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2005 ABAQUS Users’ Conference 1

Finite Element Modeling of Thermal Transport in

Composite Unit Cell using ABAQUS/CAE

J. K. Farooqi, M. A. Sheikh

School of Mechanical Aerospace and Civil Engineering, Faculty of Engineering and Physical

Sciences, University of Manchester, Manchester, M60 1QD, United Kingdom [email protected]

Abstract: FE modeling strategy for thermal transport behavior of a Woven-Fabric Composite is

sought using ABAQUS/CAE. A Ceramic Matrix Composite (CMC) has been chosen as a

convenient subject for this study as adequate physiological morphology and thermo-physical

property data is available for setting up its environment within ABAQUS/STANDARD. Limiting toa single computing platform and software in contrast to prior studies has been one of the

challenges and met successfully through ABAQUS/CAE. Concept of modeling and analyzing a

unique repetitive Unit Cell in CAE is the key to achieving representative thermal transport

character of a chosen composite. Such sophisticated materials have very complex and expensive

manufacturing routes, limited to just few capable research organizations. This fact broadens the

scope of a modeling study like this since examination of all possible material designs with various

constituent volume fractions can easily be carried out in ABAQUS/CAE with subtle manipulation

of key parameters dictated by quantitative SEM morphological data. Actual set of property data

used in Unit Cell of CMC is calculated after cumulative property degradation results extracted

from models of three observed unique porosities. Finally a comparison with experimental data is

done for establishing the health of complete modeling exercise, one to authenticate the validity of

the above scheme, secondly to open a range of modeling challenges for material designers

handicapped with lack of sophisticated manufacturing facilities. It is hoped that with more effort

in standardization of above strategy, a generic modeling scheme may evolve for virtually all

classes of composites manufactured today.

Keywords: Ceramic Matrix Composite (CMC), Damage Characterization, Finite Element

Analysis (FEA), Thermal Transport.

1. Introduction

Ceramic Matrix Composites (CMCs) have been purposefully engineered as per servicerequirement because of their capacity to retain the constituent characteristics, i.e. on one hand aid

resistance to hostile thermal and corrosive environment, as in case of aerospace and nuclear

applications, and on the other hand extend superior response to the performance aspect like

structural integrity, dimensional stability etc. Various NASA programs develop materials likeSiC/SiC composite through Chemical Vapor Infiltration (CVI) and Melt Infiltration (MI) for High

Speed Civil Transport (HSCT) engine (Brewer, 1999). CMC materials have known to be


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2 2005 ABAQUS Users’ Conference

employed in heat exchangers, industrial furnaces, gas turbines and fluidized bed combustion units.Other applications include dies and tool bits, medical implants, land-based power and transport

engines, and most importantly nuclear reactors. In the present study, only one of such CMCs has

been modeled with a detailed attention to its physical morphology and thermo-physical character.The objective is to predict this character by modeling the thermal transport in the presence of

damage history within the constituents of the CMC. Understandably, investigations for

characterization of CMCs have been the least in comparison with Polymer and Metal matrix

composites, since only limited treatment, primarily in defense and aerospace sector of the

developed countries has been possible to date. A traditional alternate adopted by the academiccommunity is modeling of these materials at micro- and meso-scale to assess their macro thermo-

physical, mechanical, electrical and chemical character within in the acceptable bounds presented

within published experimental data. Hence on similar lines, the present study involves modeling a

unit cell of CMC to develop a correlation for the constituent material’s thermal properties withthat of the macro-scale composite. During this modeling, the morphology of the CMC at micro-

scale has been developed closest to reality by replicating all features observed through SEM

micrographs. ABAQUS/CAE has been used presently for pre-processing, processing as well as

post-processing for this task.

