APPROXIMATE METHODS FOR EVALUATING MODE MIXITY IN DELAMINATEDCOMPOSITES

By

NICOLAS VIGROUX

A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2009

1

c© 2009 Nicolas Vigroux

2

To Melissa

3

ACKNOWLEDGMENTS

I owe a debt of gratitude to my advisor, sponsor, and friend, Dr Bhavani V. Sankar.

I gratefully acknowledge his support and guidance. I also thank Dr Youping Chen and

Dr Ashok V. Kumar for being part of my committee and for their valuable feedback.

Furthermore, I would like to acknowledge all my family, in particular my parents Claude

and Jean-Paul and my siblings Vincent and Marie for their unconditional support and

love. Without them, this academic and research experience would not have been possible.

Finally, I am greatly thankful to my colleagues of the Center for Studies of Advanced

Structural Composites and my classmates in the department of Mechanical and Aerospace

Engineering. Their assistance and friendship have been invaluable.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 Definition of Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Damage in Delaminated Composites . . . . . . . . . . . . . . . . . . . . . 121.3 Scope of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 LINEAR ELASTIC FRACTURE MECHANICS . . . . . . . . . . . . . . . . . . 15

2.1 Stress Intensity Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Energy Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 The J Contour Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 THE DOUBLE CANTILEVER BEAM FRACTURE TEST . . . . . . . . . . . 20

3.1 Theoretical Approach to Mode Mixity Assessment . . . . . . . . . . . . . 203.1.1 Mode I Strain Energy Release Rate . . . . . . . . . . . . . . . . . . 203.1.2 Zero-Volume Jintegral around the Crack Tip . . . . . . . . . . . . . . 233.1.3 Mode Mixity Calculations . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Algorithm Essential to FEA Data Processing: Virtual Crack Closure Technique 263.2.1 Energy Release Rate Derivation . . . . . . . . . . . . . . . . . . . . 263.2.2 Finite Element Process . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Analytical Approach to Mode Mixity Computation: Crack Tip Force Method 313.3.1 Laminated Beam Equations . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Modes of Fracture Components . . . . . . . . . . . . . . . . . . . . 343.3.3 Validation of CTFM . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 FINITE ELEMENT ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 The 2D Double Cantilever Beam Finite Element Analysis . . . . . . . . . . 374.1.1 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.3 Discrepancy Inherent to 2D Analysis . . . . . . . . . . . . . . . . . 41

5

4.2 Double Cantilever Beam Plate Element Model . . . . . . . . . . . . . . . . 444.2.1 Finite Element Procedure . . . . . . . . . . . . . . . . . . . . . . . . 444.2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 The 3D Analysis of a Symmetric Double Cantilever Beam . . . . . . . . . 474.3.1 Description of the 3D SDCB Model . . . . . . . . . . . . . . . . . . 474.3.2 Convergence of G upon the Refinement of the Mesh . . . . . . . . . 484.3.3 Delamination Front Curve for an Arbitrary Mode Mixity . . . . . . 504.3.4 Perfect Correlation among Theoretical, Analytical, and Numerical

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 MULTI-FIDELITY RESPONSE SURFACE FOR ASYMETRIC DOUBLE CANTILEVERBEAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Computer Design Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.1 The 3D FE Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.2 The 2D Analytical Method . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Low Fidelity Analysis with High Quality Surrogates . . . . . . . . . . . . . 575.2.1 Crack Tip Force Method Polynomial Response Surface . . . . . . . 575.2.2 Energy Release Rate Correction Response Surface . . . . . . . . . . 595.2.3 Phase Angle Correction Response Surface . . . . . . . . . . . . . . . 61

6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6

LIST OF TABLES

Table page

4-1 ABAQUS input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4-2 Beam theory and 2D FEA GI computation (in N.mm−1) . . . . . . . . . . . . . 41

4-3 Accuracy of ABAQUS deflection output . . . . . . . . . . . . . . . . . . . . . . 42

4-4 Output of 2D bimaterial DCB FE analysis . . . . . . . . . . . . . . . . . . . . . 44

5-1 Prediction capabilities of energy release rate correction response surfaces . . . . 59

5-2 Prediction capabilities of phase angle correction response surface . . . . . . . . . 62

7

LIST OF FIGURES

Figure page

1-1 AA 587’s tail fin recovered from Jamaica Bay, Courtesy of National TransportationSafety board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2-1 Domain of validity of major failure mechanisms . . . . . . . . . . . . . . . . . . 15

2-2 Stresses near the crack tip in an elastic material . . . . . . . . . . . . . . . . . . 16

2-3 Basic fracture mechanics modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2-4 Arbitrary contour around a crack tip . . . . . . . . . . . . . . . . . . . . . . . . 18

3-1 Cracked plate submitted to fixed loads . . . . . . . . . . . . . . . . . . . . . . . 21

3-2 Load-displacement diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3-3 Additivity of energy release rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-4 Crack tip vicnity in a 2D finite element model before the crack closure . . . . . 26

3-5 Virtual crack closure in a 2D finite element model . . . . . . . . . . . . . . . . 27

3-6 Crack front region in 2D and 3D . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3-7 Sublaminates in delaminated beam . . . . . . . . . . . . . . . . . . . . . . . . . 31

3-8 Interfaces forces at the crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3-9 Crack-tip force acting on the top sub-laminates . . . . . . . . . . . . . . . . . . 33

3-10 Energy Release Rate with respect to mode mixity . . . . . . . . . . . . . . . . . 36

4-1 The 2D finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4-2 Meshing system near the crack tip . . . . . . . . . . . . . . . . . . . . . . . . . 39

4-3 Stress field singularity in 1√r

near the crack tip . . . . . . . . . . . . . . . . . . . 39

4-4 Jintegral under plane stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4-5 Jintegral under plane strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4-6 Finite Element analysis of a cantilevered beam deflection . . . . . . . . . . . . . 42

4-7 Deflection of sublaminates in opening mode . . . . . . . . . . . . . . . . . . . . 42

4-8 Mesh refinement around the crack tip . . . . . . . . . . . . . . . . . . . . . . . . 43

4-9 A 2D plate FE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4-10 A 2D plate model with springs as connectors . . . . . . . . . . . . . . . . . . . . 46

8

4-11 Description of a beam element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4-12 Comparison of 2D plate and 3D results in an opening mode fracture test . . . . 48

4-13 SDCB Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4-14 Analysis of the state of stress in Mode I . . . . . . . . . . . . . . . . . . . . . . 49

4-15 Convergence of the energy release rate . . . . . . . . . . . . . . . . . . . . . . . 49

4-16 Delamination front in Mode I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4-17 State of stress in Mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4-18 Delamination front in Mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4-19 Application of a single dead load in a FE SDCB . . . . . . . . . . . . . . . . . . 52

4-20 Delamination front in an arbitrary Mix Mode . . . . . . . . . . . . . . . . . . . 53

4-21 Correlation of total G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4-22 Correlation of the mode mixity . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5-1 Assymetric double cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . 55

5-2 Applying couples to the ADCB legs . . . . . . . . . . . . . . . . . . . . . . . . . 57

5-3 Polynomial response surface of the energy release rate . . . . . . . . . . . . . . 58

5-4 Correction response surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5-5 Phase angle correction response surface . . . . . . . . . . . . . . . . . . . . . . . 61

9

Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

APPROXIMATE METHODS FOR EVALUATING MODE MIXITY IN DELAMINATEDCOMPOSITES

By

Nicolas Vigroux

May 2009

Chair: Bhavani V. SankarMajor: Mechanical Engineering

In laminated materials, such as composites, interlaminar damage is one of the

major modes of failure. Due to the increasing incorporation of composites into aircraft

structures, this kind of insidious damage, also called delamination, has become an air

safety concern.

Therefore, two main research fields have emerged to answer the safety requirements

of the aviation industry: the first focus onto the detection of delaminations, whereas the

second develop delamination models to understand what constraints control the interface

crack growth.

Currently sophisticated structural health monitoring tools can provide information on

damage of a structure in real time. However, the information will be most useful if one can

perform a stress analysis of the damaged structure in real-time allowing prognosis of the

life of the structure.

Thus, the aim of the study is to implement high-fidelity delamination finite

element models and compare them with speedy analytical methods to allow a gain in

computational speed without losing much accuracy.

Fracture Mechanics principles are used in the analysis of 2D and 3D finite element

models that were conducted on a basic interface problem: the double cantilever beam

fracture test. Furthermore, numerical methods, such as the Virtual Crack Closure

Technique, are implemented in those models to assess the total energy release rate

10

along with the mode mixity, which permits to fully characterize the delamination at the

crack tip. Finally, in a multifidelity approach, 2D analytical Crack Tip Force Method

that provides inexpensive computing solutions is compared to the high-fidelity 3D Finite

Element Analysis to improve its prediction capability of the criticality of the damage of

the structure.

11

CHAPTER 1INTRODUCTION

1.1 Definition of Composites

A composite material is a combination of different materials that results in a single

structure with a distinguishable interface. Since many industrial applications (e.g.,

aerospace and automotive structures) require an increasing level of performance and at

the same time weight reduction, composite materials with novel fiber performs and tough

epoxy resin are being developed.

