tows.
Athanasopoulos [43] investigated the in-plane electrical conductivity of UD carbon
performs under constant and uniform pressure and proposed mathematical expression to
describe the conduction behavior. Chung [44] and Curtin [32] have shown qualitatively
the dependence of electrical resistivity of CFRP on fiber-fiber contacts due to fiber
waviness. In this study, extended experimental characterization of the through-thickness
resistivity of dry carbon fiber tows are conducted to investigate the link between
microstructure and electrical resistivity.
Dry carbon fiber tows (instead of cured composites) are tested in this study for
two reasons: first, it is easier to control the microstructure (fiber volume fraction for
example), thus making the direct comparison of the microstructure based model and
experimental data possible; secondly, in the cured composite system, contact resistance
between carbon fibers separated by thin film of resin depends largely on the film
thickness, thus introducing uncertainties in the measurements. The absence of resin
Chapter 3
44
system in dry fiber tows reduces the uncertainties in contact resistance. Under low
current and electric field (much lower than dielectric strength of resin system), resin in
CFRP behaves as insulator and doesn’t contribute to electrical conduction. Dry carbon
fiber tows are thus comparable to CFRP in terms of electrical conduction. In this study,
contact resistance introduced by resin film between carbon fibers is demonstrated by
comparing the electrical resistivity of carbon fiber tows with and without sizing. Fiber
sizing is a thin layer of polymer deposited on the fiber surface to protect the individ ual
filaments from breaking, to improve handling of the very fine carbon filaments
(typically 5 – 7 µm diameter), and to enhance bonding between carbon fibers and resin
matrix.
In the present study, an apparatus that characterizes fiber waviness and measures
in-situ electrical resistivity of carbon fiber tows under compression is designed and
implemented. The influence of a composite fabrication pressure cycle is represented by
normal pressure applied by the MiniInstron machine to the fiber tow stack. A systematic
study of the through-thickness resistivity of carbon fiber tows under compression has
been conducted and provides further insight into the fundamental conduction
mechanisms of dry carbon fibers which is an important component of the CFRP. The
model discussed in Chapter 2 is applied in this study to describe electrical behavior of
carbon fiber tows in through-thickness direction.
3.2 Experimental Setup and Methodology
3.2.1 Setup
A computer controlled experimental setup was constructed to measure current,
voltage, load, displacement and fiber volume fraction in real time. The system allows
45
continuous measurements of the resistance and compaction state of the specimen. This
electrical and mechanical characterization setup (see Figure 3.1) includes a mini-Instron
machine with a load cell for applying and recording the force, a Teflon mold with two
copper bars serving as electrodes, a CCD camera to monitor the fiber stack height, a
digital multimeter for resistance measurement, and a data acquisition system to integrate
the results from all the components.
To eliminate the resistance introduced by the connecting wires, the resistance is
measured using a Keithley 2750 digital multimeter with a measuring resolution of 1 μΩ.
Two sets of wires were used for current and voltage measurements on the copper
electrodes, as demonstrated in Figure 3.1.
The load cell has maximum loading capacity of 500N with resolution of 1mN,
which transforms into maximum applied pressure of ~10bar over the 5cm by 1.25cm
specimen area (the dimension of copper electrode surface). A custom designed mold
was used to conduct compression testing. The mold consists of a mold base and a mold
cover, both machined from Teflon block for electrical insulation. The maximum
opening of the mold is 1 cm and the surface dimension is 5cm by 1.25cm, as depicted in
the insert of Figure 3.1. Two copper bars with the thickness of 5mm serving as
electrodes are attached to the bottom of the mold base and mold cover. The mold has
two open ends for real-time fiber volume fraction measurement using a CCD camera
(Dino-Lite ® Edge). Dry carbon fiber tows are placed in the mold cavity between the
two electrodes and pressure is applied by the Instron, ensuring good contact between the
specimen and the electrodes.
46
Figure 3.1 Experimental setup with its schematic for characterization of through-
thickness resistivity of carbon fiber tows
The mold was enclosed in an environmental chamber attached to the Instron
machine providing a constant temperature environment of 25ºC. Data was acquired at a
rate of 1Hz using a supervisory data acquisition computer and recorded on the PC for
subsequent data reduction.
3.2.2 Specimen preparation and experimental procedure
Unidirectional dry carbon tows from various manufacturers were tested. The
names of the carbon fibers were restricted from disclosing by a confidential provider,
thus general descriptions of the fibers were used in this paper (sized and unsized). All
the fibers (both sized and unsized) tested are PAN based and there is no in-house
surface modification to these raw materials received from manufacturer. Dry carbon
fiber tows are cut to 5cm length and placed in the same direction, along the longer side
of the electrodes, into the mold. 30 tows for each fiber were used in each experiment.
47
Material and setup parameters are summarized in Table 3-1. Fiber C is the sized version
of fiber B. The initial fiber volume fraction is about 20% before compression. Before
each test, the copper bar electrodes were cleaned and polished with 400 grit sandpaper
to keep a consistent surface roughness. At the final stage of compression, fiber volume
fraction was in between 60% and 70% depending on fiber type.
Table 3-1 Specimen parameters
Carbon Fiber Tow
Size
Number
of Tows
Fiber
Diameter
(µm)
Sizing Amount
(weight
percentage)
Fiber Electrical
Resistivity provided
by manufacturer
(Ωm)
Elastic
Modulus
of Fiber
(GPa)
Unsized Fiber A 12k 30 7 0% 1.70E-05 231
Unsized Fiber B 12k 30 5.2 0% 1.50E-05 276
Sized Fiber C 12k 30 5.2 1% 1.50E-05 276
Sized Fiber D 24k 30 7 1% 1.60E-05 135
Sized Fiber E 24k 30 5 0.25% 1.40E-05 170
During each test, load applied to the fiber tow specimen is increased from 0N to
450N at an increasing rate of 1N/s. A baseline test with two copper bar electrodes
touching each other is conducted before inserting the fiber tow specimen between them
into the mold. Contact resistance from copper electrodes acquired from baseline
experiment is less than 1% of the measured through-thickness resistance of the fiber
tow specimen. Specific through-thickness resistivity was obtained by multiplying the
measured resistance by the surface area and dividing it by the thickness (the mold
opening). The same loading cycle was repeated on each specimen to examine the effect
of debulking on through-thickness resistivity.
48
3.2.3 Characterization of fiber volume fraction
The reading provided by the Instron machine cannot be used to measure the
mold height due to the large compliance of the Teflon mold. A CCD camera with
microscope lens attachment recorded images of the mold and the location of each mold
surface was found using an edge detection algorithm. The difference between the two
mold surface locations provided continuous reading of the mold cavity height. Accuracy
was within one pixel and resulted in better than 10µm precision (field of view (~5mm)
divided by the camera resolution of 720 pixels).
Fiber volume fraction Fv can then be calculated using Equation 3.1.
Fv = 𝑛𝜋𝑑𝑓𝑖𝑏𝑒𝑟2
2ℎ𝑐𝑎𝑣𝑖𝑡𝑦𝑤𝑐𝑎𝑣𝑖𝑡𝑦 (3.1)
Here, hcavity and wcavity is the height and width of the mold, respectively. The
equation assumes that each tow has n fibers based on manufacturer supplied data and
that all fibers are continuous and have a constant diameter and occupy the entire length
of the mold.
The advantage of using a camera to acquire fiber volume fraction is
demonstrated in Figure 3.2 in which the fiber volume fractions calculated using the
camera to find the cavity height is compared with the mold opening obtained from the
extension data from the Instron machine. The two methods show similar fiber volume
fraction Fv when Fv is less than 50%. Loading increases significantly with increasing
fiber compaction and the mold compliance results in large error in cavity height
measurements using the Instron method. Fiber volume fraction obtained by the camera
reaches a limit at 70% under 10 bars of pressure.
49
Figure 3.2 Fiber volume fraction calculations using data from Instron and from image processing
3.2.4 Characterization of fiber waviness
The fiber-fiber contacts form the continuous conduction paths for carbon fiber
tows. The number of contact points per unit length on each fiber determines the number
of parallel conduction paths and thus determines through-thickness resistivity of carbon
fiber tows. As discussed in Chapter 2, Gutowski [25] developed a mathematical
expression correlating fiber volume fraction and the compression state to applied
normal pressure based on beam theory. The fiber network is modeled as an assembly of
50
slightly arched beams where the contact points between neighboring fibers carry the
applied force. The basic underlying assumptions are that the fibers make multiple
contacts with their neighbors along their length, that the number of contacts increase as
the bundle is compressed, and that the contact to contact length L is proportional to the
inter-fiber spacing a, which is arch height h subtracted by fiber diameter d. The fiber
waviness term β is defined as the ratio between the contact to contact length L and inter-
fiber spacing a.
Deformation of the curved fiber stack to axial and compressive loads is related
according to Equation 2.4. The fiber waviness term β can be obtained using a three
dimensional least-square optimization fitting Equation 2.4 to the experimentally
obtained relationship between compression stress σz and fiber volume fraction Vf. A set
of β, V0 and Va values is chosen such that the difference between experimental
compression results and Gutowski’s equation is minimal. Good fit to the experimental
data for all five fiber types and Equation for β, V0 and Va are summarized in Table 3-2
and shown in Figure 3.3. The small V0 values correspond to the initial states before the
upper electrode touches the fiber tows, where the fiber tows are loosely stacked and no
compressive force was applied.
Table 3-2 Fiber waviness and Gutowski fiber volume fraction terms for five fiber types
Fiber Type Fiber Waviness Term (β) V0 Va
Unsized Fiber A 620 0.02 0.89
Unsized Fiber B 365 0.06 0.89
Sized Fiber C 460 0.08 0.74
Sized Fiber D 405 0.06 0.89
Sized Fiber E 475 0.04 0.74
51
From Figure 3.3 one can note that the smaller diameter fibers (Fiber B,C,E)
require significantly higher pressure to reach a particular fiber volume fraction level
compared to the larger diameter fibers (Fiber A and D). Fiber volume fraction increases
rapidly at low pressures (below 50 kPa) followed by stiffening of the fiber stack. The
maximum fiber volume fraction at our applied experimental peak pressure of ~800 kPa
varies between 55%-75% depending on the fiber type. There is a significant difference
in the compression behavior of sized (Fiber C) and unsized fiber (Fiber B). The thin
sizing layer may act as lubricant, making the rearrangement of fiber packing easier, as
indicated by Gutowski.
52
Figure 3.3 Compaction data for all five fiber types
3.3 3D Resistor Network Model Implementation
In Chapter 2, the carbon fiber tows have been modeled as a DC circuit with an
array of electrical resistors representing the resistance from carbon fiber sections and
fiber-fiber contact resistance. The resistor network represents the equivalent 3D
microstructure of the fiber tow, as demonstrated in Figure 2.7. The 2D fiber
arrangement of the out-of-plane cross section is generated assuming random packing of
fibers and the 2D structure is extended in fiber length direction using the fiber waviness
53
term. The first contact point is randomly chosen to reflect the random nature of fiber to
fiber contacts.
The construction of the resistor network requires resistance values of the carbon
fiber sections and fiber-fiber contact resistance. Carbon fiber resistance can be
calculated from the intrinsic carbon fiber resistivity, diameter and contact to contact
length, as in Equation 3.2.
Rf = 𝜌4𝐿𝑐
𝜋𝑑𝑓2 (3.2)
Contact to contact distance 𝐿𝑐 can be calculated by multiplying fiber to fiber
spacing with fiber waviness term β, based on Gutowski’s curved beam model. For
unsized fibers, contact resistance can be estimated based on Holm’s electric contact
model and geometry of specimen, according to Equation 2.15.
The current model assumes the in-plane and through-thickness contact
resistances to be equal. This assumption is based on the fact that the fibers are
constrained by the model in both in-plane transverse and through-thickness direction.
Contact pressure in in-plane transverse direction is thus comparable to through-
thickness direction, resulting in similar contact resistance.
For sized fibers, contact resistance is calculated by fitting the experimental bulk
resistivity results as the sizing properties are often proprietary. The resistivity of the
equivalent unsized fiber assuming the same compaction behavior is calculated, and any
difference is assumed to be due to the sizing.
54
3.4 Experimental Results and Model Comparison
Through-thickness electrical resistivity results from our experiments and
modeling are compared. During the experiment, applied load (represented by normal
pressure), through-thickness resistivity and fiber volume fraction are recorded in real
time. Typical dataset is plotted in Figure 3.4; all other fiber types behave similarly. As
load is increased, fiber volume fraction increases until it reaches the compression limit.
Through-thickness resistivity keeps dropping during the compression process and
reaches a stable stage when fiber volume fraction reaches the upper limit. It’s
interesting to note that at the end of the compression stage, the pressure needed to
maintain the same fiber volume fraction drops, which suggests the reconfiguration of
fiber arrangement within fiber tows.
Figure 3.4 Typical dataset recorded during the compression process for Fiber A
55
3.4.1 Effect of fiber volume fraction
Through-thickness resistivity of carbon fiber tows depends largely on fiber
volume fraction. Fiber volume fraction determines the fiber to fiber spacing and thus the
number of contact points per unit length. A 3D resistor network that considers contact
resistance between carbon fibers [18] is used to model through-thickness resistivity of
carbon fiber tows.
It was shown by experimental investigations [45] that the through-thickness
resistivity of CFRP depends nonlinearly on fiber volume fraction. Figure 3.5 shows the
relation between through-thickness resistivity and fiber volume fraction during
compression for the two types of unsized fiber tows (Fiber A and B) with random
packing order investigated in this study. Simulation results from the proposed model are
plotted against experimental data. Two orders of magnitude drop in resistivity is
observed during the compression process. Abry [31] reported similar change in
transvers resistivity in CFRP. As fiber volume fraction increases, the through-thickness
resistivity decreases because of the increasing contacts between carbon fibers and
smaller contact resistance due to higher pressure. Good match of simulation results and
experimental data is found for the low (Vf =0.3) to medium fiber volume fraction (Vf
=0.55) range. At higher fiber volume fractions, the experimental results are lower
compared to the model predictions. This may be due to measurement errors of the
resistance at low magnitude or significant local variation of the assumed packing order
and change in microstructure leading to changes in the β term and fiber to fiber contact
distance compared to the model assumptions at higher fiber volume fractions. In all
cases, prediction errors are within 5% of the initial resistivity value at Vf = 0.3. Material
properties listed in Table 3-1 and model parameters listed in Table 3-2 are used in this
simulation. Random fiber packing order is assumed for all the simulations in this study.
56
Figure 3.5 Through-thickness resistivity of unsized Fiber A and unsized Fiber B as function of fiber volume fraction. Model describes experimental data well at volume
fraction below 60%. Fiber waviness term beta is 620 for Fiber A, and 365 for Fiber B, as listed in Table 3-2.
Resistivity of Fiber A bundle is approximately 5 times larger compared to Fiber
B. This can be explained by higher contact resistance and higher fiber resistance of
Fiber A. Modeled contact resistance RC and the resistance of fiber section between two
contact points Rf are compared in Figure 3.5. For both Fiber A and Fiber B, fiber
resistance is higher than the contact resistance at lower fiber volume fractions. Both
contact resistance and fiber section resistance drop nonlinearly with fiber volume
fraction. The fiber section resistance depends on intrinsic carbon fiber resistivity, cross
section area of the fiber and the length of fiber sections between contact points. The
intrinsic resistivity of Fiber A is slightly larger than that of Fiber B. While the cross-
57
section area of Fiber A is about 2 times of that of Fiber B, the distance between contact
points of Fiber A is about 1.7 times that of Fiber B. These two factors cancel out and
make the fiber section resistance of Fiber A and Fiber B of the same order. From Figure
3.5 we can see as both the fiber and contact resistances for Fiber A are larger than Fiber
B, the resistivity of Fiber A is greater than Fiber B as seen in Figure 3.6.
Figure 3.6 Comparison of contact resistance and fiber resistance for unsized Fiber A
and Fiber B. Both fiber resistance and contact resistance of Fiber B are larger than Fiber A in the present study.
3.4.2 Effect of fiber sizing
Fiber sizing is a thin coating layer on carbon fibers that is intended to enhance
interfacial properties between fiber surface and composite matrix. Sized fibers have
58
different electrical properties from unsized fibers due to the existence of nonconductive
sizing layer. With the thin sizing layer, sized fiber is comparable to CFRP in terms of
electrical conduction. The through-thickness resistivity’s of sized carbon fibers
measured in this study fall in the same range as the CFRP resistivity reported by Abry
[31] and Mizukami [46]. Figure 3.7 compares the experimentally obtained resistivity of
the sized fibers (C, D, and E) with modeled resistivity of the sized fibers. Parameters
used in the model is listed in Table 3-1 and Table 3-2. Random fiber packing order is
employed in the model. Since the properties of sizing are proprietary, contact resistance
is back calculated by fitting the experimental bulk resistivity.
Experimental data from unsized fiber B is plotted in Figure 3.6. Fiber C (sized
version of Fiber B with 1% sizing) resistivity is increased by a factor of ~40 times at Vf
=40% compared to the unsized fiber stack. The ratio increases with increasing fiber
volume fraction to ~200 times at Vf =60%. Resistivity of Fiber D (with 1% sizing) is
~10 times larger than that of Fiber E (with 0.25% sizing) tows due to thicker sizing
layer on Fiber D. Fiber C and Fiber D have similar sizing amounts and demonstrate
close resistivity.
59
Figure 3.7 Experimental resistivity and model results of sized fibers (Fiber C, D, and E).
Fiber C and D have same amount sizing (1%) and demonstrate similar resistivity, while Fiber E with less sizing (0.25%) demonstrates smaller resistivity.
For sized fibers the contact resistance is significantly larger than the carbon fiber
resistance and dominates the bulk resistivity. Figure 3.8 compares the predicted contact
and fiber resistance as a function of Vf for the sized fibers. Both resistance values drop
with increasing compaction but the contact resistance is an order of magnitude larger
and thus effectively determines the bulk resistivity.