1.1 Material under Investigation

It is a 10 laminate CMC (0/90o) plain weave composite material DLR-XT that has been developed

by the German Aerospace research establishment, Stuttgart, Germany for aerospace applications

involving high-temperature components of gas turbines (Krenkel, 2000). The architecture of DLR-

XT consists of T300 carbon fibers arranged in tows. These tows are used to form a plain weave togenerate a single laminate. Ten such laminates are assembled to form a sheet of the composite.

The laminate bundle is infiltrated with a polymer, which is thermally decomposed to leave a

carbon char. This is infiltrated with liquid silicon, which reacts with the carbon char to give SiCforming the matrix around the carbon fiber bundles or tows. Therefore, as a final product, the

fibers are contained in tows that are embedded in a carbon matrix, which as a whole is embedded

within the SiC matrix. Schematic of a sheet of this CMC composite is shown in Figure 1 alongwith a section of a single laminate is shown. The plain weave established using the above fiber

tows is also seen. Constituent highlights are

1. Carbon fibers bring together high strength and high modulus with high temperaturetolerance capability and also resistance to environmental attack. A suitable carbon-based

polymer (Polyacrylonitrile commonly known as PAN) is subjected to a controlled heating

to produce these T-300 carbon having orthotropic properties listed in Table 1.

2. Silicon Carbide is a very hard and abrasive material, having excellent resistance toerosion and chemical attack in reducing environments. In oxidizing environments, any

free silicon in a silicon carbide compact will be oxidized immediately. Melt Infiltration

(MI) has been employed here to transport the liquid Silicon in between the fiber tows for

curing with reaction bonding, resulting in properties listed in Table 1.

A composite’s performance characteristics are driven by manufacturing processes and the

constituents chosen above with the help of key processing parameters’ manipulation. Liquid

Silicon Infiltration (LSI) has been adopted as a successful CMC processing route which is the


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source of present study samples as well (Krenkel, 2003). But, post-production characterizationstudy shows, most manufacturing processes result in typical flaws. It is important here that we are

able to harness their physiological makeup in order to have realistic modeling.

1.2 Observation of Manufacturing Flaws

A number of optical micrographs of orthogonal planes have been used to classify four distinct

flaws, called porosities, with in this CMC. All four classes of porosity have been highlighted in aschematic shown in Figure 2 (Del Puglia, 2004a). The Figure 3 shows the x, y and z directional

othogonal views of the CMC especially highlighting the woven tow formation with in the SiC

matrix in a generic replicating fashion in Figure 3(b, c). Labelled in Figure 2, all porosities have

been briefly described here (Del Puglia, 2004a).

1. Inter-fiber micro-porosity (Class A porosity): This occurs between adjacent fiberscontained within a tow. It comprises either of a series of voids or spherical pores and

classified further as class A1 or large cracks between the fibers also classified as class A2

present within the interphase as shrinkage debonding.2. Trans-tow cracks (Class B porosity): It appears as cracks that run through the tows in

planes parallel to the fibers, which are orthogonal to principal directions.

3. Matrix cracks (Class C porosity): It is composed of cracks, embedded in the matrix,which surround the tow in planes perpendicular to the tow axes. These cracks are

contained within the SiC matrix, which encapsulates the fiber tows as an interphase.

4. Denuded matrix regions (Class D porosity): This type of porosity comprises of the largevoids, which occur at the intersection of four orthogonal tows during manufacture.

The degree of porosity has been quantified using area fraction and crack periodicity with length ofcrack and it has been utilized for outlining geometric data for the porosity sub-models as

developed earlier (Del Puglia, 2004b). Homogenization method, providing relationship between

the mechanical response and damage intensity in individual modes, has been adopted for a mixedapproach that takes into account basic strain and damage mechanism (Baste, 2001). Same

technique has been adopted here in the 2D C/C-SiC CMC for transforming and translating theindividual porosity effects on to the overall thermal behavior of the Unit Cell. Present porosity

submodels have been shown in Figure 4.