Those properties depend on the constituent materials but also on the interface.

Therefore, the optimization of the composite materials selection and manufacturing

processes has led to a wide range of applications. For instance, they can be extremely

valuable either as a core material, high strength and stiffness skins or outer protective

layers.

Because of various characteristics they can offer, they can be associated to form a

laminate structure in which laminae are joined together with precured, prepreg or wet

lay-up configurations. Among those processes, the prepreg lay-up one is the most common

in the aerospace industry, principally because it allows the production of high fiber volume

fraction composite parts (e.g., large glass/epoxy/honeycomb sandwich fairings for the

Airbus 330/340 flap tracks). However, during this mature manufacturing process, even

though squeezing rollers are used to remove air entrapped between prepreg sheets, the

fabrication of defect-free parts remains a challenge. Hence, in such aircraft structures, due

to the existence of those micro voids located at the laminar interface, stress concentration

effects can lead to debonding damage: a major failure mode in laminated composites.

1.2 Damage in Delaminated Composites

Interlaminar damage, also called delamination, represents actually an air safety

concern because this insidious kind of failure is a potential source of human tragedy.

Indeed, different factors as various as cyclic stresses, low velocity impacts, manufacturing

12

defects, lightning or bird strikes, can contribute to the propagation of interface cracks.

Under no monitoring assistance of crack extension, this mode of failure can then become

catastrophic. For instance, following the crash of a commercial airplane in Queens N.Y.,

on November 12th 2001, investigators have found that the killing of all 260 people in the

plane and five people on the ground, can be traced to such a defect in the carbon/epoxy

composite tail section of American Airlines flight 587. The failure of the attachment points

that held the vertical stabilizer to the fuselage is obvious in Figure 1-1.

Figure 1-1. AA 587’s tail fin recovered from Jamaica Bay, Courtesy of NationalTransportation Safety board

Therefore, to face that challenge, a lot of interest is focused into the detection

and characterization method of delamination. As this mode of failure is not visible,

non-destructive techniques, such as ultrasonic inspection, radiography or vibration and

thermal methods are under consideration. The ultimate goal of those studies is to give an

accurate strain mapping distribution of the whole structure. Those data can then be used

by computational codes running models of delamination behavior to assess the criticality

of each defect of the structures. Consequently, a remote monitoring assistance of aircrafts

is only possible if those computations can be realized in real time: the pilot usually needs

to react quickly!

13

1.3 Scope of the Study

The purpose of the research is to establish a relation, albeit empirical, between

the high-fidelity delamination models and fast analytical models to gain speed in the

calculation of the criticality of interlaminar defects without losing accuracy. To assess the

criticality of those static pre-existing cracks or flaws, a Linear Elastic Fracture Mechanics

(LEFM) approach is followed. Its principles will permit us to analyze the delamination

models that we are implementing.

The chapter 2 of this thesis is devoted to LEFM fundamental principles: an approach

based on energy balance, that is shown to be equivalent to classical stress analysis of

cracks, introduces global fracture parameters useful in the safety assessment of any

cracked structure. Energy criterion is preferred throughout the study as a better means

to characterize modes of fracture. As a result, those principles are heavily used in chapter

3, through the derivation of theoretical, analytical and numerical methods that allow the

evaluation of mode mixity in a basic interface problem: a double cantilever beam fracture

test.

In chapter 4, the Virtual Crack Closure Technique is implemented into different finite

element models to simulate the mechanical behavior of a cracked body. Their respective

accuracy was tested by comparing the global fracture parameters obtained to theoretical

ones. Here, limitations associated to 2D FE modeling were highlighted whereas 3D

FE analysis was proved to accurately predict the delamination behavior, but at a high

computational cost.

Such accurate prediction is used in chapter 5 to improve the prediction capability

of the speedy analytical Crack Tip Force Method. The introduction of multi-fidelity

correction response surfaces permits to gain in computational speed without losing much

accuracy.

14

CHAPTER 2LINEAR ELASTIC FRACTURE MECHANICS

The fundamental concepts of LEFM only applied to materials that obey Hooke’s law.

Indeed, the fracture toughness of composites is usually low enough to allow the collapse of

the structure before any onset or appearance of plasticity effects. Figure 2-1 shows that in

this linear domain, such structure experiences a brittle fracture.

Figure 2-1. Domain of validity of major failure mechanisms

More generally, in layered structure analysis, the Linear Elastic Fracture Mechanics

remains applicable when the failure zone is much smaller than the smallest dimension of

the specimen, which is usually the laminate’s thickness. Two equivalent approaches to

LEFM prevailed: the energy and stress intensity approach, which are described next, to

provide necessary background.

2.1 Stress Intensity Approach

When any linear elastic cracked body is subjected to external forces, a stress analysis

shows that in a coordinate axis, as defined in Figure 2-2, the stress field can be expressed

as a series in(

1√r

)n. Therefore, in the vicinity of the crack tip, as the higher order terms

vanish, a first-order asymptotic approximation (Eq. 2–1) is valid: the stress field solely

15

Figure 2-2. Stresses near the crack tip in an elastic material

depends on 1√r.

σx

σy

σz

=KI√2Πr

f1(θ)

f2(θ)

f3(θ)

+KII√2Πr

g1(θ)

g2(θ)

g3(θ)

(2–1)

The local fracture parameters Ki introduced in Eq. 2–1 are called stress intensity factors.

The subscripts i denote the fracture modes associated to specific loading conditions, as

described in Figure 2-3. f and g are functions that depend on the crack configuration. An

exhaustive list of possible configurations, along with corresponding functions are detailed

in the ”stress analysis of cracks handbook” [1].

Figure 2-3. Basic fracture mechanics modes

16

According to the principle of superposition, any load configuration can be split into

those three different modes. The latter are called modes of fracture because failure of the

structure can occur if the stress intensity factor Ki associated to a specific mode i exceeds

a limit Kic, that defines the fracture toughness of the material in a specific mode. Thus,

the structure remains safe under the conditions of Eq. 2–2.

∨ i ∈ {1, 2, 3} f(KI

KIc

,KII

KIIc

) < 1 (2–2)

where function f depends on the particular fracture criterion used. Hence, to determine

which mode of fracture is predominant, a mode mixity parameter has been introduced: the

phase angle Ψ (Eq. 2–3), which is the measure of mode II to mode I loading [2].

Ψ = tan−1

(KII

KI

)(2–3)

An analog parameter can also be defined with the out-of-plane shearing mode. The

knowledge of mode mixity along with total stress intensity factor fully characterizes the

stress behavior at the crack tip.

2.2 Energy Criterion

The energy criterion for fracture, first proposed by Griffith [3] , states that if the

energy available for a crack growth is greater than the material resistance, then fracture

occurs. Furthermore, when the crack extends, new surfaces are created in the elastic body:

the energy that has been brought to the system by the application of a set of external

forces is released. That rate of change in potential energy with respect to the crack area is

the Strain Energy Release Rate (SERR) G.

G = −dΠ

dA(2–4)

Here, to guaranty the safety of the structure the global parameter G must satisfy an

analog condition:

∨ i ∈ {1, 2, 3} Gi < Gic (2–5)

17

where Gic represents an alternative definition of the fracture toughness. Due to the

orthogonality of the fracture modes, under linear elastic assumption, G can be broken

down into its components in each mode of fracture by the linear relationship derived in

Eq. 2–6.

G = GI + GII + GIII (2–6)

Furthermore, Irwin cracks closure analysis [4] established a relationship between those two

approaches (Eq. 2–7).

G =K2I

E∗+K2II

E∗+K2III

2µ(2–7)

in which E∗ = E for plane stress, E∗ = E1−ν2 and µ is the shear elastic modulus. Eq. 2–7

is derived under the assumption of a self similar crack growth. In other words, the shape

of the crack must remain the same along its straight ahead propagation. That hypothesis

is usually not met for mixed mode fracture, in which crack kinking can occur. However, in

quasi-static state, such consideration is irrelevant. Therefore, energy and stress intensity

approaches are fully equivalent for linear elastic materials.

2.3 The J Contour Integral

The Jintegral on an arbitrary contour Γ around the tip of the crack, as shown in Figure

2-4, has been presented by Rice [5] as the path-independent integral of Eq. 2–8.

Figure 2-4. Arbitrary contour around a crack tip

J =

∫Γ

(wdy − Ti∂ui∂x

ds) (2–8)

18

where w is the strain energy density(w =

∫ εij0σijdεij

), T the traction vector (Ti = σijnj)

and u the displacement vector. For nonlinear and elastic materials, it is possible to show

under quasi-static conditions that the J contour integral is equivalent to the total energy

release rate(J = −dΠ

dA= G

); an elegant proof is derived by Anderson [6].

19

CHAPTER 3THE DOUBLE CANTILEVER BEAM FRACTURE TEST

This fracture test is a basic interfacial fracture problem that describes pretty well

the general behavior of a composite interlaminar damage because each ply of a composite

can typically be modeled as homogeneous. Furthermore, specific geometric conditions on

the crack and ligament length of that structure, respectively denoted by a and W − a in

Figure 3-1 , can reduce the size of the plastic zone, that is known to be proportional to

(KIc

σY s)2 [6]. Thus, if those two parameters are big enough, a large zone ahead of the crack

tip is K-controlled, which means that the stress intensity factor K totally characterizes

stresses near the crack tip. In addition, the external forces applied are selected to avoid

reaching the maximum load applicable to the structure. Consequently, the crack does not

propagate fast and the cracked beam can then be studied under quasi-static assumption.

