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

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<ul><li><p>A coupled analysis of mechanical behaviour and ageing forpolymer-matrix composites</p><p>Anne Schieera,*, Jean-Francois Mairea, David Levequeb</p><p>aDepartement Mecanique du Solide et de lEndommagement, Oce National dEtudes et de Recherches Aerospatiales, 29,</p><p>avenue de la Division Leclerc, 92322 Chatillon Cedex, FrancebDepartement Materiaux et Syste`mes Composites, Oce National dEtudes et de Recherches Aerospatiales, 29, avenue de la Division Leclerc,</p><p>92322 Chatillon Cedex, France</p><p>Received 17 November 2000; accepted 24 July 2001</p><p>Abstract</p><p>The use of organic-matrix composites for long lifetime aeronautics applications leads to a study of the coupling eects between</p><p>thermo-mechanical stresses and thermal ageing. Taking into account the eects of oxidation in a behaviour law requires rst, 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</p><p>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 rightsreserved.</p><p>Keywords: A. Polymer-matrix composites; B. Durability; B. Non-linear behaviour; Ageing</p><p>1. Introduction</p><p>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 theeects of thermal ageing of polymer resins in a visco-elastic behaviour law coupled with damage at the mac-roscopic level. It is rst necessary to consider themechanical, physical and chemical eects separately.The aim of this article is to present the dierent studieson these subjects at ONERA and the process that willbe henceforth followed to couple ageing with mechan-ical behaviour.The rst part describes the damageable visco-elastic</p><p>model previously developed and the second part pre-sents a computation of thermal-oxidation of polymer-matrix composites (PMCs) and their potential eects onlaminate cracking without mechanical loading. Thestudied materials (carbon-bre/polymer resin compo-</p><p>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).</p><p>2. Mechanical behaviour</p><p>The fundamentals of the visco-elastic model areestablished in the thermodynamic context which easesthe introduction of the coupling of viscous ow withdamage. The selected scale is the ply-scale and themodel describes an anisotropic non-linear viscoelasticbehaviour.Damage is then integrated, in order to:</p><p>. take into account the unilateral character ofdamage,</p><p>. integrate the anelastic strains in the damage evo-lution law and thus consider the possibility ofcrack propagation during creep process,</p><p>. distinguish propagation of cracks between normaland transverse modes (modes I and II),</p><p>0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.PI I : S0266-3538(01 )00146-4</p><p>Composites Science and Technology 62 (2002) 543549</p><p>www.elsevier.com/locate/compscitech</p><p>* Corresponding author.</p><p>E-mail address: anne.schieer@onera.fr (A. Schieer).</p></li><li><p>. take into account the residual strains occurringduring the damage process.</p><p>2.1. Visco-elastic elementary ply behaviour</p><p>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 ow.The approach is based on the existence of two poten-</p><p>tials:</p><p>. The free energy potential, described as:</p><p>2 ""a"th:C0 : ""a"thXi</p><p>1</p><p>ii :CR : i</p><p>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 timei, weighted by i; The whole strain family (i) describesa gaussian continuous spectrum.. Secondly, the dissipation potential, which denes</p><p>the internal parameters evolution laws:</p><p>2 2!i Xi</p><p>ii!i : CR 1 : !i</p><p>with !i g a i, where g() is a non-linear func-tion. The constitutive equations of the model derivedirectly from these two potentials [3]:</p><p> @ @"</p><p> C0 : " "a "tha @ </p><p>@"a "th T T0</p><p>":a @</p><p>@a g</p><p>Xi</p><p>:i</p><p>:i @</p><p>@i 1iigC1R : i</p><p>2.2. Damageable behaviour of the elementary ply</p><p>The formulation proposed to describe the damagedvisco-elastic behaviour is developed in the continuumdamage mechanics context by using a macroscopic</p><p>approach. To describe the damage in the PMCs,damage eects 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 ofbre in the material.</p><p>2.2.1. State laws and damage eects on the elasticbehaviourConcerning the elastic behaviour, the non-isothermal</p><p>formulation of thermodynamic potential, when eachcrack is opened, is:</p><p> 12" "th : C~ d; ;T : " "th </p><p>with C~ d S~ d1 S0 :d:H01Hence, a rank 4 tensor Ho is introduced and repre-</p><p>sents the damage eects 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</p><p>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</p><p>into account the unilateral property of damage. Indeed,an opened or closed crack can have quite dierenteects on the mechanical behaviour, but the introduc-tion of this phenomenon in behaviour laws remains dif-cult, particularly for a multi-axial problem wheredening 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 ).</p><p>2.2.2. Visco-elasticity-damage couplingIt has already been noted that damage plays an</p><p>important role on the asymptotic behaviour. Thus, themodel must be able to predict a damage increase duringcreep test. This eect can be introduced in the same wayas for the elasticity in the relaxed compliance tensor.</p><p>C effr S effr 1 and S effr Sr :d:Hr</p><p>where Hr corresponds to the damage eect tensor on thevisco-elastic behaviour.