A progressive damage model for unidirectional fibre-reinforced composites based on fibre fragmentation. Part II: Stiffness reduction in environment sensitive fibres under fatigue

COMPOSITES

Composites Science and Technology 65 (2005) 2039–2048

SCIENCE ANDTECHNOLOGY

www.elsevier.com/locate/compscitech

A progressive damage model for unidirectionalfibre-reinforced composites based on fibre

fragmentation. Part I: Formulation

A. Turon, J. Costa *, P. Maimı, D. Trias, J.A. Mayugo

Advanced Materials and Analysis for Structural Design, Polytechnic School, University of Girona,

EPS-PII. UdG. Campus Montilivi s/n., 17071 Girona, Spain

Received 2 September 2004; received in revised form 8 April 2005; accepted 17 April 2005Available online 17 June 2005

Abstract

Constitutive damage models for fibre-reinforced composite materials should take into account the occurrence of the differentdamage mechanisms, their interaction and their influence on the resulting mechanical properties. Fibre breakage has usually beenconsidered in damage models by means of deterministic failure criteria which thus leads to non-progressive behaviour or to a com-plete material collapse which is not realistic. This work presents a progressive damage model for fibre-reinforced composites basedon the fragmentation analysis of the fibres. The stiffness loss of a unidirectional composite comes from the parameters of the Weibulldistribution of the fibre strength and the mechanical properties of the fibre, matrix and the interface. The model has been developedfor the initial stages of damage. The model is formulated in the framework of the mechanics of the continuous media. The consti-tutive model can be employed to simulate the contribution of fibres in damage models based on the rule of mixtures.� 2005 Elsevier Ltd. All rights reserved.

Keywords: A: Glass fibres; Polymer–matrix composites; B: Fragmentation; C: Damage mechanics

1. Introduction

Fibre-reinforced polymeric composites are widelyused in structural applications because of their goodspecific stiffness and strength. However, the use of thesematerials is limited by the lack of efficient tools to pre-dict their degradation and lifetime under service loadsand environment. The inhomogeneity and anisotropyof their microstructure leads to complex damage mech-anisms (basically: fibre breakage, matrix cracking andyielding, fibre–matrix debonding and delamination).The degradation phenomenon in composites is differentfrom that of metals both in the involved mechanisms

0266-3538/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2005.04.012

* Corresponding author. Tel.: +34 972 418 383; fax: +34 972 418098.

E-mail address: [email protected] (J. Costa).

and in the effects on the mechanical properties. Indeed,the damage previous to failure in composites involvesa loss of stiffness and strength which is more relevantthan in metals. Therefore, a design tool should take thisbehaviour into account, leading to what is known as aprogressive damage model.

The development of specific design tools for compos-ites are being pursued since the early stages of theirapplication in aircraft structures. Two approaches maybe distinguished: a phenomenological approach and amechanistic approach. Phenomenological approachesare based on the empirical laws of mechanical behaviourobtained from experimental tests. These models requirea heavy experimental background and are not general inthe sense that the behaviour of a particular material andply sequence cannot be inferred from the behaviour of asimpler configuration, that is, each laminate requires a

2040 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048

complete experimental characterization. The simplest,and more extensively used, design tools assume a com-plete elastic behaviour of the material until a failure cri-terion is satisfied. Once this happens, it may beconsidered as either a complete breakage of the struc-tural element or a stiffness reduction by an arbitrary fac-tor. However, there is no agreement concerning thefailure criteria to be used in the design of compositestructures [1].

On the other hand, mechanistic approaches aim tosimulate the occurrence of damage on the constituentscale and to reproduce the interaction of the differentdamage mechanisms. Moreover, the final objective ofthese models is to establish the degraded mechanicalproperties of the composite resulting from the damagedmicrostructure. This is a complex task and, at present,computationally unaffordable. However, the powerful-ness of such a model as a design tool motivates the re-search activity in this field. Many previous works ondamage mechanisms constitute the background for thedevelopment of such models.

In addition to the intricate nature of the onset andinteraction of the damage mechanisms, a further chal-lenge is to include it in a constitutive model suitablefor use in a finite element methodology. That is, themechanistic progressive damage model should be for-mulated in the framework of the continuous mediaand thus its thermodynamic consistency proved. Inaddition, the continuous damage models should takeinto account the inherent anisotropy of composite mate-rials, as well as variability from one element to the next.