Therefore, the above fracture test will be described under the assumption of a linear

elastic material under quasi-static behavior, hypothesis that allows the equivalency

between the energy release rate G and the J contour integral Jintegral.

3.1 Theoretical Approach to Mode Mixity Assessment

3.1.1 Mode I Strain Energy Release Rate

In Figure 3-1, the geometric parameters of the facture test in which forces are applied

to each leg of the beam, are detailed. In this section, the top and bottom forces will

be assumed to have equal norms and opposite directions. As seen earlier, that loading

condition is called Mode I or opening mode.

Irwin [7] defined strain energy release rate G as the energy required to extend a crack

by a small increment (Eq. 3–1).

G = −dΠ

dA(3–1)

in which Π is the potential energy and A the area created by the extension of the crack.

Griffith Energy balance [3] states that crack extension occurs when G reaches a critical

value, also called fracture toughness of the material. Indeed, this energy balance (Eq. 3–2)

20

Figure 3-1. Cracked plate submitted to fixed loads

shows that the energy required to create new surfaces equals the strain energy that is

released.

dE

dA=dΠ

dA+dWs

dA= 0 (3–2)

where E is the total energy and Ws the work needed to create new surfaces. Therefore,

the critical strain energy release rate Gc can be expressed as

Gc =dWs

dA= 2γs (3–3)

where γs is the energy needed to create a new surface. As the crack extends, two new

surfaces are de facto created on both sides of the crack tip, which explained the presence

of the number 2 in the above equation.

As shown in Figure 3-2, the force applied to both ligaments of the double cantilever

beam is fixed: the plate is then load-controlled. Thus, the crack propagates by a

small increment at a fixed load before unloading the structure, as depicted in the

load-displacement diagram in Figure 3-2. The potential energy Π is defined in Eq. 3–4.

Π = U −WF (3–4)

21

where WF is the work done by external force and U is the strain energy stored in the

body. In this specific double cantilever beam test, WF = F∆ and U =∫ ∆

0Fd∆ = F∆

2.

Consequently, Π = −U . Substituting that expression into Eq. 3–1, we obtain Eq. 3–5.

G =

(dU

dA

)F

(3–5)

Figure 3-2. Load-displacement diagram

Figure 3-2 shows that the increase of the strain energy due to the extension of the

crack can be derived as

(dU)F =∆F

2+ Fd∆− (∆ + d∆)F

2=Fd∆

2(3–6)

According to beam theory, a relationship is established between the force F and the

displacement ∆ (Eq. 3–7).

∆ =2Fa3

3EI(3–7)

22

where I = Bh3

12and E is the Young Modulus of the homogeneous material. Substituting

that result into the expression of G derived in Eq. 3–5 gives Eq. 3–8.

G =F

2B

(d∆

da

)F

=F 2a2

BEI(3–8)

Finally, substituting the moment of inertia I, as derived above, into Eq. 3–8 permits

to reach an expression of the Mode I strain energy release rate GI in function of the

loading conditions, the material and geometric properties (Eq. 3–9).

GI =12F 2a2

B2h3E(3–9)

3.1.2 Zero-Volume Jintegral around the Crack Tip

The Jintegral is path-independent. Therefore, the contour on which the line integral

is defined can be moved very close to the crack tip. Let’s then consider three vertical

paths AB, CD and EF respectively numbered 1, 2 and 3, next to the crack tip as shown in

Figure 3-1. Those paths formed altogether a closed line along the crack tip. As nxds = dy

and Ti = σijnj, the line integral defined in Eq. 2–8 can be written as

J =

∫Γ

(wnx − σijnj∂ui∂x

ds) (3–10)

Along the path 1, using the definition of the strain energy, we derive Eq. 3–11.

J (1) =

∫ B

A

(1

2σijεijnx − σijnjui,x

)ds (3–11)

For that particular path, nx = −1 and ny = 0. Substituting the normal vector components

gives Eq. 3–12.

J (1) =

∫ B

A

(−1

2(σxxεx + τxyεxy) + σxxu,x + τxyv,x

)ds (3–12)

Using strains definitions εx = u,x and εxy = 12

(u,y + v,x) gives Eq. 3–13.

J (1) =

∫ B

A

(1

2σxxu,x +

3

4τxyv,x −

1

4τxyu,y

)ds (3–13)

23

which can be rearranged (Eq. 3–14) to make appeared the strain energy density w and the

rotation Ψt = 12(u,y − v,x) of the beam cross section at the crack tip.

J (1) =

∫ B

A

(1

2(σxxεx + τxyεxy)−

1

2τxy(uy − v,x)

)ds (3–14)

Next, a separation of the integral is realized (Eq. 3–15).

J (1) =

∫ B

A

wds−Ψt

∫ B

A

τxyds (3–15)

At this step of the computation, we identify the shear force V . Hence, the expression of

Jintegral along the path 1 is simplified (Eq. 3–16).

J (1) = w(1)L −ΨtV1 (3–16)

where w(1)L represents the strain energy per unit length along the path 1. By analogy,

along the path 2, the same form is obtained (Eq. 3–17).

J (2) = w(2)L −ΨtV2 (3–17)

For the last path, nx = 1, which results in a change of sign (Eq. 3–18).

J (3) = −w(3)L + ΨtV1 (3–18)

As the paths lie in the vicinity of the crack tip, the shear force resultants must satisfy,

by an argument of continuity, the equilibrium condition V1 + V2 = V3. Adding the

three integrals previously derived the shear forces vanish from the resulting equation. An

equation is consequently derived between the Jintegral and the sole strain energy densities,

expressed here per unit length.

J = J (1) + J (2) + J (3) = w(1)L + w

(2)L − w

(3)L (3–19)

Therefore, for elastic materials under quasi-static assumptions, the equivalency

between Jintegral and G, permits to reach a convenient way to compute the strain energy

24

release rate in a double cantilever beam fracture test: the latter is the difference between

the strain energy densities per unit length just behind and ahead of the crack tip.

3.1.3 Mode Mixity Calculations

Since we are dealing with linear elastic fracture, principle of superposition is

applicable. Thus, considering a set of forces as described in Figure 3-3, all mode mixities

can be assessed through the computation of the strain energy density along each cross

section, behind and ahead of the crack tip in a double cantilever beam fracture analysis.

Figure 3-3. Additivity of energy release rate

Behind the crack tip, the strain energy density per unit length in each cross section

of the beam is identical in both pure modes: w(1)L = w

(2)L = M2

2EIB, whereas that quantity

is, ahead of the crack tip, either null in opening mode or equal to w(3)L = M2

4EIBin shearing

mode. Substituting into Eq. 3–19 the above expression of the strain energy densities per

unit length gives us an expression for pure modes strain energy release rates (Eq. 3–20).

GI = JI =a2(F1+F2

2

)2

EIB, GII = JII =

3a2(F1−F2

2

)2

4EIB(3–20)

As a matter of fact, the above result is the same than the one previously derived in pure

Mode 1 using beam theory (Eq. 3–8). In an arbitrary mode of fracture, called mode mix,

in which F1 and F2 are respectively applied on the top and bottom half, as shown in

Figure 3-3, the energy release is computing by adding its mode components (Eq. 3–21).

GMixMode =a2

8EIB

[7

2(F 2

1 + F 22 ) + F1F2

](3–21)

25

3.2 Algorithm Essential to FEA Data Processing: Virtual Crack ClosureTechnique

The virtual Crack Closure Technique (VCCT) permits to assess with a simple

algorithm the fracture parameters for all three fracture modes. We will discuss later the

accuracy of such technique in a finite element analysis of a 3D double cantilever beam

model implemented on ABAQUS. At this point, we will focus onto the principles on

which the technique is based before detailing the various steps of the algorithm and its

implementation in the commercial software MATLAB.

3.2.1 Energy Release Rate Derivation

According to energy conservation principles, Irwin [4] assumes that the energy

released by an incremental crack extension ∆a should be equal that the amount of energy

necessary to close the crack to its initial state. In a two dimensional configuration, such as

the one shown in Figure 3-4, this work can be computed as

W =1

2

∫ ∆a

0

u(r)σ(r −∆a)dr (3–22)

where σ is the stress and u the relative displacement at a distance r behind the crack tip.

Figure 3-4. Crack tip vicnity in a 2D finite element model before the crack closure

26

Figure 3-5. Virtual crack closure in a 2D finite element model

Therefore, in quasi-static state, the energy release rate is

G = lim∆a→0

W

∆a= lim

∆a→0

1

2∆

∫ ∆a

0

u(r)σ(r −∆a)dr (3–23)

Rybicki and Kanninen [8] showed that the expression in Eq. 3–23 can be computed

numerically through finite element analysis. In finite element, since no forces are

transferred through element edges, forces need to be applied on share nodes, as illustrated

in Figure 3-4 in order to close the crack and reach the state described in Figure 3-5. The

corresponding work W needed is

W =1

2Fu (3–24)

where F is the force required to maintain together nodes i and i∗ and u the distance

between those nodes before the closure. This force can be approximated by the force at

node j of the element 2. This internal force which is equal with an opposite direction than

the force at node j∗ of the element 4 can be computed in the same step analysis from the

stress in the element. Consequently, the analysis of the finite element with the crack closed

(Figure 3-5) is not needed: this is one of the main advantage of that technique [9].

