Crack Propagation in a Three Dimensional FRC

7/29/2019 Crack Propagation in a Three Dimensional FRC

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Crack propagation in a threedimensional FRC unit cell

Three months research training preriod at Brunel University

Vincent VISSEQ Supervisor : Giulio ALFANO

August 11 2008


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Contents

1 Overview 6

1.1 Background of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 A brief introduction on composites . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Multi-scale approaches and homogenization . . . . . . . . . . . . . . . . . . . . 9

1.4 Nonlinear modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Cohesive elements in Abaqus 15

2.1 Cohesive elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Initial behavior and damage initiation criteria . . . . . . . . . . . . . . . . . . . 17

2.3 Softening onset prediction (Mixed-mode) . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Delamination propagation prediction and damage evolution law . . . . . . . . . 20

3 Representative volume element characterization 21

3.1 Size of the RVE, number of fibres and location of the crack . . . . . . . . . . . 21

3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Materials and interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Specificities of the modeling under Abaqus 24

4.1 Contact status and mesh of interface elements . . . . . . . . . . . . . . . . . . . 24

4.2 Partition of the matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Numerical simulation 26

5.1 Stress concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Damage evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Global response of the unit cell - comparison of Abaqus/Standard and Abaqus/Explicit 28

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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6 Damage modeling 30

6.1 Some damage modelings and their deficiencies . . . . . . . . . . . . . . . . . . . 30

6.2 Gradient-based damage modelings . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.3 Interpretation and tools used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Numerical simulation with a matrix damage model 33

7.1 Parameters of the Abaqus damage model . . . . . . . . . . . . . . . . . . . . . 33

7.2 Simulations for the Small Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.3 Simulations for the Bigger Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . 35

7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


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Introduction

Composite materials are now widely used in several industrial fields. The most spectacular

one is probably aviation were companies have built airliners with 50 per cent of composites.

This explains why the international research community has produced many studies on it, and

why others are undergoing. Indeed, a good knowledge upon composite materials allow us both

to predict their behavior and to design other ones which hold out properties needed.

Many different types of composite materials exist and they are increasing every day, whichis well illustrated by new reinforcements made of nanotubes. More precisely, if we focus on

reinforced composites, one can see that reinforcements are mainly fibers (long or short) or

particles (spheroid or flakes). In this study we will be particularly interested in (long) fibre

reinforced composites (FRC).

Even if a combination of both numerical simulations and experiments would have been wise,

we concentrate ourselves on an accurate computational modeling. Nowadays, even if numerical

tools allow us to perform complex simulations of mechanical problems, it is still very difficult

to accurately simulate a structural response taking into account micromechanical phenomena.

That is the reason why multi-scale approaches and homogenization techniques have been de-

veloped to link phenomena which occur at micro and macro scales.

If a multiscale approach is needed to accurately resolve mechanical problems for a whole struc-

ture, it is then obviously the case for composite ones. Indeed, composites, and (as a particular

case) unidirectional fibre reinforced composites are anisotropic materials. Furthermore, both

debonding at the interfaces between matrix and fibres and crack propagation might be taken

into account in the model to understand why, how and when micro cracks leads to a macro

crack.

In this study we will model a Representative Volume Element (RVE) of unidirectional FRC,

with a preexisting cracked fibre, up to failure. The aim is to understand the failure history,

and particularly how the stress previously carried by the broken fibre is redistributed to theunbroken fibres and to the matrix.

To this end, preliminary investigation in theoretical tools used has been made [Chapter 1

and 2].

The different steps of the study are summarized as follows:

1. The design of a representative volume element (RVE) which might be appropriate for

the work considered: ability to take into account the redistribution of stress and strain and

representativeness of a whole structure [Chapter 3].

2. A finite element modeling using the software ABAQUS [Chapter 4].

3. The simulations of fibres/matrix debonding under longitudinal load [Chapter 5].

4. The simulations of fracture types taking into account the damage of the matrix [Chapter6 and 7].

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Chapter 1

Overview

1.1 Background of this study

In this first part we are going to give an idea of the background of this study. Several conceptsand tools are needed which are related to the general theory of continuum mechanics an the

study of fibre reinforced composites.

We have decided to give explanations especially on the following points:

1. Fibre reinforced composites: products, processes and materials

2. Interface and interphase between materials

3. Homogenization techniques

4. Nonlinear finite element analysis

1.2 A brief introduction on composites

As we have already said, good knowledge on composite materials allow us both to predict their

behavior and to design other ones with properties searched. As an example, it is very useful to

know precisely what parameters are involved in energy dissipation during a crash. With those

informations, one could theoretically determine what the matrix/fibres brace which maximizes

the energy dissipated is.

Nevertheless, to have an extended idea of the properties of a composite material, it is useful

to consider the triptych : product-process-material.

1. Product: Composites materials are often called designed materials which means that

composites are designed by engineers and searchers to obtain properties which do not

exist in classical materials. For example one can have the idea to conjugate the high

ductility and strong stiffness of steel to the high strength of ceramique.

This example highlight the possibility of combining different materials. Nevertheless, the

way in which those materials are combined is determinant. According to the shape of the

structure and its main mechanical solicitations, one might adapt the kind of combination

between materials (e.g. bonded lamina, multi or uni-directional reinforcement).

2. Process: Many specific processes have been developed in the last century to obtain

reliable composites. They are mainly classified in terms of rates of production and global

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Crack propagation in a three dimensional FRC unit cell year 2008

cost. One can cite, for FRC polymers, bag molding, compression molding, poltrusion or

filament winding.

A good knowledge on those processes is needed to accurately model interfaces between

matrix and reinforcement as well as both materials. Depending on the degree, the pres-

sure and the length of cure, the process can leads to modify the behavior at mesoscopic

and structural level of the composite (e.g. prestress due to shrinkage). Furthermore,

manufacturing causes defects as voids inside the matrix or buckling of fibres.

The frequency and spacial distribution of those preexisting voids, distortions and pre-

stress are key informations to predict initiation of failure.

Finally, the process is also a constraint in the design steps.

3. Material: material properties have several facets. If the global characteristics of several

materials are well known (density, fusion temperature,...) one could also need to know

some details about smaller scales. Like the length of a polystyrene macromolecule, its

shape, the fraction of polymer cristallised, etc...As a further example, some damage modelings involve to determine a characteristic

length at mesoscopic scale (e.g. gradient based strategies to avoid mesh dependency)[1].

1.2.1 Some studies on fibre reinforced composites

A number of authors have addressed micromechanical analysis of a representative unit cell. A

brief description of some of these recent studies is reported below.