Some damage constitutive models rely on the proper-ties of a single ply and use failure criteria combined withan arbitrary factor of stiffness reduction. Therefore,these models can be considered as phenomenologicalmodels [2–5]. Other damage models for composites relyon the knowledge of the constitutive laws of themechanical behaviour of the constituents, the fibre andthe matrix, which are all incorporated in the globalbehaviour of the composite using the rule of mixturesor double-scale methods [6]. In both cases, the fibresare considered as elastic materials with deterministiclimiting values of strain and stress. Therefore, the stiff-ness loss of the composite related to fibre breakage can-not be taken into account in these models.

A mechanistic progressive damage model based onthe fibre fragmentation leading to the stiffness-loss sim-ulation of a fibre-reinforced polymer under static, sus-tained (creep) and fatigue loads is presented in thispaper. Brittle fibres, such as glass or carbon, havemarked strength scatter due to the presence of inherentsurface flaws or flaws introduced during the manufac-turing processes and handling. When a load is appliedto a composite, the fibre fails randomly at the variousflaws along them. The weaker fibres fail first whilst thestronger fibres will carry the applied load [7]. With an

increasing load, the fibre will fracture into shorter andshorter fragments until the shear stress transfer acrossthe interface is insufficient to cause further fractureof the fibre. At a fibre break, the axial load carriedby the fibre is zero and it builds up, over a certain dis-tance, to a plateau or far field value. This stress redis-tribution involves a reduction of the apparent stiffnessof the composite. Fragmentation models, however,have been extensively used to foresee the strength ofunidirectional laminates (a thorough review on thistheme can be found in Phoenix and Beyerlein [8]).Fragmentation of fibres as a general physical phenom-enon has been addressed by Curtin [9,10] and Hui et al.[11] and it is now considered as a resolved mathemati-cal problem.

This paper proposes a degradation model which al-lows the evaluation of the apparent stress for a unidirec-tional ply, by taking into account a low density ofbreaks (initial stages of damage), a lack of local stressconcentrations (global load sharing, GLS) and a parallelbehaviour between fibre and matrix. The formulation isvery flexible in order to join other degradation process,like the glass fibre degradation under moisture, which isdeveloped in Part II of this paper.

An outline of this paper is as follows. Firstly, the deg-radation of the fibre is analyzed and the apparent fibrestress resulting from the fibre fragmentation is devel-oped. Then it is introduced into the constitutive equa-tions for the composite, in a thermodynamicallyconsistent way. Some aspects, like the fibre length effectand the influence of the Weibull modulus, are analyzed.Finally, the model is compared with available experi-mental results.

2. Fibre fragmentation

2.1. Single fibre strength

The present damage model is based on the fragmen-tation modelling of a single fibre. Therefore, the staticstrength of a thin fibre is described firstly. The fibres thatact as reinforcements of composites are mostly of cera-mic materials (glass and carbon). These materials exhibita low toughness and their strength is dominated by thecrack propagation from surface flaws. In the frameworkof linear elastic fracture mechanics, the fracture is pro-duced when the stress intensity factor, KI, exceeds the fi-bre fracture toughness, KIC

KI P KIC. ð1ÞDue to the slenderness of the fibres, it is assumed thatonly crack propagation in mode I is relevant for fibrefracture. The stress intensity factor results from the axialapplied stress, r, the crack length, a, and a geometricfactor, Y, according to

A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048 2041

KI ¼ rYffiffiffia

p. ð2Þ

Among the existing flaws in the fibre surface, the largestone, amax, will determine the ultimate stress, ru, which isgiven by

ru ¼ KIC

Yffiffiffiffiffiffiffiffiffiamax

p . ð3Þ

The high scatter in the measurement of the fibre strengthis related to the size distribution of flaws. The strengthdistribution of a fibre of length L is widely describedby a Weibull distribution [12]. Using this distribution,the probability of fracture of a fibre of length L, sub-jected to a given stress, can be written as

P ðr; LÞ ¼ 1� e� L

L0rr0

� �q

; ð4Þwhere L0 is a characteristic length for which a represen-tative strength, r0, is known (P(r0,L0) = 1 � 1/e); and qis the Weibull coefficient that accounts for the spread ofthe strength values. High Weibull modulus (typicallyaround 20) leads to very low variability such as in met-als, whereas lower values, 3–8, are encountered in cera-mic materials. Eq. (4) illustrates that the longer thefibres the larger the fracture probability. This behaviouris associated with the fact that for a given stress it ismore probable to find a flaw of the critical size in a lar-ger fibre.

