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Modelling of fatigue microcracking kinetics in crossply composites and experimental validation J.F. Caron*, A. Ehrlacher Ecole Nationale des Ponts et Chausse ´es, Centre d’Enseignement et de Recherche en Analyse des Mate ´riaux, 6,8 avenue Blaise Pascal, Cite ´ Descartes, Champs-sur-Marne, 77455 Marne la Valle ´e, France Received 12 December 1997; received in revised form 18 September 1998; accepted 8 October 1998 Abstract We describe a micromechanical model of transverse cracking in ; 90 p sym composite laminates under fatigue loading. Several features are recalled which have been the subject of another article. These concern static aspects (such as stress fields, strength dis- tributions, etc.) and have permitted us to predict the number of cracks created during the first cycle of loading. This paper presents the principles of the fatigue model and also proposes several experimental aspects concerning the identification of parameters and validation of the model. A micromechanical hypothesis is suggested, which is based on a physical interpretation of damage to predict the reduction in residual strength of 90 plies during fatigue loading. The model is incremental and allows us to describe in a simple but non-linear way the increase of damage in the 90 plies of a ; 90 p s laminate. We then include this fatigue aspect in our more general tool which describes a static description of the transverse cracking kinetics of a ; 90 p sym composite. The predictions of the simulations are validated on 0 2 ; 90 2 s carbon/epoxy specimens. We have observed and counted cracks during experimental tension–tension tests and compared the results with simulations. We observe good agreement between predictions of the model and experimental results. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; B. Matrix cracking; B. Fatigue; Random behaviour; Simulations; Experimental results 1. Introduction Damage mechanisms and damage accumulation in laminates have been the subject of many investigations [1–38]. The first failure mechanism is matrix cracking in the o-axis plies. Increasing the static load level or the number of cycles leads to an increase in the number of transverse cracks/cm in the o-axis plies. This increase depends on the loading history or stacking sequence [1–5]. The damage is not directly responsible for rup- ture, but seems to be an initiator of other forms of damage like delamination or fibre breaks. In another paper we have described a model to describe the kinetics of cracking during static [6] loading in a ; 90 p s lami- nate. In this paper we extend these ideas to develop a model for fatigue loading. A common component of the two models is a micromechanical model of the micro- cracked 0; 90 s crossply composite. The laminate is dis- cretized in cells including potential cracks. The initial strengths of these cells are randomly distributed and permit progressive cracking. Furthermore each elemen- tary volume will experience a dierent loading history because each new crack will redistribute stress fields in a crossply laminate. In the second part of the paper, we review the dierent possibilities for the choice of fatigue variables. During fatigue loading, we consider the first cycle as a static loading step and for the succeeding cycles we propose a damage model based on the residual strength of the 90 plies. A micromechanical assumption will explain the decreasing residual strength of the 90 plies during fati- gue loading. We shall see that the identifications of model parameters calls for few and simple tests. Indeed, the previous studies quoted often need intensive test programmes. Finally we shall present experimental identification of the model parameters and experimental validation of the model on a 0 2 ; 90 2 s . 2. Mechanical model of a micro-cracked [0, 90] s crossply laminate Some authors [7–9] have chosen a finite-element method to study the three-dimensional problem, and defined a damaged cell between two cracks. But these Composites Science and Technology 59 (1999) 1349–1359 0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved PII: S0266-3538(98)00173-0 * Corresponding author.

Modelling of fatigue microcracking kinetics in crossply composites and experimental validation

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Page 1: Modelling of fatigue microcracking kinetics in crossply composites and experimental validation

Modelling of fatigue microcracking kinetics in crossply compositesand experimental validation

J.F. Caron*, A. EhrlacherEcole Nationale des Ponts et ChausseÂes, Centre d'Enseignement et de Recherche en Analyse des MateÂriaux, 6,8 avenue Blaise Pascal, Cite Descartes,

Champs-sur-Marne, 77455 Marne la ValleÂe, France

Received 12 December 1997; received in revised form 18 September 1998; accepted 8 October 1998

Abstract

We describe a micromechanical model of transverse cracking in ��; 90p�sym composite laminates under fatigue loading. Severalfeatures are recalled which have been the subject of another article. These concern static aspects (such as stress ®elds, strength dis-

tributions, etc.) and have permitted us to predict the number of cracks created during the ®rst cycle of loading. This paper presentsthe principles of the fatigue model and also proposes several experimental aspects concerning the identi®cation of parameters andvalidation of the model. A micromechanical hypothesis is suggested, which is based on a physical interpretation of damage to

predict the reduction in residual strength of 90� plies during fatigue loading. The model is incremental and allows us to describe in asimple but non-linear way the increase of damage in the 90� plies of a ��; 90p�s laminate. We then include this fatigue aspect in ourmore general tool which describes a static description of the transverse cracking kinetics of a ��; 90p�sym

composite. The predictions

of the simulations are validated on �02; 902�s carbon/epoxy specimens. We have observed and counted cracks during experimentaltension±tension tests and compared the results with simulations. We observe good agreement between predictions of the model andexperimental results. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: A. Polymer-matrix composites; B. Matrix cracking; B. Fatigue; Random behaviour; Simulations; Experimental results