1. Xia and al. [2] designed a three dimensional unit cell. They have studied how the load

carried by an broken fibre is then redistributed among the matrix and the unbroken

fibres. This paper is very important for us since our study is very close to it. Indeed,

the main objective is to handle the evolution of damage at the larger scale of the fibers

by results at the micro scale. In this purpose, its necessary to establish the connection

between the detailed deformation at micro scale for a multifibre damage problem and

large-scale component performance. In their paper, the stress transfer is modeled by

Coulomb friction end they are using contact (gap) elements to model interfaces.

2. Bonora and Ruggiero [3] tried to take into account the stress/strain histories due to

manufacturing. Thus they incorporate material dependency as well as damage processes

and follow the associated progressive degradation of the overall properties in advanced

special algorithms. They shown that matrix/fibre interface properties result of manu-facturing process. An example of that is the prestress due to the final cooling phase in

forming process. They developed a 2D unit cell model for SiC fiber reinforced metal

matrix composite (MMC) laminate.

3. Gonzalez and Llorca [4] formulated the project to have a precise knowledge of the lamina

behavior under transverse loading until failure to develop a robust failure criteria. The

paper shows a two dimensional representative volume element (RVE) where circular

elastic fibres are randomly distributed. They also deal with both the size of RVE (results

for 30 fibres and 70 fibres are shown) and properties of the interface (one case for a

strong interface and another with a weak interface). Those interfaces were modeled

by cohesive elements under ABAQUS and the damage of the matrix was evaluated in

term of accumulated plastic strain.

report.pdf 7 August 11 2008


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Figure 1.1: Fibres reinforced composite under transverse loading

1.2.2 Interface and interphase between materials

As in the first part of this study we are going to model debonding between matrix and fibres,

it seems to be essential to define precisely the concept of interface.

An interface is widely considered as a mathematical concept, the common boundary of

reinforcing fibre and matrix which maintain the bond in between for the transfer of loads.

This definition leads to consider the thickness of this interface as equals to zero. Furthermore,

its physical and mechanical properties are unique from the fibre and the matrix.

In opposition to this concept, some authors are using the word of interphase, which is

defined as a region of finite volume extending from the bulk fibre through the actual interface

into the matrix. Then it embraces all the volume altered during the fabrication process of

the composite. As a consequence, the chemical, physical and mechanical properties of the

interphase vary continuously or in a stepwise manner along the finite thickness.

This last concept allows us to describe the adhesion mechanisms (absorption and wetting,

electronic attraction, chemical bonding, Vander Waals forces,...) which lead to the bond

between the two materials. However, as these adhesion mechanisms are reduced to mechanical

properties, the distinction of those two concepts is often ignored.

Indeed, the interface properties are essentially reduced to:

- the shear and tension strengths

- the critical strain energy release rate for mode I, II and III (GIC, GIIC, GIIIC).

These quantities are measured by means of two kind of tests:- In the first category fall the tests in which fibres are embedded in specially constructed

blocks of matrix (single fibre compression test, fibre fragmentation test, fibre pull/push-out

test, slice compression test) which are used to determine the bond shear and tension strengths.

- The second type is represented by the interlaminar/intralaminar tests where bulk lami-

nate composites are used (Double Cantilever Beam (DCB) test to estimate GIC, End Notched

Flexure or End Loaded Slit test to estimate GIIC).

The main problem concerning these tests is the large data scattered from one laboratory

to another. J-K Kim and Y-W Mai [5] proposed an explanation of that weird fact, based onthe idea that different assumptions have been made by searchers concerning the type of failure

which had taken place without confirmation.



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1.3 Multi-scale approaches and homogenization

Multiscale approaches are necessary in mechanical modelings to have simplified methods of

resolution. Those methods derive from continuum mechanics, but are widely linked with

material(s) properties and the geometry of the structure studied. The main idea is to discretise

the bigger scale without taking into account the smaller one.

1.3.1 Example of multi-scale approach

One of the most famous two-scale method is certainly the beam theory. Because of the slen-

dernessed ratio of the structure considered, one of the three dimensions of the body is favored,

which allows us to consider its sections as rigid. This fact is taken into account by the expres-

sion of the displacement field as follows:

u(M) = u(m) + (m) X

Where m is the geometrical gravity center of the section SD(s), s the curvilinear abscissa

and X the vector mM. (See figure 1.2)

Figure 1.2: Modelisation of a beam

Remark : The classic theory based on the expression above of the displacement assumes

that an elastic deformation energy exists and the external loads derive from a potential. Under

these hypotheses one can prove that the general expression of the deformation energy can bewritten as follows:

12

: = 12

SD

[T(s) (s) + M(s) (s)]

One can see that this theory is very similar, in the general form, to the plate theory.

Indeed, this last theory is based on the same geometrical type of assumption, with two favored

directions.

1.3.2 Homogenization

Multiscale approaches concerning composites especially focus on the fact that the material isheterogeneous at the microscale but can be considered as anisotropic and homogeneous at the

macro scale.



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Homogenization is a class of theories which allows us to obtain the explicit dependence of

the bigger scale on the smaller scale.

The following two main assumptions are made:

- there exists a pattern (smaller scale) which allows us to reconstitute the whole structure

- the characteristic dimension of the pattern is negligible compared with the structures one.

Several homogenization methods have been developed based on first or second order gra-

dient theories [6]. In the first case one can advert asymptotic homogenization [7] of periodic

material and auto-coherent method [8] for randomly distributed materials.

1.3.3 Kinematic homogenization

In this section we present some details of a special type of homogenization technique called

Kinematic homogenization [9].

As other homogenization methods, the aim is to obtain an homogeneous equivalent material

which gives an approximazised response on the RVE. To this end, a series of homogeneous loadsare applied to determine the local response. Then the homogeneous operators are build using

the equivalence in deformation energy.

Kinematic homogenization define the homogeneous behavior which gives the real RVE

deformation energy for boundary conditions written as

om = ud

Let us consider the mechanical problem written upon the RVE as:

Find u(m), (m) and (m) such

u = om

div = 0 (1.1)

= K(m) (u(m))

where K(m) is the heterogeneous constitutive tensor of the RVE, (m) the local strain, and

(m) the local stress.

Remark : One can see that u(m) = (m) om is the unique solution of (1.1).Then the idea is to write the equivalence in deformation energy as

ereald =1

20

T r

(m) (m)

d0 =1

2 vol()T r

K

, (1.2)

where K defines the homogeneous behavior, and 12 vol()T r

K

the deformation

energy of the homogeneous problem.

The next step is to calculate (m)1 for given.