2.2. Density of breaks

When a single fibre is embedded in a ductile matrix,and a uniaxial stress is applied, the fibre breaks at itsweakest point (larger flaw) and there is a transfer of loadto the surrounding matrix. If the stress is further in-creased, the fibre will break at further points giving riseto a set of fragments whose length will be dependent onthe Weibull parameters and the strength of the fibre–matrix interface. In fact, fragmentation experimentshave been extensively performed to assess the propertiesof the interface (see the review presented in [13] for morereferences).

This fragmentation process is rather similar to thefibre damage in a unidirectional composite laminateeven though in this case, other damage mechanismsmay be present and local effects of stress distribution

F τ τ

τ τ

Fig. 1. At a fibre break the axial load carried by the fibre is zero. The load is cand the matrix.

may be relevant at high break density. Indeed, whena fibre breaks, the axial stress carried by the fibre iszero, and the load is transferred by the shear stress thatappears at the interface between the fibre and the ma-trix, as shown in Fig. 1. This increase of the shearstress close to the fibre break may lead to the fibre–ma-trix interface failure (debonding). The axial load car-ried by the fibre increases from zero, at a fibre break,until a constant stress (called far-field stress) at a cer-tain distance from the fibre break. The length of thiszone, where the stress is smaller than the far-field va-lue, is called the load recovery region.

The apparent stiffness of the system, matrix and fibre,decreases with the number of fibre breaks due to theirloss of ability to carry the load. The knowledge of thenumber of breaks at a given stress could provide theapparent axial stiffness of the composite supposing thatthe influence of other damage modes is neglected. Thissupposition is not completely accurate as there are the-oretical evidences that the debonding extension closeto the fibre break has an influence on the stiffness lossof the composite [14].

From Eq. (4), it could be shown that the number ofbreaks in the fibre follows a Poisson law [15]. With thisassumption, the mean number of breaks in a fibre oflength L under load r is

Nh i ¼ LL0

rr0

� �q

. ð5Þ

This result is widely used in the literature to compute theWeibull parameters (r0,q) from the experimental resultsof single fibre fragmentation tests. However, this resultis only valid at initial fragmentation stages, because asthe number of breaks increases, some flaws will be ob-scured in the load recovery region (the stresses in thiszone are smaller than the far-field stress and no morebreaks could appear in this zone).

Once the mean number of breaks is known, the dis-tance between two breaks, i.e., the length of the differentfragments into which the fibre has split, has to be com-puted. From the statistic laws, it can be shown that if thenumber of breaks follows a Poisson law, then the dis-tance between the two consecutive breaks will followan exponential law [15]

f ðxÞ ¼ Ke�Kx ð6Þ

F τ τ

τ τ

arried by the shear stress that appears at the interface between the fibre

Fig. 2. Kelly–Tyson�s shear lag model. Stress profile at a fragment ofbroken fibre. At a break the axial stress is zero and it increases until itreaches the far-field stress (EF Æ e). This region is called load recoveryregion, and has a length of lex.

2042 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048

where K is the number of breaks per unit length, and it iscomputed from Eq. (5)

K ¼ hNiL

¼ 1

L0

rr0

� �q

. ð7Þ

This assumption is valid for an infinitely long fibre andat the initial fragmentation stages. As mentioned above,at advanced fragmentation stages, some flaws will be ob-scured in the load recovery region, and then, the breakdensity will be lower than the predicted by Eq. (7).Many authors [9–11] introduce this phenomenon intotheir formulation obtaining other distribution functionswhich take into account higher break densities. Otherauthors [15,16] modify Eq. (7) and obtain an expressionof the break density without the possible obscured po-tential breaks. In fact, the exact mathematical solutionfor the problem was provided by Hui et al. [11]. Thesedistributions are more complex than Eq. (7) and, insome cases, numeric techniques are required for evaluat-ing the expression. As the purpose of the present work isto develop a stiffness degradation model for the initialstages of damage, the influence of not considering theflaws obscured in the load recovery region is rather neg-ligible. In Section 4, the numerical simulations from thepresent model based on Eq. (7) are compared to themore refined models cited above, and a very small differ-ence for the first stages of fibre breakage is obtained.Moreover, at advanced fragmentation stages, the local-ized stress transfer in the composite has an importantinfluence on the evolution of fibre breaks and the degra-dation and final failure is controlled by the formationand growth of clusters which require 2D models to beaccounted for.

In order to reach a mathematical expression of theapparent stiffness of the composite, it is necessary tocompute the average fibre stress when some fractureshave occurred.

2.3. Average fibre stress

It has been shown in previous subsections that someflaws in a fibre will grow to a fully formed crack underthe applied load. These cracks, or fibre breaks, cause anew stress redistribution along the fibre which maycause further breaks.