60
Figure 3.8 Comparison of contact resistance and fiber resistance for sized fibers. Significant drop in contact resistance Rc in Fiber E is observed, which may due to the breakage of the thin sizing layer on Fiber E at higher pressures.
While contact resistance for Fiber C and Fiber D changes little during
compaction, huge drop in contact resistance is observed for Fiber E, which has the least
amount of sizing. This may be due to the penetration of the contacting neighboring fiber
into the thin sizing layer on Fiber E under high compaction pressure. Fiber C and Fiber
D have 1% sizing on the fiber surface while Fiber E has 0.25% sizing weight fraction.
The thicker sizing layer on the Fiber C and Fiber D introduces larger contact resistance
compared to Fiber E. This is validated by the resistor network model, which shows
similar contact resistances for Fiber C and Fiber D over a wide range of fiber volume
61
fractions while the Fiber E show a much lower contact resistance, especially at higher
volume fraction.
3.4.3 Effect of debulking
Figure 3.9(a) shows the compaction results of multiple cycles of the same
specimen for Fiber A. All the other fiber types tested in this study demonstrated similar
behavior during debulking process. Similar hysteresis behavior is seen for all fiber types
and has been reported before [25]. However, there lacks reports on the impact of
debulking process on the electrical conduction of carbon fiber or CFRP. The data shows
that the compaction behavior changes as multiple debulking cycles are applied. In
context of electrical resistivity, the compaction changes the state of fiber arrangement
and modifies the fiber waviness and thus fiber to fiber contact length. Figure 3.9(b)
summarizes the β-terms for three debulking cycles and for all five fiber types. For each
fiber type, 5 specimens were fabricated and tested and the statistical results are also
presented.
62
Figure 3.9 (a) Fiber A compaction data for multiple debulking cycles and (b) β for first
three compaction cycles for all five fiber types examined. For each fiber type, 5
specimens were fabricated and tested and the average β terms and their variations are plotted.
Resistivity data for all five fiber types is plotted in Figure 3.10 for three
debulking cycles. Resistivity change is most significant between the first and second
debulking step for all fiber types tested in this study. The sized fiber show at least an
order of magnitude resistivity increase after the first debulking cycle. Debulking process
contributes to through-thickness resistivity change in two ways. First, compaction
changes the state of fiber arrangement and modifies the fiber waviness and thus fiber to
fiber contact length. It is interesting to note that β is increasing with increasing
debulking cycles as seen in Figure 3.9(b), which means the average distance between
contact points becomes larger. The resulting longer conduction path increases the
through-thickness resistivity at a given fiber volume fraction. Secondly, the increase of
electrical resistivity after debulking can be attributed to decreased pressure required to
63
compress carbon fiber tow stacks into the same volume fraction after the 2nd and 3rd
debulking cycles, as shown in Figure 3.9. Since the sizing layer is highly
nonconductive, through-thickness conductive pathways are formed through the
penetration of sizing layer creating direct fiber-fiber contact. During the 1st debulking
cycle, applied pressure is high enough for the fiber sizing to break, resulting in small
electrical resistivity. During 2nd and 3rd debulking cycle, penetration of sizing layer only
happens at limited locations due to small pressure applied, resulting in high through-
thickness resistivity. Resistivity of the sized fibers seem to converge back to first cycle
values as the compaction pressure and fiber volume fraction increases. This may be due
to the penetration of sizing layer at high pressure.
64
Figure 3.10 Through-thickness resistivity during the first three debulking cycles for 5 fiber types. The figures in the top row represent unsized fibers, while figures in the
bottom row represent sized fibers: (a) Fiber A (unsized); (b) Fiber B (unsized); (c) Fiber C (sized); (d) Fiber D (sized); (e) Fiber E (sized).
3.5 Conclusions
An apparatus was designed and implemented for in-situ measurements of
through-thickness electrical resistivity of dry carbon fiber tows as a function of
compaction. The system measures fiber volume fraction accurately using a high-
resolution CCD camera and image processing techniques. Fiber waviness was
characterized and quantified from fiber tow compression tests. The data was reduced
using Gutowski’s fiber compaction model describing fiber deformation behavior under
transverse loading. Experimentally obtained resistivity of unsized fibers (fiber type A
65
and B) compared well with a 3D resistor network model implementation. Sized fibers
(fiber type C, D and E) showed more than an order of magnitude increased resistivity
due to increased contact resistance between fibers. Repetitive debulking process
rearranged fiber contacts and reduced both the contact pressure and the fiber waviness,
resulting in larger through-thickness resistivity at latter loading cycles. Debulked sized
fiber systems showed an order of magnitude higher resistivity at lower fiber volume
fraction and tends to converge to unsized fiber resistivity at higher fiber volume
fraction. The experimental results compared well with our 3D resistor network model,
indicating that the principle conduction mechanisms are captured and modeled
accurately.
While the electrical resistivity of CFRP laminate in fiber length direction
depends on intrinsic carbon fiber resistivity and fiber volume fraction and can be well
described with rule-of-mixture (ROM) model, resistivity in transverse direction is
highly scattered. The large variation is attributed to the sparse contact points between
carbon fibers, which is the key conduction mechanism in transverse direction.
66
MODELING ELECTRICAL CONDUCTION BEHAVIOR OF COMPOSITE
LAMINATES CONSIDERING RESIN-RICH LAYER
4.1 Introduction
Chapter 2 presents a microstructure based resistor network framework to
describe electrical conduction behavior of UD laminas. In real world applications,
laminates (multiple plies) are constructed from a combination of laminas with various
orientation and material types. Desired strength and thickness can be achieved by
varying the material type and orientation in each ply.
Modeling the electrical conduction of composite laminate is further complicated
by the existence of resin rich layer between plies. Many factors may contribute to the
formation of resin-rich interface including methods of fabrication and handling issues
during layup. In some applications, particles to increase toughness are added to the resin
system for better interfacial bonding which can result in resin-rich layer, as shown in
Figure 4.1.
Inter-lamina boundaries between plies are noticeable with carbon fibers
separated by excessive resin, reducing the contacts between layers.
Chapter 4
67
Figure 4.1 Demonstration of resin rich layer; inter-lamina boundaries between plies are
noticeable with carbon fibers separated by excessive resin. Reproduced with permission [47]
Unlike in the case of a UD lamina, the conduction paths within an angle ply can
be affected by the size of the plate, especially the ratio between lamina plate length and
width (the aspect ratio). Athanasopoulos [43] conducted an extensive experimental
investigation on the impact of aspect ratio on the electrical resistivity of UD angle ply.
The reported experimental results provide a benchmark to compare with the model
implemented in current study.
In summary, from modeling point of view, a composite laminate differs from a
UD lamina in various ways:
1. Different fiber orientation and/or material type in each ply
2. Existence of resin rich layer between plies that may impact the
conduction paths, especially in the through-thickness direction where contact
between carbon fibers in neighboring plies is the key contributor to the
conduction mechanisms;
68
3. Dimensions (aspect ratio, thickness etc.) can also impact the electrical
conduction behavior of composite laminates.
In this chapter, the resistor network model framework is extended to address
these differences. The modified model applies to multi-ply laminates that may have
different fiber orientation for each ply and resin rich layers. The model is validated with
reported experimental data.
4.2 Equivalent Fiber Bundle Model
Theoretically, the same micro-structure based modeling framework can be
applied to large composite laminates with minor modifications. However, modeling
every single fiber in a multiple ply laminate is unrealistic, considering the orders of
magnitude difference between single fiber dimension and the overall laminate
dimensions. A single layer 1in by 1in by 0.25 mm lamina with 50% carbon fiber
volume fraction contains at least 10,000 carbon fibers, not to mention larger structures
with multiple laminas. Solving such a large resistor network becomes a formidable
computational challenge.
A fiber bundle model is thus utilized to achieve a balance between the degree of
details of microstructure captured and computational complexity. To overcome the
limitations of fiber level micro-structure based resistor network modeling approach, a
homogenization scheme is utilized and the resistor network is constructed at the fiber
bundle level.
Figure 4.2 shows the equivalent fiber bundle model. As discussed in Chapter 2,
a resistor network is constructed based on material properties and micro-structure (fiber
to fiber contact). Constant voltage and ground boundary surface condition is then
69
applied on the outer surfaces in the three primary directions respectively and resulting
current through each boundary is recorded. The overall resistance in three directions can
be calculated by dividing voltage difference on the two opposite boundary surfaces with
the total current flowing through them. The fiber bundle can then be represented with
three resistors in the primary directions, which will be used as the fundamental building
block for more complicated structures such as angle-ply and multiple ply with resin rich
layer.
Figure 4.2 Schematic illustration of fiber bundle model. A fiber bundle can be
represented with 3 resistors whose values can be calculated from the resistor network model with current injected from three primary directions respectively.
The overall electrical properties of a UD lamina can be represented by a
representative finite volume that contains significantly smaller number of carbon fibers.
Electrical resistivity of UD fiber bundles can be calculated through a homogenization
scheme based on resistor network constructed from a representative volume. Utilizing
this result, the computational scale for a multi-ply laminate can be reduced.
With this fiber bundle representation, UD laminae with large dimensions can be
modeled, as demonstrated in Figure 4.3. While the fundamental resistor network
70
framework remains unchanged, each resistor now represents a fiber bundle section
instead of fiber section. The statistical characteristics can be introduced by assigning
resistance values that obeys statistical distribution to the resistors representing fiber
bundles.
Figure 4.3 UD lamina represented by fiber bundle model. Each line section represents a fiber bundle, instead of single fiber as in the previous resistor network model discussed
in Chapter 2.
4.3 Angle-ply Model
An angle ply is made by cutting the UD ply at an angle θ to the fiber length
direction, as depicted in Figure 4.4. For an angle ply, there is an angle θ between the
material coordinate (𝒖 − 𝒗 ) and the structure coordinate (𝒙− 𝒚). In the material
coordinate, axis 𝒖 is aligned with the fiber length direction, while axis 𝒗 is aligned with
the transverse direction. The structure coordinate system is where the dimensions of the
composite laminate are defined: axis 𝐱 an y are aligned with the length and width
direction of the plate respectively.
71
Figure 4.4 For an angle ply, there is an angle θ between the material coordinate (𝒖 − 𝒗 )
and the structure coordinate (𝒙 − 𝒚).
To formulate a resistor network, the angle ply is discretized using an approach
that mimics the way how an angle ply is cut from a UD prepreg, as demonstrated in
Figure 4.5.
Figure 4.5 Schematic drawing of a minimum bounding rectangle (MBR) for a 45∘ ply
A minimum bounding rectangle (MBR) that defines the maximum extents of
given 2D object (in our case, the angle-ply) is created in the material coordinate system
72
(𝑢 − 𝑣). The MBR is then treated as a UD lamina and discretized into a mesh grid
composed of multiple unit cells. Each unit cell represents a fiber bundle as defined in
the previous section, and the size of the unit cell is determined from the number of
element in each direction:
Lx𝑐 =
𝐿𝑥𝐵
𝑛𝑥 (4.1)
Ly𝑐 =
𝐿𝑦𝐵
𝑛𝑦(4.2)
Lz𝑐 =
𝐿𝑧𝐵
𝑛𝑧(4.3)
Here, superscript 𝐁 denotes the bounding rectangle and superscript 𝐂 denotes
the unit cell.
The distributed resistance of the unit cell (fiber bundle) in three representative
directions can be calculated as:
Rx𝐶 = 𝜌𝑥 ∗
𝐿𝑥𝐶
𝐿𝑦𝐶 ∗ 𝐿𝑧
𝐶(4.4)
Ry𝐶 = 𝜌𝑦 ∗
𝐿𝑦𝐶
𝐿𝑥𝐶 ∗ 𝐿𝑧
𝐶(4.5)
Rz𝐶 = 𝜌𝑧 ∗
𝐿𝑧𝐶
𝐿𝑦𝐶 ∗ 𝐿𝑥
𝐶(4.6)
The next step is to crop the discretized mesh grid in the material coordinate
system with the boundary box in the structural coordinate system. Only fiber bundle
sections that fall in near proximity to the angle-ply boundaries are retained to formulate
73
the resistor network. The workflow for constructing a 3D resistor network for an angle
ply is illustrated in Figure 4.6.
Figure 4.6 Schematic illustration of the workflow for constructing a 3D resistor network
for an angle ply
While in this demonstration, identical fiber bundle properties are used from each
of the cells and thus each lamina is considered as homogeneous, the capability of
modeling the stochastic characteristics of the microstructure is retained by assuming a
statistical distribution of the resistivity, instead of a constant resistivity:
ρi ~ 𝑁(𝜌𝑖0, 𝜎𝑖), 𝑖 = 𝑥, 𝑦, 𝑧 (4.7)
74
where 𝜌𝑖0 is the nominal resistance in direction 𝑖 = {𝑥, 𝑦,𝑧}, and 𝜎𝑖 is the
variance in resistance in the corresponding direction (𝑥, 𝑦, 𝑜𝑟 𝑧).
4.4 Multi-ply Model with Resin Rich Layer
Figure 4.7 shows the ply orientations in a multi-ply CFRP laminate. A multi-ply
model can be built by connecting multiple single ply models in the through-thickness
direction. To quantitatively describe the conduction behavior, models for the contact
resistance between plies are needed. Hence, number of inter-ply connections and the
contact resistance values are needed.
Figure 4.7 Ply orientations in a multi-ply CFRP laminate. Carbon fiber tows are
schematically shown with black lines.
4.4.1 Number of inter-ply connections
For a [0∘|90∘] laminate it would be misleading to use a straight rigid rod model
to determine the inter-ply connections. If the fibers are assumed to be straight and
evenly distributed in both layers as depicted in Figure 4.8(a), the nominal contact
75
density (defined as number of connections per unit area) is 1
𝐿𝑠2, with 𝐿𝑠 being the
distance between the center of two neighboring fibers. This will give extremely huge
number of connections between layers, considering the small value of 𝐿𝑠 (normally of
the order of sub micrometers).
Figure 4.8 Reduction in number of contacts due to fiber undulation. (a) contacts
between fibers in [0-90] layup assuming fibers are straight; (b) reduced contacts between fibers in [0-90] layup considering fiber undulation; (c) contacts between fibers
in [0-45] layup assuming fibers are straight; (d) contacts between fibers in [0-45] layup considering fiber undulation.
76
As demonstrated from Chapter 2 and 3, carbon fibers are undulated even in a
UD lamina. Let’s first consider the case where only fibers in one ply are undulating
using the same fiber arch assumptions as in Chapter 2. The connection density dC in this
special case becomes
dC =1
𝐿𝑐𝐿𝑠(4.8)
Now consider the case where carbon fibers in both layers are undulating with
average contact span of 𝐿𝑐, as depicted in Figure 4.8(b). Red dots in Figure 4.8(b)
denote the inter-ply connections. The connection density is further reduced to
dC =1
𝐿𝑐𝐿𝑐(4.9)
For a two-ply laminate with difference of θ in fiber orientation, the estimation of
contact density is conducted in a similar approach, as demonstrated in in Figure 4.8(c)
and (d). In general, connection density is a function of θ as defined in Equation 4.10.
dC(𝜃) =1
𝐿𝑐2 sin(𝜃)
(4.10)
With connection density dC(𝜃) determined, the number of inter-ply connections can be
calculated as in Equation 4.11.
NCo = dC(𝜃)𝐿𝑥𝐿𝑦 (4.11)
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The superscript 𝑜 denotes the case without resin rich layer. 𝐿𝑥 and 𝐿𝑦 are the
dimensions of the laminate plate defined in structural coordinate system.
Existence of resin rich layer further reduces the contacts between plies in
addition to the impact of fiber waviness. In this study, an inter-ply connectivity term is
introduced to describe the quality of the inter-ply interface:
C =Nc𝑁𝑐𝑜
(4.12)
Where Nc is the number of actual contacts between plies, and Nco is the nominal number
of contacts if resin rich layer doesn’t exist, as defined in Equation 4.11.
Given an inter-ply connectivity term, the actual number of contacts Nc can be
back calculated from Equation 4.12. The plies are then connected by randomly picking
Nc points in the virtual contact layer, and connecting these points to the closest points in
the grids representing the two plies correspondingly. While the calculations of inter-ply
contact resistances are based on fibers, these resistances are grouped in the fiber tow
representation.
4.4.2 Inter-ply contact resistance
4.4.2.1 Composition of contact resistance
The inter-ply contact resistance Ri depends on the mechanisms that drive the
connections (doped resin, partial discharges, added conducting inclusions for example
nanotubes et al.). In this study, Ri is assumed to be contributed by two sources: (1)
Constrictive resistance between carbon fibers as discussed in Section 2.2.7. The value is
in the range of 10 ~ 100 Ohm for carbon fibers investigated (AS4, IM7, T700, T800).
78
(2) Tunneling resistance induced from the carbon fibers separated by thin resin layer.
The interface resistance can be easily changed should other conduction mechanisms
need to be considered.
Calculation of contact resistance is complicated, since in an actual composite
material it may be affected by a number of factors, such as the type of sizing and resin
system, processing pressure, precondition of fiber surface, and the tunneling gap. For a
specific contact, it is often difficult to know the thickness of an insulating film and to
determine the exact value of the contact resistance. Fortunately, average sizing
thickness can be back calculated with limited information provided.
Sizing is the thin coating layer applied to carbon fibers to enhance the fiber-resin
interface quality. Although the detailed properties of the sizing material are usually
confidential and unavailable, it is made from polymers and can be regarded as
insulators. Sizing thickness depends on the density and weight fraction of the sizing,
and the density and diameter of the carbon fiber, as expressed in Equation 4.13. For two
typical carbon fibers IM7 and T700 with the properties listed in Table 4-1, the sizing
thickness is plotted in Figure 4.9. Sizing thickness ranges from 3nm to 30nm.
tsizing =𝑑𝑓2∗ (√
𝑤𝑡𝜌𝑓𝑖𝑏𝑒𝑟𝜌𝑠𝑖𝑧𝑖𝑛𝑔
+ 1− 1) (4.13)
Table 4-1 Properties of IM7 and T700 carbon fiber
Property IM7 T700
Specific Heat [ kJ/kg∙K ] 0.879228 0.753624
Electrical Resistivity [Ωm ] 1.5 1.6 × 10-5
Thermal Conductivity [W/(mK)] 5.4 9.196
79
Density [g/cm3 ] 1.78 1.8
Modulus [GPa] 276 230
Figure 4.9 Sizing thickness as a function of sizing weight fraction and fiber radius
During the curing process, not all sizing material is dissolved into the resin
matrix, leaving some of the carbon fibers separated by thin layer of resin/sizing. In
addition to coated sizing material on carbon fiber, excess resin that get trapped between
the fibers also contribute to the separation of carbon fibers. Formation of such insulating
polymer layers were also reported by other researchers [48].