1. Introduction

Damage mechanisms and damage accumulation inlaminates have been the subject of many investigations[1±38]. The ®rst failure mechanism is matrix cracking inthe o�-axis plies. Increasing the static load level or thenumber of cycles leads to an increase in the number oftransverse cracks/cm in the o�-axis plies. This increasedepends on the loading history or stacking sequence[1±5]. The damage is not directly responsible for rup-ture, but seems to be an initiator of other forms ofdamage like delamination or ®bre breaks. In anotherpaper we have described a model to describe the kineticsof cracking during static [6] loading in a ��; 90p�s lami-nate. In this paper we extend these ideas to develop amodel for fatigue loading. A common component of thetwo models is a micromechanical model of the micro-cracked �0; 90�s crossply composite. The laminate is dis-cretized in cells including potential cracks. The initialstrengths of these cells are randomly distributed andpermit progressive cracking. Furthermore each elemen-tary volume will experience a di�erent loading history

because each new crack will redistribute stress ®elds in acrossply laminate.

In the second part of the paper, we review the di�erentpossibilities for the choice of fatigue variables. Duringfatigue loading, we consider the ®rst cycle as a staticloading step and for the succeeding cycles we propose adamage model based on the residual strength of the 90�

plies. A micromechanical assumption will explain thedecreasing residual strength of the 90� plies during fati-gue loading. We shall see that the identi®cations ofmodel parameters calls for few and simple tests. Indeed,the previous studies quoted often need intensive testprogrammes.

Finally we shall present experimental identi®cation ofthe model parameters and experimental validation ofthe model on a �02; 902�s.

2. Mechanical model of a micro-cracked [0, 90]scrossply laminate

Some authors [7±9] have chosen a ®nite-elementmethod to study the three-dimensional problem, andde®ned a damaged cell between two cracks. But these

Composites Science and Technology 59 (1999) 1349±1359

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved

PII: S0266-3538(98)00173-0

* Corresponding author.

Page 2: Modelling of fatigue microcracking kinetics in crossply composites and experimental validation

three-dimensional models are generally very cumber-some and many authors have proposed simpli®edapproaches. By taking into account the thin aspect ofthe laminates, it is also possible to use classical thinlaminate theory, but these methods are not useful forexploring the stresses in the vicinity of cracks and edges.

We have chosen a plate approach which we havecalled multi-particular (see Ref. [10] for details) becausewe consider a bi-dimensional geometry (independenceof z) and several particles (one for each ply) per geome-trical point. In this paper we use the ID multi-particularmodel of Cox [11], often called shear-lag analysis, whichis the most commonly used [2,4]. This model considersthe crossply laminate to be like a uniaxial material andmakes a hypothesis for the form of the stress or strain®elds in the thickness of each ply to arrive at `average'displacement ®elds. The model then introduces animportant hypothesis concerning the behavior: inter-facial shear stresses that balance axial stresses are pro-portional to the di�erence in the displacements of twoadjacent layers for any x. This leads to the existence of aconstant, K0;90

�xz , called the interface sti�ness. With thishypothesis, and writing the equilibrium of a 90� crackedcell, we can calculate the axial stress N90

xx�x� for any x.We use the terms E90, E0, Ec for the 90�, 0� and

�0; 90�s longitudinal Young moduli, F for the appliedloading per unit width (N/m), b and 2d for the 0� and90� thicknesses, �� and �90 for the linear expansioncoe�cients, and �T for the temperature change fromcuring temperature to room temperature. Finally wehave the expression for the axial stress in the 90� direc-tion as a function of x, the distance to the crack (x � 0at the center of the cell, x � �h on the crack):

N90xx�x�d� ��0 ÿ �90��T

1

E90� d

bE0

� �ÿ1�E90

Ec

F

b� d

� � !

1ÿ ch �xd

ch �hd

" #�1�

where

�2 � K0;90�xz d

bE0 � dE90

bE0E90

We have identi®ed K0;90�xz by a change of scale (see Ref.

[6] for more details) by making a three-dimensional®nite-element calculation of a �0; 90�s cracked cell, withthe following assumptions. The displacement Ui�x� in alayer of the uniaxial multi-particular model is equal tothe average thickness over a section x of the micro-dis-placement in the same layer of a 3D model. The axialstress, Ni

xx�x�, in a layer of the multi-particular model isequal to the Integral over the section x of the axial stress

of a 3D model in the same layer. The interface shearstress, �i;i�1xz , of the multi-particular model is deducedfrom the equilibrium equations. It is also the interfacialshear stress of the 3D model.