When a fibre breaks, the load carried by the fibredrops down to zero at the position of the break andthe load is carried by the shear stress between the fibreand the matrix (see Fig. 2). The stress in a broken fibre,rF, as a function of the distance from the break can bewritten as

drF

dz¼ 2s

R; ð8Þ

where R is the radius of the fibre, s is the maximumshear stress and z is the distance from a break.

This causes a stress redistribution near fibre breaks(see Fig. 2) which has been widely studied. Cox [17]was the pioneer to predict the real stress near the breaksby using a shear-lag model. The formulation of Cox�smodel is quite complex and other simplified shear-lagapproaches have been derived. One of the most widelyused is the shear-lag model which was first introducedby Kelly and Tyson [18], and which assumes a linear in-crease of the axial stress from a fibre break, until a cer-tain distance from it. At this distance, called the loadrecovery region, the stress reaches the far-field stress,see Fig. 2. According to the Kelly–Tyson shear-lag mod-el, the length of this load recovery region, lex, is obtainedfrom the far-field stress, EFe (where EF is the fibreYoung�s modulus and e the composite strain), the radiusof the fibre, R, and the maximum shear stress, s, be-tween the fibre and the matrix before fibre debondingor matrix yielding occurs

lex ¼RsEFe2

. ð9Þ

From this stress redistribution, the average fibre stressalong the fibre, rm, can be computed by integratingthe axial stress over all of the fibre fragments alongthe fibre length

rm ¼ N RðLÞh i ¼ N1

L

ZxRðxÞf ðxÞdx; ð10Þ

where f(x) is the fragment length distribution given inEq. (6), and R(x) is the average stress in a fibre of lengthx. This integral is worked out in two steps. First, theaverage stress corresponding to the axial stress profilealong a fibre fragment of length x is computed. Then,this axial average stress is integrated over all fibre frag-ments. In order to compute the axial average stresses fora fibre fragment of length x, it is necessary to distinguishwhether the fibre fragment is greater to two times thelength of the load recovery region (2lex), or not.

(a) Average stress in a fibre of length 2lex 6 x. In a fi-bre of length x, greater than two times the stress recov-ery region, the stress profile assuming a linear shear-lag

A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048 2043

model, is shown in Fig. 2. Then, the average stress is gi-ven by the following equation:

rm ¼ EFe 1� lexx

� �. ð11Þ

(b) Average stress in a fibre of length x < 2lex. In afibre shorter than two times the stress recovery regionthe far-field stress cannot be reached. Then

rm ¼ EFex

4lex. ð12Þ

(c) Average fibre stress along the whole fragmented fi-

bre with initial length L. In order to compute the averagefibre stress along the whole fragmented fibre, it is neces-sary to define R(x). Three possibilities arise:

(c.1) If the distance between breaks is shorter than twotimes the stress recovery region, then R(x) is givenby Eq. (12).

(c.2) If the distance between breaks is greater than twotimes the stress recovery region but shorter thanthe initial fibre length L, then R(x) is given byEq. (11).

(c.3) If the distance between breaks is greater than theinitial fibre length, L, then R(x) is equal to thefar-field stress.

With these assumptions, the piecewise function R(x) isformulated as follows:

RðxÞ ¼EFe x

4�lex ; x 6 2lex;

EFe 1� lexx

� �; 2lex 6 x 6 L;

EFe; x P L.

8><>: ð13Þ

Using Eqs. (7) and (13), Eq. (10) reads

rm ¼ EFeKZ 2�lex

0

x2

4lexf ðxÞdxþ

Z L

2�lexðx� lexÞf ðxÞdx

�

þZ 1

Lxf ðxÞdx

�; ð14Þ

where f(x) is given by Eq. (6). The analytical solution ofthe previous equation is given by

rm ¼ EFe1� e�2lexK

2lexKþ Klexe�LK

� �. ð15Þ

Using Eqs. (7) and (9), the product 2lexK reads

2lexK ¼ EFerR

� �qþ1

; ð16Þ

where rR is a reference stress widely used in the litera-ture [9]

rR ¼ 2L0srq0

d

� � 1qþ1

. ð17Þ

Associated with the reference stress rR, there exists acharacteristic in situ gauge length dR [9]. These twoquantities are inter-related, with rR being the character-istic fibre strength at gauge length dR. dR equals to twicethe stress recovery region at stress rR

dR ¼ RsrR. ð18Þ

Using Eqs. (16)–(18), Eq. (15) can be written as

rm ¼ EFe1� e

� EFerR

� �qþ1

EFerR

� �qþ1þ 1

2

EFerR

� �qþ1

e� L

dREFerR

� �q

0BB@

1CCA.

