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Page 1: A coupled analysis of mechanical behaviour and ageing for polymer-matrix composites

A coupled analysis of mechanical behaviour and ageing forpolymer-matrix composites

Anne Schieffera,*, Jean-Francois Mairea, David Levequeb

aDepartement Mecanique du Solide et de l’Endommagement, Office National d’Etudes et de Recherches Aerospatiales, 29,

avenue de la Division Leclerc, 92322 Chatillon Cedex, FrancebDepartement Materiaux et Systemes Composites, Office National d’Etudes et de Recherches Aerospatiales, 29, avenue de la Division Leclerc,

92322 Chatillon Cedex, France

Received 17 November 2000; accepted 24 July 2001

Abstract

The use of organic-matrix composites for long lifetime aeronautics applications leads to a study of the coupling effects between

thermo-mechanical stresses and thermal ageing. Taking into account the effects of oxidation in a behaviour law requires first, anunderstanding of every mechanism of degradation and the interactions between them. The objective of this paper is to presentcurrent studies of these subjects and, in particular, our own approach, in order to include oxidation in a macroscopic behaviour

law. For the oxidation section, studies have led to establishment of kinetics of thermal-oxidation for several polymer matrixes. Forthe mechanical section, a visco-elastic model coupled with damage is presented and allows to obtain the multiaxial creep behaviourof any composite laminate from few short-time tests conducted on elementary laminates. # 2002 Elsevier Science Ltd. All rights

reserved.

Keywords: A. Polymer-matrix composites; B. Durability; B. Non-linear behaviour; Ageing

1. Introduction

This study takes place in the general context of long-term use of composite materials for supersonic applica-tions (submitted to high temperature and severemechanical loading). The aim is to take into account theeffects of thermal ageing of polymer resins in a visco-elastic behaviour law coupled with damage at the mac-roscopic level. It is first necessary to consider themechanical, physical and chemical effects separately.The aim of this article is to present the different studieson these subjects at ONERA and the process that willbe henceforth followed to couple ageing with mechan-ical behaviour.The first part describes the damageable visco-elastic

model previously developed and the second part pre-sents a computation of thermal-oxidation of polymer-matrix composites (PMCs) and their potential effects onlaminate cracking without mechanical loading. Thestudied materials (carbon-fibre/polymer resin compo-

sites) belong to the materials which could be used forsome structural parts of the future European supersoniccivil aircraft plane: T800H/F655-2 (bismaleimid resin)and IM7/977-2 (epoxy resin).

2. Mechanical behaviour

The fundamentals of the visco-elastic model areestablished in the thermodynamic context which easesthe introduction of the coupling of viscous flow withdamage. The selected scale is the ply-scale and themodel describes an anisotropic non-linear viscoelasticbehaviour.Damage is then integrated, in order to:

. take into account the unilateral character ofdamage,

. integrate the anelastic strains in the damage evo-lution law and thus consider the possibility ofcrack propagation during creep process,

. distinguish propagation of cracks between normaland transverse modes (modes I and II),

0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0266-3538(01 )00146-4

Composites Science and Technology 62 (2002) 543–549

www.elsevier.com/locate/compscitech

* Corresponding author.

E-mail address: [email protected] (A. Schieffer).

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. take into account the residual strains occurringduring the damage process.

2.1. Visco-elastic elementary ply behaviour

Unlike some visco-elastic models based on an integralformulation [1], the model chosen here is based on aspectral formulation and uses a non-linear function [2].Besides, multiaxial loading can be simulated and theanisotropy aspect is introduced in the viscous and elas-tic behaviour and thermal stresses are taken intoaccount due to the non-isothermal formulation of themodel [3]. The equilibrium state of the material is givenby the knowledge of external variables (temperature Tand the total strain ") and internal variables such as theanelastic strain ("a) and a family of rank 2 tensors (�i)homogeneous with a strain and corresponding to theelementary mechanisms of viscous flow.The approach is based on the existence of two poten-

tials:

. The free energy potential, described as:

2� ¼ ð"�"a�"thÞ:C0 : ð"�"a�"thÞþXi

1

�ið�i :CR : �iÞ

where �o is the relative density, Co and Cr are rank 4tensors which describe respectively the elastic and theviscous anisotropy, "th describes the thermal strain.Each mechanism �i is associated with a relaxation time�i, weighted by �i; The whole strain family (�i) describesa gaussian continuous spectrum.