27

If the finite element mesh is refined enough to allow the convergence of the limit in

Eq. 3–23, the strain energy release rate can be computed for the Modes I and II, only

modes available in 2D. This calculation is led using Eq. 3–25.

Gi =1

2B∆aFi∆ui (3–25)

in which the subscript i indicates the type of loading conditions. In opening mode (i=I),

the force is normal to the crack plane whereas in shearing mode (i=II), the force is

parallel. In both cases, the direction of the displacement is collinear to the corresponding

force applied.

Shivakumar extended the range of application from 2D to 3D [10], which allows not

only a gain in accuracy but also a Mode III assessment. The integral formulation takes the

following form

Gi = lim∆a→0

1

2wi∆a

∫ δi+1

δi−1

∫ ∆a

0

u(r, s)σ(r −∆a, s)drds (3–26)

where the couples (s,wi) and (r,∆a) are the distance and the element length respectively

along and normal to the crack front; σ is the stress distribution ahead of the crack front

and u the relative displacement behind the crack front.

3.2.2 Finite Element Process

Figure 3-6. Crack front region in 2D and 3D

28

In a 3D solid finite element model comprising 20-nodes brick element (CRD20R), we

consider the association of two elements located above the crack plane, on both sides of

the crack tip. The nodes of the crack tip interface elements are shown in Figure 3-6. The

application of Eq. 3–26 in such structure gives Eq. 3–27.

G =1

2∆2

(3∑i=1

Fci(ai − Ai) +2∑i=1

Fdi(bi −Bi)

)(3–27)

where ∆ is both the width and the length of the quadratic element. ai, Ai, bi and Bi are

the position vectors of the brick-element nodes, as depicted in Figure 3-6. Therefore, the

differences computed in Eq. 3–27 permits to assess the opening and shearing distances at

a distance ∆ and ∆2

from the crack tip, respectively along the direction z for opening and

along x and y for in-plane and out-of-plane shearing. Those distances are multiplied by

the forces Fci and Fdicalculated at nodes ci and di respectively in order to evaluate the

work needed to extend the crack by the length of the element ∆. Each force is computed

using ABAQUS NFORC output that is the nodal force caused by the stress in a specific

element. When a node is located on the crack tip, the associated force presented in Eq.

3–27 is the sum of the nodal force from both top elements ahead and behind the crack tip.

Another way to do the calculation is to consider two halves of elements on both sides

of the crack tip. The origin of that new reference element is thus shifted by ∆2

in the

direction of the crack tip. In Figure 3-6 , its middle is the node c3 and its length is the

distance ∆ = c4 − c2. With this new reference, the energy release rate is given by

G =1

2∆2

(Fc3(a3 − A3) + Fd2(b2 −B2) +

1

2(Fc2(a2 − A2) + Fc4(a4 − A4))

)(3–28)

In that case, as the reference element does not coincide with the elements of the

finite element model, the force Fc2 is computed adding the nodal forces of the 4 FE

elements, located on the top of the crack plane, that surround this node. Due to the

fundamental FE equation (KU = F ) in which internal forces are unloaded to satisfy

the equilibrium, such nodal force is opposite to the one that one could compute taking

29

into consideration the set of 4 elements located below the crack tip plane. Those two

different approximations give sensibly the same results, especially if a fine mesh has been

implemented.

In order to split the energy release rate into the three modes of fracture, it is required

to derive each of them using the respective force and displacement components of each

direction of the coordinate axis defined in Figure 3-6. Such approximation of the energy

release rate components is expressed in Eq. 3–29 [11].

GI =Fz∆w

2B∆aGII =

Fx∆u

2B∆aGIII =

Fy∆v

2B∆a(3–29)

• Implementation of the VCCT in MATLAB

The implementation of the Virtual Crack Closure Technique has been done using

the commercial software MATLAB. The input necessary to run such code are the output

of the finite element model, i.e. the nodal forces and the displacements that we have

defined for each unit cell. Those ABAQUS outputs are accessible through a History

output request. Then, it is possible from ABAQUS to store all forces in a vector column,

where each line is associated to the node of an element. Understanding in which order

ABAQUS list the 20 nodes of each element allows the extraction of the force components

that appear in Eq. 3–27 and Eq. 3–28.

To compute the displacement components, an analogous procedure is followed: the

position of each node that is used to compute displacement components are extracted

before being rearranged in a convenient way. Finally, the computation of the energy

release rate is done using the data stored for each element along the crack tip. This

algorithm permits us to obtain the delamination front curve, i.e. the strain energy release

rate in function of the width of the beam.

30

3.3 Analytical Approach to Mode Mixity Computation: Crack Tip ForceMethod

In this section, we describe simple and closed-form solution, first proposed by Sankar

[12], of the mode I and mode II energy release rates for the general delaminated beam

problem, as depicted in Figure 3-7.

Figure 3-7. Sublaminates in delaminated beam

The energy release rates are expressed in terms of all axial loads, bending moments

and shear forces applied at the cross section of the crack tip.

3.3.1 Laminated Beam Equations

We are here considering, in the derivation of the set of equations that permits to

compute energy release rates and mode mixity, the xy-plane as the reference plane for

both top and bottom layers; this assumption permits us not to introduce any offset values

along the computation. According to the classical beam theory, the displacement field for

laminated composite beam is of the form

u(x, z) = u0(x) + zΨ(x) w(x, z) = w0(x) (3–30)

In the above in-plane displacement equation, u0(x) is the axial displacement of

points in the reference plane and, Ψ(x) the rotation about the y-axis. Furthermore,

the transverse displacement is supposed to stay constant through the thickness. Strain

31

relations derived from the previously described displacement field are

εxx ≡ u,x = u0,x + zΨ,x (3–31)

where ,x denotes partial differentiation with respect to x.

For each orthotropic layer of laminated composite beam, the stress-strain relations are

σxx = Q11εxx τxz = Q55γxz (3–32)

in which Q11 and Q55 are stiffness terms. For plane strain with a fiber in the same

orientation than the direction of the beam, those terms are defined by Q11 = E11

(1−ν12ν21)and

Q55 = G13.

For the top layer, the force and moment resultants are defined as

FT = bP,M, V c = b

∫ h1

0

bσxx, zσxx, τzxcdz (3–33)

in which P , M and V are respectively the axial, bending and shear force resultants; b and

h1 are the width and the thickness of the top layer. The stress resultants are related to

strains via F = [S]e where the compliance matrix [S] is given in Eq. 3–34.

S =

a11 b11 0

b11 d11 0

0 0 a55

(3–34)

(e)T =

⌊εx0 κx γxz

⌋(3–35)

The stiffness terms appearing above are defined as

ba11, b11, d11, a55c =

∫ h1

0

⌊Q11, zQ11, z

2Q11, κQ55

⌋dz (3–36)

κx and κ are respectively the beam curvature and the shear correction factor that can

be assumed as 5/6 according to Sankar et al [13]. For the bottom layer, those equations

32

are still valid, except for the boundary of the integral that needs to take into account the

different thickness.

Figure 3-8. Interfaces forces at the crack tip

Near the crack tip, a free body diagram analysis permits to write the equilibrium

between force and moment resultants

F3 + F4 = F1 + F2 (3–37)

Since the sublaminates 3 and 4 are intact ahead of the crack tip, the strains are identical.

This equivalence is stated in Eq. 3–38.

(e3) = (e4) = [Ct](F3) = [Cb](F4) (3–38)

In Eq. 3–38, [Ct] and [Cb] are respectively the stiffness of the top and bottom layers

and they are equal to [S−1t ] and [S−1

b ], respectively.

Solving for F4, we obtain Eq. 3–39.

(F4) = ([Ct]−1[Cb] + I)−1(F1 + F2) (3–39)

Figure 3-9. Crack-tip force acting on the top sub-laminates

33

From the free body diagram equilibrium of Figure 3-9, an expression of the crack tip

force is derived in Eq. 3–40.

Fct = F1 − F3 (3–40)

Using the relationships derived in Eq. 3–40 and Eq. 3–37, a new expression of the

crack tip force Fct is reached (Eq. 3–41). That new form is advantageous because the only

variables are the material properties expressed in the stiffness terms Ct and Cb and the set

of external loads applied to the body ( F1 and F2 ).

Fct = F4 − F2 = ([Ct]−1[Cb] + I)−1(F1 + F2)− F2 (3–41)

3.3.2 Modes of Fracture Components

Assuming quasi-static conditions and elastic properties for the material considered,

the energy release rate G is equal to Jintegral. Therefore, using the expression of the

zero-volume Jintegral around a closed contour in the vicinity of the crack tip for a double

cantilever beam, stated in Eq. 3–19, a new relation (Eq. 3–42) is established for the total

energy release rate.