ð19Þ

3. Continuum damage model

The next step is to show how the fibre damage modeldeveloped in the previous sections is introduced in theformulation of the constitutive laws. For this purpose,two questions should be addressed: how a fibre break af-fects the surrounding fibres and which law is used toassemble the mechanical behaviour of the fibre and thematrix phase.

When a fibre breaks, the load is transferred to theother fibres of the section. There exist two basic ap-proaches for the study of this phenomenon. The firstone is to consider that the load is transferred homoge-neously to the other fibres of the section. This assump-tion is called global load sharing (GLS) and is usefulat the first fragmentation stages, when the formationand growth of clusters is not important. Another ap-proach is to consider the stress concentrations of the fi-bres near the broken one. This is called local loadsharing (LLS) and the earliest model following this ap-proach was a fully elastic model developed by Hedgep-eth and van Dyke [19].

Since the formulation of the damage model for the fi-bre has been made for initial fragmentation stages, aGLS situation is considered for the formulation of thecomposite damage model. This assumption discardsthe ability of the model to simulate the compositestrength, because a GLS hypothesis cannot predict theformation of clusters and their growth, which will causethe failure of the composite.

On the other hand, the assembly of the mechanicalbehaviour of the different constituents of the compositeis done by considering a linear addition of the behaviourof each component relative to its volume fraction, thatis, by using the mixity law or rule of mixtures assuminga serial–parallel behaviour [20].

The elastic strain fields in matrix and fibres are func-tions of material morphology. If constitutive behaviouris in parallel, strains are equal in matrix and fibre,

2044 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048

whereas, if the constitutive behaviour is in serial, strainis distributed according to constituents� relative stiffness.To determine the strain in every constituent, the influ-ence tensors Tijkl have to be defined in the followingway:

eNij ¼ TNijklekl; eij ¼

XvNeNij ; I ijkl ¼

XvNTN

ijkl;

N ¼ F;M; ð20Þ

where eNij is the strain in the fibre (N = F) and in the ma-trix (N = M), eij is the strain of the composite, and Iijkl isthe identity matrix. It is assumed that the tensors Tijkl donot depend on the damage evolution.

The free energy of the model is obtained by addingthe free energy given for the fibre damage model andthe free energy given for any matrix damage model

w e; dF; dM� �

¼ 1� dF� �

wF0 eF� �

vF þ 1� dM� �

wM0 eM� �

vM;

ð21Þwhere dF and dM are the fibre and matrix damage vari-able, respectively, vF and vM are the fibre and matrix vol-umetric fraction, and

wN0 eð Þ ¼ 1

2eNij C

Nijkle

Nkl ¼

1

2emnTN

mnijCNijklT

Nklopeop; N ¼ F;M;

ð22Þis a convex function which represents the free energy perunit volume for the undamaged material.

The rate of dissipation, N, can be written as

N ¼ rij _eij � _w ¼ rij �owoeij

� �_eij �

X ow

odN_dNP 0;

N ¼ F;M. ð23Þ

Then the rate of external energy ðrij _eijÞ has to be equalto the rate of free energy ð _wÞ if process is elastic andgreater if damage evolve.

The constitutive equation has the simple form

rmn ¼owoemn

¼X

1� dN� � 1

2TN

mnijCNijklT

Nklopv

N

� eop; N ¼ F;M.

ð24Þ

Using Eqs. (24) and (21) to obtain the derivate of thefree energy respect to the damage variables, Eq. (23)can be written as

N ¼ vFwF0_dF þ vMwM

0_dMP 0. ð25Þ

From the previous equation, it can be shown that forproving the thermodynamic consistence of the damagemodel the derivate of the damage variables must be po-sitive, i.e., _d

FP 0 and _d

MP 0. The damage model

developed here takes into account the fibre degradationby means of the fragmentation model, whereas matrixdegradation is not considered (it could easily be intro-

duced). Thus, the derivate of the damage variable thatrepresents the fibre degradation must be positive.

3.1. Fibre damage model

In order to set up the fibre damage model, the defor-mation, e, is considered the free variable of the problem,the break density, r, the internal variable and, finally,the fibre damage variable, dF(r) the dependent variable.

For the formulation of the constitutive damage modelit is required to define a suitable norm (a.1), a damagecriterion (a.2) and the evolution law for the damage var-iable (a.3) [21].