The local deformation depends linearly from as (m) = L(m) .1Engineering notations are used further for simplicity



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= i i is associated with , where, for example 1 = [1, 0, 0, 0, 0, 0] and 4 =[0, 0, 0,

2, 0, 0].

More generally, we have : = [1, 2, 3,

2 4,

2 5,

2 5]Then one solves six typical problems as:

= 1 ud =1 0 00 0 0

0 0 0

xyz

= x00

where1(m) is related to 1.

Finally we have 1(m) = L(m) 1 as follows:

1(m) = 1(m); 2(m); 3(m); 4(m); 5(m); 6(m)

1

0

0

00

0

Then, with the knowledge of L(m) (matrix 6 6), one can determine the homogeneousbehavior of the RVE. For any the local deformation solution upon the RVE equals to:

(m) = L(m) (1.3)

That implies, assuming a linear constitutive law (m) = D (m) :

(m) = H(m) (1.4)With : H(m) = D(m) L(m)Then one can write the local deformation energy as:

elocald =1

2 (m)t (m)

And energy equivalence as:

1

2

t L(m)t D L(m)

d =

1

2 vol () t

D , (1.5)

Finally, the homogeneous constitutive matrix is written as followsD = L(m)t D L(m) (1.6)Where = 1vol()

d has been used.



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1.4 Nonlinear modeling

Consider the mechanical problem schematized in figure 1.3:

Figure 1.3: General simple mechanical problem

and defined by the following conditions of admissibility:

Stress admissibility

is a symmetric matrix

div + fd = 0 in all end n = Fd upon 2

Displacement admissibility

u = Ud in 1

1.4.1 Linear case

Let us consider the linear constitutive law (engineering notations)

= D and the linear strain-displacement relationship (Finite Element Method (FEM))

= B uwhere u is the vector of nodal displacements.

Then one can write the internal forces as follows

qint = B

t di.e.

qint = B

t D B udAs u is independent from x,y,z, one can write

qint = B

t D Bd uWhere

B

t D Bd is the stiffness matrix of our problem.Finally we have the quasi static equilibrium equation

K u = qextWhere K is the tangent stiffness matrix end qext the external forces.



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1.4.2 Nonlinear case

Let us consider the nonlinear constitutive law:

= ()

In addition we assume that the strain-displacement relationship remain the same as before,i.e.

= (B u)

Accordingly, the internal forces have the following expression:

qint = B

t (B u)d

Then we introduce a residual function g = qint-qext. This function is useful in order to use

an iterative method (Newton-Raphson, arclength, etc...). Indeed it allows us to estimate the

error committed and to check the convergence of the problem.Now, let us consider the derivative of g with respect to u

dg

du=

dqint

du=

Btd

dBd

The matrix B

t dd Bd = Kt is the tangent stiffness matrix of the nonlinear problem.

This kind of tangent stiffness matrix is then computed at each iteration (or increment).

1.4.3 Newton-Raphson method

Considering a one-dimensional problem, for a generic increment the force will change from aninitial value qexti to the the value at the end of the increment, q = qexti + q. Let us denote

u0 the value of the displacement at the end of the increment. The equilibrium equation is

generally not satisfied and a residual g is computed, g = f(u) qext = 0. Then the Taylorseries gives:

y = g(u0) +dg

du(u0)(u u0)

and y=0 leads to:

q = u0 dgdu

(u0)1g(u0) if

dg

du(u0) = 0

Then, if we set

u0 = dgdu

(u0)1g(u0) = K10 g(u0)

the next solution u1 will be written as

u1 = u0 + u0

This procedure is then iterated until ui is sufficiently small.

The error committed with this approximation is of the order of u2i (Taylor series expanded

up to the first order).

This Newton-Raphson method is called full because we calculate at each iteration anew tangent stiffness matrix. Some simplified method are frequently used where the tangent

stiffness matrix is computed for the first iteration only.



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1.4.4 Dynamic explicit method

The dynamic explicit method proposed in Abaqus is based on a central-difference operator. Let

us consider an increment at the time t. First, the dynamic equilibrium equations are satisfied

at the beginning of the increment. Then the accelerations at time t are used to advance the

velocity to time t + t/2 and, finally, the displacement to time t + t.Practically, the accelerations at the beginning are computed as follow:

uNi =

MNJ1 PJi IJi

With MNJ the diagonal (lumped) mass matrix, PJi is the applied load vector and IJi the

internal force vector at the increment i.

Remark: IJ is assembled from the contributions of each elements such that a global stiffness

matrix need to be formed. However, the explicit procedure requires no iterations and no

tangent stiffness matrix which leads to a high efficiency of the procedure.

The velocities and displacements are then computed as:

uNi+ 1

2

= uNi 1

2

+ti+1 + ti

2 uNi

uNi+1 = uNi + ti+1 uNi+ 1

2

The main disadvantage of an explicit analysis is that the global equilibrium is never checked,

so that an error is inevitably introduced. As a consequence, in order to keep this error within

an acceptable range, very small time increments are necessary. In particular, a maximum time

increment is evaluated, which is equal to, or an estimate of, the time needed by a dilatational

wave to cover the smallest element of the mesh.



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Chapter 2

Cohesive elements in Abaqus

The finite-element code Abaqus, which has been used in this project includes special cohesive

elements to model adhesive joints or bonded interfaces taking into account degradation of

material properties. This model is widely based on the work of PP. Camanho and C.G Davila[12] and [13].

It is possible to define three types of responses of these elements: continuum, traction/

separation and gasket. In our particular case, as we are dealing with interfaces between fibres

and matrix, the constitutive thickness of the elements is basically very close to zero. This

directly implies the use of cohesive element behavior defined in terms of traction separation

law which allows us to:

- Model delamination at interfaces

- Specify materials data (as fracture energy) as a function of the ratio of normal to shear

strength at the interface

- Characterize failure by progressive degradation of the material stiffness, which is driven bydamage process

- Take into account multiple damage mechanisms.

Moreover, cohesive elements are fully nonlinear (finite strain and rotation) and can have

mass so that they can be used in dynamic analysis (i.e. they are available in Abaqus/Standard

and Abaqus/ Explicit).

Failure mechanisms can generally be modeled in two steps:

- First : A damage initiation criterion

- Second : A damage evolution law.

In Abaqus, a third step is used with the possibility to remove elements upon reaching acomplete damage state.

2.1 Cohesive elements

In this section, a description of the three dimensional cohesive elements available in Abaqus

[11] is given.