80
Depending on whether a thin resin layer exists between contacting fibers, inter-
ply contact resistance comes from two parts: 1) constriction resistance coming from the
direct contact between carbon fibers; and 2) tunneling resistance induced from thin resin
or sizing layer between carbon fibers. As demonstrated in Figure 4.10(a), under low
processing pressure, the thin resin/sizing layer between carbon fibers is compressed but
not penetrated, leaving the carbon fibers separated by a very small distance. The yellow
area denotes the area where the separation distance is in the sub 100 nm range, when
tunneling resistance starts to show an impact. Under high processing pressure, sizing
can be penetrated and direct contacts between carbon fibers is formed, denoted by the
red area in Figure 4.10(b).
Figure 4.10 Schematic illustration of the parts of contact resistance. (a) tunneling resistance is dominant when a thin resin layer exists between carbon fibers; (b)
constriction resistance becomes dominant if direct contact between carbon fibers is formed.
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4.4.2.2 Critical processing pressure
Critical pressure is the processing pressure under which the resin layer is
penetrated and direct contact between carbon fibers starts to form. Below certain critical
processing pressure, sizing alone can carry the load through plastic deformation.
Critical load Lc carried by resin layer can be calculated from Equation 4.14.
FC = 𝐻𝐴𝑐 (4.14)
Where 𝐻 is the hardness of sizing, and 𝐴𝑐 is the critical loading carrying area. Relation
between the radius of loading carrying area (yield radius 𝑎) and the indentation depth d
is defined in Equation 4.15.
a = √df𝑑 (4.15)
c
Combining Equation 2.2, 2.14, 4.13, 4.14, and 4.15 yields Equation 4.16.
P =4Hπd
(πηvf−2√
πηvf)β
(4.16)
For the fibers to have direct contact, indentation depth d is twice the sizing
thickness tsizing. Combining 4.13 and Equation 4.16, the critical processing pressure as
function of sizing weight fraction can be calculated, as plotted in Figure 4.11.
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Figure 4.11 Critical processing pressure for fibers in direct contact. Below critical pressure, thin sizing layer exists between carbon fibers, and the dominant conduction mechanism is tunneling conduction. Above critical pressure, direct contact between
carbon fiber becomes the dominant conduction mechanism.
4.4.2.3 Constriction resistance for fibers with direct contact
For carbon fibers with direct contacts, the constriction resistance is calculated
using the same method as in the case for unsized carbon fibers discussed in Chapter 2.
The formula is given in Equation 2.15.
4.4.2.4 Tunneling resistance between fibers with small separation distance
One of most widely used formula for calculating tunneling resistance is
proposed by Simons [49], as stated in Equation 4.13. Simmons’s equations predict the
electrical tunneling resistance between two planar electrodes separated by a thin
83
insulating layer. In this study, the carbon fibers are modeled as two planar electrodes,
due to their large surface curvature compared to the dimension of resin film thickness,
separated by a polymer film. The film thickness dependence of the tunneling current
can be expressed as:
J = (6.2 ×1010
Δs2){𝜑1 exp(−1.025𝛥𝑠𝜑1
0.5) − (𝜑1 + 𝑉)exp(−1.025𝛥𝑠(𝜑1 +𝑉)0.5)},(4.13)
where
Δs = s2 − 𝑠1, (4.14)
φ1 = 𝜑0 − (𝑣
2𝑠)(𝑠1 + 𝑠2) − [
5.75
(𝐾(𝑠2 − 𝑠1))] ln [
s2(s−s1 )
s1(s−s2 )] (4.15)
s1 =6
𝐾𝜑0(4.16)
s2 = s [1 −46
3φ0𝐾𝑠 + 20− 2𝑉𝑘𝑠] +
6
𝐾𝜑0(4.17)
where φ0 is the height of the rectangular potential barrier (in volt). 𝑠 is the
insulating layer thickness (in angstrom), K is the dielectric constant of the insulating
film material, V is the applied voltage difference across the thin resin film. According to
Li et al. [50], φ0 is assumed to be 5.0 eV and polymer matrix dielectric constant K is
assumed to be 3.98.
The tunneling resistance is calculated as
Rtunneling =𝑉
𝐽𝐴𝑐 (4.18)
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Where 𝐴𝑐 is the contact area.
Figure 4.12 plots tunneling resistance as function of film thickness, where
contact area is assumed to be of the same order as the contact area from direct carbon
fiber contacts (~1e-12 m2). It shows that film thickness smaller than 0.7 nm yields a
tunneling resistance comparable to fiber resistance and fiber-fiber contact resistance.
This implies that without loading, sizing acts as an insulator while loading may
compress or damage sizing, increasing its conductivity.
Figure 4.12 Tunneling resistance as function of separation distance
The impact of constriction and tunneling resistance is further investigated for
carbon fibers with various sizing weight fraction, as shown in Figure 4.13. Below the
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critical pressure (denoted with the red dashed line), contact resistance is calculated with
the tunneling resistance formula, while constriction resistance formula is used under
pressure higher than the critical pressure.
Figure 4.13 Combined constriction resistance and tunneling resistance as function of processing pressure. Below the critical pressure (denoted with the red dashed line),
contact resistance is calculated with the tunneling resistance formula, while constriction resistance formula is used under pressure higher than the critical pressure.
4.4.3 Construction of resistor network
Figure 4.14 presents the workflow for constructing a 3D resistor network for
multi-ply CFRP laminate with resin rich layer. First, each ply is treated as UD lamina
and modeled using the approach for angle ply discussed in Section 4.3. Inter-ply
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contacts are modeled using the approach discussed in Section 4.4. A 3D resistor
network can be constructed by connecting multiple plies with the inter-ply contact
resistance.
Figure 4.14 Schematic illustration of the workflow for constructing a 3D resistor
network for multi-ply CFRP laminate with resin rich layer. [0/0] layup is presented for clarity; the model can also consider a random ply orientation.
The following figure shows some examples of 3D resistor network model of
multi-ply laminates with various fiber orientation and inter-ply connectivity.
Figure 4.15 Demonstrations of model for multi-ply laminate. For the sake of simplicity,
only one layer of resistors is plotted for each ply, while in real calculations, multiple
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layers of resistors are used for each ply. (a) [0/90] two ply laminate with 60% inter-ply
connectivity; (b) [0/45] two play laminate with 40% inter-ply connectivity.
Once the resistor network has been formulated, current can be injected at the
opposing surfaces to calculate resistance in three primary directions. At one surface,
current source is introduced at a single injection point and then distributed over the
boundary surface through resistors connected in parallel. These resistors represent
discretized conductive electrodes and have small resistance values that can be
calculated from the electrical resistivity of the electrode material, copper for example.
At the other surface, electrode is modeled in the same way, except that the distributed
resistors are connected to ground, instead of the current source.
𝑅𝑠𝑦𝑠𝑡𝑒𝑚 =𝑈𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝐼𝑖𝑛𝑗
(4.19)
The electrode resistance is much smaller than that of the composite. Thus, the
resistance of the composite panel can be approximated by the resistance of the system
including electrodes:
𝑅𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 ≅ 𝑅𝑠𝑦𝑠𝑡𝑒𝑚 = 𝑈𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑𝐼𝑖𝑛𝑗
(4.20)
The inclusion of electrode resistance not only eliminate the requirement for
further homogenization of the voltage on the boundary surface to calculate equivalent
resistance of the system, but also provide the capability to investigate the impact of the
contact quality between carbon composite and electrodes on measured resistance.
Parametric studies are conducted and the results are reported in Section 4.6.
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4.5 Model Convergence Tests
The diameter of carbon fiber (5 − 7 μm) is small compared to the size of a
normal test coupon (a few inches), resulting in huge number of fibers. One would
require a large number of nodes to represent the fibers in the present model. If chosen
properly, a fraction of the microstructure can be a good representative of the overall
structure in terms of modeling electrical conduction behavior. A representative volume
element (RVE) is chosen to calculate the equivalent resistivity of the overall material
system. To evaluate the effect of RVE size, a series of microstructures were constructed
using increasing RVE size. The length, width and thickness of RVE is chosen such that
the number of nodes in the 3 primary directions within the generated resistor network is
similar. Number of nodes is increased by simultaneously increasing the dimensions in
three directions to keep similar node increasing rate for all three primary directions. All
the simulations in this study are run with Dell Precision 1500 workstation with 4-core
Inter Xeon 5600 series, 2.0 GHz processor and 16 GB of DRAM.
Researchers [36] have noticed that the size of the model (number of nodes)
could affect the calculated results however the simulated results tend to converge with
increasing number of nodes. Romanov [36] did a convergence test using 2D fiber
arrangement and revealed that a representative volume element (RVE) consisting of
around 200 fibers gives a good representation of a real UD composite laminate
structure, the results were confirmed by our simulations in Chapter 2. In this study,
convergence tests of electrical resistivity of CFRP laminate in three primary directions
were conducted using 3D resistor network. The material properties are the properties of
HTA-7 epoxy composite laminate. Properties of HTA-7 and model parameters are listed
in Table 4-2.
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Table 4-2 Properties of HTA-7 fiber and model parameters Parameter Value
Filament diameter 7 μm
Elastic modulus 238 GPa
Electric resistivity 1.6 e-5 Ωm
Specific heat 1.13 kJ/(kg*K)
Thermal conductivity 6.83 W/(m*K)
Fiber waviness term β 350
Figure 4.16 shows the results of convergence tests for both resin-rich interface
and resin-lean interface configurations: resin-rich layer is considered with inter-lamina
connectivity value of 0.1, while inter-ply connectivity of 1.0 represent a laminate with
resin-lean interfaces. Resistivity values are normalized with respect to the first
resistivity value in each series, to clearly demonstrate the percentage change in
resistivity values. Ten simulations were conducted using the same model size and
parameters, considering the stochastic characteristics introduced in the current model,
the standard deviation between the calculated results are also plotted.
90
Figure 4.16 Convergence tests for two cases: (a) connectivity = 0.1; (b) connectivity = 1.0. Large variations are observed for resistivity in Z direction, especially for laminate
with resin-rich interface (inter-lamina connectivity = 0.1)
The first thing to notice is that the variance of resistivity values of resin-rich
interface specimens (inter-ply connectivity value of 0.1) are larger than that of resin-
lean interface specimens (inter-ply connectivity value of 1.0), especially in the through-
thickness direction. For both cases, variation of resistivity is the largest in Z direction,
followed by Y direction, while the variation in X direction resistivity is rather small.
The same pattern is confirmed by recent experimental observations reported by
Hirano[47], who concluded that the electric conductivity of toughened CFRP (with
resin-rich interface) is not a material property but depends on resin flow during
fabrication. The large variance can be explained by the randomized locations of the
reduced number of contact points at the inter-lamina interface due to reduced inter-ply
connectivity, introducing more uncertainty into the calculations.
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In general, the variance tends to decrease with increased model size. As the
model size increases, the variations induced by the unconnected resistors at the edges of
the network is reduced, as demonstrated in Figure 4.17. Model calculations are
relatively steady at 6000 nodes (corresponds to about 300 fibers) and above for all
configurations. Hence all computational models used in the following study used model
size that ensured convergence of results.
Figure 4.17 Schematic illustrating unconnected fiber at the edge of resistor network.
4.6 Model Validation
4.6.1 Through-thickness resistivity compared with reported experimental data
for CFRP
The model is first validated with reported experimental data for electrical
resistivity of UD CFRP laminates under low current density, with and without resin rich
layers. Multiple researchers have reported electrical resistivity of CFRP laminates.
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While the X direction resistivity/conductivity values are quite consistent and Y
resistivity a little more scattered, large discrepancies can be found in the reported
through-thickness direction electrical resistivity (from ~0.05 Ωm to ~100 Ωm) [47],
[51]. This discrepancy can be partially explained by the impact of resin rich layer with
the current model, as discussed in the next section.
For this study, the experimental data reported by Abry [51] is chosen for
comparison for two reasons. Firstly, in addition to the easily accessible electrical and
mechanical properties of the fiber and resin systems used in the measurement, Abry also
provided cross-sectional micrographs for the laminates tested, which are handy in
determining model parameters such as inter-ply connectivity and fiber volume fraction.
Also, the same dataset was compared against a 3D microstructure based resistor model
in Chapter 2 and demonstrated the model’s capability to describe electrical conduction
behavior of composite laminates without resin rich layer but not for those with resin
rich layers. The comparison in this study shows the added feature of inter-lamina
connectivity which captures the difference in resistivity of Y and Z direction for CFRP
laminate with resin rich layers.
The way Abry varied the fiber volume fraction is by intentionally adding resin
rich layer in-between adjacent plies, rather than uniformly compressing the fiber
preforms as performed in CFRP laminate processing. In the simulations,
microstructures are created in a similar way to achieve the desired fiber volume
fraction. Fibers at the interface were partially removed to create a resin-rich interface.
The model parameters are listed in Table 4-2. The comparison for resistivity in the X
and Y directions were conducted in Chapter 2, hence this study focuses on the electrical
conduction in Z direction.
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Figure 4.18 summarizes the comparison between simulation results and
experimental data. The three horizontal solid lines represent experimental data
corresponding to three fiber volume fraction values. Since inter-lamina connectivity is
not quantitatively accessible, simulations were conducted with a series of inter-lamina
connectivity values ranging from 0.02 (resin-rich interface) to 1.0 (resin-lean interface).
The intersections between simulation results (dashed lines) and experimental data (solid
lines) indicate inter-lamina connectivity values that provide good matches between
experimental data and simulation results. Inter-lamina connectivity values for laminates
with fiber volume fraction of 0.43 and 0.48 are significantly smaller than of 0.59,
reflecting the fact that resin-rich interface was manually created for specimens with
lower fiber volume fractions, as demonstrated from the microscopic graph of the cross
section of the specimens in Figure 4.19.
94
Figure 4.18 Comparison between simulation results and reported experimental data
from Abry [51]
95
Figure 4.19 Cross-section of the unidirectional specimens. (a) Vf=0.43; (b) Vf=0.59.
Reproduced with permission [51]
4.6.2 Resistor network model compared with FEM, analytical and experimental
results
In this study, the developed resistor network model is validated with models
from various sources as well as experimental data, using carbon preforms with various
aspect ratios as test scenarios. Aspect ratio (denoted by λ) of a laminate plate is the
length to width ratio of the specimen, as demonstrate in Figure 4.20. The length
direction is aligned with the test direction (the direction in which current/voltage is
applied). An λ smaller than 1 means width is larger than the length.
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Figure 4.20 Schematic illustration of specimen aspect ratio (λ). λ is defined as the length to width ratio of a laminate plate, where length direction is aligned with the test
direction (direction in which current/ voltage is applied).
Weber and Kamal [52] have shown the dependence of the measured
conductivity on the aspect ratio of CFRP samples. Tse et al. [53] have also shown
dependence of the measured electrical conductivity on the width of CFRP sample
transverse to the fiber direction. This aspect ratio dependence of CFRP resistivity was
clearly confirmed by Athanasopoulos [43] with extensive experimental
characterizations.
In this work, virtual experiments that mimics Athanasopoulos’ experimental
investigations are conducted, and the simulation results are compared with the reported
experimental data. A finite element (FE) model is also developed to examine the
accuracy of the resistor network model using a commercial FE software package
COMSOL®. The carbon fiber preform is modeled as a simplistic block with
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homogeneous material properties. The effect of ply orientation is considered by rotating
the material coordinate.
In both the resistor network model and the FE model, length of the virtual
specimen is 0.1 m and thickness is 0.18 mm, equal to the specimen dimensions used in
the reported experiments. The width is varied to achieve the desired aspect ratio. Inter-
ply connectivity is assumed to be 100% to reflect that carbon fiber preforms were used
in the experiments, thus there was no inter-ply resin-rich layer.
Figure 4.21 plots the experimental data and theoretical results of carbon
preforms as function of aspect ratio (λ) and ply orientation. At small λ, conduction
mechanism is dominated by the direct connection between electrodes with continuous
carbon fibers, thus the resistivity is reduced and reaches a steady state value at similar
levels to the fiber tow resistivity in the X direction. As λ increases, less connections
between electrodes through continuous carbon fibers can be found. The contact between
carbon fibers in transverse direction dominate the conduction behavior. Consequently,
resistivity increases and converges to a steady state value at similar levels to fiber tow
resistivity in Y/Z direction.
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Figure 4.21 Experimental and theoretical results as function of aspect ratio (λ) and the
fiber direction (θ) of the UD preform for thickness h = 0.18 mm. Reproduced with permission [43]
Figure 4.22 plots the resistivity as a function of λ and fiber orientation, using
simulation results from the virtual tests. Dashed lines and solid lines in the figure
represent results from FE model and the resistor network model respectively. Small
discrepancy between the two models can be noticed. In general, the difference between
the resistor network model and FE model is less than 10% for all configuration,
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indicating high fidelity of the developed resistor network model. A good match is also
achieved when comparing the simulation results with experimental data reported in
Figure 4.21. Not only do the resistivity changes with λ and fiber orientation follow the
experimental trends, in addition the absolute values of resistivity also match well. These
extensive virtual experiments validate the capability of the developed resistor network
for considering the effect of geometrical parameters on electrical conduction behavior
of CFRP.