(a.1) A suitable norm: k is defined in the strain spacefrom Eq. (7)

k ¼ K ¼ 1

L0

EFer0

� �q

. ð26Þ

(a.2) Damage criterion:

�F kt; rtð Þ :¼ G ktð Þ � G rtð Þ 6 0 8t P 0; ð27Þwhere t indicates the actual time and rt is the damagethreshold for the current time. If r0 denotes the initialdamage threshold, then it must be accomplished thatrt P r0 at every time. G(Æ) is a suitable monotonic scalarfunction and ranges from 0 to 1

GðrÞ ¼ 1� 1� e�2lexr

2lexrþ rlex e�Lr

� �. ð28Þ

Damage initiation is produced when the norm, k, ex-ceeds the initial damage threshold, which is a materialproperty.

As the fibre break density, r, is the internal damagevariable, the initial damage threshold r0 correspondsto the intact material, when no fibre break has been in-duced, i.e., r0 = 0.

(a.3) Evolution laws: The evolution laws are definedby the rate expressions

_r ¼ _l;

_dF ¼ _l

o�F k; rð Þok

¼ _loG kð Þok

;ð29Þ

where _l is a damage consistency parameter used to de-fine loading/unloading conditions according to theKuhn-Tuchker relations

_l P 0; �F kt; rtð Þ 6 0; _l�F kt; rtð Þ ¼ 0. ð30Þ

From the above expression, it is easy to prove [22] thatthe evolution of the internal variables may be explicitlyintegrated to render

rt ¼ maxfr0;maxfrsgg 0 6 s 6 t;

dt ¼ GðrtÞ.ð31Þ

This fully describes the evolution of the internal vari-ables for any loading/unloading/reloading situation.

A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048 2045

As mentioned, the thermodynamic consistence of themodel is proved by checking that the derivate of thedamage variable is always positive _d

FðrÞ P 0 [23].The fibre damage variable is a monotonic function

which always increase with an increasing number ofbreaks, then the inequality will be satisfied wheneverthe derivation of the break density is greater or equalto zero, i.e., _r P 0. This means that the number ofbreaks cannot decrease, which it is always accomplishedbecause a decrease in the number of breaks would meanthat two fragments had joined, which is physicallyimpossible, so the thermodynamic consistence is proven.

4. Experimental validation and discussion

4.1. Local strain distribution in a single fibre

In a recent paper, the axial fibre strain in a modelcomposite was measured with a very high spatial resolu-tion by means of Raman spectroscopy [24]. In that pa-per, the authors studied the fragmentation process ofglass fibres (optical glass fibres from the Fibre OpticsCentre, UK) in an epoxy resin matrix (AralditeLY5052). They showed the variation of strain along a14 mm long, copolymer-coated HNO3-treated glass fi-bre, of diameter 120 lm, at a matrix strain of 0.27%,0.45%, 0.69% and 1.10%. The tensile strength of these fi-bres for a length of 10 mm was 300 MPa and the inter-facial shear strength, s, 15 MPa. Young�s modulus of thefibre was assumed to be 73 GPa. For each one of thesampled matrix strains, they showed the axial fibrestrain distribution along the fibre.

In order to correlate their results to the present model,the average fibre strain for each deformation has beencomputed by integration of the axial strain along the en-tire fibre length from the data that appeared in the fig-ures of the article.

Fig. 3 shows the experimental values extracted from[24] and the results given with the present numeric modelfor a different Weibull modulus. Although the Weibull

Fig. 3. Evolution of the apparent stiffness of the fibre (rm/EFe) with

the composite strain.

modulus was not specified in their paper, the typical val-ues for glass fibres range between 3 and 5. A good qual-itative agreement between the numeric and experimentalvalues is observed.

4.2. Stress–strain behaviour

A progressive damage model should be able to repro-duce the on-axis non-linear behaviour of a unidirec-tional composite where the mechanical properties arefibre-dominated. Prior to the comparison of the modelwith experimental data, it is interesting to compare itto other fragmentation models with a more refined ap-proach aimed to account for the fact that the load recov-ery region obscure the flaws therein.

4.2.1. Comparison with other authors’ models

Curtin [10] presented the following equation for sim-ulating the stress–strain behaviour:

rm ¼ EFe� qEFelex

þ qEFeZ 2lex

0

P x;EFe� �

lex � xþ x2

4lex

� �dx; ð32Þ

where P(x,EFe) is the fragment length distribution. Cur-tin developed an approach to estimate the fragmentlength distribution for the single fibre composite prob-lem that is quite accurate but not exact. Indeed, it wasassumed to be accurate by many authors until the pub-lication of the work of Hui et al. [11] in which the exactsolution for P(x,EFe) was presented. Using the fragmentlength distributions, P(x,EFe), obtained by Curtin [9]and Hui et al. [11], Eq. (31) cannot be solved analyti-cally, and only can be evaluated numerically.