The functions Nk used to interpolate displacements from the nodal ones are standard

lagrangian shape functions. We can write (in the [ , , ] coordinate system) those functionsfor the 8-nodes cohesive element as :

N1 = (1)(1)(1); N2 = (1)(1); N3 = (1); N4 = (1)(1);

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Figure 2.1: 3D cohesive elements

N5 = ( 1)( 1); N6 = ( 1); N7 = ; N8 = ( 1)The main differences between cohesive elements (defined in terms of traction separation

law) and classic finite elements is that they are supposed to symbolize interfaces. Quantities,

therefore, which represent the internal state of stress are no longer stress a tensor but a vector

of normal tn, tangential tt and shear ts components (figure 2.2).

Figure 2.2: Definition of stress quantities along the interface

The definition of these quantities implies the use of a particular direction for each element,

called direction of through-thickness behavior, along which tn is aligned. Obviously, this

direction is perpendicular to the interface. Practically, one needs to construct a bottom and

a top face of the element. To do that, a particular meshing method called sweep mesh -

advancing front is used. The idea is to define a first surface which will be meshed (two

dimensional mesh). Then this mesh will be extruded to the top face.

This procedure allows us to obtain an element defined as shown in figure 2.2.

These particularities of cohesive elements in Abaqus imply, in the visualisation step, to

reconsider the signification of the stress values [33; 32; 31] and the corresponding strains

[33; 32; 31]. Indeed we have no longer stress and strain tensor but vectors of stresses and

relative displacements at the interface. That leads to identify [33; 32; 31] with [tn; ts; ts]and also [33; 32; 31] with [n; s; t], where i, i = n,s,t is the relative displacement at the

interface (or separation displacement).



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2.2 Initial behavior and damage initiation criteria

2.2.1 Initial behavior

Initially, the interface behavior is defined by a linear elastic traction separation law. This law

is written in terms of an elastic constitutive matrix K that relates the nominal stress t to thenominal strain across the interface:

t =

Enn Ens EntEns Ess EstEnt Est Ett

ns

t

= E = K =Knn Kns KntKns Kss Kst

Knt Kst Ktt

ns

t

Where the corresponding strains n, s and t are derived from displacements on the inter-

face and from its thickness as follows :

n =

n

T0 ; s =

s

T0 ; t =

t

T0

Remark 1: The constitutive behavior presented above is a coupled law. To obtain an un-

coupled one it is sufficient to define the off-diagonal terms of K as equal to 0.

Remark 2: T0 is, in Abaqus, an artificial constitutive thickness of the cohesive element

which is, by default, defined equal to 1. That allows us to assume that, initially, the nominal

strains equal the separation displacements (i.e. = ).

Remark 3: Nevertheless, the real constitutive thickness Tc is used to define the stiffness, as

proposed by L. Daudeville, O. Allix and P. Ladevese [14] :

Knn =E33Tc

, Kss =2G13Tc

and Ktt =2G23Tc

.

Kij i, j = n,s,t are users defined values.

As the real thickness is close to zero, K is associated to a penalty stiffness. One needs to

define carefully this value because a very large penalty stiffness is detrimental to the stable

time increment and may result in ill-conditioning of element operation.

Remark 4: To have a positive defined matrix one might consider the condition that :

Kii i=j

K2ij

2.2.2 Damage initiation criteria

Damage initiation criteria are based on the schematic traction-separation response (shown

in figure 2.3) for the uncoupled delamination mode [11]. (The first linear part is explicated

subsection above.)

One enters the second linear part once the damage criterion used is met.

The criteria proposed in Abaqus are :

- Two criteria based on stress ratio, i.e. tit0i

called Maxs and Quads damage criterion.

- Two criteria based on strain ratio, i.e. ii0

with i = [n,s,t] called Maxe and Quade damage

criterion.

They are written as follow:



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Figure 2.3: Typical traction-separation response

max

tnt0n

;tst0s

;ttt0t

= 1

MaxsDamageCriterion

;

tnt0n

2+

tst0s

2+

ttt0t

2= 1

QuadsDamageCriterion

(2.1)

And

max

n0n

;s0s

;t0t

= 1

MaxeDamageCriterion

;

n0n

2+

s0s

2+

t0t

2= 1

QuadeDamageCriterion

(2.2)

Remark:

.

symbolize the fact that only positive value are taken, i.e. pure compressive

stress or deformation does not initiate damage.

2.3 Softening onset prediction (Mixed-mode)

In the case of a mixed mode delamination [12] and [13] propose a formula to obtain the value

of the total displacement corresponding to 0n (delamination in mode I only). A demonstration

of this formula is proposed below.

In mixed mode one might define the total displacement at the interface as:

m = n2 + 2s + 2t (2.3)Let us consider the quadratic damage initiation criteriatn

t0n

2+

tst0s

2+

ttt0t

2= 1 (2.4)

and assume that Knn = Kss = Ktt = K. Immediately we have the fact that ti = Ki and

t0i = K0i , i = n,s,t.

Then it leads to:

n0n2 + s

0s2 + t

0t2 = 1 (2.5)



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Case 1: n = 0:In the paper, the authors proposed to consider that 0s =

0t and to introduce the mode

mixity ratio as:

(n = 0) = 2s +

2t

n (2.6)so that we can rewrite (2.5) as:

2n

1

(0n)2 +

0s

2= 1 (2.7)

With (2.7) one can express the value of2n which corresponds to the mixed mode initiation:

2n =

0n

2

0s

2 1

(0s)2 + (0n)

2

2

(2.8)

Indicating by n the value of n corresponding to damage initiation in a mixed mode

decohesion:

n = 0n 0s

1

(0s)2 + (0n)

2 2 (2.9)

we can see that generally 0n = n, but in the case where 2s + 2t = 0, and then = 0 weare able to retrieve that 0n = n, which is the onset softening in mode I only.

With this value of n, the mixed mode relative displacement corresponding to the onset

softening is:

0m =

2n +

2s +

2t (2.10)

According to the definition of :

=

2s +

2t

n=

2s +

2t

n(2.11)

Then:

0m = n 1 + 2 (2.12)Finally the result is:

0m = 0n 0s

1 + 2

(0s)2 + (0n)

2 2 (2.13)

Case 2: n = 0.

By (2.5) and the assumption 0s = 0t one obtains immediately that:

s2 + t2 = 0s = 0m (2.14)Hence we retrieve the formula given in [12].



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2.4 Delamination propagation prediction and damage evolu-

tion law

The criteria used to predict delamination propagation under mixed-mode loading conditions

are usually established in terms of the energy release rates (GI; GII; GII I) and fracture

toughness (GIC; GIIC; GIIIC).