100
Figure 4.22 Results from the virtual tests. Solid represent results from FE model lines (denoted with “FEM” in legend), while dashed lines represent resistor network model (denoted with “ResNet” in legend). The solid black line denotes the “critical aspect
ratio” 𝜆𝑐𝑟, and the two dashed black lines denote the rough boundary for 𝜆 ≪ 𝜆𝑐𝑟 and
𝜆 ≫ 𝜆𝑐𝑟 respectively.
The graph can be divided into three regions. The solid black line denotes the
“critical aspect ratio” 𝜆𝑐𝑟, and the two dashed black lines denote the rough boundary for
𝜆 ≪ 𝜆𝑐𝑟 and 𝜆 ≫ 𝜆𝑐𝑟 respectively. In Region I, which is located between the two
dashed lines, resistivity is sensitivity to 𝜆. In Region II, which is located to the left of
the boundary line 𝜆 ≪ 𝜆𝑐𝑟, there exists direct connections between electrodes through
101
continuous fiber, thus conduction in fiber length direction is the main conduction
mechanism. In Region III, which is located to the right of the boundary line 𝜆 ≫ 𝜆𝑐𝑟,
there is no direct connections between electrodes through continuous fiber, thus current
is forced to conduct in the in-plane transverse direction through contact points between
fibers. The dominant conduction mechanisms can be demonstrated with current density
plot from FE model for an angle ply with 45∘ fiber orientation, as shown in Figure 4.23,
although it cannot show the localized current concentration through contact spots. Color
on the surface represents electric potential, while thin red lines represent current
streamlines.
Figure 4.23(a) shows a specimen with 𝜆 of 0.2, representing conduction in
Region II. Figure 4.23(b) shows a specimen with 𝜆 of 5, representing conduction in
Region III. In Region II, current streamlines are mostly straight from one boundary to
the other, indicating the current is conducted in the fiber length direction. Resistivity in
this region is in the similar level as fiber tow resistivity in fiber length direction, as seen
from Figure 4.22. In Region III, current streamlines are distorted and forced into the
transverse direction. Resistivity is close to the in-plane transvers resistivity of fiber
tows.
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Figure 4.23 Current streamline plot from FE model for an angle ply with 45∘ fiber
orientation. (a) aspect ratio 𝜆 = 0.2, representing conduction in Region II; (b) aspect
ratio 𝜆 = 5, representing conduction in Region III.
4.6.3 Parametric study of the impact of resin rich layer
The aim of these studies is to demonstrate the impact of geometrical parameter
(length to width ratio, ply thickness and inter-ply connectivity) on the electrical
conduction behavior of UD carbon composites with various ply orientations.
4.6.4 Impact of inter-ply connectivity on resistivity in the three principal
directions
One major advantage of the current model over the model discussed in Chapter
2 is its capability of considering resin-rich interface layer. The aim of this study is to
investigate the impact of resin-rich layer (describe with inter-ply connection term) on
the electrical resistivity in all three principal directions.
With the existence of resin-rich interface layer, electrical conduction of CFRP
laminates in through-thickness direction demonstrates different behavior compared to
that in the fiber length direction and in-plane transverse direction. To start with, large
variance of through-thickness resistivity can be observed even within specimens
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processed under same conditions. Hirano [47] noticed the through-thickness
conductivity varied from 164.74 Ωm to 47.62 Ωm for IMS60/133 UD CFRP laminates
during the fabrication process where resin can flow around, resulting in changing inter-
lamina connectivity. A model that considers stochastic inter-ply interface quality can
address this microstructure and fabrication process dependent electrical property. Due
to the stochastic nature of inter-lamina interface quality, it’s not a trivial job to
experimentally create inter-lamina interface with desired connectivity values.
Numerical parametric analysis was thus conducted in this study to reveal the impact of
resin-rich layer on electrical resistivity of CFRP laminates in the three principal
directions. The same material properties and model parameters from Table 4-2 were
used. Inter-ply connectivity was varied from 2% to 100% and 10 simulations were
conducted for each inter-ply connectivity and direction configuration. The calculated
resistivity values and their variations are plotted in Figure 4.24. All resistivity values are
normalized by dividing the resistivity with the resistivity value in Z direction at inter-
lamina connectivity of 2%.
The impact of inter-ply connectivity on X direction resistivity is negligible,
while the through-thickness resistivity is sensitive to changes in inter-lamina
connectivity. This implies that the electrical resistivity of CFRP laminates should be
interpreted statistically, rather than deterministically. As for the experimental
characterizations, large number of measurement repetitions are needed to accurately
capture the stochastic electrical resistivity, especially in the through-thickness direction.
It can also be noted that electrical resistivity of CFRP laminate is not isotropic in
transverse (Y and Z) directions, with resistivity in Z direction at least one order of
magnitude larger than in the Y direction. Resistivity in the Z direction drops rapidly
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with increasing inter-lamina connectivity (better interface quality), and tends to
converge to resistivity value in the Y direction. Similar behavior was noticed by
Abry[51] that with higher fiber volume fraction (0.59) and good inter-lamina interface
resistivity in Z direction (0.0482 Ωm) is comparable to that in the Y direction (0.0416
Ωm).
Figure 4.24 Impact of inter-lamina interface. Three levels of inter-lamina connectivity are demonstrated with the inserts. Resistivity in Z direction is sensitive to changes in
inter-lamina connectivity especially in lower connectivity range, while the influence of inter-ply connectivity term is negligible on resistivity in the X and Y directions.
4.7 Summary and Conclusions
Extending the modeling work described in Chapter 2, a resistor network that
uses fiber bundle as the minimum modeling unit is implemented to achieve a balance
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between details of the micro-structure considered and computational burden. Ply
orientation other than 0∘ can also be considered.
Stochastic characteristics of CFRP microstructures including fiber arrangement,
fiber waviness, and inter-lamina contacts are also considered in the current model which
not only explains the large through-thickness resistivity as noticed by other researchers
in their experimental measurements, but also explains the abnormally large variance
noticed in the measured through-thickness resistivity values of specimens processed
under the same conditions.
The impact of specimen aspect ratio is investigated using the angle ply model
and compared with experimental results reported by other researchers. Good match
between simulation results and experimental data indicates that the model captures the
major conduction mechanisms for angle plies.
Special attention has been paid to the impact of resin rich layer. It’s found that
depending on processing pressure, inter-ply contact resistance can be dominated
between two conduction mechanisms: tunneling conduction and direct conduction.
Formulas for calculating contact resistance under various conditions are derived.
Tunneling resistance is orders of magnitude higher than constriction resistance. An
inter-ply connectivity term is introduced to describe the severity of resin-rich interface.
A parametric study is carried out to demonstrate the impact of inter-ply connectivity on
resistivity in three principal directions. It’s been found that while in-plane resistivity is
barely affected by inter-ply connectivity, the through-thickness resistivity largely
depends on inter-ply connectivity, a 90% drop in through-thickness resistivity can be
observed if inter-ply connectivity increases from 0.1 to 1.0. The impact of resin-rich
layer is two-fold. Firstly, existence of resin-rich layer reduces the number of conductive
106
paths between adjacent laminas, resulting in higher through-thickness resistivity.
Secondly, randomized location of limited inter-lamina contact points due to existence of
resin-rich layer introduces uncertainties into the network, leading to larger variance of
the overall resistivity.
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MODELING HIGH ELECTRIC CURRENT IMPACT
5.1 Introduction
CFRP exposed to high current environment such as lightning strike can undergo
severe damage. Of all the damage mechanisms induced due to high electric currents,
joule heating is the most obvious one. While lightning strike with 200 kA peak current
presents an extreme case of high current conditions, other applications where the overall
current applied doesn’t seem to be high at first glance can also impose detrimental
effects on CFRP, due to the localized current concentration effect to be discussed in this
chapter.
Several publications [29], [54], [55] have considered the issue of material
property change under high temperatures at the macro scale. Joule heating is the key
cause for substantial increase in the interconnect temperature and the reduction of
overall resistivity of the composite laminates. However, the effect of poor inter-ply and
intra-ply connections at micro-scale involving single fiber or fiber tows, which are
subject to much higher current density than the carbon fibers, has not been adequately
addressed in reported simplified finite element models treating CFRP as homogeneous
materials [8], [19].
Although knowing the temperature at each intra-ply and inter-ply connection
points is the first step toward any approach of further thermal effect analysis, the
temperature profile of a single interconnect is difficult to obtain experimentally due to
its microscale. The traditional approach has been to use the temperature of the entire
Chapter 5
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laminate to update the temperature- dependent electrical resistivity in the reported
research studies to date. However, it can provide neither the local temperature rise
profile at the connection points nor the resistivity change due to material degradation
mechanisms specific to certain connection points such as resin breakdown. In addition,
failing to consider the localized Joule heating at contact points will underestimate the
temperature rise. Consequently, the predicted resistivity change of the CFRP will be
much lower than actually observed.
In the present work, current concentration at two levels (at fiber and cross
contact points) are introduced, followed by discussion of its impact on the thermal
development in the CFRP at the microscale level. Possible temperature and electric
field dependent material properties and degradation mechanisms are investigated, based
on which a modified resistor network is implemented to capture the nonlinear
conduction behavior of CFRP under high current density.
5.2 Current Concentration at Micro-Scale Level
5.2.1 Current concentration within carbon fibers
While it would be unrealistic to characterize the current density or temperature
rise at each contact point, analytical model can be utilized to demonstrate the impact of
excessive Joule heating at the connection points.
The current distribution within CFRP laminate is uneven due to the fundamental
conduction mechanism that only carbon fibers are conductive and carry the current. To
demonstrate the current concentration effect, let’s consider a simplified UD lamina
which is subjected to a current in the through-thickness direction, as shown in Figure
5.1. A square packing order is assumed to simplify the analysis. With this assumption,
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the composite lamina can be divided into multiple layer, with carbon fibers in the same
layer exhibiting the same potential patterns.
110
111
Figure 5.1 Schematic illustration of current concentration at intra-ply contact points. (a)
current applied to top surface of CFRP laminate; (b) RVE containing two contacting fiber sections; (c) current path through carbon fibers and contact points. Current is
concentrated at the contact points due to small contact area compared to carbon fiber cross section area.
While at the macro-scale level, the current seems to be evenly distributed over
the entire top surface, at micro-scale, current can only enter the next layer through
limited locations where carbon fibers contact other carbon fibers from the adjacent
layer. Figure 5.1(b) schematically depicts the RVE view of two fibers in contact. The
RVE length (in fiber direction) 𝐿𝑅𝑉𝐸 is the average distance between two contact points,
as given by Equation 2.3 from section 2.2, and its width 𝑊𝑅𝑉𝐸 is given by Equation 5.2.
Current 𝐼𝑐𝑜𝑚𝑝 is applied on the top surface, and is divided into two equal parts and
passes through the contact spots at the two ends, as shown in Figure 5.1(c).
Current density through carbon fibers is calculated by dividing the current
flowing through carbon fibers with the cross-section area of the fiber:
𝜎𝑓𝑖𝑏𝑒𝑟 =𝜎𝑐𝑜𝑚𝑝𝐿𝑅𝑉𝐸𝑊𝑅𝑉𝐸
2(𝜋𝑅𝑓𝑖𝑏𝑒𝑟2 )
(5.1)
From the discussions in section 2.2.1, the RVE width for square packing order is
given by Equation 5.2:
𝑊𝑅𝑉𝐸 = 𝑅𝑓𝑖𝑏𝑒𝑟√𝜋
𝑉𝑓(5.2)
112
Length and width of RVE can be related using the fiber waviness term defined
as the fiber arc height over contact span:
𝛽 =𝐿𝑅𝑉𝐸
𝑊𝑅𝑉𝐸 − 2𝑅𝑓𝑖𝑏𝑒𝑟(5.3)
Combining Equation 5.1-5.3 gives
𝜎𝑓𝑖𝑏𝑒𝑟 = 𝐾𝑓𝑖𝑏𝑒𝑟𝜎𝑐𝑜𝑚𝑝 (5.4)
where
𝐾𝑓𝑖𝑏𝑒𝑟 =
(
1
𝑉𝑓−
2
√𝜋𝑉𝑓)
𝛽 (5.5)
𝐾𝑓𝑖𝑏𝑒𝑟 is the current concentration factor within the carbon fiber. The fiber
waviness term 𝛽 is in the range of few hundreds to a few thousands for commonly used
carbon fiber such as IM7 and AS4. For IM7 UD lamina with fiber volume fraction 𝑉𝑓 of
55%, and 𝛽=1000, the current concentration factor 𝐾𝑓𝑖𝑏𝑒𝑟 is around 1000. It means the
actual current flowing through carbon fibers is 1000 times as large as the apparent
current density applied to the surface of laminate plate. If a current of 40 A is applied to
a 1 inch by 1 inch plate in through-thickness direction (overall current density of 62000
A/𝑚2), the current density through carbon fibers is around 24.8 𝑀𝐴/𝑚2 (about 400
times as large as overall current density applied to composite surface), assuming fiber
volume fraction of 0.55 and fiber waviness term of 400.
Figure 5.2 plots the dependence of 𝐾𝑓𝑖𝑏𝑒𝑟 on fiber volume fraction 𝑉𝐹 and on
fiber waviness term 𝛽. Difference between current density with carbon fibers and the
overall surface density grows with increasing 𝛽 but reduces with increasing Vf. As 𝛽
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increases, the distance between contact points increase, indicating less current injection
points on the fiber, thus the current concentration through carbon fiber is enhanced.
Increasing Vf, on the contrary, will increase the number of contact points, as the distance
between fibers reduces, thus smaller distances between contact points.
Figure 5.2 𝐾𝑓𝑖𝑏𝑒𝑟 as function of fiber volume fraction 𝑣𝑓 and on fiber waviness term 𝛽.
5.2.2 Current concentration at contact spots
The current is further concentrated at the contact spots due to the smaller contact
area compared to fiber cross sectional area. As depicted in Figure 5.1(c), current is
passed from one carbon fiber to another through limited contact points. Current density
across the contact points depend on their contact area.
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A micromechanics based method is developed in section 2.2 to calculate the
equivalent contact area. Radius of the contact spot 𝑎𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is determined by structural
parameters as well as processing parameters as described in Equation 5.6 (see
discussions in section 2.2).
𝑎𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = √3𝐹𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑅𝑓𝑖𝑏𝑒𝑟
8𝐸
3
(5.6)
Where E is the elastic modulus of carbon fiber, and Rfiber is fiber radius.
𝐹𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is the contact force given by Equation 5.7. It’s assumed that the load is
only carried by carbon fibers.
𝐹𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝑃 ∗ 𝐿𝑅𝑉𝐸 ∗𝑊𝑅𝑉𝐸 (5.7)
Where P is the processing pressure.
Combining Equation 5.2,5.3,5.6, 5.7 gives
𝑎𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝑅𝑓𝑖𝑏𝑒𝑟 ∗√3𝑃𝛽2 (
𝜋𝑉𝐹− 2√
𝜋𝑉𝐹)
8𝐸
3
(5.8)
Current density at the contact spot can then be calculated:
𝜎𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝜎𝑐𝑜𝑚𝑝 ∗𝐿𝑅𝑉𝐸 ∗ 𝑊𝑅𝑉𝐸𝜋𝑎𝑐𝑜𝑛𝑡𝑎𝑐𝑡
2= 𝐾𝑐𝑜𝑛𝑡𝑎𝑐𝑡 ∗ 𝜎𝑐𝑜𝑚𝑝 (5.9)
Where 𝐾𝑐𝑜𝑛𝑡𝑎𝑐𝑡 =𝐿𝑅𝑉𝐸∗𝑊𝑅𝑉𝐸
𝜋𝑎𝑐𝑜𝑛𝑡𝑎𝑐𝑡2 , a dimensionless number, is the current concentration
factor at the contact spot. Using Equation 5.8, 𝐾𝑐𝑜𝑛𝑡𝑎𝑐𝑡 can be expressed as in Equation
5.10:
115
𝐾𝑐𝑜𝑛𝑡𝑎𝑐𝑡 =4
𝜋∗√𝛽𝐸2(
𝜋𝑉𝑓−2√
𝜋𝑉𝑓)
9𝑃2
3
(5.10)
Figure 5.3 plots the dependence of 𝐾𝑐𝑜𝑛𝑡𝑎𝑐𝑡 on processing pressure and fiber
waviness term 𝛽. As processing pressure increases, contact area between carbon fibers
increase, resulting in less concentrated current density at the contact spots. As fiber
waviness term 𝛽 increases, the average distance between contact points increase,
resulting in less contact points, and thus more concentrated current density at the
contact spots.
Figure 5.3 Kcontact as function of a )processing pressure (other parameters are fixed: 𝑉𝑓 = 0.55,𝐸 = 273 𝐺𝑃𝑎,𝛽 = 400, 𝑅𝑓𝑖𝑏𝑒𝑟 = 3.5 𝜇𝑚); and b) fiber waviness term β
(other parameters are fixed: 𝑉𝑓 = 0.55, 𝐸 = 273 𝐺𝑃𝑎,𝑃 = 800,000 𝑃𝑎, 𝑅𝑓𝑖𝑏𝑒𝑟 =
3.5 𝜇𝑚).
116
With the above simplified analytical solutions in mind, in the following sections,
we will quantify the temperature increase due to Joule heating, and evaluate if the
localized heating plays an important role in the conduction behavior of CFRP.
5.3 Joule Heating Effect
5.3.1 Within carbon fibers
Considering the current concentration at the contact interface, one important
question to ask is: what is the temperature developed within the carbon fibers and at the
contact spots during current flow?
Neglecting the heat transfer between carbon fibers and the surrounding less
thermally conductive resin matrix, the carbon fibers can be regarded as an adiabatic
system. Equation 5.11 describes the temperature increase in carbon fibers.
𝐼𝑓𝑖𝑏𝑒𝑟(𝑡)2𝑅𝑓𝑖𝑏𝑒𝑟 = 𝑚𝑓𝑖𝑏𝑒𝑟𝐶𝑓𝑖𝑏𝑒𝑟
𝑑𝑇𝑓𝑖𝑏𝑒𝑟𝑑𝑡
(5.11)
Where 𝐼𝑓𝑖𝑏𝑒𝑟(𝑡) is the transient current, 𝑅𝑓𝑖𝑏𝑒𝑟 , 𝑚𝑓𝑖𝑏𝑒𝑟 and 𝐶𝑓𝑖𝑏𝑒𝑟 are the electrical
resistance, mass and thermal capacitance of the carbon fiber respectively.