We de®ne from the 3D calculation, the interfacebehaviour K0:90

�xz �x� as follows:

K0;90�xz �x� �

U90�x� ÿU0�x��0;90xz �x�

�2�

�0;90�xz seems to be independent of x (numerical errorsnear the crack expected) and independent of the cellsize. �Kmean � 2:5� 1013 Pa=m�.

3. Fatigue of a [0,90]s laminate

There are two main possibilities for introducing thefatigue behaviour in damaged laminates. The ®rst takesplace on the macroscopic scale. By introducing internaldamage variables into the macroscopic behavior of theply, the ply becomes a damaged equivalent homo-geneous material. For transverse cracking these vari-ables can be associated with a decrease in transverse andshear sti�nesses of the material [14,15]. The choice ofdamage parameters can be guided by physical con-siderations like cracks density [16,17]. To determine thedevelopment law for the damage parameters, thesetechniques need tests on each crossply studied becausethey do not allow estimation of local stresses in thevicinity of the transverse crack which is necessary todetermine further cracking. We prefer to work on themicro-scale, with the help of linear-elastic fracturemechanics.

To represent the fatigue behaviour of a material,engineers use commonly S/N curves, also namedWoÈ hler curves, S being the stress and N the number ofcycles to failure. This representation is obtained bymaking several fatigue tests, according to a certainloading program, which lead to the rupture of the spe-cimen. The ®rst point on the curve gives the monotonicrupture stress of the specimen, while the rest of thecurve consists of points showing the applied stress leveland the number of cycles to cause specimen failure atthe stress. The S/N curve de®nes a domain where thespecimen is not destroyed. In the remainder of thispaper we shall denote this curve S(N) and the inversecurve N(S).

If the loading is periodic, one loading cycle can becharacterised by a frequency, f, a maximum stress level,S, and by the di�erence between this maximum and theminimum stress level, �S. The stress-ratio, r, is oftenintroduced, and it is de®ned as:

r � �Sÿ�S�=S � Smin =S

1350 J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359

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We use the data pair (�S; r) to describe one cycle. ThenN��S; r� is the number of cycles to failure at ��S; r�.However, the damage state of the material is not takeninto account in this description, only its failure. Damageis generally described by a scalar or tensorial variable,D, which varies between 0 (no damage) and 1 (failure).This damage depends on loading ��S; r� and number ofcycles, frequency, but also humidity, temperature, etc.

There are many di�erent models in the literature. Themost common is the Miner law [18] which describesdamage state after n cycles at the stress level S0, as:

D � n

N�S0�

where N�S0� is the number of cycles to cause specimenfailure at S0. For more complex loading history, an ele-mentary step �Si; ni� can be de®ned. Usually a linearcumulative law is used to predict the damage state:

D �Xi

Di �Xi

niN�Si�

But such a linear law is rarely pertinent to describecomposite materials behaviour. Many models followfrom Miner's law. However, even non-linear laws suchas those of the sophisticated models listed in Hwang andHan [19] do not give a physical sense to the damage.

A contrasting approach attempts to link the damagestate of the material with a mechanical characteristicwhich can be measured, like the tangential modulus[20], secant modulus [21], or fatigue modulus [19,22].When the material studied shows a plastic behaviour,the residual strain [23] after loading can be used. Theresidual strain can also be an appropriate way to quan-tify damage if some internal stresses (thermal stressesfor example) decrease as a result of microcracking inthe material, as is often the case for composite materi-als. But for the kind of crossply we have studied inthis paper (carbon/epoxy ��; 90p�sym in tension±tensioncyclic loading) these damage indicators do not changesigni®cantly.

By contrast, if we consider the residual strength ofthese laminates, we measure a decrease during loadingwhich is the reason why several authors have used afatigue model of laminates based on residual strength[21,24±26]. We have also chosen this method, and in thispaper we propose a model which issues from linear-elastic fracture mechanics, which predicts the decreasein residual strength of the 90� plies in a �0; 90�s laminate.

3.1. The residual strength, Re, as a damage indicator

The objective is to ®nd an analytical law which mod-els the decrease in residual strength of a 90� sectionduring a fatigue loading ��S; r�, where �S is the

amplitude SmaxÿSmin of applied loading and r its classi-cal fatigue ratio Smin=Smax.

We denote as dRe

dN the decrement of Re during oneloading cycle. Making the assumption that fatigue sig-ni®cantly a�ects the strength of the section, we proposethe more general following law as possible:

dRe

dN� g�Re;�S; r�50 �3�

where Re is the residual strength of a section of 90� plybefore the cycle. Re is characteristic of the damage stateof a section of 90� ply.

3.1.1. Residual strength

Calling Rinite the initial strength before the loading, we

de®ne Re as the residual strength after N cycles at��S; r� loading.

Eq. (3) can be viewed as a di�erential equation withseparate variables on the residual strength Re functionof the variable N. If we integrate such a di�erentialequation, Re appears as a function of Rinit

e , �S, r and N.We can write Re as Re�Rinit

e ;�S; r;N�.