. Secondly, the dissipation potential, which definesthe internal parameters evolution laws:

2’� ¼ 2’�ð!iÞ ¼Xi

�i

�ið!i : CR

�1 : !iÞ

with !i ¼ gðÞ a þ i, where g() is a non-linear func-tion. The constitutive equations of the model derivedirectly from these two potentials [3]:

¼ �@

@"¼ C0 : ð"� "a � "thÞ

a ¼ �@

@"a¼ � "th ¼ �ðT� T0Þ

":a ¼ �

@’�

@a¼ gðÞ

Xi

�:i

�:i ¼ �

@’�

@i¼1

�ið�igðÞC

�1R : � �iÞ

2.2. Damageable behaviour of the elementary ply

The formulation proposed to describe the damagedvisco-elastic behaviour is developed in the continuumdamage mechanics context by using a macroscopic

approach. To describe the damage in the PMCs,damage effects on elastic properties and viscous beha-viour have been considered simultaneously [4]. Con-cerning the damage nature, a scalar variable was chosensince it is admitted that cracking has a preferentialdirection of propagation as a result of to the presence offibre in the material.

2.2.1. State laws and damage effects on the elasticbehaviourConcerning the elastic behaviour, the non-isothermal

formulation of thermodynamic potential, when eachcrack is opened, is:

¼1

2"� "thð Þ : C~ ðd; �;TÞ : "� "thð Þ

with C~ ðdÞ ¼ S~ ðdÞ�1 ¼ ðS0 þ �:d:H0Þ�1

Hence, a rank 4 tensor Ho is introduced and repre-sents the damage effects on the elastic behaviour. So isthe initial compliance tensor, d is the damage variable(scalar) and � the damage deactivation index [5].This description of the thermodynamic potential

derives from the choice which was made for Ho and forthe compliance tensor equivalence. This potential pro-vides the elasticity law and introduces the thermo-dynamic force associated with parameter d.In the case of complex loading, it is necessary to take

into account the unilateral property of damage. Indeed,an opened or closed crack can have quite differenteffects on the mechanical behaviour, but the introduc-tion of this phenomenon in behaviour laws remains dif-ficult, particularly for a multi-axial problem wheredefining a tensile and a compression domain is not tri-vial. Moreover, it is necessary to introduce a crackclosing condition depending on both loading anddefects orientation. Our formulation allows to take intoaccount the progressive closing of defects (role of �).

2.2.2. Visco-elasticity-damage couplingIt has already been noted that damage plays an

important role on the asymptotic behaviour. Thus, themodel must be able to predict a damage increase duringcreep test. This effect can be introduced in the same wayas for the elasticity in the relaxed compliance tensor.

C effr ¼ ðS eff

r Þ�1 and S eff

r ¼ Sr þ �:d:Hr

where Hr corresponds to the damage effect tensor on thevisco-elastic behaviour.

2.2.3. Damage evolution lawIn order to take into account the possible evolution of

damage during creep process, we choose to make thedamage depend on visco-elastic strains [6]. The damageevolution variable d is given by:

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fðy�Þ ¼ dc: 1� exp �y� � y0

yc

� �m� �and d ¼ sup

�<tð f ð y�ÞÞ

where dc, yo, yc and m are four constants to determine, y�depends both on elastic and viscous strains in mode Iand II. Every details about this damage visco-elasticmodel are presented in [5].It is worth noting that a generalised laminate theory

has been developed to obtain any laminate behaviourfrom the elementary ply.

2.3. Limitations of the visco-elastic model with damageand its identification

An identification method has been established allow-ing distinction between the different parameters of themodel. Nevertheless, some difficulties appeared duringthe identification on our two materials: indeed, distin-guishing the viscous behaviour from the damage effectsis very difficult since the non-damaged domain remainstoo limited. Some validations of the model are pre-sented in Figs. 1 and 2; for complex laminates and long-term creep tests.Even though simulations obtained are satisfactory for

most cases, some simplifications of the model should bereconsidered, such as:

. the non-uniformity of damage in the ply thickness,

. the influence of the adjacent plies behaviour,

. or that the damage evolution may be time-depen-dant.

Thanks to the model, we were able to determine theresidual strength of a quasi-isotropic laminate after a60,000 h creep period. It turns out that decrease of per-formances due to the viscous and damageable beha-viour of different plies remains very small (less than5%). Thus, we point out that the effects of ageing onviscous behaviour and damage but also on the failure

criterion (we have used an admissible fibre strain criter-ion) should be taken into account to evaluate correctlythe residual performances.