It should be noted that nonlinear effects such as temperature dependent capacity
is not considered in this estimation.
Two current waveforms are considered in this study: a constant current and a
current ramp with constant increasing rate (ramp), as shown in Figure 5.4.
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Figure 5.4 Typical current waveforms: a) constant current; and b) current ramp.
In the constant current case where current is fixed as 𝐼𝑓𝑖𝑏𝑒𝑟 for the time
considered, temperature increasing rate can be calculated from
dT
dt= 𝐼𝑓𝑖𝑏𝑒𝑟
2𝜌𝑓𝑖𝑏𝑒𝑟
𝐷𝑓𝑖𝑏𝑒𝑟𝐶𝑓𝑖𝑏𝑒𝑟(5.12)
Using properties of AS4 carbon fiber, temperature increase rate can be estimated
as 66∘/s if 40A is applied to the 1’’ by 1’’ CFRP plate.
In the current ramp case where current is expressed in Equation 5.13, the
temperature development is calculated by integrating Equation 5.11, and the resulting
solution is given in Equation 5.14.
Ifiber = 𝐶1𝑡 (5.13)
T(t) =𝐶12𝜌𝑓𝑖𝑏𝑒𝑟𝑡
3
𝐷𝑓𝑖𝑏𝑒𝑟𝐶𝑓𝑖𝑏𝑒𝑟(5.14)
118
If a current ramp with a peak of 40A is applied to a 2’’ by 1’’ CFRP specimen
within 100ms, temperature increase is 6.54 ºC. If 200 KA is applied to the same
specimen within 0.5ms (lightning strike current waveform A), temperature increase can
be estimated as 1.6×109 ºC not considering damage. Obviously, CFRP will decompose
under such high temperatures.
5.3.2 At contact spots
The contact spots between carbon fibers may be subjected to more severe
heating because of the constriction resistance and the larger current concentration effect.
By comparing the equations for thermal and electrical conduction within the
contacting material, Holm [41] derives a simple relation between the temperature T of
the contact spot and the voltage drop 𝑈 across the contact, as expressed in Equation
5.15.
∫ 𝜌(𝑇)𝜆(𝑇)𝑑𝑇 =𝑈2
8
Tcs
T0
(5.15)
Where ρ(T) is temperature dependent electrical resistivity, and λ(T) is temperature
dependent thermal conductivity.
Most good conductors satisfy the Wiedermann-Franz law (Equation 5.16),
which states that good electrical conduction and thermal conduction usually go hand in
hand.
ρ(T)λ(T) = 𝐿𝑇 (5.16)
Where L=2.34e-8 (V/K)2 is the Lorenz number, and temperature T is in K.
119
Using Equation 5.16, integration of Equation 5.15 can be performed and
𝑇𝑐𝑜𝑛𝑡𝑎𝑐𝑡 becomes
𝑇𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝑇0√1+ (𝑈
𝑈0)2
(5.17)
with 𝑈0 = 2𝑇0√𝐿, and 𝑇0 is the ambient temperature or the temperature of the carbon
fiber far away from the contact spot. In Figure 5.5 , the contact spot temperature
𝑇𝑐𝑜𝑛𝑡𝑎𝑐𝑡 is plotted as a function of the voltage drop U across the contact spot for three
different ambient temperatures 𝑇0. The temperature at the locations far away from the
contact spots (ambient temperature 𝑇0) is seen to barely have any influence on the
temperature of the contact spot. It indicates a localized heating is generated at the
contact spots.
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Figure 5.5 Temperature at the contact spot according to Equation 5.17 plotted as a function of the voltage drop over the contact region for three different ambient
temperatures
5.4 Temperature Dependent Electrical Resistivity
Multiple researchers [56-57] have reported electrical resistivity of graphite
based carbon fiber reduces at elevated temperatures. This is due to activation of
electrons from valence bands to conduction band. Carbon fiber behaves as a semimetal,
a material with a small overlap in the energy of the conduction and valence bands. With
fewer charge carriers than metals, semimetals usually have lower electrical and thermal
conductivities, with a negative temperature coefficient of conductivity. Arrhenius plots
are often used to analyze the nonlinear temperature induced effects such as the rates of
chemical reactions, and in this case the intrinsic electrical resistivity of carbon fiber.
The Arrhenius equation is given in the form:
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ρ = C ∗ e EakT (5.18)
Where ρ is the intrinsic electrical resistivity of carbon fiber, 𝐸𝑎 is the activation energy,
C is a constant, k is Boltzmann constant with value of 8.617342x10-5 eVK-1 and T is the
temperature in K. Activation energy, with the unit of electron-volt (eV, the amount of
energy gained by the charge of a single electron moved across an electric potential
difference of one volt), is defined as the minimum amount of energy required to trigger
a temperature-accelerated failure mechanism.
The value of activation energy indicates the relative tendency of a failure
mechanism to be accelerated by temperature, i.e., the lower the 𝐸𝑎, the easier it is to
trigger a failure mechanism with temperature.
Taking log operation on both sides of Equation 5.18 yields
ln(𝜌) = ln 𝐶 +𝐸𝑎𝑘
1
𝑇(5.19)
Plotting ln(ρ) against 1/T allows one to determine the constants in the Arrhenius
plot, as demonstrated in Figure 5.6. For a thermally activated process, an Arrhenius plot
give a straight line. The slope of the linear approximation in Arrhenius plot will be
equal to Ea/k according to Equation 5.19.
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Figure 5.6 Arrhenius plot for IM7 and T700 carbon fiber. Activation energy can be back calculated from the slope of the linear fit.
Activation energy for cured CFRP in three primary direction is reported by
Takahashi [58] and plotted in Figure 5.7. The activation energy is lowest in the fiber
length direction, since electrons can be transferred along graphite fibers. In the
transverse and through-thickness directions, the electrical current flows through
contact between neighboring graphite fibers, requiring more energy to activate
electrons.
As discussed in Chapter 4, resin interface plays a larger role in these two
directions in terms of electrical conduction.
123
Figure 5.7 Activation energy in three primary directions for typical carbon composite
laminates. Activation energy has the unit of micro electronvolt (meV), the amount of energy gained by the charge of a single electron moved across an electric potential
difference of one volt, and is defined as the minimum amount of energy required to trigger a temperature-accelerated failure mechanism.
5.5 Temperature and Electric Field Induced Material Degradation
CFRP laminates subjected to high current density such as lightning strike are
expected to experience a series of physical or/and chemical changes including matrix
decomposition, charring and ablation at different temperatures. Carbon fibers will
sublimate and ablate completely at the temperature of 3590 K.
There has been a lot of research on the mechanisms of material degradation in
CFRP [59]. It was recognized that factors such as carbon fiber breakage, fiber ablation,
and carbonization of resin matrix could have remarkable influences on the electric
conductivity of CFRP materials [19], [60], [61]. The present work has focused on the
temperature and electric field induced resin degradation, while the consideration of fiber
degradation is limited to the temperature dependent intrinsic resistivity. Carbon fiber
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ablation and breakage are out of the scope of current research and are not considered in
current model.
The two known kinds of polymer breakdown are thermal and electric
breakdown as discussed in the following sessions.
5.5.1 Thermal breakdown
Thermal breakdown is the result of excessive heating of the insulator by the
electric current which causes, at a certain voltage, the polymer to melt or char. In this
case, the dielectric strength is proportional to the square root of the ratio of thermal and
electrical conductivity of the plastic. Carbonization of resin system occurs at
temperature of about 1500 K. As the resin system inside the laminate is heated under
inert (oxygen-free) environment, it begins to lose its non-carbon atoms. As the non-
carbon atoms are expelled, the remaining carbon atoms form tightly bonded carbon
crystals, which are electrically conductive[13].
5.5.2 Electric breakdown
An important electrical property of polymers working as insulators is their
dielectric strength. If a voltage is applied to an insulator and steadily increased, a point
will be eventually reached where the polymers break down and causes a significant
decrease in resistance. Much experimental work has been done on the dielectric
breakdown of polymers to characterize their dielectric strength. The dielectric strength
will depend on the type and shape of the plastic and electrodes, the rate with which the
field is increased, and even the temperature. The dielectric strength of polymers is
generally in the range of 1 to 9 MV/cm at 20°C and these values fall at elevated
temperature.
125
The dynamics of thermal and electric breakdown is complicated by various resin
types and temperature history, and modeling of this process remains an active topic
which is beyond the scope of the current study. A simplistic ON-OFF model is used to
describe the degradation behavior of resin systems under high temperature, as
demonstrated in Figure 5.8. The electrical behavior of contact points in CFRP material
system was partitioned into two distinguished zones, one was highly resistive and the
other one is electrically conductive. At a critical temperature or electric field threshold,
resin system behaves as an insulator, while at temperatures and electric fields higher
than the threshold values, it behaves as a conductor with same properties as carbon.
Figure 5.8 ON-OFF model for resin breakdown
126
5.6 Modeling Approach
5.6.1 Model overview
In order to provide robust electrical analysis for CFRP under high current
impact, it is very desirable to have an efficient 3-D simulation methodology to estimate
the temperature profiles in the carbon fibers as well as contact spots and evaluate the
thermal-electrical coupling.
The 3D resistor network framework is adopted, with modifications to
incorporate material degradation due to high current or electric field. Instead of using
constant resistance values for resistors representing carbon fibers and contact points,
temperature and electric field dependent resistances are used to model the high current
impact such as Joule heating and material degradation. The coupling between thermal
and electrical properties are achieved by designing a thermal-electrical RC unit circuit
that simulates electrical and thermal conductions simultaneously, as discussed in later
sections. Other details that can be easily added into the modeling framework include
contact resistance between carbon fibers that consider the surface roughness, and fiber
breakage due to high temperature.
The model is implemented with Matlab® and SPICE[62], a circuit simulation
package that solves the generated resistor network. The workflow of implementation of
the model is illustrated in Figure 5.9. Discretized 3D microstructure of CFRP laminate
is constructed using Matlab script and transformed into a netlist file that describes the
connections and values of resistors and capacitors in the resistor network model
developed in Chapter 2. Boundary conditions such as voltage difference and ground
positions are also assigned in Matlab script and written to netlist file. The netlist file is
then input into the SPICE package to conduct the thermal and electrical conduction
127
analysis. Temperature profile, current flowing through and voltage at each node at every
time step can be extracted and input into Matlab for further data reduction.
Figure 5.9 Workflow for implementing the 3D resistor capacitor network work with thermal-electrical coupling
5.6.2 Thermal-electrical RC circuit
When the composite is subjected to small DC currents, the joule heating effect is
negligible and hence not necessary to include in the model. However, in present study
we do integrate the joule heating effect and temperature dependent material properties
to more accurately capture the conduction behavior under high current density.
128
To consider temperature dependent material property, thermal conduction within
the CFRP laminate should be considered and the temperature at each node needs to be
predicted. Thermal conduction is similar in nature to electrical conduction and the
variables used to describe the thermal conduction process are analogous to those that
describe the electrical conduction process. Based on the thermal-electrical analogy
(Table 5-1), a 3-D RC distributed thermal-electrical circuit model has been developed,
as shown in Figure 5.10.
Table 5-1 Analogy between thermal and electrical conduction
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Figure 5.10 Coupled thermal electrical resistor capacitor network model.
Based on the least thermal resistance path, heat is conducted within the CFRP
laminate through the contacts between carbon fibers, due to the difference between
thermal conductivity’s of carbon fiber and the resin system. Using the thermal-electrical
analogy, thermal conduction with the CFRP laminate can thus be modeled as a network
of thermal resistors and capacitors. Thermal capacitors are considered in the present
model to represent the fact that thermal conduction are orders of magnitude slower than
electrical conduction. Figure 5.10 demonstrates the coupling between electrical and
thermal conduction models with a minimal resistor network that consists of two
contacting fibers. Figure 5.10(a) shows the formulation of electrical conduction model,
with red arrows representing the direction of current flow. Figure 5.10(b) shows the
130
network model for thermal conduction, with diamond signs representing Joule heating
and red arrows representing thermal gradient directions. Joule’s law dictates heat is
generated in conductors with current passing through, thus Joule heating occurs
everywhere along the electrical conduction paths. In the present model, the location of
Joule heating source is simplified as the center of each resistor. Joule heating within
carbon fibers is considered in the current model but is not plotted in the schematic
drawing for clarity. Although the magnitude of current that flows through contact points
and the carbon fiber are of the same order, current density at contact points is
significantly larger than that within carbon fiber, due to the smaller contact area
compared to cross section area of carbon fiber, resulting in localized heating at the
contact points.
The two conduction networks were combined to address both conduction
mechanisms, as demonstrated in Figure 5.10(c). Each resistor in the pure electrical or
thermal conduction network is replaced by a unit circuit that is implemented with a
four-port sub-circuit: two electric ports (e1 and e2) and two thermal ports (t1 and t2).
This thermal network can be easily implemented and simulated using SPICE in the
same manner as an electrical circuit network by simply employing the proper
counterparts as illustrated in Table 5-1. The voltage of thermal ports (t1 and t2)
correspond to temperature, while current flowing through these thermal ports
correspond to the heat conducted.
Figure 5.11 shows the flowchart for the electrical-thermal coupling for one time
step. Electrical and thermal conduction models are coupled through the Joule heating
term in thermal conduction model and temperature dependent resistor term in electrical
conduction model.
131
Figure 5.11 Flowchart showing the Coupling between electrical and thermal conduction
networks
Temperature dependent resistor behavior is expressed as an Arrhenius type
function:
𝑅𝑡𝑒𝑚𝑝 = 𝑅𝑖𝑛𝑖𝑡 ∗ exp (𝐸𝑎𝑘𝐵∗ (
1
𝑉(𝑇𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟 )−
1
𝑉(𝑇𝑎𝑚𝑏))) (5.20)
Where 𝑅𝑡𝑒𝑚𝑝 is the temperature dependent electrical resistance, and 𝑅𝑖𝑛𝑖𝑡 is the
initial resistance at room temperature.
132
Note that the temperature of resistor is extracted from the thermal conduction
model by referring to the voltage of node T_node (see Figure 5.10), which gives the
temperature of the node based on thermal-electrical analogy.
Current flowing through the resistor can be calculated using the following
formula once resistance (R_temp) and voltage difference between the two electrical
ports are determined:
𝐼 =𝑉
𝑅𝑡𝑒𝑚𝑝=𝑉(𝑒1) − 𝑉(𝑒2)
𝑅𝑡𝑒𝑚𝑝(5.21)
The heating source in thermal conduction model comes from Joule heating (Qjh)
that depends on voltage difference between and current flowing through the two
electrical ports:
Qjh = (𝑉(𝑒1) − 𝑉(𝑒2)) ∗ 𝐼 (5.22)
To compute the equivalent resistance, a voltage difference is applied across the
electrode, in the direction d (d ∈{x,y,z}) , in which the resistivity is being calculated.
Solving this network yields the corresponding currents, allowing the equivalent
resistance and hence conductivity of the entire structure to be determined from
geometry parameters, as expressed in the following formula:
ρi=𝑈
𝐼∗𝑥𝑗𝑥𝑘𝑥𝑖 , ( 𝑖, 𝑗, 𝑘 ∈ {𝑥, 𝑦, 𝑧}|𝑖 ≠ 𝑗 ≠ 𝑘) (5.23)
133
Where, ρi (𝑖 ∈ {𝑥, 𝑦, 𝑧} is electrical resistivity in one of the three principal directions,
𝑥𝑖(𝑖 ∈ {𝑥, 𝑦, 𝑧} is the specimen dimension in the corresponding direction, U and I are
the measured voltage difference and current respectively.
5.7 Results and Discussions
5.7.1 Variations among simulations using same modeling parameters
The stochastic characteristics of CFRP is captured in the developed model,
through randomized initial resistances and also randomized contact locations. In this
study, five simulations were run using the same model parameters listed in Table 5.1.
The simulated through-thickness resistivity’s in these five repetitions are compared in
Figure 5.12. A linear current ramp with peak current of 100 A and duration of 100 ms is
applied.
Table 5-2 Model parameters
Parameter Value
Carbon fiber electrical resistivity 1.5 × 10-5 Ωm
Carbon fiber thermal conductivity 5.4 W/(mK)
Carbon fiber thermal capacity 0.879228 kJ/kg∙K
Inter-ply connectivity 10%
Fiber volume fraction 0.55
Critical temperature for resin degradation 1500K
Critical electric field for resin degradation 1× 109 V/m
The only difference between these five simulations is the locations of the inter-
ply connections, which is randomized during the construction of resistor network. It has
been seen that large variations in initial resistance can be observed among the
repetitions. The randomized connection points affect conductive paths, introducing
uncertainties in overall through-thickness resistivity. After current application, there is
134
little difference between through-thickness resistivity from these simulations. Thermal
breakdown of resin-rich interface happens due to excessive heat from Joule heating, and
the resistivity of the inter-ply resistance drops to the level similar to direct carbon fiber
contact. The residue resistivity is reduced to that of a UD lamina without resin-rich
layer.
Consideration of the stochastic characteristics of CFRP microstructure is not
limited to the locations of inter-ply connections. Other factors that can be considered in
current model implementation includes randomized fiber bundle resistivity in three
primary directions, and the values of inter-ply contact resistances.
Figure 5.12 Variations of simulated through-thickness resistivity using same model
parameters
135
5.7.2 Impact of resin-rich layer
While there is increasing reported literature on modeling the mechanical and
thermal response of CFRP laminates subject to high current density such as simulated
lightning strike [19], [54], there are few reports on changes in electrical conduction
behavior during and after application of high current density. During lightning strike,
through-thickness resistivity determines how deep the current can penetrate and the
Joule heat accumulated. The existence of resin-rich layer may drastically change
conductive paths, resulting in different material damage patterns. The high temperature
and current density presented in a lightning strike determines that temperature
dependent material properties and material degradation mechanisms need to be
considered.