3.1.2. Residual lifetime

Rupture occurs when the residual strength hasreached the maximum applied stress, so that from Eq.(3), we can give an expression for the residual lifetimeNr�Rinit

e ;�S; r�, the number of cycle at ��S; r�, that thematerial can still support before rupture.

So:

Nr�Rinite ;�S; r� �

�SRinit

e

1=g�Re;�S; r�dRe �4�

It will be noted that Eq. (4) can represent the classicalN��S; r� curve. Now, in order to avoid doing a largenumber of tests to identify experimentally the law offatigue dRe

dN � g�Re;�S; r�, it seems necessary to proposean expression for g�Re;�S; r�.

We suppose that the residual strength is connected tothe maximum size of initial defects in the section of 90�.This defect becomes a crack, growing when the numberof loading cycles increases. Many researchers havealready made these assumptions, often with the help ofthe linear-elastic fracture mechanics (LEFM). Theyhave used the stress-intensity factor or the energy-release rate concepts to predict the behaviour of a crackin the 90� plies [27]. Some propagation criteria are alsointroduced [3], also in fatigue, as in Wang et al. [1] whochoose a classical Paris law and de®ne a critical size ofdefect for propagation. They then assume a randomdistribution of defects and simulate the development ofcracking.

We have followed a similar approach, used the sametools, and have obtained an analytical expression for

J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359 1351

Page 4: Modelling of fatigue microcracking kinetics in crossply composites and experimental validation

g�Re;�S; r�. We have also considered the work ofBoniface and Ogin [28] who have shown experimentallythat it is only the range, �S, of loading and not r whichdrives the crack growth. We therefore suggest:

dRe

dN� g�Re;�S� � ÿCR3ÿ�

e �S� �5�

where C and � are parameters of the Paris law.The choice of such a particular form for g�Re;�S; r�

is not really important for the development of our ideas.Several propositions could be made, which is the reasonwhy we develop in an appendix the details of operationsto obtain such a law.

It will be noted that we do not approach the problemdirectly. The purpose is just to propose a global physicalexplanation for microcracking of laminates. In thedevelopment of the model there appear parameters suchas C and � which will be experimentally identi®ed in theunidirectional 90� ply.

3.2. Residual strength and residual lifetime of 90� pliesfor fatigue loading: de®nitions

A particular form of g�Re;�S� � dRe

dN is suggested in(5). We may now express Re�Rinit

e ;�S� after N cycles ofa ��S; r� loading. Rinit

e is the residual strength beforethis loading.

3.2.1. Residual strength after N cycles of a ��S; r�loading

Indeed, integration of 1=g�R;�S� between Rinite and

Re�Rinite ;�S;N�, the strength after N cycles, gives an

analytical expression:

N � �Sÿ�

C��ÿ 2� Rinite

ÿ ��ÿ2ÿRe Rinite ;�S;N

ÿ ��ÿ2� ��6�

and so:

Re�Reinit;�S;N� � �ÿC���ÿ 2�N�S� � �Rinite ��ÿ2�1=�ÿ2

�7�

3.2.2. Residual lifetime

We have called the residual lifetime of a specimen(with a strength equal to Rinit

e ) loaded with ��S; r�, thenecessary number of cycles Nr�Rinit

e ;�S; r�, to break it.From Eq. (6)

Nr�Rinite ;�S; r� � �Sÿ�

C��ÿ 2� �Rinit�ÿ2e ÿ S�ÿ2� �8�

where

S � �S=�1ÿ r�:

4. Random strength of cracked layers

We now return to the main purpose of this paper, theevolution of the microcracking of a ��; 90p�sym in fati-gue. We have discretized the 90� plies in a ®nite but highnumber of potential microcracks.

To allow a progressive micro-cracking we introduce arandom strength for the 90� layers strength. This part isdetailed in the previous paper [6]. We give here againthese principles and results for a better understanding ofthe rest of the text.

Peters [29] and Thionnet [30] introduced the prob-ability of rupture of a volume of material. This prob-ability is identi®ed experimentally from tension tests onthe crossply lay-up under consideration, and also withthe help of ®nite elements method [31].

We adopt a similar method here. Doing tests on dif-ferent size of sample (width, thickness, length) [32], wehave observed that for our material, the size e�ects seemto be controlled by the edges area of specimens. So, wehave made the assumption that the critical defects areon the edges of the specimen and we have introducedthe concept of sections. These sections symbolize pre-ferential sites of cracking. The probability law of therandom strengths of these sections is identi®ed on theunidirectional material �90q�. By taking into accountscale e�ects we deduce the sections strengths of 90� lay-ers in the ��; 90p�sym laminate. (Only one programme oftests for each type of prepreg.)

Hypothesis

1. The static strength of the �90q� material is linked tothe most critical defect present in the specimen. Assoon as failure starts in the vicinity of this defect, itpropagates brutally leading to specimen rupture.