3. OMC oxidation and effects on mechanical behaviour

Nowadays, thermal degradation is still a complex andlittle-known phenomenon. Indeed, over long periods oftemperature exposure, ageing effects added to mechan-ical damage can considerably alter the material natureby chemical modifications or by micro-crackingappearance. Several models have already been estab-lished to understand ageing mechanisms. In this section,we will first describe the acquired knowledge aboutpolymer matrix materials oxidation on an experimentalas well as a theoretical point of view. Then we will try tounderstand how thermal ageing can influence themechanical behaviour.

3.1. Experimental point of view

Previous studies have been conducted at ONERA[7,8] to describe the cracking mechanism of compositessubmitted to a severe thermal and oxidising environ-ment over a long time. It was notably observed thatcracks clearly appears in unidirectional composites (butno cracking under vacuum) although no crack isobserved for the resin only. Besides, the oxidised layerdo not undergo uniform modifications at the molecularscale. The oxidised matrix becomes stiffer and morebrittle than the safe matrix and a stress field appears dueto the volume variation in the oxidised layer. Fibresplay an important role in the crack onset since the stressfield induced by the different thermal expansion betweenfibre and matrix is added to the field induced by thechange of volume. The combination of these effects lead

Fig. 1. Long-time creep test of a [02,�453]s T800H/F655-2 laminate.

Fig. 2. Creep test of an IM7/977-2 [�45]4s laminate: loading at (50–

80–100–140–170) MPa.

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to initiate cracking in composites. This will be con-firmed by the numerical simulations that will be pre-sented in the next section.

3.2. Theoretical point of view

3.2.1. Resin caseA model describing oxidation in F655-2 and 977-2

resins has been chosen among the different modelsdeveloped [8].This model predicts both the concentration profiles

and the oxygen consumption across the specimenthickness by coupling diffusion kinetics and oxygenchemical reactions in polymer chains. The constitutiveequation of the model is:

@C

@t¼ Dm:

@2C

@x2� RðCÞ

with the boundary condition:

CS ¼ S P02

where:

C: the oxygen concentration,t: the ageing time,x: the depth of penetration for reactive products,Dm: the coefficient of oxygen diffusion into the

material,R(C): the local oxygen consumption rate,S: the coefficient of oxygen solubility,PO2: the oxygen partial pressure of the oxidising

environment.

The oxygen diffusion is expressed owing to Fick’ssecond law. This kinetic model differs from the othersby the choice made for the analytical expression of thelocal oxygen consumption rate R(C), which depends onchemical reactions between oxygen and polymer com-ponents. The equation above has been solved in steady-state and validated on polymer resins.

3.2.2. Unidirectional caseThe oxidation behaviour of unidirectional composites

is more complex. Indeed, it has been observed that theoxygen diffusion in the longitudinal and transversalfibre directions is different (Fig. 3). In the transversaldirection, no crack appears and the diffusion coefficientDT can be deduced from the matrix one (Dm) by takinginto account the fibre volume fraction.However, in the longitudinal fibre direction, the oxi-

dation process is considerably modified and acceleratedin comparison with the resin case. The oxidised layergrowth with ageing time is not stabilised anymore. Thisprocess can either be due to high level stresses located at

fibre/matrix interfaces that can accelerate the oxygendiffusion by opening out inter molecular bond and por-osity, or to cracks onset in the oxidised matrix andinterfaces which allows oxygen to propagate instanta-neously towards the material core.According to experimental data on unidirectional

composites, the second solution is the most likely to bebut it’s yet difficult to know exactly the time of cracksonset and the place where they are located (free surfacesor core/oxidised layer interface). In order to answerthose questions, finite-element simulations have beenconducted.For the first time, a macroscopic stress threshold of

interface debonding has been determined from tensiletests (or creep tests) on 90� laminate specimens. Thecorresponding interfacial stress has been then calculatedwith a two-dimensional computation and depends onthe local fibre-volume fraction. By varying the loadingangle, we induced a variation in the distance betweenfibres without changing the mesh (see Fig. 4). Two dif-ferent configurations have been investigated (square andhexagonal) for a global fibre volume fraction (Vf) equalto 60%.

Fig. 3. Oxidised and cracked layer of a T800H/F655-2 unidirectional

composite after 4000 h of ageing at 180 �C.

Fig. 4. Square configuration for Vf=60%.