In an uncoupled law the final displacements (or displacement at failure) are obtained as

fn = 2GIC/t0n,

fs = 2GIIC/t

0s,

ft = 2GIIIC/t

0t .

However, in mixed-mode, an interaction between the energy release rates appears. Hence,

two predictive criteria are defined as:

GI

GIC

+

GII

GIIC

= 1

PowerLawCriterion; GIC + (GIIC GIC)

Gshear

GT

= GC

BKCriterion(2.15)

With Gshear = GII + GII I and GT = GI + Gshear; equals 1 or 2; [0.5, 2].

Then, the energy release rates corresponding to the total decohesion are obtained from:

GI =

nfm0

tndn, GII =

sfm0

tsds, GII I =

tfm0

ttdt (2.16)

Using the irreversible, bi-linear, softening constitutive behavior (which summarize the dif-

ferent steps exposed in this chapter) :

ti =

K

i

maxi

0i

(1 di)K i 0i < maxi fi0 maxi fi

(2.17)

and the linear softening law:

d =fm

maxm 0m

maxm (fm 0m)

, d [0, 1] (2.18)

also than (2.3) and (2.6) in equations (2.16 a., b. and c.) and substituting in (2.15.a) or in

(2.15.b), the criterion for total decohesion can be established in terms of m and . Solving

the equation for m, the mixed-mode displacements corresponding to a total decohesion, fm,

are obtained for the power law criterion as:

fm =

2(1+2)K0m

1

GIC

+

2

GIIC

1/ n > 0

(fs )2 + (ft )

2 n 0(2.19)

and for the B-K criterion as:

fm =

2K0m

GIC + (GIICGIC)

2

1+2

n > 0(fs )2 + (

ft )

2 n 0(2.20)

Remark: An exponential damage evolution law is proposed in Abaqus which leads to asimilar definition of fm.



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Chapter 3

Representative volume element

characterization

The peculiarity of our study is that we consider a fiber reinforced composite with one crackedfibre. That is why one might neither use a RVE with just one fiber (figure 3.1) nor a two

dimensional one. Indeed, the aim is to model the propagation of cracks and the redistribution

of the stress previously carried by the unbroken fiber to the matrix and the other fibers of the

composite. Furthermore, the stress concentration which exists around the crack is subject to

move along the three axes when debonding phenomenon or damage appears.

Figure 3.1: Classical RVE for fibre reinforced composite

As a first step of modeling this type of behavior, we concentrate ourselves on the aim to

proof that such studies are possible and accurate (or not) with a software as Abaqus. That is

why we are beginning with a relatively simple RVE which is subject to change regarding to

the matrix modeling, the number of fibres and their location, some size criterion or even the

crack shape and location.

3.1 Size of the RVE, number of fibres and location of the crack

Many studies have been made to establish a critical size of RVE with criteria like Hill condition,

Effective properties or Coefficient of correlation [15]. Nevertheless the main information which

is of interest for us is the global magnitude of the fibers radius, according to the aim expressed

above. (For example, the radius of AS4 carbon fibers is approximately 2 .5m.)

From this information, one of the simplest case which takes into account the redistribution

of stress in the matrix and some fibers can be thought as a cube with one central broken fibre

and four quadrant cylinders [figure 3.2].

21


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Figure 3.2: Representative Volume Element

Concerning the location and the shape of the crack we will assume that:

- the central fibre is totally broken

- the broken fibre can be modeled as two perfect cylinders (crack shape defined as z = Cte)

- Those two cylinders have the same length (the crack appears at the very middle of the

central fibre)

The last assumption derives straightly from the hypothesis of a RVE, because of the in-

finitesimal signification of it. Concerning the first two assumptions, it is clear that alternative

modeling could have been chosen. For example, the shape of the crack can be randomly chosen

as well as the average of the broken part of the central fiber.

Remark: One can consider that in a longitudinal load, we could have used the symmetries

of the problem to simplify the model. But, even if in this step we are applying this type of

load, the idea is to build, in the long term, a RVE able to support every kind of loads. Fur-

thermore, the assumption made of representativeness of the whole structure implies to keep a

cubic (or a cobble) shape.

3.2 Boundary conditions

The boundary conditions that we are applying to the unit cell are very similar to those used

in kinematic homogenization, i.e. : ud = OM.Then for the particular case of longitudinal load we have, with respect to the coordinatesystem used in our Abaqus model:

=

0 0 00 0 00 0 1

Then:

ud = [0, 0, ] on top, and ud = [0, 0,] on bottom ()

Furthermore, kinematic homogenization boundaries leads to define:

ud = [0, 0,z] on Slat1 Slat2



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The other possibility is to impose, for every lateral surfaces (apart from the top and bottom

ones, where () remains valid), that the displacement following e1 and e2 of every nodes arethe same, i.e.:

(qn, qm) Slati, u(qn) ei = u(qm) ei , i = 1, 2

3.3 Materials and interfaces

In the very first step of the modeling we have used an isotropic linear elastic behavior for the

matrix and fibres. The materials and their properties are:

Fibre Carbon (AS4) : E = 210000M P a and = 0.3

Matrix Epoxy (914) : E = 10000M P a and = 0.2

Moreover the unit cell count seven interfaces (six between the matrix and the fibres and

one between the two parts of the broken central fibre). The first six are modeled by cohesive

elements and the seventh by frictionless contact.

Remark 1: One can see that from the first increment of the calculus (in the case of a lon-

gitudinal pull load), there is no longer contact between the two parts of the central fibre. In

this sense, the seventh interface is not taken into account for the calculus.

Remark 2: Concerning the cohesive element properties, as observed earlier (Chapter 2),

one has to specify the characteristics used to define the fracture initiation criteria and the

damage evolution law, i.e. the constitutive matrix K, the vector of relative displacement at

softening offset 0 and the interlaminar fracture toughnesses GI, GII, GII I

In the case of a longitudinal pull load, cohesive elements are stressed in both mode I and

mode II (opening and sliding). Indeed, sliding is given by both crack and differences of Youngs

modulus, and opening by differences of Poissons coefficients (the presence of the crack changes

the values of mode I relative displacements compared with the uncracked unit cell).

Even if mode III is theoretically inexistent, we can see that Abaqus gives us a (very low)

value of relative displacement corresponding to this mode.

Remark 3: For utility, the length unit used is the m, which leads to define the Youngs

modulus in T P a. Accordingly, the stress values results will have to be reinterpreted.



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Chapter 4

Specificities of the modeling under

Abaqus

The construction of the three dimensional numerical model leads to some difficulties we aregoing to expose as well as the solutions we have proposed.