G = w(1)L + w

(2)L − w

(3)L − w

(4)L (3–42)

where w(i)L is the strain energy density per unit length for a specific sub-laminate, as

defined in Eq. 3–43.

wL =1

2FT [C]F (3–43)

in which F is the vector force as described in Eq. 3–33 and [C] is the stiffness of the

sublamintes. [C] is computed taking the inverse of the compliance matrix [S] (Eq. 3–34),

which depends of the sublaminate. Substituting that expression of the strain energy

density into Eq. 3–42 gives Eq. 3–44.

G =1

2

(FT

1 [Ct]F1 + FT2 [Cb]F2 − FT

3 [Ct]F3 − FT4 [Cb]F4

)(3–44)

34

The use of the strain identity relationsip derived in Eq. 3–38 and equilibrium between

forces presented in Eq. 3–37 allows us to establish a new relation (3–45).

G =1

2

((F3 − F1)T [Ct + Cb](F3 − F1)

)(3–45)

Substituting Eq. 3–40 into Eq. 3–45, we obtain a more compact form (Eq. 3–46).

G =1

2FTct(Ct + Cb)Fct (3–46)

If the force at the crack-tip is represented by the column vector FTct =

⌊Fx C Fz

⌋, the

energy release rate can be decomposed into different fracture modes.

G = 12

⌊Fx C Fz

⌋c11 c12 0

c12 c22 0

0 0 c66

Fx

C

Fz

= 1

2c11F

2x + c12FxC + 1

2(c22C

2 + c66F2z )

(3–47)

The components of the energy release rate can be split into the two fracture modes.

The mode III doesn’t intervene here because our model is planar. The force Fx provokes

the two laminates of the double cantilever beam to slide on each other: it has then a

shearing effect. Moreover, the couple C and the force Fz are responsible for the opening

of the crack: those components belong to Mode I effect. Therefore, the components of the

energy release rate can be expressed as in Eq. 3–48.

GI =1

2(c22C

2 + c66F2z ) , GII =

1

2c11F

2x + c12FxC (3–48)

Finally, from the energy release rate mode components, the mode mixity Ψ can be

computed under the assumption of a linear elastic material (Eq. 3–49).

Ψ = tan−1(KII

KI

) = tan−1

√GIIGI

(3–49)

35

Figure 3-10. Energy Release Rate with respect to mode mixity

3.3.3 Validation of CTFM

The energy release rate components stated in Eq. 3–48 have been computed using

Matlab for different load configurations. To check the accuracy of the results obtained, the

same calculation has been led using an other analytical method detailed by Hutchinson

and Suo in their review on ”Mixed mode cracking in layered materials” [2].

The total strain energy release obtained by both methods is identical: the standard

deviation of those two distributions, which are plotted in figure 3-10, is indeed around 0.2.

Moreover, an even better equivalency is reached for the phase angle Ψ: the standard

deviation is in this case closed to 3.10−4.

36

CHAPTER 4FINITE ELEMENT ANALYSIS

An interlaminar damage study was conducted in a double cantilever beam. The

purpose of the finite element analysis, realized using the version 6.7 of the software

ABAQUS, is to assess with high accuracy the mode mixity for different types of loading

conditions, geometric and material properties. First, we proposed 2D finite element

models in order to show the inherent discrepancy of either plane stress or plane strain

2-dimensionnal analysis. Then, a 2D plate model is introduced. The application of the

VCCT algorithm on that model permits to obtain the energy release rate components

along the width. Finally, a 3D finite element model of an homogeneous, isotropic,

symmetric double cantilever beam is created. The mode mixity is then computed through

the implementation of the same numerical method: the Virtual Crack Closure Technique.

That high fidelity analysis will be compared to analytical techniques such as the

Crack Tip Force Method whose main advantage is based on its low computation cost. As

a matter of fact, combining high fidelity of 3D finite element analysis with expensiveness

analytical methods will allow us to benefit from both computations.

4.1 The 2D Double Cantilever Beam Finite Element Analysis

The purpose of the 2D analysis is to assess the effects of the loading forces on a

cracked plate and the accuracy of mode mixity calculation in modeling including mesh

elements that allow a low number of computation steps in comparison with cumbersome

3D finite element models.

4.1.1 Finite Element Model

First, let us consider a symmetric double cantilever beam test in opening mode.

For that case, the same surface traction is applied on both legs of that beam in which a

crack has been created in the mid-plane. Moreover, we are assuming a straight extension

of the crack. In the implementation on a finite element code, that assumption permits

us to model only half of the beam and then reduce the CPU time approximately by

37

a factor of two . This model reduction is possible thanks to the application on the

remaining structure of a set of boundary conditions, as shown on the finite element model

of Figure 4-1, which also introduces all the parameters of this fracture test: a and W are

respectively the crack and ligament length, B is the width and F is a shear force applied

on each leg of the double cantilever beam whose height is h.

Figure 4-1. The 2D finite element model

The material chosen for that test is an high strength aluminum alloy that is assumed

to be homogeneous, linear, elastic and isotropic. Thus, only two engineering constants

are required to characterize its mechanical properties: the Young’s modulus E and the

Poisson’s ratio ν. Those input to the finite element code implemented on ABAQUS, along

with geometric properties and load conditions are detailed in Table 4-1.

Table 4-1. ABAQUS input

load geometric properties material propertiesF=100 N a=10mm, W=20mm, h=1mm E = 70.103MPa, ν = .35

In order to reach the most accurate result possible, a special attention has been

devoted to the type of elements and to the meshing system. Therefore, around the crack

tip, as shown in Figure 4-2, the mesh has been refined and the sweep algorithm has been

used to link the surface meshing scheme adopted for the crack tip area with the structured

quadrilateral elements that are used to mesh the rest of the beam.

4.1.2 Results and Discussion

The assessment of the crack tip stress field is here realized using a path constituted

of nodes that lie on the mid-plane of the double cantilever beam, where the interlaminar

38

Figure 4-2. Meshing system near the crack tip

damage occurs. Plotting in log scale (Figure 4-3) the stress σy normal to the crack

propagation direction with respect to the distance from the crack tip highlights the fact

that the stress field is submitted to a singularity in 1√r

in a large linear elastic zone near

the crack tip.

Figure 4-3. Stress field singularity in 1√r

near the crack tip

As a result, the linear elastic fracture mechanics can be applied. Due to the elastic

behavior of the material, the strain energy release rate G is equal to Jintegral which is

directly computed by an ABAQUS plug-in in the version 6.7-1 used to run the simulation.

Thus, a request of that history output permits to access that data. The analysis has been

made for plane stress and plane strain elements and the associated Jintegral results along

different contours near the crack tip are shown in Figure 4-4 and Figure 4-5.

39

Figure 4-4. Jintegral under plane stress

Figure 4-5. Jintegral under plane strain

Those graph demonstrate that, except for the first contour which probaly lies in the

plastic zone, the Jintegral is path-independent. To assess the accuracy of the computation

of the strain energy release rate via 2D Finite Element analysis, we compute, using the

same parameters than the one inputted in the FE model, the theoretical expression of GI

derived in an earlier section and stated in Eq. 4–1.

GI =12F 2a2

B2h3E∗(4–1)

where E∗ = E for plane stress and E∗ = E1−ν2 for plane strain. Table 4-2 shows the

existence of a non negligible discrepancy.

40

Table 4-2. Beam theory and 2D FEA GI computation (in N.mm−1)

Beam theory 2D FE Analysis differential (%)plane stress GI = 171.43 GI = 195.32 13.93plane strain GI = 150.43 GI = 171.39 13.93

Consequently, this 2D analysis allowed us to assess its inaccuracy even though

the meshing system involved thousands of elements in the vicinity of the crack tip.

Furthermore, at that point, other analysis established that refining the mesh does not

convey better results: the inaccuracy in the computation of the energy release rate in a

cracked plate is probably inherent to the 2 dimensional aspect and to the type of elements

used. But, another origin of that discrepancy, which is exactly the same for both plane

stress and plane strain conditions, could also lie in the approximation done by the finite

element software ABAQUS while implementing this model. To remove this suspicion

regarding the accuracy of that commercial software, the energy release rate has been

computed using only ABAQUS deflection output, whose accuracy has been proved to be

excellent compared to beam theory.

4.1.3 Discrepancy Inherent to 2D Analysis

• Accuracy of ABAQUS deflection calculation

In order to check the accuracy of the finite element software used, we have compared

the ABAQUS output of the maximum deflection of a simple cantilevered beam (Figure

4-6) with the theoretical result derived from beam theory (Eq. 4–2).

∆ =Fl3

3E∗I(4–2)

in which F is the force applied at the edge of the beam, l the length of the beam and

E∗ = E for plane stress and E∗ = E1−ν2 for plane strain, where E is the Young’s modulus

and ν the Poisson’s ratio. The FEA simulation has been executed under both plane stress

and plane strain conditions. Moreover, the shearing force F has been modeled as a surface

41

Figure 4-6. Finite Element analysis of a cantilevered beam deflection

traction. The comparison of the results (Table 4-3) shows an excellent correlation: the

ABAQUS deflection output are thus accurate.