As discussed in section 5.3, the transient thermal-electrical study requires
activation energy term that describes Arrhenius type temperature dependent electrical
resistivity. Activation energy values for carbon fiber (8.5 meV) taken from [58] is used
in this study. All the other model parameters were kept the same as in Table 5-2. Resin
charring temperature of 1500 K was chosen as the critical temperature for thermal
breakdown, and the dielectric strength of epoxy resin (1e9 V/m) was chosen as the
critical potential gradient for electric breakdown. Inter-lamina connectivity of 0.1 and
1.0 were chosen to demonstrate the different nonlinear electrical conduction behaviors
for CFRP laminates with and without resin rich layer.
The current density was increased linearly from 0 to 65000 A/m2 in 100 ms. The
peak current density is chosen to be comparable to the current density used in our
ongoing experimental characterizations. It’s equal to the current density due to
application of 40 A of electric current (the maximum current the power source can
deliver) into composite with surface area of 1 inch by 1 inch (25.4mm by 25.4mm,
136
typical dimension of specimens tested). Figure 5.13 plots the resistivity change over
time. Resistivity values are normalized with the initial resistivity before application of
high current.
Figure 5.13 Resistivity change over time. Laminate with small inter-lamina connectivity (resin-rich interface) undergoes quicker and larger resistivity drop in through-thickness
direction. Sudden drop in resistivity around 10ms can be explained by the localized heating
137
Figure 5.14 shows the representative temperature history at a selected contact
points between carbon fibers, at intra-lamina contact, and at an inter-lamina connection
points. The variations of temperature at each location group due to conducting ten
separate simulations is represented by the colored area. For laminates with resin-rich
interface, temperature at inter-lamina connection points is one order of magnitude
higher than other locations, indicating the influence of the localized Joule heating. For
laminates without resin-rich interfaces, temperature at inter-lamina connection points is
of the same order as at fiber-fiber contact points. The temperature increase in carbon
fibers is not significant in both cases. Although the local temperature at inter-lamina and
intra-lamina contact points are high, it has little impact on the overall temperature
increase of the laminates since the contact points occupy a very small fraction of the
total volume of the laminates. It indicates that even if no significant overall temperature
increase is detected in the composite, the local region of contacts can experience a
significant increase in temperature causing a drastic change in material property at that
location. It should be noted that the current model may overestimate the temperature
increase due to two assumptions in the model: 1) heat loss to the ambient air is
neglected considering the short duration of current application; and 2) phase change of
material that consumes energy is not considered in the present model.
138
Figure 5.14 Temperature profile at selected location: contact between carbon fibers, at fiber-fiber contacts, and at inter-lamina connection points for two types of composites:
a) with resin-rich interface and b) without resin-rich interface.
5.7.3 Parametric studies on the impact of model parameters
The model developed in this study introduced new parameters, compared to
those discussed in previous chapters. Due to the intricate and stochastic nature of
composite microstructure, it’s often hard to experimentally exam the impact of these
parameters. Parametric study provides a handy tool to investigate how the electrical
conduction changes with these parameters. It has been demonstrated that fiber waviness
term 𝛽 has dominating impact on the through-thickness resistivity of UD carbon
139
composites under low DC current. In this study, electrical conduction behaviors for
CFRP under high current density for CFRP with low (𝛽 = 200) and high (𝛽 = 1000)
are compared. Inter-ply connectivity is fixed to 100%, and inter-ply resistance is set to
10 𝛺, which represents good inter-ply connections. Other model parameters are the
same as in Table 5-2 except for fiber waviness term and activation energy. Resin
breakdown is not considered. Thus, any resistance change is attributed to the
temperature dependent intrinsic resistivity of carbon fibers. A linear current ramp with
peak current of 40A and duration of 100ms is applied to the specimen with 1 inch by 1
inch surface area.
Figure 5.15 shows the through-thickness resistivity responses for these model
configurations. The impact of the newly introduced parameter, activation energy, is
also plotted in the same figure. In general, larger activation energy leads to larger
resistivity drop under same current application. Increase in fiber waviness beta indicates
more sparsely distributed contact points and larger local current density at the contact
spots, resulting in larger drop in resistivity. The simulation with the largest activation
energy (13 mV) and the largest fiber waviness term (1000) yields most significant drop
in through-thickness (~20%) resistivity. This also demonstrates that without
consideration of resin breakdown, resistivity reduction is limited even under large fiber
waviness term and activation energy.
140
Figure 5.15 Parametric study on fiber waviness term and activation energy.
Another set of virtual tests is conducted to investigate the impact of inter-ply
resistance. Inter-ply resistance is varied from 1000 𝛺, which represents carbon to
carbon contact resistance, up to 20000 𝛺 that is in the same level of tunneling
resistance, representing multi-ply laminate with resin-rich inter-ply layer. Inter-ply
connectivity is maintained at 50% for all simulations. Other model parameters are same
as from Table 5-2. A linear current ramp with peak current of 100A and duration of 100
𝜇𝑠 is applied to the specimen with 1 inch by 1 inch surface area. Figure 5.16 shows the
simulation results.
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A large inter-ply resistance not only affects the absolute resistivity value before
current application, but also changes the resistivity reduction after current application.
A 60% drop in through-thickness resistivity is found for specimen with 20000 𝛺 inter-
ply resistance, while the resistivity reduction for specimen with 1000 𝛺 inter-ply
resistance is about 36%. Specimens with inter-ply resistance smaller than 10000 𝛺
demonstrate similar resistivity change throughout the whole duration of current
application. Difference between resistivity of these specimen is negligible, indicating
that the inter-ply resistance’s drop to the same level due to Joule heating from the large
current application within short duration.
Figure 5.16 Impact of inter-ply resistance. A large inter-ply resistance not only affects
the absolute resistivity value before current application, but also changes the resistivity reduction after current application.
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5.8 Summary and Conclusions
This research proposes a fast SPICE based 3D electro-thermal simulation
methodology to characterize thermal effects due to Joule heating in CFRP for
application that are subjected to high current densities. The concept of current
concentration within carbon fibers and through fiber-fiber contact spots is introduced,
and its impact on localized heating is quantified with an analytical model. It’s been
found that the local current density flowing through carbon fibers can be orders of
magnitude larger than the overall surface current density, while the current density at
contact spots is even larger due to smaller contact area. It is essential not to
underestimate the impact of temperature rise at contact spots due to Joule heating.
Therefore, accurate temperature estimation is extremely important to perform a more
realistic electrical and thermal analysis of CFRP.
The effect of resin-rich layer on resistivity change and temperature development
in carbon fibers and contact points is included in our analysis for the first time, to
provide more accurate and realistic description of CFRP’s nonlinear electrical
conduction behavior. It shows that the existence of resin-rich interface layer increases
current concentration, enhancing the local Joule heating. A further investigation on the
temperature profiles shows that the inter-ply contact points suffer much higher
temperature rises than intra-ply contact points due to the limited number of contact
points between plies and thus higher current concentration. It is observed that
temperature at the contact spots in the CFRP with resin-rich layer easily reach the
critical temperature where ablation of carbon fiber and thermal decomposition of resin
will initiate. Resin-rich layer must be considered in the thermal and electrical analysis
of composite structures. The developed model is able to capture the large resistivity
drop due to resin breakdown and charring.
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The simulation methodology has also been applied to quantify the effect of
structural and material parameters on the electrical conduction behavior, including fiber
waviness term, inter-ply resistance and activation energy. In general, larger activation
energy leads to larger resistivity drop under same current application. Increase in fiber
waviness beta indicates more sparsely distributed contact points and larger local current
density at the contact spots, resulting in larger drop in resistivity.
It should be noted that the simulation methodology developed here is quite
general, and can be easily extended to consider other nonlinear conduction mechanisms
by replacing the formulas for temperature dependent material properties. Stochastic
nature of CFRP micro-structure can be described with statistical distributions of
resistance values, instead of using one single resistance for each type of resistor.
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EXPERIMENTAL INVESTIGATION OF HIGH CURRENT IMPACT
This chapter present experimental investigation of the electrical conduction
behavior of both dry fiber tows and cured CFRP laminates subjected to high current
densities. A modularized characterization apparatus is designed to which one can input
the desired current/voltage waveform. By switching specimen fixtures, the apparatus
can be used for electrical characterization of dry carbon fiber tows and cured
composites in each of the three principal directions. Electrical responses of cured
composite under simulated lightning strike are also discussed. The coupled thermal-
electrical resistor network model developed in Chapter 5 is utilized to explain the
experimental results.
6.1 Electrical Characterization of Dry Fiber Tows
While this dissertation focuses on the electrical conduction behavior of cured
composite laminates, the dry fiber tow systems eliminate the uncertainties introduced by
the curing process and make it easy to control the micro-structure parameters such as
sizing amount and fiber volume fraction. In this study, an experimental apparatus is
designed to investigate the difference in conduction behavior of sized and unsized dry
fiber tow systems under high current density (up to 7e4 A/m2). This study serves as a
first step towards understanding the role of resin property changes under high current
density in the conduction behavior of cured composite laminates.
Chapter 6
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6.1.1 Materials and preparation
This study utilized dry fiber tows made from two carbon fiber types: IM7 from
Hexcel and T700SC from Toray. The Hexcel IM7 fiber is without sizing, and the Toray
T700SC fiber has sizing amount of 1.25% (by weight). The fibers come in roll form and
are cut into desired length. Sizing of fibers are provided by carbon fiber manufacturers
and no other surface treatment in done in-house. Properties of the fibers used in the tests
are listed in Table 6-1.
Table 6-1 Properties of fiber groups for high current density tests
Property
IM7 T700SC
Density [g/cm3 ] 1.78 1.8
Modulus [Gpa] 276 230
Electrical Resistivity [Ωm ] 1.5 e-5 1.6 e-5
Sizing amount 0 1.25%
Fiber diameter (𝝁𝒎) 7.1 7.0
6.1.2 Experimental setup
Figure 6.1 shows the schematic illustration of the characterization apparatus. A
Sorenson DCR-B 2700-watt power supply is used in order the supply the required
current to the fiber tows or CFRP laminate. The power supply operates in a constant
current mode, in the range 0-100 A, and is controlled by a command signal from the
function generator. Due to the limitations of the power line in the laboratory, maximum
current can be applied is about 50 A. The current flow, voltage drop and surface
temperature of the laminate are monitored and recorded using an A/D converter and in-
house developed software written in LabVIEW. The current flow through the CFRP
laminate is monitored with current channel of Keithley 2700 multi-meter, utilizing an
internal current shunt in the power supply. The voltage drop across the laminate is
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measured with the voltage channel of Keithley 2700 multi-meter. Resistance of the
composite laminate can then be calculated from the measured voltage and current.
Figure 6.1 Schematic illustration of electrical characterization apparatus.
The apparatus is modularized with changeable specimen fixtures. By switching
specimen fixtures, this setup can measure resistance response under various current
waveforms in both in-plane and through-thickness directions, while the main current
modular can remain unchanged.
The same specimen fixtures used for dry fiber tows in the low DC current test as
discussed in Chapter 3 is used in this study.
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6.1.3 Typical resistance response
The electrical characterization apparatus can operate in both current and voltage
application modes. Figure 6.2 presents typical measurements of voltage, current and
resistance during the dry fiber tow tests in voltage application mode. Specially designed
voltage waveform is used as input and the resulting current is recorded. A small
constant voltage is maintained at the initial stage, followed by a voltage ramp up. The
voltage waveform ends with a plateau.
Little resistance change can be observed at the initial low applied voltage.
Increased applied voltage results in sudden drop of resistance, indicating breakdown of
the sizing layer. The sizing breakdown is irreversible and the resistance drops to a low
level after the voltage application. At the final stage, voltage is kept constant; resistance
continues to drop slowly. This continued small drop in resistance is probably due to the
heating of the carbon fibers. The increase in conducted current accumulates heat over
time which heats the fibers.
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Figure 6.2 Typical resistance response for dry fiber tow under a voltage ramp. Voltage ramp and the corresponding current response are also plotted.
Figure 6.3 shows the electrical responses for both sized and unsized carbon
fibers in the experiments in which the current ramp is applied. Instead of controlling the
voltage, current waveform is introduced in the through-thickness direction through two
copper bars serving as electrodes. Compression load is kept as 500N for both cases.
Different resistance responses can be observed form the two carbon fiber types.
For unsized IM7 fiber tows, less than 5% drop in resistance is observed, while for sized
T700SC fiber tows, resistance drops by 18% at the end of the current waveform. This
large difference in resistance is attributed to the breakdown of the thin sizing layer due
to Joule heating during the current application mode.
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Figure 6.3 Resistance response for unsized IM7 and sized T700SC fiber tows: a) unsized IM7 fiber tows see less than 5% drop in resistance; b) sized T700SC fiber tows
yields larger resistance drops (18%) at the end of the current waveform.
6.1.4 Influence of processing pressure
Parametric study using the computational resistor network model considering
the impact of resin rich interface developed in Chapter 5 indicates that inter-ply
interface plays an important role in nonlinear resistivity change under high current
density, especially in the through-thickness direction. In this study, two types of carbon
fibers (with and without sizing) are tested subjected to various loading conditions, to
demonstrate the influence due to presence of thin resin layers. As indicated in Table
6-1, the unsized fiber is Hexcel IM7, and the sized version is Toray T700SC with 1.25%
(by weight) sizing amount. Although manufactured by different suppliers, these two
carbon fibers have similar electrical and mechanical properties, making them suitable
for the comparison.
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Multiple electrical characterization of carbon fiber tows under high current
density are conducted under constant compressive load. In each test, compressive load
is adjusted and maintained constant during the characterization by a mini-Instron
machine. Figure 6.4 shows the resistivity response for both sized and unsized fiber
tows. Resistivity is normalized with the first measured value.
Figure 6.4 Resistivity response under various load amount for unsized and sized fibers.
Resistivity is normalized with the first measured value. a) unsized IM7 fiber tows: no noticeable change in resistivity under high compressive force; a) sized T700SC fiber
tows: drop in resistivity decreases with the increase of compressive load. Drop in resistivity is still noticeable (~15%) even under high compressive force (1000 N)
For unsized IM7 fiber tows, there is no noticeable change in resistivity under
high compressive force; under lower compressive force (150 N), a mere 5% drop in
resistivity is observed. Under higher compressive force (800 N and 1000 N), there is no
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clear drop in resistivity and larger variations in the normalized resistivity is observed.
As discussed in Chapter 2 and 3, high compressive force yields smaller resistivity; thus,
the measurement error induced from the characterization apparatus becomes more
significant, causing larger deviations in the normalized resistivity. Since there is no
resin or sizing between the fibers, direct contact between carbon fibers is the dominant
conduction mechanism at the contact spots, yielding small contact resistance. Joule
heating is thus limited in this case, leading to smaller temperature rise and ultimately
smaller change in resistivity.
Resistivity response of the sized T700SC fiber tows, on the other hands,
demonstrates a distinct dependence on compressive load. Reduction in resistivity is
most significant under small load, and the change in resistivity decreases with the
increase of compressive load. Drop in resistivity is still noticeable (~15%) even under
high compressive force (1000 N). Under small load, carbon fiber tows are not tightly
packed. Contacts between carbon fibers are sparse and limited. In addition to the limited
number of contacts, contact area is also smaller under smaller compressive load,
contributing to enhanced localized current concentration and the resulting excessive
Joule heating. The large drop mainly comes from thermal breakdown of sizing layer. As
load increases, contact resistance drops significantly as discussed in Chapter 2 and
Chapter 3. The conduction mechanism is dominated by direct contact between carbon
fibers. The small drop in resistivity during current application is attributed to the mild
temperature rise.
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6.1.5 Resistivity change after repetitive current application
In our model, the impact of Joule heating can be decomposed into two
categories: 1) temperature dependent carbon fiber resistivity, which is reversible once
temperature returns to normal value; and 2) thermal breakdown of the sizing layer,
which has irreversible impact on the micro-structure of fiber tows or cured composites.
Resistivity change due to this mechanism remains even after the temperature returns to
the initial value.
The aim here is to compare the contributions of irreversible sizing breakdown
and reversible temperature dependent carbon fiber resistivity to the overall resistivity
drop under high current density. Sized T700SC carbon fiber tows are subject to
repetitive current waveforms according to Table 6-2. The current waveforms have the
same shape and peak current (40A) but with different durations. It consists of a linear
ramp up to the peak current within the first 10% of duration, hold the peak current for
80% of duration, and then drops linearly in the remaining 10% of duration.
Table 6-2 Current waveforms used in the repetitive current application tests
Cycle Current waveform
1-3 10ms ramp-up, 80ms hold, 10ms ramp-down
4-6 50ms ramp-up, 400ms hold, 50ms ramp-down
7-9 100ms ramp-up, 800ms hold, 100ms ramp-down
Figure 6.5 plots the through-thickness resistivity versus accumulated time. There
is at least 5 minutes time gap between the tests to allow temperature of the specimen to
drop. For the sake of simplicity, time gaps between the tests are omitted in the plot.
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Significant differences in the resistivity response can be observed for the
unsized IM7 fibers and sized T700SC fibers. Initial resistivity of sized T700SC is more
than 20 times larger than that of unsized IM7 fibers.
Change in the residue resistivity for unsized IM7 fibers after current applications
is limited. 15% drop in resistivity is found after 9 cycles of current application. For
sized T700SC fibers, it can be clearly noticed that resistivity is partially recovered after
each test. Small difference in resistivity is observed between the tests, except between
the first two tests, where irreversible resistivity is significant. After the first cycle, most
of the thin sizing layers are broken down and charred and become conductive, which
explains the irreversible resistivity drop during the initial cycle. In the subsequent
cycles, direct contact between carbon fibers becomes the dominant conduction
mechanism. Resistivity drop during these cycles are attributed to decrease in the
intrinsic carbon fiber resistivity at elevated temperature from Joule heating, which is
confirmed from the recovery of resistivity when temperature drops. Resistivity after the
9th cycle is only about 1/10 of the initial resistivity before current applications, and falls
close to resistivity of unsized IM7 fibers.