1.3 Thermal Properties

Thermal conductivity is one of the driving forces in designing materials for thermal applications.In a material, heat flow is proportional to the temperature gradient with the constant of

proportionality being the thermal conductivity. Its general form is

i

iji

xd

T d k q −= (1)


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where iq is the heat flux and ijk is the thermal conductivity. This is a second-order tensor and in

most cases a symmetric one. Another important parameter for controlling thermal transport is

thermal diffusivity α . It is defined as the ratio of a material’s capacity to conduct heat versus its

capacity to store it. It is related to thermal conductivity k , specific heat pC and density ρ as

pC k ρ α = (2)

Relevant thermal properties of CMC constituents, shown in Table 1, along with air for the pores or

cracks are employed for the thermal transport models.

2. Experimental Measurements

Experimental work conducted earlier at Material Science Center has been used for validation

purposes (Sheikh, 2001). The theory and experimental procedure for the thermal diffusivitymeasurement by the flash method, introduced earlier (Parker, 1961), has been described in detail

(Sheikh, 2000). It requires the surface of a small sample being irradiated with a laser pulse, and the

temperature response at the opposite surface recorded. The recorded data would be the

temperature- time profile of the rear face. The plot for temperature rise maxT T ∆∆ measurement

against time would highlight the peak value of 1 at a certain time. Considering the initial

temperature condition of 20oC, the time for half temperature rise value 0.5 would be noted as half-

rise time 21t . For each test these are obtained to calculate the thermal diffusivity α using the

appropriate expression in Equation 3.

For a thin disc specimen with one face uniformly irradiated (1D heat flow) the thermal diffusivity

α is given by the simple relation

L 0.139

21

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ =

t α (3)

where L is the sample thickness and 21t is the experimentally obtained half-rise time.

Finite-element methods for determining the thermal transport properties of solids are based on the

two thermal analyses; Steady State and Transient. The steady state thermal analysis using FEM

involves applying a temperature gradient xT ∆∆ across the composite section in a 1D heat flow

simulation. Using Fourier Law, xk , is given as

T

xqk x x ∆

∆= (4)


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where xq is the overall heat flux in the x -direction calculated by integrating the nodal flux values

across rear face. In the current analysis, xq is obtained using the nodal flux values (in x -

direction) given by the FE solution on one of the faces where the temperature boundary condition

is applied as in Figure 5(a). However, due to a high degree of mesh non-uniformity and asignificant difference in the thermal properties of the matrix and fiber, the nodal flux values vary

quite considerably across that surface, and the summation employed to calculate the overall flux as

∑∑==

= N

i

i

N

i

ii x A Aqq11

(5)

where N is the number of elemental faces, i A is the surface area ofthi element and iq is the

average of the nodal flux values for thethi element. The mesh size dictates N , which is arrived at

after experimenting with a range of mesh densities for convergence and accuracy. In the case of

transient thermal analysis, a heat flux is applied to one face of the composite section for a shorttime and the temperature history is recorded on the opposite face to simulate the experimentalconditions. The temperature profile is obtained by averaging temperature across the complete rear

face. This is obtained from an expression similar to Equation 5, as here iq is replaced by the

nodal temperatures iT as

∑∑==

= N

i

i

N

i

iiav A AT T 11

(6)

avT , the average temperature, is calculated for each time step through the transient analysis and a

temperature history is recorded. In case of ABAQUS/CAE, the output database has generated the

field data for temperature for all elements. The rear surface has too many elements to manuallycalculate the average temperature. Hence, temperature output is exported to a spreadsheet, where it

is ordered as suggested for finding the area of a planar surface from known coordinates (Braden,1986). Assuming 1D uniaxial heat flow, the half rise time related to this average temperature value

is then used in Equation 3 to calculate thermal diffusivity α and thermal conductivity k is found

from Equation 2. The remaining scalar properties, density ρ and specific heat pC for the

composite are determined using the rule of mixtures in Equation 7 shown in Table 3. It isimportant to emphasize that two different volume fractions are involved here. One is 75% Carbon

fiber with in the Carbon matrix, forming the fiber tow. The other is the 65% fiber tow in the

composite with remaining being SiC matrix surrounding it. Using rule of mixtures, specific heat

pC and density ρ as calculated as

)( ) ⎭

⎬⎫

−+=−+=

m f f f C

m f f f C

CpV V CpCp

V V

1

1 ρ ρ ρ (7)