Table 4-3. Accuracy of ABAQUS deflection output

plane stress plane strainFE

analytical1.0015 1.0003

• Discretization of G

Figure 4-7. Deflection of sublaminates in opening mode

42

In a double cantilever beam, the strain energy density of any sublaminates can be

expressed through its deflection from the mid-plane.

U (a) =1

2(F1δ

a1 + F2δ

a2) (4–3)

Such expression, stated in Eq. 4–3, can be inserted into Eq. 4–4 to obtain the energy

release rate from a finite difference derivation, which is valid under frictionless assumption

of the interface [14].

G =∂U

∂A crack=

∆U

B∆a=

1

B

U (a+δa) − U (a)

∆a(4–4)

The deflection output required have been obtained through a plane stress analysis of a

bi-material double cantilever beam with 2 linear isotropic materials: Steel and Aluminum.

The respective Young’s modulus, necessary to describe the linear elastic behavior of the

structure are EFe = 200GPa and EAl = 70GPa. Furthermore a straight ahead crack

has been inserted and a seam has been defined to allow the cross section behind the crack

tip to slide on each other: the stress is concentrated at the crack. Moreover, quadratic

eight-nodal elements have been used, and a refinement of the mesh was done defining

a sweep region near the crack tip, as shown in Figure 4-8. Elsewhere, the elements are

structured.

Figure 4-8. Mesh refinement around the crack tip

43

Table 4-4. Output of 2D bimaterial DCB FE analysis

a = 10mm (a+ δa) = 10.5mmδ1 (µm) 2.598 2.972δ2 (µm) 6.667 7.663U J.mm3 463.24 531.79Jint (N.mm−1) 128.55 142.12Gth (N.mm−1) 115.71 127.58error (%) 11.1 11.4

Through an history output request, the deflection of the beam but also the Jintegral

have been obtained for two distinct configurations: the crack length has been increased in

the second one from 10 mm to 10.5 mm. Table 4-4 shows the theoretical value of G was

derived using Eq. 4–5.

Gth =F 2a2

2BI

(1

EFe+

1

EAl

)(4–5)

Substituting the values of the strain energy densities into Eq. 4–4, we obtain a value

closed by 1.3% from the arithmetical mean of the two Jintegral output. Therefore,

ABAQUS Jintegral output are in good correlation with a computation based on deflection

results, which have been proved to be accurate. Hence, as the total energy release rate

can’t be correctly assessed through that too simplistic double cantilever beam modeling,

an other FE model involving shell elements have been implemented in ABAQUS and is

described in the next section.

4.2 Double Cantilever Beam Plate Element Model

4.2.1 Finite Element Procedure

In this model, we are still trying to benefit from the speedy computing ability of two

dimensional analysis, but here a better description of the bonding between laminates is

reached. Indeed, the two sub-laminates are going to be modeled by two distinct plates and

the crack is not going to be implemented using any pre-configured ABAQUS plug-in.

Behind of the crack tip, there is no connection between the two plates, whereas ahead

of the crack tip, there is a continuous elastic body that needs to be modeled. Therefore, as

we can use only discrete connectors to establish a link between the two plates, a fine mesh

44

is implemented on the part of each plate that needs to be connected together. For each

pair of nodes, a connector, such as a spring permits to bond the structure.

Furthermore, this structure is solid, the cohesion between molecules is strong; thus

it is necessary to model the springs with a huge stiffness. Moreover, an offset value has

been incorporated in the decription of the shell elements in order to dispose the nodes

of both top and bottom plates on the mid-plane. This offset introduction permits to

get connectors defined with a distance null. An alternative approach to bond the plates

altogether is to consider beam elements, whose kinematics characteristics are detailed in

Figure 4-11. To model thousands of springs or beams, as shown in Figure 4-9, a MATLAB

code has been incorporated to the ABAQUS python script.

Figure 4-9. A 2D plate FE model

• Execution of the VCCT in the spring model

Between each pair of nodes ahead of the crack tip, three springs have been modeled:

one in each direction of the Cartesian coordinates. Each of those pairs defines a unit cell

in the VCCT algorithm. Furthermore, the springs are defined by a stiffness K. Hence,

the only output required from ABAQUS, once the simulation complemented, are the

displacements between the nodes where the springs have been tied up and just behind.

The strain energy release rates can be computed in each unit cell along the crack tip

through the expression derived in Eq. 4–6.

Gi =1

2B∆aFiui(a−∆a) (4–6)

in which Fi = Kui(a) is the force in the springs. ∆a and B are respectively the distance

between two rows of springs, and two columns in that matrix filled with more than three

45

thousands springs. Moreover, i represents the direction associated to a specific mode

of fracture. An example of such simulation, which has been run in an opening mode, is

presented in Figure 4-10.

Figure 4-10. A 2D plate model with springs as connectors

The main difficulty in the data processing lies in the choice of the stiffness. Indeed,

the best approximation would be to define stiffness as big as the finite element software

allows, but in that case, as the force remains constant, the displacement between each

pair of nodes at the crack tip are so small that it is not always possible to quantify them

through any history output request. Indeed, the number of digits needed overflows the

limit number possible to allocate. To overcome that hurdle, another model has been set up

with a rigid connection instead.

• Implementation of the VCCT in the beam element model

The modeling of a rigid connection through beam element, which ABAQUS

description is presented in Figure 4-11, is possible because such connector allows the

evaluation of the forces and moments that are applied at the nodes which define its

edges. Indeed, the nodes of the top plate are in virtual contact with those of the

bottom one thanks to the offset of the planes on which the nodes have been defined: a

zero-displacement condition must then be satisfied.

Finally, substituting the forces and moments output into an expression analog to Eq.

4–6 permits to reach an approximation of the energy release rate.

46

Figure 4-11. Description of a beam element

4.2.2 Results and Discussion

In this model where the sublaminates of that symmetric double cantilever beam

fracture test have been defined by 2D shell elements and the connectors used to join the

structure that has not yet been split by the growth of the crack are beam elements, the

necessary ABAQUS output are connector forces and moments along with displacements

behind the crack. The fracture analysis has been conducted in different mixity modes but

the assessment of the total energy release rate through that technique has revealed to have

a significant discrepancy. For instance, in mode I loading conditions and under the same

geometric parmeters, Figure 4-12 shows a non negligible gap between the resulting 2D

beam element and 3D solid element models in the energy release rate distribution along

the width.

4.3 The 3D Analysis of a Symmetric Double Cantilever Beam

4.3.1 Description of the 3D SDCB Model

First, a 3D solid deformable part has been created from the extrusion of a sketch

defining one lateral face of the beam. This part has been partitioned in order to define

a specific cell for the sublaminates interface. Furthermore, a solid homogeneous section

whose material is an isotropic aluminum alloy characterized by a set of two engineering

47

Figure 4-12. Comparison of 2D plate and 3D results in an opening mode fracture test

Figure 4-13. SDCB Finite Element Model

constants (Young’s modulus E = 70.103MPa and Poisson’s ratio ν = 0.35) has been

assigned to that part. Then, 20-node quadratic brick (CRD20R) elements, which belong to

the quadratic 3D stress family, have been used to mesh the hexagonal structured system

with a reduced integration element control. Before meshing the instance, the global size

of the seed has been refined. Moreover, a contour integral crack has been implemented

defining its front and extension direction ~q. A seam has also been assigned to the surface

between the two sublaminates, behind the crack tip. Finally, a static step has been

created. In the latter, boundary conditions and loads have been configured, as shown in

Figure 4-13.

4.3.2 Convergence of G upon the Refinement of the Mesh

In a mode I configuration, the ABAQUS finite element model typically gives the

characteristic sate of stress shown in Figure 4-14. Depending on the seed global size

48

Figure 4-14. Analysis of the state of stress in Mode I

selected, the Jintegral value extracted for each unit cell along the crack tip through an

history output requested varies, as Figure 4-15 illustrates.

Figure 4-15. Convergence of the energy release rate

Indeed, these graphs show that the curved delamination front along the width is

smoother when the number of elements, which is inversely proportional to the seed

selected, increased. Testing our model with a finer mesh, was incompatible with the

computer memory allocation possibilities. But, the curve obtained with a refined seed of

0.25, gives accurate enough results since it appears in Figure 4-15 that the latter curve

seems to have already converged.

It is interesting to note that in order to reach the convergence of the energy release

rate; it is not required to refine the mesh specifically in the crack tip area compared to

49

the rest of the model because we made a static assumption for the crack analysis. A finer

mesh in the region surrounding the crack tip is especially needed in dynamic models to

allow the propagation of the crack by a smaller increment.

As displayed in Figure 4-15, the Jintegral reached a minimum at the edge of the beam

and a maximum at the center. In mode I, the energy comes principally from longitudinal

bending and the non uniform trend is caused by the anticlastic curvature that occurs in

each leg of the double cantilever beam. During the delamination growth, a change in the

specimen bending rigidity occurs, as the constraint conditions vary from plane strain for

short delamination to plane stress for long one. This variation of constraints controls the

shape of the distribution of the energy release rate. Furthermore, a quantification of that

variation in the specimen bending rigidity can be assessed through a ratio, introduced by

Davidson [15], involving DCB properties.