Indeed, the specificities of cohesive elements concerning geometrical, status and meshing

methods have to be treated with a particular care.

4.1 Contact status and mesh of interface elements

The constitutive thickness of interface elements is basically very thin as regards to the whole

model dimensions. In our case, we defined a thickness equals to 0.1m which leads to a finer

mesh than that of the matrix or the fiber. This fact implies to give a slave status of the

cohesive elements in the tie (completely fixed) interactions.Furthermore, compatible meshes between cohesive and surrounding elements have to be

carefully constructed.

Moreover, the specific meshing method used to solve those problems for central interface

elements (where the most important issues appear) are as follows.

First, one needs to partition the cylinder in two parts and to define a new meshing path,

as the automatic one leads to ill definition of bottom and top faces of the interface.

Figure 4.1: Partition and path definition

24


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Moreover, without partition no user defined path is allowed for cylindric bodies. Then the

proposed model leads to well defined elements as shown figure 4.2:

Figure 4.2: Bottom and top definition. 4.2.b: Local coordinates.

where the bottom face is colored in purple and the top in brown.

Remark: The local coordinate system linked with every cohesive elements is here : e3 the

through-thickness direction, e2 parallel for every element with the z global axis and e1 = e3e2(where symbolize the vectorial product).

The figure 4.2.b shows the local coordinate systems for two elements Cn and Cn+m :

4.2 Partition of the matrix

Because of the relatively complex geometry of the matrix, the automatically computed mesh

is not satisfying. Then the partition of the matrix seems to be recommended.

To improve the quality of the mesh we performed the partition as shown figure 3.2, which

partition can be performed by the partition tool in the mesh module or, more simply, by

constructing the part in two times (the central cylinder and the remaining volume) with the

option of keeping internal boundaries.



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Chapter 5

Numerical simulation

The results shown in this chapter are mainly useful to prove the accuracy of a three dimen-

sional model for a mixed mode delamination. As we do not have the possibility to compare

them with experiments, we will focus on qualitatively regain the classical phenomena (stressconcentration, redistribution of stress from the fibre to the matrix) which appear in cracked

reinforced composites under longitudinal pull load.

The materials properties used are exposed in chapter 3. Nevertheless, numerical values of

interface properties and boundary conditions are detailed below:

Properties of the interface:

Relative displacement values at onset softening 0n = 0t = 0s = 0.1m, which leads(formula (2.13)) to 0m = 0.1m.

K is diagonal and Knn = Kss = Ktt = 1GP a.

The interlaminar fracture energy GI = GII = GII I = 5.104kJ/m2, which correspondto a displacement at failure equals to 1m.

Displacement control:

The boundary conditions are: Ud = [0, 0, 2m] on top, Ud = [0, 0,2m] on bottomand Ud = [0, 0, 0] on every lateral faces.

Where:

=

1 0 0

0 1 0

0 0 0

5.1 Stress concentration

Firstly, the numerical simulation allows us to follow the displacement of the stress concentration

from the outskirts of the crack and then along the central fibre outline.

Figure 5.1 gives an idea of the phenomenon. As recalled in chapter 2, the stress quantity

23 represent the longitudinal shear stress upon the interface (linked with Mode II).

Remark: In the case of strong interface between fibre and matrix, the stress concentration

can not move, as it happens here, from the outskirts of the crack to the bottom and top facealong the interface between the crack fibre and the matrix. Indeed this phenomenon supposes

the degradation, up to failure, of the cohesive elements where the stress concentration was

26


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Figure 5.1: Longitudinal shear load / S23 - Step 1, 10 et 20

previously located. To simulate the displacement of stress concentration in this next case, one

need to model the matrix as a damageable material.

5.2 Damage evolution

In the corresponding in figure 5.2 of damage values in cohesive elements, the location of the

stress concentration widely depends on the interface softening.

The SDEG value is the value of D in the relation i = (1 D) Kii ti. Then SDEG = 1(red in the color classification) signify the fact that the materials are no longer bonded.

We can see that mode II and mode I occurred in the simulation. Then one need to be

careful in the interpretation of the stress/displacement curve for one cohesive element.

Figure 5.2: Damage variable at Step 13. 5.2.b: Strain/Stress curve in mode II

For one element located near the crack , the figure 5.2.b show the strain/stress curve in

mode II.

The strain/stress curve corresponding to mode II is decomposed, as expected, in:

0 s < 0.098m the linear initial behavior

0.098 s < 0.98m the linear softening part

0.98 < s the fully damaged behavior

Remark: If we mark, as proposed chapter 2, s = 0.098, i.e. the value of s at onset

softening in mixed mode delamination, we obtain that

n

2+ s

2+ t

2= 1.



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5.3 Global response of the unit cell - comparison of Abaqus/Standard

and Abaqus/Explicit

The mean stress (of 33) on the central fiber (top face) versus the displacement is presented

in figure 5.3 for several types of resolution. The idea is to compare the results given by

Abaqus/Standard and Abaqus/Explicit for the same problem. In chapter 1, section 4, we have

explained the main differences between implicit and explicit resolution methods. It is now the

place to show how both methods are available for our study.

Indeed, the use of an explicit method take into account the density of materials (to build

the mass matrix). However, the fact that the length unity used is the micro meter leads to

define the density in 1018Kg/m3. Then, the time increment used for the explicit resolution

tends to zero as :

t =Lmin

cd= Lmin

+ 2

In our case, Lmin

1 for the coarse mesh,

1.1018Kg/m3, E = 0.21T P a and = 0.3.

Thus t 1.109, which implies to compute one million iterations to solve the simulation.This trouble leads to use a pseudo mass, which is an arbitrary value of the density. Because

of the quasi-static problem we are solving, this procedure is frequently used. However, it is

important to be sure that no dynamic effects are appearing during the process.

Figure 5.3: Stress-displacement curve of the central fibre (top face) for implicit and explicit

method

The first curve (serie 1) is the result of the Abaqus/Standard resolution. Series 2, 3 , 4

and 5 are respectively the resolution of the same mechanical problem under Abaqus/Explicit

with = 1.1010, = 1.108, = 1.107 and = 1.106. Moreover, their time increments

are respectively t = 2.105, 2.104, 7.104 and 2.103.

Dynamic effects inter into account in the last two series, where oscillations appears. This

gives us the condition to use a value of 1.108 for a coarse mesh (Lmin 1). However,

even with a refined mesh we are going to use this condition.Remark 1: One identify time and displacement in our case as we applied displacement of

the top and the bottom face linearly in function of the time.



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Remark 2: The load carried by the central fiber is clearly dependent from the level and the

number of damaged elements of interface. Obviously, the load carried when all the cohesive

elements are fully damaged tends to zero.