2. To associate, statistically, the ruptures of the �90q�laminate and the cracking in ��; 90p�sym laminates,we have adopted the hypothesis that most criticaldefects are on the specimen edges and we thereforehave a surfacic defects distribution (edge surfaces).

3. It seems in our tests that during monotonic loading,cracks propagate across the laminate width instan-taneously. We have then adopted that hypothesis.

4. We suppose the same initial density of defects inthe 90� plies for the two because they are manu-factured by the families of specimen (�90q� and��; 90p�sym), because they are manufactured in thesame way.

4.1. Law of strength distribution for the virgin sectionsof ���; 90p�sym specimen

4.1.1. Stage 1Rupture tests on a �90q� specimen give the statistical

distribution law C�x� of these strengths Ro. C�x� ischosen as a Weibull law.

1352 J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359

Page 5: Modelling of fatigue microcracking kinetics in crossply composites and experimental validation

Tests are made on ¯at T300-174 carbon/epoxy and wehave identi®ed Weibull law by ®tting experimental point(Fig. 4).

C�x� � Prob�Ro4x� � 1ÿ exp ÿ x

� ��� ��9�

4.1.2. Stage 2

We de®ne a link of �90q� which is an elementaryvolume of the sample.

The elementary volume has the thickness of a ply, thewidth of the specimen, and the length 1=N cm, where Nis the number of potential crack/cm we want to use inour simulation.

If we note Nmail the number of elementary volume inthe �90q� specimen, and 1 its length, we have for the ¯atspecimen:

Nmail � Nlq �10�

4.1.3. Stage 3

In our simulation of the ��; 90p�sym specimen, we havecalled a section a set of s elementary volumes de®ned in�90q�. Hence, we have for ¯at specimen:

s � 2p

We can note that s is quite independent of the width ofthe specimen.

With the hypothesis that failure of the weakest linkleads to the rupture of the section, we can determine thelaw of strengths S�x� of a ��; 90p�sym section:

S�x� � 1ÿ �1ÿM�x��s �11�

where M�x� is the law of strengths of an elementaryvolume, that we can connect (failure of the weakest linkleads to rupture of the specimen) to C�x�, the strengthlaw of the specimen give in Eq. (9).

1ÿ C�x� � �1ÿM�x��Nmail

Hence:

S�x� � 1ÿ �1ÿ C�x��s=Nmail �12�

From (10), Eq. (12) leads to:

S�x� � 1ÿ �1ÿ C�x��2p=Nlq �13�

2p=Nlq is the edges surfaces ratio between the sample�90q� and a cell in the model of a ��; 90p�sym laminate.

4.1.4. Stage 4The random strength Rs, for each section is randomly

determined according to the normal law S�x�:

S�Rs� � random; 0 < random < 1

Rs � ��ÿ ln�1ÿ random�Nlq=2p�1=� �14�

5. The kinetics of transverse cracking for a [�,90p]slaminate: the algorithm

The algorithm for cracking is, for a part, the same asthe one which is detailed in Ref. [6]. There is no changein the preliminary part. In this part, the 90� plies in a��; 90p�sym laminate is discretized in sections. These sec-tions represent potential cracks. The main choice is todetermine the density N of such sections. We haveworked between 50 sections/cm and 1000 sections/cm.Then, the initial strength of each section is randomlydetermined according to (14). Finally, we de®ne a celllike a set of sections between two broken sections.Initially there is only one cell. Each cell can divide itselfinto two new cells when an internal section is broken(creation of crack). In the iterative part, there are twodi�erent steps: the ®rst cycle, and the followings For themaximum level Smax of the ��S; r� fatigue loading, thestress in 90� is analytically calculated in each section inrespect to Eq. (4). A test is made to determine the max-imum di�erence between strength and stress. If thatdi�erence is negative, the concerned section is brokenand the geometry (cells, etc.) is modi®ed. The newstresses in sections are calculated, and we iterate theprocess as long as new cracks can appear under Smax,that is to say, when the strength±stress di�erence in allthe sections become positive.

The fatigue model starts only after this ®rst cycle.From Eq. (8) we know which section have the short-

est lifetime Ns. We break it and fatigue the others withtheir own maximum level of stresses and during Ns.That is to say that we calculate the residual strength ofall the non broken sections after each new crack.

These levels of stresses are calculated by Eq. (1) wherethermal stresses are put to zero ��T � 0�. Indeed, wehave seen before that it is not the level of stresses but thespan of stresses which pilots the decrease of strength.Since thermal stresses are constant, they do not con-tribute to the fatigue of the sections.

Then the geometry (cells, etc) is modi®ed. The newstresses in sections are calculated (without thermal con-tribution), and we iterate the process until a ®xed num-ber of cycles.