4.3.3 Delamination Front Curve for an Arbitrary Mode Mixity

• VCCT reference element selection

In the VCCT description of the chapter 3, two different reference elements were

introduced in order to compute the front delamination curve of a straight ahead crack.

Implementing both of them in Matlab gives sensibly the same result: the continuity of

the stress, allowed by the fine mesh selected, permits those two approximations to be

equivalent. However, a slightly difference is noticeable at the edge of the delamination

front because, for the unit cell that has been offset by half of the distance separating two

consecutive nodes along the crack front, the result is erroneous at both ends. Indeed, a

mid-side node of the quadrilateral meshing system used is missing.

• An excellent matching with Jintegral

To assess the accuracy of the Virtual Crack Closure Technique implemented in

the 3D SDCB, the resulting total energy release rate has been compared to ABAQUS

computation of Jintegral. Such comparison is allowed by the elastic behavior of the

material.

50

Figure 4-16. Delamination front in Mode I

From the analysis of several SDCB finite element models covering a wide range

of mode mixity possibilities an excellent matching have been noticed either for mode I

(Figure 4-16), mode II (Figure 4-18), or an arbitrary mix mode (Figure 4-20).

In Figure 4-17, are shown the state of stress of the finite element model when the

same force has been applied in the same direction to both legs of the symmetric double

cantilever beam. In that case, the legs slide on each other due to the frictionless contact

allowed by the implementation of the seam in the model.

Figure 4-17. State of stress in Mode II

Applying the VCCT algorithm to the set of forces and displacements described in

Chapter 3, highlights again the excellent correlation of Gtotal with Jintegral but also conveys

the presence of mode III components at the edge of the beam (Figure 4-18). Therefore,

the shearing forces applied through a surface traction on the edge of the cracked beam

legs, create components slightly out of the reference plane. As this edge effect is minor,

51

it will be neglected in the comparison with the other theoretical and analytical methods

that have been derived in chapter 3 for a 2D configuration. Furthermore, such a graph

demonstrates that the shearing forces are responsible for the mode II fracture component,

which is almost uniform across the whole width of the beam.

Figure 4-18. Delamination front in Mode II

In a mix mode configuration, where the bottom half of the SDCB has been submitted

to a dead load, the effects noticed in both pure modes are combined (Figure 4-20).

Figure 4-19. Application of a single dead load in a FE SDCB

4.3.4 Perfect Correlation among Theoretical, Analytical, and NumericalMethods

For any arbitrary mix mode, the total energy release rate, obtained through the

Virtual Crack Closure Technique, match not only with the J contour integral but also with

the resulting equations of both derivation of Jintegral from strain energy densities of each

52

Figure 4-20. Delamination front in an arbitrary Mix Mode

cross section of the beam(Eq. 3–21) and Crack Tip Force Method (Eq. 3–48). Indeed, the

three distributions of the total energy release rate overlay (Figure 4-21).

Figure 4-21. Correlation of total G

In addition, the phase angle computation, which is a measure of the mode mixity,

gives sensibly the same results in either method. The small gap (Figure 4-22), between

the expression of the phase angle, computed through the VCCT, in function of the one

53

Figure 4-22. Correlation of the mode mixity

computed in the same load configurations but with the CTFM can be partially imputed to

the presence of a shearing component out of the plane of delamination.

54

CHAPTER 5MULTI-FIDELITY RESPONSE SURFACE FOR ASYMETRIC DOUBLE

CANTILEVER BEAM

As previously shown, the study of a symmetric double cantilever beam fracture test

demonstrates that the low-fidelity Crack Tip Force Method can accurately predict the

global fracture parameters associated with crack propagation. Nevertheless, a change in

the selected design space, for instance a variation in the thickness ratio h1

h2, as introduced

in Figure 5-1, reveals that there are discrepancies between the two methods.

Figure 5-1. Assymetric double cantilever beam

In this Chapter, our goal is to combine the high-fidelity 3D FE model to the fast

Crack Tip Force Method to obtain high accuracy at low computational cost in the

determination of the global fracture parameters G and Ψ associated with the asymetric

double cantilever beam fracture test for any given load configuration. First, a correction

surface technique is presented to couple the two methods of analysis at various points

of interest in the design space. Then, the latter is implemented to fit the values of the

ratio of the energy release rates and the difference of the phase angles of the analytical

and numerical methods at specific design points. Finally, the predictive capability of

the response surface is assessed. Moreover, particular attention is given to the design of

experiments by detailing the respective load configurations that are implemented in the 3D

FE and 2D analytical models in order to capture the effects of the opening and in-plane

55

shearing modes. The effects of the out-of-plane shearing modes can be neglected as the 3D

FE model reveals that the energy release rate in this fracture mode is much smaller than

the two other mode components.

5.1 Computer Design Experiments

Given the load configuration applied to both arms of the double cantilever beam, as

depicted in Figure 5-1, the principle of superposition permits to split this double cantilever

beam fracture test into 3 distinct tests. In each of the latter, only a single kind of force is

applied: axial loads, couples or shearing forces.

Those fracture tests allows us to generate computer design experiments in both 3D

FE and 2D analytical model by selecting different thickness ratio.

5.1.1 The 3D FE Modeling

To implement the three previously described fracture tests into the ABAQUS

finite element software, the 3D model introduced in chapter 4 is used. In the latter,

the thickness ratio will be changed while building the model. The thickness ratios selected

are 1, 1.5, 2 and 3. We don’t focus on extreme thickness ratios because a crack that

would propagate at the surface of the body is not likely to cause the failure of the whole

structure.

The three sets of loading conditions are created by applying a surface traction in the

case of a shearing force and by applying a pressure otherwise. The pressure is uniformly

applied to the edge surface to create the axial force whereas to create a couple on one

arm, the pressure is applied according to a linear distribution whose center is positioned

at the middle of the leg: a set of forces having opposite directions is therefore created, as

depicted in Figure 5-2.

When applying a couple on one ligament, the 3D FE model needs to have the same

number of rows of elements at each side of the ligament’s middle. Otherwise, the set of

forces can’t correctly create the desired couple. Furthermore, a violation of the contact

boundaries is also observed when the previous condition is not satisfied.

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Figure 5-2. Applying couples to the ADCB legs

5.1.2 The 2D Analytical Method

The crack Tip Force Method described in chapter 3 is implemented to compute

at low computational cost the global fracture parameters denoted GCTFM and ΨCTFM

for a given set of loads (P,M, V ) applied to each ligament of the double cantilever

beam. Except in the case of the application of moments to the double cantilever beam

arms, which is similar in high-fidelity 3D FE model and low-fidelity 2D analytical one,

the application of shearing forces and axial loads presents some problems. Indeed, to

produce an equivalent loading condition as in the 2D FE model where shearing forces

are applied at the edge of each arm, it is necessary to translate the shearing force applied

into a moment. Therefore, both shearing force and moments need to be inputted in

the analytical method. Furthermore, the axial loads cannot be directly applied in the

2D analytical model: a moment is necessary to take into account the fact that in the

analytical model the axial load P is not applied at the middle of each ligament but at the

delamination plane.

5.2 Low Fidelity Analysis with High Quality Surrogates

5.2.1 Crack Tip Force Method Polynomial Response Surface

Due to the low computational cost of the Crack Tip Force Method, large number

of low fidelity analysis could be performed to generate high-order polynomial response

surfaces. In the case of application of moments, a fifth-order polynomial response surface

of the energy release rate has been generated in Figure 5-3 using the data of 64 design

points whose variables are the thickness ratio h1

h2and the phase angle ΨCTFM , which

characterizes the ratio of mode II to mode I energy release rate. The design variables are

57

Figure 5-3. Polynomial response surface of the energy release rate

normalized to improve the stability of the MATLAB code used to generate the polynomial

response surface.

The performance of that fit is evaluated through the computation of R2, that

measures the fraction of the variation in the data captured by the response surface.

Such predictor is defined as the ratio of the variation SSr of the response surface z from

the average of the data points z and the variation SSz of the data from its average. SSr,

SSz and R2 are respectively presented in Eq. 5–1, Eq. 5–2 and Eq. 5–3.

SSr =nz∑i=1

(zi − z)2 (5–1)

SSz =nz∑i=1

(zi − z)2 (5–2)

where nz is the number of data points

R2 =SSrSSy

(5–3)

58

For the response surface of Figure 5-3, this ratio is very closed to 1: R2 = 0.998465.

Furthermore, since the adjusted form of R2, defined in Eq. 5–4 is also found to be much

closed to 1 (R2a = 0.998209), we are sure about the prediction capability of our model.

R2a = 1− (1−R2)

(nz − 1

nz − nβ

)(5–4)

where nz and nβ are respectively the number of data points and the number of coefficients

of the polynomial response surface. For a fifth-order, this coefficient is equal to 21.

5.2.2 Energy Release Rate Correction Response Surface

The high-fidelity analysis gives a few number of accurate energy release rate values

that are going to be used to correct the low fidelity response. For each thickness ratio

(h1

h2= 1, 1.5, 2, 3), 4 values of G are obtained for various phase angles. Then, a quadratic

correction response surface is plotted in Figure 5-4 using those 16 design values by fitting

the value of the ratio of the high-fidelity energy release rate G3D and the approximate

energy release rate values that are computed for the same phase angle Ψ and thickness

ratio using the 5th order polynomial response surface.