Remark 3: This phenomenon leads to a redistribution of the stress in the matrix. Hence,

the stress/displacement curve for an element of the matrix close to the central fibre and to the

top face is shown (figure 5.5) .

Figure 5.4: Stress/displacement in an element of the matrix

5.4 Discussion

The crack propagation in fibre reinforced composites is theoretically allowed to expand in the

three space dimensions. Several types of failures are commonly listed as: opening of the matrix,

debonding at the interfaces, failure of fibres which are surrounding the crack. Obviously, a

real failure of the material can be a combination of several failure types.

In this numerical simulation, we tried to qualitatively represent sliding between fibre and

matrix. The stress concentration which is theoretically predicted has been shown. Further-

more, behavior of interface elements accurately correspond to the results expected.

In that sense, this simulation validates the use of cohesive elements in a three dimensional

problem involving a fibre reinforced composite unite cell.

However, it could be wise to:

Define a bigger RVE which involves more fibres to reduce the effects of the boundaryconditions on the propagation of the crack.

Have a damage matrix model to allow the crack to expand in three dimensions. Thenit could be possible to consider the competition between debonding along the interface

and opening of the matrix.

In the next chapters, we are going to expose the main tools and the work related to those ends.



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Chapter 6

Damage modeling

6.1 Some damage modelings and their deficiencies

Damage appears in materials as a consequence of the presence of voids. Their growth and

coalescence under loading is characterized by localized cracks and softening behavior.

In function of the type of material, the plastic strain at onset softening and the rate of loss

of load-carrying capacity is more or less important. That is the reason why several kinds of

damage modeling have been developed.

But even for a quasi-brittle material (for example), computational models can roughly be

separated into three categories :

- Micromechanical models where materials are modeled by lattice structures or braces of

constituents and interfaces. However, because of the high level of detail considered, these

methods lead to large computational costs.- Damage modeled by dominant macroscopic crack, using the concept of cohesive zones,

is also proposed. Its main disadvantage consists in an extensive remeshing of the structure to

follow the propagation of the crack.

- The third damage model is based on continuum mechanics and the use of a damage quan-

tity defined at each points of the structure. This method leads to a pathological dependence

on the fineness of the spatial discretisation as well as on its orientation. Mathematically this

deficiency is related to a loss of ellipticity of the equilibrium equation. Owing to the local tran-

sition of these partial differential equations (elliptic to parabolic and finally hyperbolic) the

tangential material stiffness change of sign. Then the mechanical problem becomes ill-posed.

Practically, this deficiency leads to have a localized damage, with the consequence that thefinner the mesh is, the smaller the energy needed to damage the material becomes.

In the next section, we are going to briefly expose some modified (isotropic) damage models

which were designed to avoid this deficiency.

6.2 Gradient-based damage modelings

6.2.1 Method based on the principle of virtual work

One of the proposed methods to avoid this problem is based on the rewriting of the principleof virtual work taking into account the damage quantity and its gradient [16].

Then the power of internal forces and the power of external forces are written as:

30


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Pi

u,

d

dt

=

: D (u) d

B d

dt+ H grad

d

dt

d

Peu,d

dt = f u d + F u d + A d

dt d + b

d

dt d

With:

(X, t): the macroscopic damage quantity

B: the internal work of damage

H: the flux vector of internal work of damage

A, b: the volumetric and surfacic external source of damage work

Then for a quasi-static problem one obtains that:

, v, , Pi (v, ) + Pe (v, ) = 0

6.2.2 Gradient-enhanced damage

An isotropic damage model (which is implemented in CAST3M) has been developed by R.H.J.

Peerlings et al. in [17-19]. The idea is to link the classical constitutive law for damage (6.1)

to a set of equations ( and ) defining the new mechanical problem.

= (1 D) H (6.1)

div + f = 0 (

)

eq c eq = eq ()

With:

eq: the local equivalent strain defined at each point of the body

eq: the corresponding non-local equivalent strain

c: the constant gradient parameter with the dimension of length square

The link between D and eq is performed by the fact that D = D(), where is an historic

parameter which appears in the Kuhn Tucker relations as below:

0 , eq

0 ,

( eq

) = 0

After the establishing of the weak forms of () and () (for the boundary conditionsn = FD on 2 and eqn = 0 on ), the iterative method is initiated by the differentiationof (6.1):

= (1 D) H D H (6.2)

The Kuhn Tucker relation gives us that, for the increment i, i = eq,i if = eq or

i = 0 otherwise. Furthermore, the linearization of D() gives us that D = (D/) .

Then the weak formulation of () can be written, for the increment i, with eq,i and ui asonly unknown.

Finally, the new finite element problem to solve is written as:



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Kuui1 Kui1Kui1 K

ui eq,i

=

fuextfi1

fuint,i1

K eq,i

Remark 1: As Kui1 = Kui1, the generalized stiffness matrix is non-symmetric.

Remark 2: The iterative method proposed by the authors is based on the idea that for a

quantity , its value at the increment i is : i = i1 + i.

6.3 Interpretation and tools used

One can see that an advanced damage model (even for an homogeneous isotropic material)

leads to deeply change the mechanical problem to solve. Moreover, the underlying idea of the

models above is that non-local effects are acting at each point of the material. Then those

models are questioning the fact that stress and strain are locally defined.

The damage modeling (proposed in Abaqus/Explicit) we are going to use, simply em-

ploy the microscale characteristic length of the material to reduce the mesh dependency phe-

nomenon. Thus, the concept of equivalent plastic displacement is define as:

upl = L pl (6.3)

For solid elements, L is taken as the cube root of the integration point volume.

One can see that damage modeling is very close to the one applied for cohesive elements

degradation. Indeed, damage occurs after that a specified criterion has been reached and the

very same softening laws (linear or exponential) are available.

However, this model is called plastic-damage because of the fact that the damage occurs

after yielding of the material. This type of modeling is drawn figure 6.1 [11].

Figure 6.1: Plastic-damage model under Abaqus

Remark: The fact that the damage models are available for Abaqus/Explicit only is the

reason why most of the simulations with this resolution method also than the developments

given concerning the theory of explicit procedures (chapter 1) and its validity using a pseudo-

mass (chapter 5 and 7).



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Chapter 7

Numerical simulation with a matrix

damage model

In this chapter, we are mainly using the boundary conditions and material properties definedpreviously, apart from the parameters of the Plastic-Damage model of the matrix.