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Figure 6.5 Through-thickness resistivity of (a)unsized IM7 and (b)sized T700SC carbon
fiber tows after repetitive current applications. After each test, resistivity is partially recovered. Smaller residue resistivity change can be observed for unsized IM7 fibers,
while large change (~90%) in residue resistivity can be observed for sized T700SC fibers.
Figure 6.6 gives a close-up view of the electrical response of sized T700SC
fibers in the first three cycles. Current waveform follows the desired pattern well, with
only small variations in current waveform found among the three tests. The initial
resistivity in the third cycle is close to that in the second cycle, while huge difference
between the first and second cycles can be observed.
Difference in the initial resistivity between two cycles represents the irreversible
resistivity change, which is mainly attributed to the thermal breakdown of sizing
reducing its resistivity. Irreversible resistivity reduction is significant after the first
cycle but negligible in subsequent cycles, indicating the destructive change in
composite microstructure mainly happens in the first cycle.
Difference between the resistivity at the end of one cycle and the beginning
resistivity in the subsequent cycle represents the reversible resistivity, which is
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attributed to the temperature dependent carbon fiber resistivity. Reversible resistivity
drop after the first current cycle is similar to that after the second cycle.
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Figure 6.6 Electrical response of sized T700SC in the first three 100ms current cycles.
Most significant difference is observed between the first and second cycle, while subsequent cycles demonstrate little difference in current and resistance response.
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6.2 Electrical Characterization of Cured Composite Laminates under Medium-
High Currents
The previous section shows promising results for understanding the dominant
factors contributing to reduction in resistivity of dry fiber tows under high current
density. The next step is taken to investigate the electrical conduction behavior of CRFP
laminates under similar current conditions.
6.2.1 Materials and preparations
The same aerospace grade carbon fibers IM7 from Hexcel and T700 from Toray
are used as fiber reinforcements. The carbon fibers are received in prepreg form.
Prepreg sheets were stacked based on desired layup sequence and were cured in an
autoclave according to the temperature profile recommended by the prepreg
manufacturer (i.e. 2 oC/min heating and cooling ramps and a 2-hour isothermal dwell at
180 oC for IM7).
After the curing process, the laminate plates were cut into coupons with desired
dimensions using a water jet cutter. For in-plane measurements, the coupon size is 5-
inch by 1-inch, while for through-thickness measurements, the coupon is 1-inch by 1-
inch. The dimensions of the coupons are chosen such that the measured resistance falls
into the most accurate measuring window of the equipment. Thickness of the coupons
depend on the number of plies stacked. The nominal thickness of IM7 prepreg is about
0.25mm; thus a 4-ply laminate has a nominal thickness of 1mm.
It was especially important to minimize contact resistance during the
experiments with high electrical current levels as high resistance will cause arcing and
the burning of samples. Coupon surfaces were polished to remove the thin layer of
excessive epoxy on the surfaces in order to expose the conductive fibers, as shown in
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Figure 6.7. The polished surfaces were then covered with conductive paint to ensure
uniform current density over the surface.
Figure 6.7 Polished specimen surface. Carbon fibers are exposed for better contact with
the electrodes
6.2.2 Setup and specimen fixtures
The same current application module used for characterization of dry fiber tows
(see Figure 6.1 for schematic illustration) is used for this study. Special specimen
fixtures are designed for mounting cured composite specimens.
6.2.2.1 Specimen fixture for resistance characterization in the in-plane direction
Figure 6.8 demonstrates the specimen fixture for in-plane characterization. It
consists of two sets of copper electrodes sitting on two nonconductive bases made of
Teflon. The two bases are connected with a screw rod to adjust the span between
electrodes. The electrode set at each end consists of an L-shaped copper block that
provides support to laminate specimen and also space for wiring, and a rectangular
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copper bar with screws to clamp the specimen and provide additional electrical contact.
The upper copper bar is 1-in in length, 1/4-in in width, and 1/8-inch in thickness. Before
each test, the bar is sanded with 400 grit sand paper to maintain clean contact surfaces.
To improve the conductivity between the CFRP panel and the copper electrodes silver
paste is applied at the interfaces between the copper electrodes and the CFRP laminate.
The same procedure holds for the tests in through-thickness direction as well.
Figure 6.8 Specimen fixture for mounting composite specimens in the in-plane tests
6.2.2.2 Specimen fixture for resistance characterization in the through-thickness
direction
The fixture consists of two copper rods attached to a mini-Instron machine,
serving as electrodes, as shown in Figure 6.9. The rods were machined into square
shape at the end to fit the square shape of laminate specimens. The dimension of the
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square surface is 1-inch by 1-inch, same as the dimension of laminate specimens. Wires
for current and voltage measurements are attached to the side of the copper rods.
Figure 6.9 Specimen fixture for through-thickness tests
During tests, laminate specimens are sandwiched between the two square
surfaces and mini-Instron machine was used to provided consistent contact pressure to
help reduce contact resistance between specimens and copper electrodes. Copper
electrodes are insulated from the mini-Instron machine by two Teflon blocks. A
baseline test with only the two copper bars touching (no composite specimen in
between) was conducted before each test to monitor the resistance of the electrodes. It
was checked that the resistance from electrodes is smaller than 1% of the measured
resistance of composites.
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6.2.3 Current waveform
Due to the safety requirements of the power supply, maximum current can be
applied is about 50 A. Linear current ramp waveform is used in this study with various
durations. Considering the small area on which the current is distributed, the current
density in the composite laminate is high enough to induce noticeable resistance
changes. These waveforms are denoted as “medium-high” current, to distinguish them
from the ultra-high current induced from lightning strikes.
The differences between the “medium-high” and “lightning” current waveforms
are two-fold: firstly, current duration of the lightning current waveform (~50 s) is
much shorter than the medium-high current waveforms (~100 ms). Secondly, their
magnitudes also differ, with lightning current in the order of 200 kA, and the medium-
high current under 50 A.
6.2.4 Typical resistance response (first observations)
Figure 6.10(a) and Figure 6.10(b) demonstrate the resistance change in relation
to voltage change in the fiber length direction and through-thickness direction
respectively. The voltage is changed manually and resultant current is recorded.
Resistance is calculated from the recorded voltage and current values.
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Figure 6.10 Electrical response for 4-ply T700 CFRP, in fiber length direction (X) and through-thickness direction (Z) respectively.
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In the fiber length direction, current in fiber length direction increases quickly,
following the increase in voltage, up to about 48 A, which is the safety threshold set by
the power supply. Current stays unchanged until material breakdown happens (at ~140
ms), where current drops quickly, indicating loss of conductive pathways due to fiber
breakage. In the experiments, a material breakdown event can be indicated from flash
coming from the laminate or burning smells.
In the through-thickness direction, voltage is increased gradually to detect the
critical voltage where material breakdown can be noticed. The current follows well with
voltage increase. Small reduction in through-thickness resistance is noticed as voltage
increases, before a sudden dramatic drop at around 90s, indicating initialization of
material breakdown.
In the following sessions, simulation results using the coupled thermal-electrical
RC network model discussed in Chapter 5 are compared with experimental results in
both in-plane and through-thickness directions, to investigate the applicability of the
model in various scenarios.
6.2.5 In-plane resistance compared with simulation results
Figure 6.11 shows the fiber length direction electrical resistance change of the 4-
ply [0]4 IM7-977/3 composite laminate during medium to high level current
application. The abscissa in Figure 6.11(a) is the time in micro-seconds, and the
ordinate is the electrical resistance normalized using the reference resistance at the
initial time. The linearly increasing current is also included in the figure. Five
specimens cut from the same laminate plate are tested and the variations are represented
as error bars in the figure.
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Figure 6.11 Comparison between simulated and measured resistivity response under
high current density for [0]4 IM7-977/3 cured composites. The green vertical line and arrow in (b) denotes the range of current density used in the tests.
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Figure 6.11(b) shows the simulation results using the material properties and
model parameters listed in Table 6-1. Both experimental data and simulation results
show that little change in resistivity can be noticed in the current range applied. While
current applied in the experiments is limited by the equipment capacity (denoted by the
green vertical line in Figure 6.11(b)), the model can investigate the impact of current
density beyond the experimental range, as demonstrated from Figure 6.11(b). From the
simulation results, resistance drop is barely noticeable, which compares well with the
experimental data. Resistivity drop will still be within 10% in the fiber length direction,
even if the current density is increased to 6.5𝑒6 𝐴/𝑚2. It should be noted that the
reduction in resistivity is attributed to the temperature increase in the carbon fibers and
contact points due to Joule heating. Fiber breakage under super high temperature is not
considered in the model. Thus, the model will not predict the resistivity increase due to
fiber breakage. To investigate the drastic increase in resistivity after material breakdown
(fiber breakage in this case), a detailed model describing the dynamic breakdown
behavior of carbon fiber as function of time and temperature is needed, which is beyond
the scope of this thesis.
6.2.6 Through-thickness resistance compared with simulation results
Composite laminates using IM7 carbon fiber as reinforcement are tested based
on the layup configurations listed in Table 6-3.
Table 6-3 Specimen layup
Specimen Layup
A [0]2
B [0/45]
C [0]8
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D [0]2 with thermoplastic powders
E [0]8 with thermoplastic powders
As demonstrated from the parametric study in Chapter 5, inter-ply quality is
vital to the nonlinear electrical conduction behavior of CFRP. To investigate the effect
of resin rich layer, special groups of specimen (D and E in Table 6-3) were prepared
with extra layer of thermoplastic powers added between carbon fiber prepregs. The
cross-section micrograph of the specimen with added particles is shown in Figure 6.12.
It can be clearly seen that carbon fibers from two adjacent layers are separated by
intentionally introduced resin rich layer.
Figure 6.12 Microscopic image of the cross-section of specimen E. Thermoplastic powers were added between plies, creating resin-rich layers.
Error! Reference source not found. Figure 6.13 shows the results for
specimens without manually added resin rich layer and the corresponding simulation
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results. The lines in each figure represents multiple specimens cutting from the same
composite laminate. Large variations in measured resistance among the specimens with
same material and fabrication methods demonstrate the stochastic nature of carbon
composite laminates. Although the starting resistance value varies, they show similar
trends of resistance change with current application. In the 2-ply [0/0] case, mild
resistance drop is observed over the whole current duration, which is well captured by
the model as demonstrated from the simulation results. The 2-ply [0/45] specimens
demonstrate larger variations among the specimens, which may be attributed to the fiber
misalignment during cutting or stacking the plies with different fiber orientation. Also,
one of the specimens shows larger resistance drop during the current application. The
extra reduction in resistance not captured by the Joule heating indicates material
degradation starts to contribute to the nonlinear electrical conduction behavior. This can
be demonstrated more clearly from the resistance drop patterns for 8-ply specimens.
Increase in laminates thickness introduced more uncertainties in material
microstructure. The “kinks” from resistance response curves in Figure 6.13Error!
Reference source not found.(c) indicates occurrence of material degradation,
breakthrough of resin for example. The 8-ply specimens show larger resistance than 2-
ply specimens in general, indicating worse inter-ply contact qualities. The extra
resistance drop is suspected to be attributed to the resistance drop due to material
degradation at the inter-ply interface.
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Figure 6.13 comparison between simulation results and experimental data for IM7 specimens without thermoplastic powders as listed in Table 6-3. (a) Specimen A with [0]2 layup; (b) Specimen B with [0/45] layup; (c) Specimen C with [0]8 layup.
169
This hypothesis is further supported by the resistance responses for the
specimens with intentionally added powder, as shown in Figure 6.14. Figure 6.14 (a)
and (b) plot the resistance response for [0]2 Specimen D and [0]8 Specimen E. The first
thing to notice is the larger resistance drop after the current application, as compared to
the specimens without embedded powder (resin rich) interfaces. Furthermore, the 8-ply
specimens show larger resistivity than the 2-ply specimens, indicate even worse inter-
ply contact quality induced from the increased difficulties to get the excess resin out of
the thicker laminates.
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Figure 6.14 Characterization of resistance response for (a) [0]2 Specimen D, and
(b) [0]8 Specimen E. The 8-ply specimens show larger resistivity than the 2-ply specimens, indicate even worse inter-ply contact quality induced from the increased difficulties to get the excess resin out of the thicker laminates
First attempts were made to relate the resistance change to temperature increase
due to Joule heating. Simulation results are shown in Figure 6.15 using the model
considering Joule heating effect of carbon fibers, but without resin breakdown, along
with experimental data from Figure 6.14(b). The two dotted lines represents two sets of
model parameters that give the upper and lower bound of the resistance values.
However, the trend for resistance change during current application is off. The model
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fails to predict as large resistance drop as observed from the experiments for 8-ply [0]8
carbon laminates (Specimen E). If only the temperature dependent carbon fiber
resistivity is considered as the major contributor to the resistance drop, a 30% drop in
resistivity is expected, while the experimental data yield more than 70% reduction in
through-thickness resistivity. This large discrepancy between simulation results and
experimental data indicates that reduction of the intrinsic carbon fiber resistivity with
the increase of temperature is not the key contributor to the overall resistivity drop.
Figure 6.15 Simulation results considering only the effect of temperature dependent
material properties. Predicted resistivity drops are smaller than observed from experiments.
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A parametric study is then carried out to demonstrate the limitations of a model
only considering the temperature dependent carbon fiber resistivity as the nonlinear
effect. Figure 6.16 plots the through-thickness resistivity normalized with the initial
resistivity before current application, for inter-ply connectivity ranging from 5% to
100%. In general, Joule heating at the inter-ply contact points becomes more severe as
the inter-ply connectivity decreases, causing higher temperature at the contact points,
thus lower electrical resistivity. However, it should also be noted that the reduction in
resistivity is limited, even if the inter-ply connectivity is considerably low: a 5%
connectivity yields 30% drop in resistivity, and a mere 8% drop in resistivity for inter-
ply connectivity of 5%.
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Figure 6.16 Limitations of a model considering only temperature dependent resistivity,
but not considering resin breakthrough. Maximum resistivity drop is only about 30% even for a small inter-ply connectivity (5%).
The failure to describe the resistivity change of CFRP under high current with a
simplistic model only considers the impact of temperature dependent intrinsic resistivity
indicates that the dominant mechanisms have yet been captured.
The high temperature induced from Joule heating not only affect the intrinsic
resistivity of the carbon fibers, which is nondestructive and reversible, but can also
impact the material structural in destructive ways. First, carbon fibers may break under
the ultra-high temperature at the contact points. This will lead to loss of conductive
paths, thus increasing resistivity. Secondly, resin matrix may degrade under the ultra-
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high temperature and/or the high electric field across the contact points. Once a thermal
or electrical breakdown occurs, the resistivity of the resin matrix may decrease
drastically, enabling it contribute to the conductive paths. The fact that the through-
thickness resistivity always drop more with higher current density indicates that the
second scenario is more likely the case. Careful inspection of Specimen E under
microscope shows that cracks have developed after the high current test around resin-
rich interface, as shown in Figure 6.17. This discovery further supports our hypothesis.
Figure 6.17 Micrograph showing the crack found in Specimen E after high current
application
For composites with resin rich interface, it’s thus favorable to use a model that
considers resin breakdown. The “ON-OFF” behavior model of resin breakdown
discussed in section 5.2.1 is adopted and parametric study using the model
configuration as demonstrated Figure 6.18 is carried out to demonstrate the impact of
resin breakdown. For clarity, a 2D configuration is presented, while the model considers
the 3D structure of CFRP. demonstrated Figure 6.18 (a) presents the case where
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conduction through thin resin layer is not considered, with inter-ply connectivity of
~15%. Model in demonstrated Figure 6.18 (b) considers both direct carbon to carbon
contact, and tunneling conduction through thin resin layer at the spots where direct
carbon to carbon contact may not be present. At low DC current condition, there should
be little difference in the resistivity calculated using these two models, since the
tunneling resistance is very high compared to the direct contact resistance, thus direct
contact between carbon fibers dominate the conduction mechanism. As Joule heat
accumulates, resin may break down and become conductive, this is when resin starts to
contribute to electrical conduction of CFRP comparable to carbon fibers.
Figure 6.18 (a) CFRP model without electrical conduction contributed by resin matrix; (b) CFRP model considering both direction carbon to carbon contact, and tunneling
conduction through thin resin layer.
Figure 6.19 plots the results from the parametric study. 10% of direct contact is
assumed for all model configurations, and tunneling conduction through thin resin layer
is increased from 0% to 90%. Same current waveform as used in experiments is applied
176
in the models: current ramps up linearly to peak current of 40A within 100ms. As the
ratio of contact through resin layers increases, larger reduction in resistivity can be
observed from the same current waveform. Resistivity reduction predicted by the model
without considering resin breaktdown (0% dielectric breakdown) can be attributed to
temperature dependent intrinsic resistivity, and is reversible should the temperature
drops. The extra reduction in resistivity as observed from simulation results given by
models that considers contributions from resin breakdown demonstrates the dominant
role of resin breakdown. If only 10% of the contacts are considered as tunneling
conduction, in addition to the fixed 10% direct carbon fiber contacts, a total drop of
40% in resistivity can be found. Model configuration with 90% of tunneling conduction
will yield a significant 65% drop in resistivity, which is comparable to the resistivity
reduction seen from the experiments (see Figure 6.14).
177
Figure 6.19 Parametric studies on the effect of resin breakthrough on resistivity.
6.2.7 Impact of current duration
CFRP specimen with T700 carbon fiber as reinforcement is tested using the
specimen fixture for through-thickness characterization described in section 6.2.2.
Three current waveforms with various current durations are applied: a) 100ms, b)
1000ms, and c) 2000ms. Voltage, current, load, and resistivity response over time are
recorded and presented in Figure 6.20. Temperature is not recorded in a continuous
way, but at discrete time points with a hand-held IR camera.
178
In the 100 ms cycle, resistivity drops almost linearly with the increase of
current. In the 1000 ms cycle, resistivity first drops with the increase of current, up to a
point where resistivity starts to increase at higher current. The increase of through-
thickness resistivity during current application were also noticed and reported by other
researchers [63]. It can be explained by the separation of contact points due to
expansion of resin matrix at high temperature from excessive Joule heating.