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where f V is the fiber-volume fraction and the subscripts c , f and m denote the composite,

fiber and matrix, respectively. Given the composite weave complexity and fiber tow anisotropy,

no attempt has been made here to apply the rule of mixtures for calculating thermal properties ofthe composite, α and k . The results obtained during the whole modeling process are listed inTable 4. It is imperative that the final results obtained conform fairly close to the experimental

results (Sheikh, 2001), as compared in Table 5.

3. Modeling

Figure 6 shows a schematic view of the unit cell that is created with the help of SEM micrographs

of the DLRXT composite sections in the X, Y and Z planes shown in Figure 3. Bright areas denoteSiC and dark areas denote segments of the fiber bundles in the composite of 0/90 o configuration

set in plane weave pattern. From within these features a 3D unit cell is identified, which on

replicating itself through mirror translation in the three spatial directions produces the macro-

structure of the CMC. Very recently researchers have adopted similar approaches for structuralanalysis only (Nicoletto, 2004), (Woo, 2004), (Zako, 2003). A brief but specific outline of the

modeling undertaken with ABAQUS/CAE is given here. This entails the appropriate role of

various segments of the FE code ABAQUS/CAE called ‘modules’ for one complete cycle of

analysis as listed;

1. A quarter geometric part of CMC unit cell is created in PART module by extruding a basic sketch into a 3D feature. Further modification renders it exactly one-quarter volume

of the complete unit cell shown being assembled in Figure 7 in the ASSEMBLY module.

It also depicts the strategy used for one-quarter part being employed four times to

construct the whole, after precise translation, rotation and alignment. Each quarter part isfurther divided to identify unique fiber tow and matrix regions with the help of dissecting

planes in vertical and horizontal directions.

2. The material properties calculated for the unit cell at the end of Table 2 (and in Table 3for transient analysis) are assigned in PROPERTY module to appropriate compartmentsidentified for each constituent material. The sub-models used for predicting sequential

degradation due to each of the three porosities shown in Figure 4 have not been discussed

in detail due to limited space, but still the results are posted in Table 2. It is important to

highlight that contrary to the Class A, B and C porosities, the class D porosity is modeled

within the CMC unit cell as a central cavity as shown in Figure 7 with its materialconveniently chosen as air. A set of custom datum local coordinate systems is defined for

the quarter part to which orthotropy is linked. This is retained with in the part irrespective

of its assembly orientation. This is important for fiber tow orthotropic bais. For the

assembled regions to thermally interact and allow unhindered heat flow, master and slavesurfaces are defined on mating faces and INTERACTION module is used to establish a

TIE constraint amongst them in unique pairs.

3. DC3D4 tetragonal elements have been selected for meshing the unit cell in order toharness all complicated geometric features generated while dissecting volumes. Anoptimal mesh density was attained with 186,754 elements shown in the Figure 8 after


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and for each element area calculation (Braden, 1986) for finding avT using Equation 6.

Thermal diffusivity is calculated using Equation 3 using the half rise time 21t from the

rear face average temperature history obtained.