7.1 Parameters of the Abaqus damage model

The previous chapter gives us an idea of the different types of damage modeling. Practically,

the Plastic-Damage model under Abaqus leads to define some specific parameters which are

the Yield Stress, the Fracture Strain, the Deformation at Fracture (or Fracture Energy) and

the Fracture ratio.

Figures 7.1 and 7.1.b show the influence of the Fracture Strain and Deformation at fracture

for a simple example (one 3-dimensional element fixed on one side and pulled on the oppositeside) for a Yield Stress given. The Fracture Ratio is not taken into account because the results

are insensitive to its changes.

Figure 7.1: Variation of damage model parameters

Those two parameters can be considered as means to adapt the damage model for the type

of material used. A small fracture Strain gives a damage model well fitted for a quasi-brittle

type of material and a large one to a ductile material.

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7.2 Simulations for the Small Unit Cell

Mesh of the small unit-cell:

Figure 7.2: Mesh of the small Unit Cell

Parameters of the damage model:

Fracture Strain = 1 ; Yield Stress = 100M P a ; Fracture Energy = 1.105kJ/m2

With those parameters we are obtaining the case where the crack is propagating inside the

matrix to lead to its opening.

Figure 7.3: Values of the damage initiation criterion - opening of the matrix

For this simulation, the dynamics effects linked with the use of a pseudo-mass are controlled

using two values of (figure 7.4).

Figure 7.4: Dynamic effects

Hence we can see the differences of results between these two simulations for some particular

quantities (evolution of the normal stress in a cohesive element and evolution of the average



35/38


of normal stress upon the central fibre top face). Figure 7.4, serie1 correspond to = 7.108

and serie2 to = 7.1010.

Remark : The other case (sliding at the interfaces without failure of the matrix) is easily

obtained in changing the parameters of the Plastic-Damage model or the properties of the

interface.

7.3 Simulations for the Bigger Unit Cell

Mesh of the small unit-cell:

To have a more accurate modeling of the failure, a bigger unit-cell has been build. Thus,

thirteen fibres are involved and the dimensions of the RVE become 40 40 40m. Figure7.5 and 7.5.b show this unit-cell and the mesh used for the simulations.

Figure 7.5: Bigger unit-cell and its mesh

With this representative volume element, several simulations have been made for different

values of Fracture Strain which allows us to obtain the case of debonding of the interface and

the opening of the matrix. These two type of failure are shown figures 7.6 and 7.6.b.

Figure 7.6: Values of the damage initiation criterion - a. opening of the matrix - b. debonding

at the interface



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7.4 Discussion

The prediction of failure and its propagation in a composite material is frequently based

on Linear Elastic Fracture Mechanics (LEFM). In this simulation we have proposed a finite

element approach of this phenomenon, at the micro scale. This allowed us to represent two

types of fracture depending to the properties of the materials and interfaces.However, one of the main questions of this project can be expressed as : is Abaqus accurate

to model this unit cell and the phenomena of crack propagation? This question can now be

answered in three principal view angles:

The possibility to use special types of interface elements, called cohesive elements (Abaqus/Standard and Abaqus/Explicit), to represent debonding at the interface between fibre

and matrix, accurately model the physical phenomenon. This model is also very close

to other recents works as Alfanos ones [20, 21].

However, some work has been done to implement the model presented in [21], where the

friction at the interface is added to the damage model in the cohesive zone. Regrettably,

we did not have the time to solve all the troubles linked with the implementation of the

model as a user subroutine Abaqus/Explicit. Although some existing user-subroutines

implemented for Abaqus/Standard (implicit) were available, the implementation of the

code for Abaqus/Explicit resulted in some error at run-time, which was not possible to

eliminate in the little time available. That is why no results with this Friction-Damage

Model are presented in this report.

The fact that we are modeling the unit-cell at the micro scale leads to a stunning difficulty.Indeed, the use of a damage model implies to run the computation under Abaqus/Explicit

and then to define the density of the materials. Thereby, the time increment (whichrepresent that in one increment a mechanical wave cover the smallest element of the

mesh) tends dramatically to zero.

As proposed chapter 6, it is possible to use a pseudo mass in order to avoid this difficulties.

Nevertheless, a particular study is necessary to establish the validity of the results.

Finally, we can estimate that the damage models proposed under Abaqus are quiteinsufficient. Indeed, the mesh dependency troubles are solved by the use of characteristic

length of the material, without taking into account the gradient of the damage quantity.

It seems that the use of a more advanced model, as the Gradient-Enhanced Damage

Model could improve the accuracy of the simulations.



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Conclusion

In this project we have simulated the propagation of fracture in a Representative Volume Ele-

ment, using an industrial finite element software. The simulation of different type of fracture,

opening of the matrix, delamination at the interfaces and a combination of both types, have

been shown.

To this end, two types of RVE have been built (five fibres and sixteen fibres involved re-

spectively). Special interface elements and a damage model proposed in Abaqus were used.Furthermore, we began the implementation of a Friction-Damage Model to improve the be-

havior of the Cohesive Elements.

Along this report, several special studies have exposed the background of the project (Fibre

Reinforced Composites and some recents works related, Multi-Scale approaches, Non-linear

modeling - chapter 1 - and, finally, some current damage models - chapter 6).

This work leads to several research lines. One could summarize them as:

- Complete the implementation of the Friction-Damage Model for the interface elements

in Abaqus/Explicit,

- Implement other better damage models or discrete-crack models (X-FEM) also for the

implicit analysis,- Include a brittle damage model for the fibre,

- Incorporate stochastic approaches concerning the position or/and the properties of the

fibres,

- And, last but not least, investigate on the link between micro and macro scales, to develop

a multi-scale strategy.

Acknowledgments

I really thank Mr Giulio Alfano, lecturer and researcher at Brunel University, who guided me

during these three months research training period. I am also grateful to Mr Ali Bahtui andMr Fiorenzo Decicco who gave me their support and precious advice.

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References

[1] Mallick. 1993. Fibers Reinforced Composites. Marcel Dekker, inc..

[2] Z. Xia, W. A. Curtin and P.W. Peters. 2001. Multiscale Modeling of Failure in Metal

Matrix Composites. Acta Materialia 273-287.

[3] N. Bonora, Andrew Ruggerio. 2006. Micromechanical Modeling of Composites Inter-

face Part 1 and 2. Composites Science and Technology 214-322.

[4] C. Gonzalez, J. LLorca. 2007. Mechanical Behavior of Unidirectional Fiber ReinforcedPolymers Under Transverse Compression Composites Science and Technology 2795-2806.

[5] J.-K. Kim and Y.-W. Mai. 1998, Engineered Interfaces in Fibre Reinforced Compos-

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