This algorithm is extremely simple and rapid. So itpermits to well studied the random aspects of the simu-lations. Indeed, making 1000 di�erent simulations takes

J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359 1353

Page 6: Modelling of fatigue microcracking kinetics in crossply composites and experimental validation

only a few seconds. It has been implanted on PC. Thein¯uence of the degree of discretisation on the simula-tions is negligible in the range 50±1000 sections/cm. Theintroduced random aspect needs several simulations foreach loading, but we have observed that with only fewsimulations, average results are very similar. However itdepends on the set of Weibull parameters and especially� which characterizes the scattering of strengths. If � isgreat (>10) the scattering is weak and the simulationsbecome quasi deterministic. If � is small it will benecessary to make many simulations to ®t the randombehaviour.

A study of sensitivity of the model to the di�erententry parameters has been led. Thus, variations of 10%on the value of the interface shear sti�ness, or onexperimental Weibull parameters, or on the parameters� and especially C of the fatigue law, modify slightly theresults. Concerning C, a variation of 50% is acceptable.It is important because we shall see that the identi®ca-tion of C is made with such a precision.

A great hope of this work in view of the results ofsimulations, is that it seems possible to extract threeinteresting parameters which should be su�cient todescribe the kinetics of cracking. Indeed, if we representthe number of cracks versus the log(number of cycle)for several similar simulations we observe that it is pos-sible to ®t the results with two straight lines, an hor-izontal one, and an other with a slope P (Fig. 1).

Nmean is the mean number of cracks appeared duringthe ®rst cycle for several simulations. Nons is the meannecessary number of cycles to reach the onset of thefatigue cracking. P is the slope of the curve. We shall seein the following that such representations ®t well theexperimental tests.

6. Experimental identi®cations

We have carried out our own tests to have a completeentry data set, rupture statistics on �90q� specimens(Table 1), modulus, etc. The two Weibull parameters, �,�, which govern the random distribution of initialstrength and the two fatigue parameters � and C aredetermined by testing ¯at �908� specimens during staticand fatigue loading.

The material is a carbon/epoxy T3001174. prepregmanufactured by Brochier Company.

We give mechanical data in Table 1.

6.1. Experimental identi®cation of a and b (thestatistical distribution of strengths C�x�

Tests are made [32] on ¯at T300-174 carbon/epoxyspecimens (thickness=1.12mm, length=80mm, width=10mm), and we identi®ed Weibull law by ®tting experi-mental points (Fig. 2).

We have found:

� � 11� � 57:3� 106

Let us make a few remarks concerning the precisionof these parameters determination. Foret [41] hasshown that 1000 tests are necessary to have a reallygood determination and we have only made 25tests. But this lack of precision does not a�ect theideas of our model. Moreover, the study of sensitivityhas proved that 10% of variation on Weibull para-meters does not modify signi®cantly the results ofsimulations.

6.2. Experimental identi®cation of the fatigue lawparameters � and C

We propose to identify � and C with fatigue tests on�90q� specimens. On such a specimen, it is easy toobserve the rupture of the ®rst section because it is thesame as the specimen rupture.

The residual lifetime of a section under ��S; r� isgiven by (8), where R0 is the random initial resistance

Nr�Ro;�S; r� � �Sÿ�

C��ÿ 2� �R�ÿ2o ÿ S�ÿ2� �15�

where S � �S=�1ÿ r�Consequently, Nr�Ro;�S; r� is also random.

6.2.1. The statistical distribution law F�x� of the �90q�lifetime

F�x� is the probability that the lifetime of a specimenis lower than x.

F�x� � prob�Nr4x�

Using (15) to express Nr versus Ro, we obtain a newexpression for F�x�:Fig. 1. Representation of the result of several simulations.

1354 J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359

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F�x� � prob R04�C�S���ÿ 2�x� S�ÿ2� 1

�ÿ 2

� ��16�

This expression is the same as (9), wherex � �C�S���ÿ 2�x� S�ÿ2� 1

�ÿ2.We can rewrite F�x� as a Weibull with 3 parameters

�f, �f and xo, functions of �, �, � and C:

F�x� � 1ÿ exp ÿ C�S���ÿ 2�x� S�ÿ2

��ÿ2

� � ��ÿ2

!

� ÿ exp ÿ xÿ xo

�f

� ��f� � �17�

where

�f � �=�ÿ 2�f � ��ÿ2=C�S���ÿ 2�xo � ÿ�Sÿ2=C��ÿ 2��1ÿ r��ÿ2

8<: �18�

6.2.2. Experimental identi®cation of � and C

The knowledge of �f, �f and xo enables the determi-nation of the Paris parameters � and C.