4. Results

From Table 4, it can be seen that the dominant effect of Class B (trans-tow cracks) is on the

transverse k -value and of Class C porosity (matrix cracks) is on the longitudinal k -value. Thelatter value is also affected by shrinkage debonding or Class A2 porosity. But it is largely the

Class C porosity that has a major role in reducing the overall spatial thermal conductivity profile

of the CMC when these values are utilized in the unit cell. The percentage error of the through

thickness thermal conductivity value is almost approximately 5%, which is considered acceptablein such simulations. An even larger error within the In-plane thermal conductivity values can be

due to the fact that the material morphology observed in micrographs has been modeled to thenearest possible feature with the exception that the curved plane visible in Figure 2(b, c) betweentwo interwoven fiber tows containing matrix within is modeled here with a 60o slope and sharp

cornered edges that turns on both flat surfaces on top and bottom of the unit cell, instead of the

being curved. Class D porosity quantification has been adopted as done earlier by morphologicalmodification in the unit cell (Del Puglia, 2004a).

5. Conclusions

It has been successfully demonstrated that the thermal transport character of a CMC can be

predicted by modeling, through ABAQUS/CAE, a representative unit cell developed from themicrographs obtained through SEM observations. A further development of a realistic set of

property values for the CMC constituents has also been done through sub-models for different

manufacturing porosities. This has resulted in the conviction of following this same sequence ofmodeling methodology for material designers for thermal characterization and other studies

employing this FE package. A much faster assessment tool for material designers is evolving for

validating theoretical predictions in absence of rigorous manufacturing experimentation.

6. References

1. Baste S., "Inelastic behaviour of ceramic-matrix composites," Composites Science and

Technology, vol. 61, no. 15, pp. 2285-2297, 2001.

2. Braden B., "The surveyor's area formula," The College Mathematics Journal, vol. 17, no. 4,

pp. 326-337, 1986.

3. Brewer D., "HSR/EPM combustor materials development program," Materials Science andEngineering A, vol. 261, no. 1-2, pp. 284-291, 1999.


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4. Del Puglia P., M. A. Sheikh, and D. R. Hayhurst, "Classification and quantification of initial porosity in a CMC laminate," Composites Part A: Applied Science and Manufacturing, vol.

35, no. 2, pp. 223-230, 2004a.

5. Del Puglia P., M. A. Sheikh, and D. R. Hayhurst, "Modelling the degradation of thermaltransport in a CMC material due to three different classes of porosity," Modelling and

Simulation in Materials Science and Engineering, vol. 12, no. 2, pp. 357-372, 2004b.

6. Krenkel W., "C/C-SiC samples from DLR (DLR-Institute of Structures and Design, Stuttgart,

Germany)," Stuttgart, 2000.

7. Krenkel W., "Microstructure Tailoring of C/C-SiC Composites." In 27th International Cocoa

Beach Conference on Advanced Ceramics and Composites: B, Cocoa Beach Florida, pp. 471-476, 2003.

8. Nicoletto G., and E. Riva, "Failure mechanisms in twill-weave laminates: FEM predictions vs.experiments," Composites Part A: Applied Science and Manufacturing, vol. 35, no. 7-8, pp.

787-795, 2004.

9. Parker W. J., R. J. Jenkins, C. P. Butler, and G. L. Abbott, "Flash method of determiningthermal diffusivity, heat capacity and thermal conductivity," Journal of Applied Physics, vol.

32, no. 9, pp. 1679-1684, 1961.

10. Sheikh M. A., A. S. C. Taylor, A. D. R. Hayhurst, and R. A. Taylor, "Microstructural finite-

element modelling of a ceramic matrix composite to predict experimental measurements of its

macro thermal properties," Modelling and Simulation in Materials Science and Engineering,vol. 9, no. 1, pp. 7-23, 2001.

11. Sheikh M. A., S. C. Taylor, D. R. Hayhurst, and R. Taylor, "Measurement of thermal

diffusivity of isotropic materials using a laser flash method and its validation by finite element

analysis," Journal of Physics D: Applied Physics, vol. 33, no. 12, pp. 1536-1550, 2000.

12. Woo K., and N. S. Goo, "Thermal conductivity of carbon-phenolic 8-harness satin weave

composites," Composite Structures, vol. 66, no. 1-4, pp. 521-526, 2004.