The quadratic order has been chosen for the correction response surface between

the high-fidelity and low fidelity analysis because a higher polynomial order does not

improve the prediction capability of the correction factor. Indeed, R2a decreases between

the quadratic and cubic polynomial order, as shown in Table 5-1.

Table 5-1. Prediction capabilities of energy release rate correction response surfaces

R2 R2a

quadratic CRS 0.69 .53cubic CRS 0.78 .47

The multi-fidelity energy release rate Gmultifidelity at a given design point is derived in

Eq. 5–5.

Gmultifidelity = G2D.CRS (5–5)

59

Figure 5-4. Correction response surface

The procedure is to find the 2D approximated value G2D using the 5th order

polynomial CTFM response surface, then multiply the latter by the value of the correction

response surface at the same design points. To validate this multi-fidelity approach,

this previous procedure has been implemented at around 20 design points that have not

been used in the construction of the correction response surface. At those points, the

Gmultifidelity values can be compared to the high-fidelity values resulting from the 3D

Finite Element Analysis. This comparison has shown that the introduction of the high

quality surrogates in the low-fidelity analysis permits to reduce the average error, as

defined in Eq. 5–6.

eav =1

nz

nz∑i=1

|zi − zi| (5–6)

where nz is the number of design points, zi is the high-fidelity energy release rate value at

design i and zi is the 2D CTFM approximated value.

Using the 5th order polynomial response surface to get the 2D anlytical approximation

of the energy release rate, the average error with the 3D high-fidelity 23 design points is

60

around 11%. The multi-fidely design reduced that average error to 6.1%. Consequently,

this multi-fidelity design approach of the asymetric double cantilever beam improves the

prediction capability of the total strain energy release rate using only a few number of

high quality surrogates 3D FE design points.

5.2.3 Phase Angle Correction Response Surface

Implementing the same approach to the phase angle Ψ, a correction response surface

is shown in Figure 5-5. The difference of the high-fidelity and low-fidelity phase angles

is here used to predict the 3D FE value ΨV CCT using a large number of analytical

phase angles derived in function of two design variables: the thickness ratio and the

2D analytical phase angle that results from the application of a couple at the top ligament

of the double cantilever beam. The couple applied at the bottom ligament is chosen to be

constant and equal to 1 in both 2D analytical and 3D FE computer experiments run.

Figure 5-5. Phase angle correction response surface

Table 5-2 shows an excellent prediction capability of the polynomial response

surface of the phase angle difference expressed in radians with respect to the normalized

61

analytical phase angle. Changing the mode mixity bounds from [0, π2] to [−1, 1] permits a

stabilization of the optimization process.

Table 5-2. Prediction capabilities of phase angle correction response surface

R2 R2a

quartic CRS 0.97 .93

As a result, using the energy release rate and phase angle correction response surface,

it is possible, given a thickness ratio and a set of couples applied at both ligaments of

the asymetric double cantilever beam, to approximate those global fracture parameters

in real time. Indeed, the computation cost involved in the calculation of the low-fidelity

fracture parameters and in the application of the correction factors computed for the

specific design selected is low. Furthermore, those global fracture parameters are assessed

without losing much accuracy as we demonstrate that a quadratic correction response

surface can already improve the predictive capability of the 2D analytical approximation

by a considerable amount.

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CHAPTER 6CONCLUSIONS

Our study shows the assessment of the total energy release rate Gtotal along with the

mode mixity Ψ, which measures the mode II (shearing) to mode I (opening) loading, via

the successful implementation of an algorithm, the Virtual Crack Closure Technique, in

2D plate, and 3D solid finite element models of a double cantilever beam fracture test.

Under quasi-static and linear elastic behavior, the delamination front curve obtained is in

excellent agreement with the Jintegral curve computed by ABAQUS. The latter has also

been found in agreement with an expression of the energy release rate obtained through a

finite difference of its definition itself (i.e., the change in potential energy with respect to

the area created by an extension of the crack).

Furthermore, the Crack Tip Force Method, first proposed by Sankar [13] has been

validated from the excellent correlation shown by its comparison with the general standard

solution, described in Hutchinson and Suo’s review [2]. Then, this analytical method

has been compared to the high fidelity 3D Finite Element Analysis of a homogeneous,

isotropic and symmetric double cantilever beam: the difference noticed was small. In

addition, the results obtained with both methods were very close to theoretical results,

which identify the strain energy release rate in a double cantilever beam as the difference

between the strain energy densities of each cross section of the legs that are located just

behind and ahead of the crack tip.

A study of the variation of thickness ratios shows that there is no longer an agreement

between the results derived using the Crack Tip Force Method and the 3D Finite Element

Analysis, when the double cantilever beam is not symmetric. Yet, in a multi-fidelity

approach, it is shown that a large number of low-fidelity 2D analytical values of the

fracture parameters combined with a few number of high-fidelity surrogate 3D FE values

can predict at low computational cost the fracture parameters for a specific design without

losing much accuracy.

63

Finally, it would be interesting to study the effect of the variation of the material

properties. The logical next step would be to consider orthotropic materials before

considering plies with different orientations. As the energy release rate is also a function of

the crack length when the latter becomes small, it would also be advantageous to complete

our correction response between high-fidelity and low-fidelity analyses to integrate the

crack length as a design variable in the expression of the delamination front curve.

64

REFERENCES

[1] H. Tada, P. C. Paris, G. R. Irwin, The stress analysis of cracks handbook, 2ndedition, Paris production, Saint-Louis, MO, 1985.

[2] J. W. Hutchinson, Z. Suo, Mixed mode cracking in layered elastic materials, Adv.Appl. Mech. 29 (1992) 63–191.

[3] A. A. Griffith, The phenomena of rupture and flow in solids, Philos. T. Roy. Soc. A221 (1920) 163–198.

[4] G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate,J. Appl. Mech. 24 (1957) 361–364.

[5] J. R. Rice, A path independent integral and the approxiamte analysis of strainconcentration by notches and cracks, J. Appl. Mech. 35 (1968) 379–386.

[6] T. L. Anderson, Fracture Mechanics, Fundamentals and Applications, CRC Press,2005.

[7] G. R. Irwin, Onset of fast crack propagation in high strength steel and aluminiumalloys, Sagamore research conference proceedings 2 (1956) 289–305.

[8] E. F. Rybicki, M. F. Kanninen, A finite element calculation of stress intensity factorsby a modified crack closure integral, Eng. Fract. Mech. 9 (1977) 931–938.

[9] A. Leski, Implementation of the virtual crack closure technique in engineering fecalculations, Finite Elem. Anal Des. 43 (2007) 261–268.

[10] K. N. Shivakumar, P. W. Tan, J. C. Newman, A virtual crack-closure technique forcalculating stress intensity factors for cracked three dimensional bodies, Int. J. Fract.36 (1988) 43–50.

[11] D. Xie, S. B. Biggers, Progressive crack growth analysis using interface element basedon the virtual crack closure technique, Finite Elem. Anal Des. 42 (2006) 977–984.

[12] B. V. Sankar, A finite element for modeling delaminations in composite beams,Comput. Struct. 38(2) (1991) 239–246.

[13] O. Park, B. V. Sankar, Crack-tip force method for computing energy release rate indelaminated plates, Compos. Struct. 55 (2002) 429–434.

[14] X. Sun, B. D. Davidson, A direct energy balance for determining energy releaserates in three and four point bend end notched flexure tests, Int. J. Fract. 135 (2005)51–72.

[15] B. D. Davidson, R. A. Schapery, Effect of a finite width on deflection and energyrelease rate of an orthotropic double cantilever specimen, J. Compos. Mater. 22(1988) 640–656.

65

BIOGRAPHICAL SKETCH

Nicolas Vigroux was born in Paris, France, in 1985. He attended public schools in

Chamalieres, a gateway to the marvelous Chaıne des Puys of the Massif Central in France.

After graduating from high school with honors in July 2003, he successively went to

Lycee Louis Le Grand, Paris and Lycee Lakanal, Sceaux, where he received an intensive

training principally in mathematics, physics and engineering science to compete in the

French nationwide selective exam. In September 2005, he was admitted to one of the most

prestigious Grandes Ecoles: the Ecole Nationale Superieure des Mines de Saint-Etienne

from which he received in 2008 the title of Ingenieur Civil des Mines (master’s degree)

with the highest honors (Mention Tres Bien). Alongside the engineering and managerial

formation he experienced there, his interest in economics drove him to attend classes

delivered by the Economic, Administration and Accounting Superior Institute of the

University Jean Monnet in Saint-Etienne, from which he graduated, in 2007, with a

Bachelor of Science in Economics. In summer 2007, he performed nanoprecipitates

characterization by transmission electron microscopy in new aeronautical alloys at the

National Research Center in Toulouse. In August 2007, he joined the Department of

Mechanical and Aerospace Engineering at the University of Florida where he has been

pursuing his Master of Science on fracture mechanics.

66