Twenty-four tests have been made on �908� carbon/epoxy. There were fatigue tests with S � 35 MPa andr � 0:1. We can deduce the values of Weibull para-meters by ®tting experimental points. We have found:

�f � 2:38�f � 5� 104 cyclexo � ÿ1:14� 104 cycle

8<:

Fig. 3 shows this ®tting.Now we can give numerical values for � and C.It should be noted that we have three equations for

only two unknowns. From Eqs. (18a) and (18b) we ®nda ®rst set of Paris parameters:

� � 5:36C � 1:89� 10ÿ20�Pa2 � Cycle�ÿ1

From Eqs. (18b) and (18c):

� � 5:62C � 3:5� 10ÿ20�Pa2 � Cycle�ÿ1

Concerning �, the two values are quite similar. We haveshown before that a gap of 10% does not modify resultsof simulations. Concerning C, the di�erence is moreimportant, but the model is really not sensible to thisparameter. As long as the values are of the same order,the simulations are not a�ected.

It is interesting to underline that the ideas which arebehind the model seem pertinent. Indeed, we determineexperimental parameters from di�erent ways, and it isalways a hard challenge for a model. In the following wechoose arbitrary to consider the mean values.

7. Experimental validation for a [02,902]s carbon/epoxycomposite

Several authors have made test studies to observe thekinetics of cracking during a fatigue loading. For

Fig. 2. Rupture probability of 90� ¯at specimens: test results and two

parameters Weibull law (� � 11, � � 57:3� 106 Pa).

Table 1

Mechanical data for a T300/174 carbon/epoxy

Exx (GPa) Eyy (GPa) �xy Gxz (GPa) Gyz (GPa) �x � 10ÿ5 (/�C) �y � 10ÿ5 (/�C) e � layer thickness (mm)

Carbon/epoxy T300/174 130 10 0.3 5 3.86 ÿ0.12 3.4 0.14

Fig. 3. Residual lifetime probability of �908� specimens. Fatigue load-

ing (S � 35 MPa, r � 0:1): test results and three parameters Weibull

law.

(18a)(18b)(18c)

J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359 1355

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example, the experimental work of Wang et al. [1] is veryoften used by the researchers to ®t their models (Fig. 4).

Daniel and Charewicz [35] and others [36,37] havealso made experimental studies. The observations arethe same. But we were not able to simulate theseexperiments because we do not have a complete set ofdata, which is the reason why we have made our owntests. The main di�culty is the observation of thecracks. We have chosen an apparatus (Fig. 5) whichallows us to count and observe cracks on the polishededges of specimen during the tests. To con®rm theseobservations, we have an acoustic system composed of apiezo-electric microphone and a recorder. With such asystem we are able to count the number of characteristicevents which are linked to the appearance of transversecracks. We hope that, at least for high stress levels, thecracks propagate through the entire width of the speci-men. Other tests carried out on two polished �02; 908�sedges seem to con®rm this fact.

Tests are tension±tension tests. The waveform issinusoidal at a ®xed frequency (10Hz), the fatigueparameter r is equal to 0.1 and we change only the Smax

value.It can be seen from Figs. 6±8 di�erent test results and

the ®t of several simulations. It seems that 500MPa isthe threshold stress at which cracks occur under quasi-static loading. We have not made fatigue tests at lowerstress level. We have represented, delimited by twostraight lines the scattering of simulations. We note agood agreement. Moreover, we give for each test thevalues of the three parameters Nons, Nmean and P, whichare de®ned earlier, in the part of this text concerning thealgorithm of cracking. Nmean is the mean number ofcracks appeared during the ®rst cycle for several simu-lations. Nons is the mean necessary number of cycles toreach the onset of the fatigue cracking. P is the slope of

Fig. 6. The kinetics of cracking of a �02; 902�s. S � 670 MPa, r � 0:1,f � 10 Hz.

Fig. 4. Transverse crack density in a �02; 902�s carbon/epoxy versus

number of cycles [1].

Fig. 5. Test and crack-observation apparatus.

1356 J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359

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the curve. Nons and P are determined by least squaremethod. Such a description ®ts the tests well.

8. Conclusions

A multi-particular model similar to shear-lag-analysisis chosen to compute stresses in the 90� plies of a��; 90p�s laminate. The 90� plies are discretized in cells(sections) including potential cracks. The initialstrengths of these sections were randomly distributedaccording to a law which is experimentally identi®edfrom tests on unidirectional samples �90q�. The residualstrength Ri

e of a section i is our fatigue variable. Foreach step of loading ��S; r;N�, we propose a decreasinglaw for Ri

e versus Rieinit (the previous residual strength of

the section i before the step of loading):

Re�Reinit;�S;N�;� �ÿC��ÿ 2�N�S� � �Rinit

e ��ÿ2�1=�ÿ2

The model is incremental and permits in a simple butnon-linear way to describe the increasing of damage in90� plies of a ��; 90p�s laminate.

A random behaviour is introduced on initial strengthsof sections, permitting a simulation of a progressive (so,realistic) transverse cracking kinetic during loading.Furthermore, we notice that because of the mechanicalmodel construction (redistribution of stresses after eachnew cracking) each section sees a di�erent level ofloading �S for each step of loading, and so each strengthsection decreases di�erently. This leads to a progressivetransverse cracking kinetic during fatigue loading: anew crack is created when the residual strength of asection becomes lower or equal to its loading.