In order to obtain a more simple expression of thestress–strain behaviour, Curtin [10] obtained a simpleanalytic form for Eq. (31) neglecting the third term onthe right-hand side of the equation, i.e., assuming that,with few breaks, it is unlikely to find small fragmentsof a length shorter than two times the exclusion length.The obtained expression is close to the non-simplifiedsolution for Weibull modulus larger than 3

rm ¼ EFe 1� 1

2

EFerR

� �qþ1 !

. ð33Þ

In the same way, Hui et al. [11] proposed approximateanalytic relationships for the stress–strain curve for var-ious regimes of the Weibull�s modulus, providing anapproximation for the apparent stress at the fibre

rm ¼ EFe 1þ 1

2

EFerR

� �qþ1

þHEFerR

� �2qþ2!e� EFe

rR

� �qþ1

1�18k

EFerR

� �qþ1� �

; ð34Þ

2046 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048

where

k ¼ qqþ 1

;

H ¼ 7qþ 12

24ð2qþ 3Þ .ð35Þ

However, the results obtained with Eq. (33) were asaccurate as the results presented by Neumeister [25],who developed an analytic model for the stress–strainbehaviour that accounted for the exclusion zonesformed by slip and for the stresses carried by the shortfragments (x < 2lex)

rm ¼ EFe

1þ EFerR

� �qþ11þ1

2

EFerR

� �2qþ2

ln 1þ EFerR

� �q� �1þ EFe

rR

� �qþ1� �

2664

3775.

ð36ÞIn order to compare these models with the approach gi-ven by Eq. (19), the stress–strain dependence in a com-posite of an epoxy-reinforced with AS4 carbon fibreshas been numerically simulated. The diameter of the fi-bres was 7.1 lm and Young�s modulus of 234 GPa. Thefibre strength for a fibre of 12.7 mm in length is 4275MPa, with a Weibull shape modulus of 10.7 [26]. Forthe interfacial shear strength an epoxy matrix Epon828matrix cured with mPDA was taken intro consideration[27]. The interfacial maximum strength is taken as themaximum matrix shear strength using the Von Misescriterion 70=

ffiffiffi3

p¼ 40 MPa.

Fig. 4 shows a good agreement between the results gi-ven with the present and other models for low strain val-ues. Taking the evolution of the fragmentation model ofHui et al. as the exact solution (it is extremely accuratefor q = 10.7), the relative error in the stress is 5% for a2.5% strain. It should be taken into account that the cor-responding stiffness loss at this strain is of 12% whichwould have lead to complete failure of the unidirectionalply. For higher strains, significant differences appear be-tween the models. Therefore, it may be concluded thatthe simplified expression for the break density given in

Fig. 4. Comparison of the fibre stress–strain behaviour for AS4-graphite epoxy composites.

Eq. (5) is accurate enough to simulate the first stagesof fibre breakage damage.

4.3. Comparison with experimental stress–strain

behaviour

In order to compare the proposed model with exper-imental results, tensile tests were carried on a unidirec-tional composite material. The laminate material usedwas glass fibre reinforced epoxy matrix (E-glass andbisphenol A(/F), amine cured, fibre weight fraction51%). Specimens with dimensions 9 · 29.5 · 300 mm3

where cut from the laminate along of the principal direc-tions. The tests were performed under a standard cross-head displacement rate of 0.015 mm/s, with a Universalhydraulic test machine MTS 810 Material Test System(250 kN maximum load). Young�s modulus of the fibrewas taken as 70 and 3 GPa for the matrix. The diameterof the fibres was 10 lm, and a Weibull reference param-eters, for a 10mm long fibre, of r0 = 2300 MPa andq = 3.6 were taken [28]. A reference shear strength, s,of 23MPa was assumed.

The damage model has been implemented in a FEMenvironment (ABAQUS [29]) by means of a user-writtenconstitutive equation. The influence tensors, Tijkl, havebeen defined from the classical mixity law for serial–par-allel morphology and with some correction in the pois-son ratio. Under plane stress and in compact form,they can be expressed as

T F ¼

1 0 0�vM cF

12�cM

12ð ÞvMcF

22þvFcM

22

cM22

vMcF22þvFcM

22

0

0 0cM66

vMcF66þvFcM

66

26664

37775; ð37Þ

TM ¼

1 0 0vF cF

12�cM

12ð ÞvMcF

22þvFcM

22

cF22

vMcF22þvFcM

22

0

0 0cF66

vMcF66þvFcM

66

26664

37775; ð38Þ

where cNij are the terms of the constitutive matrix for thefibre and for the matrix.