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These different simulations give a first estimation of amicroscopic stress threshold of interface debonding innormal mode (Fig. 5).Next, mechanical effects of volume variations in the

most oxidised layer have been computed for each con-figuration from data of strain variations obtained by theoxidation model (see Fig. 6). Thus the stress distributioncomputed was compared with the threshold determinedby tensile tests (see Fig. 7).This simulation is for the moment quite qualitative

since values of mechanical properties in the oxidisedlayer are still not precisely determined as well as theinterface debonding criterion. The different chemicalevolutions of the matrix are not taken into account butthe oxidised layer has been considered stiffer than thematrix core, according to the first results of micro-indentation tests.However, in spite of all uncertainties, such a model

allows the authors to assert that cracking appears veryquickly in the oxidised fibre/matrix interface and is onlyinduced by oxidation effects. Indeed, by consideringthat the most likely damaging mode is the interface

opening in the normal direction (the effects of shearstresses have been neglected), the stress field induced byoxidation is sufficient enough to initiate interfacedebonding where the distance between fibres is maximal(the macroscopic threshold for the IM7/977-2 systemaged at 200 �C is reached between 30 and 50 h).A three-dimensional simulation has been performed

to give an estimation of the stress distribution across theentire oxidised layer (Fig. 8). The medium layer betweenthe totally-oxidised layer and the non-oxidised area hasbeen neglected. Besides, we have considered that theoxidised layer was formed instantaneously and did notchange with time until the first cracks appeared. Indeed,compared to the ageing time scale, the oxidised layer forresin becomes stable very quickly for high temperatures.Moreover, we have considered that the matrix shrink-

Fig. 7. Interface normal stresses after 30 h of ageing at 200 �C.

Fig. 5. Interfacial normal stresses depending on loading angle.

Fig. 6. Strain variation in the oxidised layer obtained with the oxida-

tion model at 200 �C (977-2 epoxy resin).

Fig. 8. 3-D interface normal stresses after 30 h of ageing at 200 �C

(IM7/977-2 system).

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age was uniform in the totally-oxidised layer (butdecreased with time) and only took place in the matrix(no oxidation of the fibres). Actually, volume variationsare not uniform but decrease from the free edge of thecomposite into the safe area. Thus, it is worth notingthat cracks are less likely to appear at the interfacebetween matrix core and oxidised layer, the stress fieldbeing very low compared to the stress field obtained inthe oxidised layer and near the free edges (Fig. 9).Those simulations clearly show that a volume varia-

tion (resin shrinkage) is sufficient enough to initiateinterface debonding in normal mode after few hours ofageing. Nevertheless, the oxidation model describes anoxygen concentration gradient in the oxidised layerdecreasing from the free edges of composite laminatetowards the matrix core which induces a volume varia-tion gradient. Thus, the cracking of interfaces will occurrather near the free edges than in the oxidised layer andwill proceed in the parallel direction of fibres.In order to have a more accurate idea on the appear-

ance time of first debondings and crack propagation,better data are required about both interface strengthfor the two loading modes (normal and shear) andelastic properties of the oxidised layer. In this way, somemicro-mechanical tests will be performed to improvethis information (elastic properties determined bymicro-indentation tests [9]).

3.3. Study outlook

Taking into account damage mechanisms in the dif-fusion coefficient appears now necessary to complete theoxidation model. Indeed, as seen previously, the oxida-tion process is accelerated owing to the fibre presenceand can be simulated by an increase of the apparentdiffusion coefficient, which governs the oxygen penetra-tion in the material. Besides, previous studies on hygro-thermal behaviour of polymer composites pointed out

the interface role by introducing an interfacial diffusioncoefficient [10]. Indeed, it seems logical to take intoaccount a different oxidation kinetic in the interfacialarea since the chemical composition differs from thematrix one. These two points will constitute the purposeof the next study and will allow the authors to integratemore easily the thermal-oxidation process in mechanicalbehaviour laws. Besides, experimental studies are inprogress in order to emphasise the ageing effect on bothvisco-elastic and damageable behaviour.

4. Conclusion

First of all, the understanding of different damagemechanisms occurring during long-time use of polymercomposites without mechanical loading was investi-gated. This step was necessary to better understand howthermal ageing can considerably reduce composite life-time by decreasing the cracking onset threshold. Thus, afirst idea to introduce effects of ageing in the mechanicalbehaviour could be to take it into account through thecracking threshold. Nevertheless, the phenomenoninduced by oxidation remains very complex: as has beenpreviously shown, the oxidised layer is located near thefree edges of the composite but can rapidly proceedtowards the material core under the effects of bothmechanical loading and oxidation. Hence, a ply scalehomogeneous behaviour should be reconsidered and itwill probably be necessary to investigate a more accu-rate analysis at the components scale (fibre/interface/oxidised layer/matrix).

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