Finally, tension±tension fatigue tests (�S; r) are madeon �02; 902�s ¯at specimens of carbon/epoxy with acount of cracks versus the number of cycles. These ®rstobservations validate our simulations. Furthermore, thekinetics seem to be described by only three parameters:the number of cracks after the ®rst cycle, the necessarynumber of cycles to start the fatigue cracking and theslope of a straight line ®tting the fatigue curve, numberof cracks � f�log�N��. It seems to be a simple way tointroduce such a model in an engineer software.

Appendix. Proposal for an analytical expression ofstrength decrease

The objective is to ®nd an analytical law which pro-vides the decrease of residual strength of a 90� sectionduring a fatigue loading ��S; r�. We suppose that initialdefects in the resin between the ®bers of a 90� sectionbecome cracks during fatigue loading. Moreover, wecall Kc the toughness of the resin and decide that thepropagation of the crack will be described by a Parislaw.

Fatigue of a microcracked media: a Paris law

A general form of such a law may be:

da

dN� f�Kmax;�K� �A1�

where Kmax is the maximum stress intensity factor dur-ing the cycle, and �K the span of this intensity factor.

Boniface and Ogin [28] show experimentally that Eq.(A5) for transverse cracking of laminates can be reducedto:

da

dN� f��K� �A2�

The same dependence is chosen in own study. Physi-cally, it is possible to justify this result.

Fig. 7. The kinetics of cracking of a �02; 902�s laminate. S � 530 MPa,

r � 0:1, f � 10Hz.

Fig. 8. The kinetics of cracking of a �02; 902�s laminate.

S � 1000 MPa, r � 0:1, f � 10Hz.

J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359 1357

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Several experimental studies [38,39] have shown thatduring a monotonic loading, if a crack appears then itpropagates across the laminate width instantaneously.On the contrary, other works [40] have proved thatduring a cycle loading cracks propagate slowly. We canconsider that a cyclic loading is composed of a ®rstcycle, called static cycle, and of following cycles. OnlySmax (and so Kmax) pilots the number of cracks duringthis ®rst cycle. For next cycles, a new crack should be afatigue crack, and �S (or �K) becomes the main con-trol parameter of the cracking kinetics. In this frame-work, we choose the simplest and the most current lawfor the crack propagation:

da

dN� B��K�� �A3�

where B and � are characteristics of the studied mate-rial.

Strength of a section of 90�

We use concepts of linear-elastic fracture mechanics(LEFM).

This situation attempts to represent locally a trans-verse crack of size 2a in the resin between the ®bers in a90� ply loaded in mode 1 with a stress S. We have:

K � S�������ap �A4�

where K is the stress-intensity factor at the tip of thecrack.

The strength of a section of the 90� ply will be deter-mined by the length of the greatest crack in the resinnear this section. We note this critical length 2acrit. If theloading S makes that K reaches the toughness Kc of thematerial, then the crack grows and this leads (the load-ing S is constant) to the rupture of the section. S is thenequal to the strength Re of the section.

Re � Kc������������acritp �A5�

acrit is a random value, so Re will be random too. For aslong as S is lower than Re, the section doesn't break. Ifwe know the toughness Kc of the resin and the strengthRe of the section we can propose an estimation of acrit inthis section:

acrit � 1

Kc

Re

� �2

�A6�

Fatigue of a section

In fatigue, for a ��S; r� loading, we can extand (A4):

�K � �������ap

�S �A7�

During a cyclic loading, the crack propagates throughthe thickness (following a Paris law as we suppose)

da

dN� B��S

�������ap �� �A8�

If S reaches Re, we have for acrit and from (A5):

dRe

dN� ÿ 1

2

Kc����p

acrit3=2

dacrit

dN�A9�

using (A8):

dRe

dN� ÿ 1

2

KcB����p

a3=2�S

������������acritp� ��

�A10�

eliminating acrit with the help of (A6) gives for (A10):

dRe

dN� g�Re;�S� � ÿCR3ÿ�

e �S� �A11�

with:

C � B

2

K�2ÿ��c

With the model, the level of damage of a section of a 90�

ply is given at a micro-scale by the length of the greatestdefect acrit in the resin, or in a similar way but at a moremacro-scale [see (A11)] by the strength of the section(we suppose that Kc is known).

This is of course, a simpli®ed model (for example theform of stress ®elds in the resin doesn't take intoaccount the presence of ®bers) but it gives a simple

1358 J.F. Caron, A. Ehrlacher / Composites Science and Technology 59 (1999) 1349±1359

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expression capable to connect at a macro-scale and for asection of material, the decreasing of strength, the levelof loading, and the initial state (initial strength Re) ofthe section of a 90� ply.

It will be possible, from this ®rst approach to proposea more general form, including fewer simpli®cations, as:

dRe

dN� g�Re;�S� � ÿCR e �S� �A12�

Then, three parameters must be identi®ed.

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