Fig. 5 shows the deviation of the experimental behav-iour from a linear relation and the ability of the pro-posed model to capture the experimental results.

4.4. Influence of weibull modulus

As a result of the formulation of the progressive dam-age model, the analysis of the dependence of the com-posite behaviour on the Weibull parameters isstraightforward. For instance, the effect of the Weibullmodulus, q, on the stress–strain curve is shown in Fig.6. The higher the Weibull modulus the narrower thestress–strain curve, and, as a consequence, the area un-der the stress–strain curve, the fracture energy, is smal-

Fig. 7. Comparison of the fibre stress–strain behaviour for AS4-graphite epoxy composites.

Fig. 5. Comparison of the fibre stress–strain behaviour for AS4-graphite epoxy composites.

Fig. 6. Influence of the Weibull modulus in the stress–strainbehaviour.

A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048 2047

ler. However, the maximum admissible stress increaseswith q. This behaviour is a consequence of the widenessof the strength distribution for small Weibull moduluswhich causes the early breakage of the larger flaws anda high resistance at large strain due to the smaller flaws.For large q (narrow flaw size distribution) all flawswould break all at once at approximately one stressvalue.

Nevertheless, it should be remembered that theassumption of global load sharing for the present modellimits the validity of these curves to a low break density,i.e., small deviations of the linear behaviour. Failureprediction is out of the scope of the present modeland, certainly, it would occur shortly after the deviationfrom linear behaviour as a consequence of the formationof clusters of breaks (see Phoenix and Beyerlein [8]).

4.5. Fibre length effect

Most of the models published in the literature are de-rived for infinite fibre length. There exist a few numberof models which take into account the effect of the finitefibre length. For instance, Andersons et al. [30] dealtwith this problem and proposed a fragment length dis-tribution function that takes into account the size effectat initial fragmentation stages. On the other hand, thereis a large amount of evidence of size effects in the ulti-mate properties of composites [31]. A consequent depen-

dence of the damage evolution on size (i.e., fibre length)is expected.

In this work, the effect of the fibre length is taken intoaccount in the computation of the apparent stress alongthe fibre, from a probabilistic point of view, as it hasbeen summarized in Eq. (14).

The explicit inclusion of the fibre length in the formu-lation of the present model (Eq. (19)) is one of the differ-ences between most of the previously published models(see for example Eqs. (34), (35) and (37)) and the presentmodel.

The longer the fibre length the larger the stiffness loss,as it is shown in Fig. 7. However, this effect is very smalland it is only evident in a strain range. This behaviourmay be explained as the balance between two oppositeconsequences of an increase of the fibre length. On theone hand, a longer fibre has a higher probability of fail-ure which tends to increase the number of breaks and, asa consequence, causes a stiffness loss. On the other hand,the stiffness loss due to the presence of a break is smallerin a larger fibre than in a shorter one.

5. Conclusions

A progressive damage model for unidirectional com-posite laminates based on the fibre fragmentation hasbeen developed from a physical point of view and thenformulated in a thermodynamically consistent mode.The model is valid at the initial stages of fibre fragmenta-tion and it assumes a parallel behaviour of fibre and ma-trix and the absence of local load sharing in the vicinity ofa broken fibre. It has been shown that the model is able toaccount for the stiffness loss of a unidirectional compos-ite. In addition, the sensitivity of the model to theWeibullparameters and fibre length has been presented.

As it has been formulated, the damage model can betaken as representative of the fibres in constitutive mod-els based on a rule of mixtures. This fact represents animprovement with respect to traditional models thatconsider the fibres as a pure elastic constituent until anominal strength is reached. Therefore, the non-linear

2048 A. Turon et al. / Composites Science and Technology 65 (2005) 2039–2048

behaviour of fibre-dominated composites can be repro-duced with the proposed model.

The present formulation may be refined in severalways. For instance, stiffness loss due to the debondingnear the fibre break could improve the accuracy of themodel. However, the formulation presented is flexibleenough to permit the introduction of other degradationprocesses such as the static fatigue of glass fibres inmoist environments. This question is addressed in PartII of this paper.

Acknowledgements

The authors acknowledge the useful comments ofProfessor Pedro Camanho from the University of Porto(Portugal). This work has been supported by theSpanish government through DGICYT under thecontract: MAT 2003-09768-C03-01. The first authorthanks the University of Girona for the Grant BR01/09.

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