13. Zako M., Y. Uetsuji, and T. Kurashiki, "Finite element analysis of damaged woven fabriccomposite materials," Composites Science and Technology, vol. 63, no. 3-4, pp. 507-516,

2003.

Table 1. Standard thermal property values of constituent materials.

Material k (W m-1 K-1) ρ (kg m-3)

pC (x10-6 J kg-1 K-1)

Carbon Fibre Transverse 4 1928 921

Carbon Fibre Longitudinal 40 1928 921

Carbon Matrix 10 1800 717

SiC Matrix 70 3200 1422

Air 0.001 1 1


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Table 2. Degradation of k with sequential introduct ion of each class of porosity.

C Tow SiC MatrixLongitudinal ( || ) 32.4075 70Thermal Conductivity values for Virgin

Material (W m-1 K-1) Transverse ( ⊥ ) 5.086 70

Longitudinal ( || ) 32.3444 70Thermal Conductivity values forMaterial with Class A Porosity(W m-1 K-1) Transverse ( ⊥ ) 5.0368 70

Longitudinal ( || ) 32.24 70Thermal Conductivity values forMaterial with Class A and B Porosity(W m-1 K-1) Transverse ( ⊥ ) 4.44125 70

Longitudinal ( || ) 29.683 29.683Thermal Conductivity values forMaterial with Class A, B and CPorosity (W m-1 K-1) Transverse ( ⊥ ) 4.44125 70

Table 3. pC and ρ for CMC from constituent materials’ property values.

Material Property Carbon Fiber Carbon Matrix SiC Matrix Composite

1928 1800

1832 (fiber tow) 3200Density

ρ (kg m-3)2310.000

921 717.48

870.12 (fiber tow) 1422Specific Heat

pC (x 10-6 J kg-1 K-1)

1063.278

Table 4. k values for Through-Thickness and In-Plane heat flow direction along

with its overall sequential degradation in the unit cell.

C Tow SiC Matrix Macro Unit Cell

Thermal Conductivity(W m-1 K-1)

|| ⊥ || ⊥ In PlaneThrough

Thickness

Values for Virgin Material 32.4075 5.086 70 70 34.89 16.29

Values for Material with Class APorosity

32.3444 5.0368 70 70 34.85 16.23

Values for Material with Class Aand B Porosity

32.24 4.44125 70 70 34.55 15.52

Values for Material with Class A,B and C Porosity

29.683 4.44125 29.683 70 26.94 15.09

Values for Material with Class A,B, and C Porosity used in ClassD Porosity affected unit cell

29.683 4.44125 29.683 70 25.06 13.64


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Table 5. Comparison of 1D experimental results vs. FE analyses results fo r thermal

conductivity, k (W m-1 K-1).

FE ModelingDirection Experimental

Steady-State Transient

Through Thickness 14.39 13.639 11.55

In-Plane 22.45 25.06 25.16

Acknowledgment

Thanks are in order for Commonwealth Scholarship Commission for their financial support to the

principal author for the research conducted.

Figure 1. Schemmatic drawing of a laminate and a single lamina.

FE ModelingDirection Experimental

Steady-State Transient

Through

Thickness14.39 13.639 11.55

In-Plane 22.45 25.06 25.16


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Figure 2. Schematic drawing of a general plane orthogonal to either the X- or Z-direction il lustrating four classes of porosit ies: A, B, C, and D.

Figure 3. Optical micrographs of DLR XT C/C-SiC CMC with conversion to unit cellmodel; (a) y-direction, (b) x-direction and (c) z-direction.


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Figure 4. Porosit y sub-models along with their respective FE meshes.

(a) Class A Model

(b) Class B Model

(c) Class C Model


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Figure 7. Unit cell geometry being assembled from 4 quarter parts having fibre-volume fraction 65%.

Figure 8. Meshed unit cell with 186,754 DC3D4 tetrahedral elements .

class D porosity

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