</p><p>2.2.3. Damage evolution lawIn order to take into account the possible evolution of</p><p>damage during creep process, we choose to make thedamage depend on visco-elastic strains [6]. The damageevolution variable d is given by:</p><p>544 A. Schieer et al. / Composites Science and Technology 62 (2002) 543549</p></li><li><p>fy dc: 1 exp y y0yc</p><p> m and d sup</p></li><li><p>to initiate cracking in composites. This will be con-rmed by the numerical simulations that will be pre-sented in the next section.</p><p>3.2. Theoretical point of view</p><p>3.2.1. Resin caseA model describing oxidation in F655-2 and 977-2</p><p>resins has been chosen among the dierent modelsdeveloped [8].This model predicts both the concentration proles</p><p>and the oxygen consumption across the specimenthickness by coupling diusion kinetics and oxygenchemical reactions in polymer chains. The constitutiveequation of the model is:</p><p>@C</p><p>@t Dm: @</p><p>2C</p><p>@x2 RC</p><p>with the boundary condition:</p><p>CS S P02</p><p>where:</p><p>C: the oxygen concentration,t: the ageing time,x: the depth of penetration for reactive products,Dm: the coecient of oxygen diusion into the</p><p>material,R(C): the local oxygen consumption rate,S: the coecient of oxygen solubility,PO2: the oxygen partial pressure of the oxidising</p><p>environment.</p><p>The oxygen diusion is expressed owing to Fickssecond law. This kinetic model diers 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.</p><p>3.2.2. Unidirectional caseThe oxidation behaviour of unidirectional composites</p><p>is more complex. Indeed, it has been observed that theoxygen diusion in the longitudinal and transversalbre directions is dierent (Fig. 3). In the transversaldirection, no crack appears and the diusion coecientDT can be deduced from the matrix one (Dm) by takinginto account the bre volume fraction.However, in the longitudinal bre direction, the oxi-</p><p>dation process is considerably modied 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</p><p>bre/matrix interfaces that can accelerate the oxygendiusion 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</p><p>composites, the second solution is the most likely to bebut its yet dicult 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, nite-element simulations have beenconducted.For the rst time, a macroscopic stress threshold of</p><p>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 bre-volume fraction. By varying the loadingangle, we induced a variation in the distance betweenbres without changing the mesh (see Fig. 4). Two dif-ferent congurations have been investigated (square andhexagonal) for a global bre volume fraction (Vf) equalto 60%.</p><p>Fig. 3. Oxidised and cracked layer of a T800H/F655-2 unidirectional</p><p>composite after 4000 h of ageing at 180 C.</p><p>Fig. 4. Square conguration for Vf=60%.</p><p>546 A. Schieer et al. / Composites Science and Technology 62 (2002) 543549</p></li><li><p>These dierent simulations give a rst estimation of amicroscopic stress threshold of interface debonding innormal mode (Fig. 5).Next, mechanical eects of volume variations in the</p><p>most oxidised layer have been computed for each con-guration 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</p><p>since values of mechanical properties in the oxidisedlayer are still not precisely determined as well as theinterface debonding criterion. The dierent chemicalevolutions of the matrix are not taken into account butthe oxidised layer has been considered stier than thematrix core, according to the rst results of micro-indentation tests.However, in spite of all uncertainties, such a model</p><p>allows the authors to assert that cracking appears veryquickly in the oxidised bre/matrix interface and is onlyinduced by oxidation eects. Indeed, by consideringthat the most likely damaging mode is the interface</p><p>opening in the normal direction (the eects of shearstresses have been neglected), the stress eld induced byoxidation is sucient enough to initiate interfacedebonding where the distance between bres 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</p><p>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 rst 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-</p><p>Fig. 7. Interface normal stresses after 30 h of ageing at 200 C.</p><p>Fig. 5. Interfacial normal stresses depending on loading angle.</p><p>Fig. 6. Strain variation in the oxidised layer obtained with the oxida-</p><p>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).</p><p>A. Schieer et al. / Composites Science and Technology 62 (2002) 543549 547</p></li><li><p>age was uniform in the totally-oxidised layer (butdecreased with time) and only took place in the matrix(no oxidation of the bres). 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 eldbeing very low compared to the stress eld obtained inthe oxidised layer and near the free edges (Fig. 9).Those simulations clearly show that a volume varia-</p><p>tion (resin shrinkage) is sucient 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 bres.In order to have a more accurate idea on the appear-</p><p>ance time of rst 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]).</p><p>3.3. Study outlook</p><p>Taking into account damage mechanisms in the dif-fusion coecient appears now necessary to complete theoxidation model. Indeed, as seen previously, the oxida-tion process is accelerated owing to the bre presenceand can be simulated by an increase of the apparentdiusion coecient, which governs the oxygen penetra-tion in the material. Besides, previous studies on hygr...</p></li></ul>


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