Temperature measurement with hand held IR camera shows that the temperature
increased by about 60 ∘C. Expansion of resin can also be reflected from the increased
load, as denoted by the golden lines in Figure 6.20.
In the 2000 ms cycle, resistivity response follows the trend as in the 1000 ms
cycle, until breakdown of resin occurs, as indicated by the sudden drop of resistivity as
well as load. Almost 100 ∘C increase of temperature was observed on the specimen
surface.
179
180
Figure 6.20 Current, voltage, resistance and load recordings during application of
current waveform with three durations: (a) 100ms, (b) 1000ms, and (c) 2000ms. Blue, orange, grey and gold lines represent voltage [V], current [A], normalized resistivity
and normalized load respectively.
6.2.8 Residue resistivity change after repetitive current applications
Repetitive current waveforms were applied to the same T700SC composite
specimen with dimension of 1’’ by 1’’. The current waveform used are listed in Table
6-4. In both cycles, current is increased in a linear way with peak current of 40A, which
is equivalent to about 60000 A/m2 expressed in current density. In Test A, the same
100ms linear ramp current waveform was used in both cycles, while in Test B, current
duration is 100ms in the first cycle, followed by current duration of 1000ms in the
second cycle.
Table 6-4 Current waveforms used in the repetitive current application tests.
Cycle Current Waveform
Test A Test B
1 90 ms ramp up, 10 ms ramp down 90 ms ramp up, 10 ms ramp down
2 90 ms ramp up, 10 ms ramp down 900 ms ramp up, 100 ms ramp down
Figure 6.21 shows the accumulated resistivity changes after two cycles of
current application. Resistivity is normalized by the initial resistivity before current
applications,
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Figure 6.21 Accumulated resistivity response for two types of tests: (a) same current waveform with 100ms current duration applied in the two cycles; (b) current duration is
100ms in the first cycle, and 1000ms in the second cycle.
In Test A, resistivity decreases with increasing current density, and recovers
partially after withdraw of current. Permanent resistivity change after first cycle was
observed, while additional cycles do not change the residue resistivity significantly.
In Test B, resistivity almost recovers to the initial value in the 100 ms cycle,
with small change in residue resistivity (less than 5%), indicating little destructive
change in material micro-structures. Resistivity response of the 1000 ms cycle follows
well with the 100 ms cycle until Joule heating becomes significant and causes resin
expansion, resulting in increase in through-thickness resistivity, as denoted from the
light blue window in Figure 6.21(b). After the 1000 ms cycle, irreversible resistivity
increase is observed, indicating permanent change in material micro-structure.
Extensive tests with 33 cycles in total was carried out. In all cycles, current is
increased in a linear way with the same peak current 40 A. Current duration in each
cycle varies according to Table 6-5. At least 5 min time gap was kept between cycles to
allow the specimen to cool down.
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Table 6-5 Current durations in the 33 cycles.
Cycle Current Duration
1-16 100 ms
17-32 1000 ms
33 5000 ms
Figure 6.22 plots the residue resistivity change as the test cycle progresses.
Temperature on the specimen surface was checked with a hand-held IR camera. It
shows a 7% reduction in residual resistivity after applying a high electric current for
100ms for the first time, further reducing to 91% after 16 cycles. There was no
significant heating during the 100ms current cycles. For longer duration currents (1000
ms), an increase in resistivity is observed in the initial cycle as well as noticeable
temperature increases. The following cycles shows gradual decrease in residue
resistivity. At the end of the 1000 ms cycles, resistivity returns to the similar level as the
end of the 100ms cycles. During ultra-long current duration cycle (5000ms), significant
increase in temperature (~ 120 oC) was observed, together with the largest drop (25%)
in residue resistivity after the current application.
183
Figure 6.22 Residue resistivity change as the test cycle progresses. A 7% reduction in residual resistivity after applying a high electric current for 100ms for the first time,
further reducing to 91% after 16 cycles. Total reduction in resistivity is about 35% after the last current cycle, where excessive heating is observed.
6.3 Resistance Response under Simulated Lightning Impulses
6.3.1 Descriptions of the experimental data
CFRP specimens are fabricated in the Center for Composite Materials (CCM) at
University of Delaware, and characterized using a simulated lightning strike apparatus
at our industrial collaborator’s facility. Unlike the well-controlled current waveform
used in previous studies, a recurrent impulse generator capable of delivering up to 500
V was used for the impulse tests. CFRP using AS4 and IM7 carbon fibers as
reinforcement are tested. The specimen dimensions and fabrication methods are the
same as those used in the tests described in Section 6.2.
184
Repetitive impulses with increased peak voltages applied to the same specimen.
Figure 6.23 shows the recorded voltage, current and resistivity under the increasing
peak voltage for the same IM7/9773 composite specimen. The impulse duration is
typically about 500μs. The kinks in the resistivity response curve in Figure 6.23(c)
indicates material degradation that has detrimental effect on electrical conduction in the
composite. It’s also observed that most significant changes happen in the first cycle,
which is in line with the observations from the dry fiber tow tests discussed before.
185
186
Figure 6.23 Typical voltage (a), current (b), and resistivity (c) response of 8-ply 1’’ by
1’’ IM7/-773 composite specimen. It also represents typical voltage, current, and resistivity response for other carbon composites tested in this study. Most significant
changes in resistivity normally happen in the first cycle.
6.3.2 Comparisons between simulation results and experimental data
With the added capability of considering resin breakdown under high
temperature or high electric field, the developed model in this study can be used to
describe the nonlinear conduction behavior of CFRP under simulated lightning strike
current.
Figure 6.24 presents comparison between simulation results and experimental
data. Model parameters are defined in Table 6-6 Parameter values used for modeling
resistivity of [0/90]2s AS4 laminate. Inter-ply connectivity is chosen such that the trend
of resistivity reduction is close to that from experimental observations. In this specific
case, inter-ply connection is set to 60%.
Table 6-6 Parameter values used for modeling resistivity of [0/90]2s AS4 laminate.
Parameter Value
Carbon fiber electrical resistivity 1.7×10-5 Ωm
Carbon fiber thermal conductivity 6.83 W/(mK)
Carbon fiber thermal capacity 1.13 kJ/kg∙K
Inter-ply connectivity 60%
Fiber volume fraction 0.55
Critical temperature for resin degradation 1500K
Critical electric field for resin degradation 1×109 V/m
Fiber waviness term 850
In this case, the high initial contact resistance generated sufficient power at the
contacts to enhance localized Joule heating and eventually induce current crowding
effects to bring the contact resistance to a greatly reduced value similar to that observed
in the dry fiber tow system.
187
Resistivity was significantly reduced by 57% in the first 20 𝜇𝑠 of current
application, namely, 3.5 𝛺𝑚 to 1.5 𝛺𝑚. After 40 𝜇𝑠, resistivity almost reached a steady
state value, with a slightly increasing trend. The model captures the drastic drop in
resistivity in the first 20 𝜇𝑠 and also the steady state value after long period of current
application. However, the increasing trend in resistivity at the end of current application
is not captured by present model. This trend is also noticed and reported by other
researchers [64-65], as suspected to be due to electromagnetic forces between carbon
fibers, pushing the fibers away from each other.
The five simulations utilized the same set of model parameter values, but yet
they give different resistivity response. The variations between these five simulations
comes from the stochastics terms embedded in present model: 1) randomized inter-ply
connection locations; 2) inter-ply contact resistance obeying statistical distributions. In
this study, a normal distribution for inter-ply contact resistance is used. This distribution
can be easily adjusted should other distribution types are proven to be more accurate.
This test case demonstrates the capability of the present model for considering
stochastic characteristics of CFRP micro-structures.
188
Figure 6.24 Comparison between simulations results and experimental data for a 8-ply AS4 composite laminate with layup of [0/90]2s, and size of 1 inch by 1 inch. Experimental voltage waveform is extracted and used as input in the model. Five
simulations are run using the same model parameters.
6.3.3 Residue resistivity change after repetitive current applications
Similar to the repetitive current tests conducted in Section 6.1, repetitive
lightning strike impulses are applied to the same AS4 [0/90]2s specimen, with desired
peak voltage in each cycle listed in
Table 6-7.
Table 6-7 Desired peak voltage in each cycle.
Cycle Desired peak voltage (V)
189
1 100
2 150
3 200
4 250
5 300
6 350
7 400
8 425
To show the contributions to the reduction in resistivity by two mechanisms,
temperature dependent intrinsic resistivity change and resin breakdown under Joule
heating induced high temperature, accumulated resistivity response during repetitive
current applications is plotted in Figure 6.25. Similar trend is observed as discussed in
Figure 6.5(b) for sized T700SC fibers under repetitive current applications.
190
Figure 6.25 Accumulated resistivity response during repetitive current applications. Irreversible resistivity reduction (denoted by blue arrows) is significant in the first cycle and decreases in the following cycles, while reversible resistivity change (denoted
by red arrows) is similar in all cycles.
Yellow arrows denote the initial drop as current increases, while green arrows
denote partial recovery in resistivity when current drops, which is not captured in the
current model. Red arrows denote resistivity recovery between tests, which is the
difference in the resistivity at the end of one cycle and the beginning resistivity in the
191
subsequent cycle. This reversible resistivity change is attributed to the temperature
dependent carbon fiber resistivity. Reversible resistivity is similar in all cycles.
Blue arrows denote the permanent reduction in resistivity after one current
cycle, which is the difference in the initial resistivity between two cycles. This
irreversible resistivity change is mainly attributed to the thermal breakdown of thin
resin rich layer. Irreversible resistivity reduction is significant in the first cycle and
decreases in the following cycles.
6.4 Summary and Conclusions
In this chapter, electrical characterizations under high current density are carried
out for dry fiber tows and cured composites experimentally. A modularized
characterization apparatus is designed in which one can input the desired
current/voltage waveform. By switching specimen fixtures, the apparatus can be used
for electrical characterization of dry carbon fiber tows and cured composites in each of
the three principal directions.
The influence of resin rich layers on the resistivity of carbon composites under
high current density is investigated with specially designed specimens that were
prepared with extra layer of thermoplastic powers added between carbon fiber prepregs.
The coupled thermal-electrical resistor network model developed in Chapter 5 is
utilized to explain the experimental results. Good agreements between simulation
results and experimental data indicate that the developed model captures most
characteristics of the electrical conduction behavior.
The contributions of reversible and irreversible resistivity change are identified
with carefully designed repetitive current tests for both dry fiber tows and cured
composites. It is found that for dry fiber tow with sizing and cured composites, thermal
192
breakdown of the thin resin/sizing layer contributes significantly to the nonlinear
conduction behavior under high current density conditions.
193
CONCLUSIONS, CONTRIBUTIONS, AND FUTURE WORK
7.1 Conclusions
This work has focused on developing a model that captures major electrical
conduction mechanisms of CFRP under various current conditions. A comprehensive
literature review revealed that most methods used to model electrical conduction of
CFRP fail to capture the impact of micro-structure of CFRP, especially the fiber-fiber
contact, and resin-rich layer between plies, which can drastically change the conduction
pattern.
The model is developed in an incremental way, based on cross-validations from
experimental discoveries. First, this study formulated a resistor network framework for
describing electrical conduction behavior of UD laminas and fiber tows under low DC
current. The experimental characterization of dry fiber tows under compressive loading
conditions is conducted. Electrical resistivity of dry fiber tows can be captured well
with the developed model. While applying the model to reported resistivity of CFRP in
three primary directions, the model predicts the resistivity in fiber length and in-plane
transverse direction correctly, but show large discrepancy between the predicted
through-thickness resistivity and that from reported experimental data. This
discrepancy motivated the consideration for CFRP with presence of resin rich layers.
Hence, the specific features at the microscopic level of a multi-ply laminate
were introduced and their influence on resistivity were investigated. The features
explored were varying ply orientation, existence of resin-rich layer, and dependence on
Chapter 7
194
geometric parameters of resistivity. Severity of resin-rich layer is described with an
inter-ply connectivity term. Formulas for estimating contact resistance from multiple
sources including direct fiber-fiber contact and tunneling resistance across thin resin
layer are derived. The modified model is compared against experimental results and
finite element model, while investigating the impact of specimen geometry parameters.
Good agreement was found between the developed model and experimental results as
well as FE model.
Finally, this work investigated the impact of high current density both
numerically and experimentally on the resistivity of carbon composites. Simplified
analytical model examining the impact of localized Joule heating revealed that current
concentrations due to microstructure constraints can introduce excessive Joule heating
at contact spots. It’s thus vital not to under-estimate the temperature rise at contact
spots, even at seemingly small overall applied currents. Based on these analysis, the
model is further improved to consider impact of Joule heating. Both reversible change
in resistivity such as temperature dependent resistivity and irreversible change such as
thermal and electric breakdown of resin matrix can be considered in the present model.
Electrical characterizations under high current density are carried out for dry
fiber tows and cured composites experimentally. The contributions of reversible and
irreversible resistivity change are identified with carefully designed repetitive current
tests. It’s found that for dry fiber tow with sizing and cured composites, thermal
breakdown of the thin resin/sizing layer contributes significantly to the nonlinear
conduction behavior under high current density. The developed model captures most
characteristics of the electrical conduction behavior. For the part where the model fails
195
to capture, possible explanations were given with support from other researchers’
findings.
7.2 Unique Contributions
Main contributions of this thesis work are as follows:
1. A micro-mechanics based resistor network model that correlated the
micro-structure parameters of fiber tows and UD CFRP with its electrical
property. To be specific, a fiber waviness term is used to describe the
distribution of contact points between carbon fibers, and the contact
resistance is linked to processing pressure, elastic modulus of carbon fiber,
and the intrinsic resistivity of carbon fiber. Parametric study shows that fiber
waviness term is the most important parameter in determining the through-
thickness resistivity of UD laminas when there is no resin-rich interface
present within the laminate.
2. A more generalized resistor network model that uses fiber bundle tow as
the basic modeling unit is developed to consider the impact of structural
dimensions (aspect ratio for example), stacking sequence, and ply
orientations.
3. The impact of resin-rich layer is addressed quantitatively. An interface
connectivity term is defined to describe the severity of resin-rich layer.
Parametric study shows that interface connectivity not only largely impact
the resistance of CFRP laminate (orders of magnitude change in through-
thickness resistivity can be achieved by changing the connectivity from 1%
to 100%), but also introduces variations into the observed resistance.
196
4. Current concentration at multi-scales in CFRP is discussed in detail for
the first time. A computational model to further investigate the Joule heating
effect is formulated. The computational work resulted in increased
understanding of the electrical behavior of CFRP under high current density.
5. Fundamental experimental work was performed to study the in-plane and
through-thickness resistivity’s of dry carbon fiber tows with and without
sizing subject to high current density. This work leads to the design and
implementation of a micro-mechanics based model that links the mechanical
properties and loading to the electrical behavior of carbon composites.
6. The use of repetitive current application to separate the contribution of
two major nonlinear conduction mechanisms – the temperature dependent
intrinsic resistivity of carbon fibers, and the material degradation under high
temperature induced from Joule heating – is also an innovative idea. With
the help of the modified model considering resin breakdown, it’s found in
the initial cycle, resistivity change is mainly irreversible, which can be
attributed to detrimental changes to material microstructure due to high
temperature or high electric field, while in the following cycles, resistivity
changes tend to recover after withdrawal of voltage impulses, which can be
explained by the temporary change in the intrinsic material resistivity due to
elevated temperature.
7.3 Future Work
The intellectual merit of this study is formulation of the micro-mechanics based
RC network framework to describe thermal-electrical conduction behavior of carbon
composites. Assumptions about nonlinear conduction mechanisms are utilized in the
197
current model, due to lack of information or for simplification of computation. One may
find it valuable to relax some of the assumptions used in the present model for more
accurate descriptions of the conduction behavior. For example, the resin breakdown is
represented with a simplistic “ON-OFF” behavior based temperature and electric field,
as demonstrated in Figure 5.8; with more knowledge on the thermal degradation
behavior of resin available, one can replace the simple formulas in the thermal-electrical
unit circuit, with more sophisticated ones. This can be easily modified without changing
the computation framework. Other details that can be integrated into the current model
include estimation of contact resistance between carbon fibers considering their surface
roughness.
Another direction of future work can be the application of the developed model
to optimal design of composite for better lightning strike protection. Although this
dissertation discussed the key factoring impacting conductive paths in carbon
composites, the optimal design of carbon composite for effective lightning strike
protection is not yet solved. With better understanding of nonlinear electrical
conduction mechanisms in carbon composites, more beneficial guiding principles for
stacking laminas and optimal placement of lightning strike protection strips or meshes
can be integrated with the mechanical design of composite structures.
Additional future work could be based on broadening the modeling scopes. For
example, one could include the capacitive properties of carbon fibers and resin system
to study how the carbon composite responds to AC current. Work reported by [55],
[66], [67] has mentioned the use of electrical properties of carbon fibers in the inductive
heating of CFRP. With the capability of considering localized heating using the micro-
198
structure based modeling framework developed in this study, interesting localized
heating patterns may be discovered.
Another example for extending the modeling framework is to consider damage
propagation within carbon composites. Damages in carbon composite can be thermally
or mechanically induced. Figure 7.1 shows the approach to model fiber breakage with
the developed resistor network framework. A broken fiber can be modeled with a super
large resistance. Using algorithms for parameter estimation, the resistance’s in the
resistor network can be estimated and correlated to the state of fiber breakage (very
large fiber resistance indicated large probability of fiber breakage).
Figure 7.1 Modeling broken fiber with resistor network
Figure 7.2 schematically demonstrates the crack growth initiated from localized
Joule heating. If the correlation between temperature profile and degradation of carbon
fiber or resin system is established, state of material can be estimated from the
temperature profile. Material damage state information can then be fed back into the RC
network model to update the electrical conduction behavior. With this iterative scheme,
damage propagation within CFRP due to Joule heating can be predicted.
199
Figure 7.2 Schematic illustration of Joule heating induced damage propagation in CFRP
200
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