Modelling of the elasto-viscoplastic damage behaviour of glassy polymers

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International Journal of Plasticity 24 (2008) 945–965 www.elsevier.com/locate/ijplas

Modelling of the elasto-viscoplastic damage behaviour of glassy polymers F. Zaı¨ri a,*, M. Naı¨t-Abdelaziz a, J.M. Gloaguen b, J.M. Lefebvre b a

Laboratoire de Me´canique de Lille (UMR CNRS 8107), USTL, Polytech’Lille, Avenue P. Langevin, 59655 Villeneuve d’Ascq Cedex, France b Laboratoire de Structure et Proprie´te´s de l’Etat Solide (UMR CNRS 8008), USTL, Baˆt. C6, 59655 Villeneuve d’Ascq Cedex, France Received 30 March 2007; received in final revised form 25 July 2007 Available online 9 August 2007

Abstract Constitutive equations are proposed in order to describe the elasto-viscoplastic damage behaviour of polymers. The behaviour is well accounted for by a modified Bodner–Partom model comprising hydrostatic and void evolution terms. The true stress–strain and volumetric strain behaviour of typical rubber-toughened glassy polymers (RTPMMA and HIPS) were experimentally determined at constant local true strain rate by using a video-controlled technique. Successful agreement is obtained between experimental results and the proposed model. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Ductile damage; Cavitation; Crazing; Growth of voids; Nonlinear viscoplasticity; Rubber-modified glassy polymers

1. Introduction It is now commonly accepted that the assumption of constant inelastic volume in the modelling of rubber-toughened glassy polymers is generally violated. Indeed, inelastic dilatation in such materials can be significant and may be attributed to many damage processes such as crazing in the matrix, interfacial debonding and cavitation in rubber parti*

Corresponding author. Tel.: +33 328767460; fax: +33 328767301. E-mail address: [email protected] (F. Zaı¨ri).

0749-6419/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2007.08.001

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cles. In recent years, constitutive equations for polymers have attracted a lot of investigations (Boyce et al., 1988; Arruda et al., 1995; Wu and Van der Giessen, 1995; Bardenhagen et al., 1997; Chaboche, 1997; Sobhanie et al., 1997; Tervoort et al., 1997; Krempl and Ho, 2000; Frank and Brockman, 2001; Khan and Zhang, 2001; Van der Sluis et al., 2001; Ho and Krempl, 2002; Anand and Gurtin, 2003; Goldberg et al., 2003; Krempl and Khan, 2003; Al-Haik et al., 2004; Colak, 2005; Anand and Ames, 2006; Colak and Dusunceli, 2006; Khan et al., 2006; Saramito, 2007). However, these models do not take into account the major role played by damage processes on inelastic properties of most polymers. Prediction of ductile damage thus remains an important challenge in the modelling of polymer mechanical behaviour. In order to describe the nonlinear viscoplastic behaviour (strain softening and strain hardening) of glassy polymers, two approaches can be considered: physical and phenomenological. Regarding strain softening little is known about its origin, and modelling relies on phenomenological assumptions. On the contrary, the physical origin of strain hardening is better understood, and ascribed to the orientation of the macromolecular chains. Physical models derived from the rubber-elasticity theory have thus been proposed. The pioneering work by Haward and Thackray (1968) was further extended to a three-dimensional representation by Boyce et al. (1988). In this approach an Eyring type, defect-driven description of yielding (Eyring, 1936; Argon, 1973) is connected to the rubber-elastic response due of the chain network during strain hardening. The anisotropic hardening, finding common origins in the behaviour of cross-linked rubbers, is described by the three-chain (Boyce et al., 1988), eight-chain (Arruda et al., 1995), full-chain (Wu and Van der Giessen, 1995) and neo-Hookean approximation (Tervoort et al., 1997). Although these physical models brought significant progress, the viscoplastic deformation mechanisms in glassy polymers are not fully elucidated in view of the experimental results. Alternatively and to overcome this difficulty, another approach based on phenomenological considerations can be used. Models initially developed for metals are considered for that purpose. Van der Sluis et al. (2001) used the well-known Perzyna (1966) viscoplastic model to describe the nonlinear behaviour of polycarbonate. The latter material was also modelled by Frank and Brockman (2001) by adapting the Bodner and Partom (1975) viscoplastic constitutive equations. Ho and Krempl (2002) used the Krempl (1984) model to describe the viscoplastic nonlinear behaviour of polymethylmethacrylate. The model was also used (Krempl and Ho, 2000; Krempl and Khan, 2003; Colak, 2005; Colak and Dusunceli, 2006) to describe the inelastic behaviour of some polymers which exhibit a stress–strain behaviour that closely resembles the one found in metals, namely polyamide 66, high density polyethylene and polyphenylene oxide. One may notice that this phenomenological approach has also been used to model the nonlinear inelastic behaviour of polymers prior to yielding (Chaboche, 1997; Goldberg et al., 2003). In the same phenomenological way, inelastic equations were developed from rheological models combining springs and dashpots. Khan and Zhang (2001) used such an approach to capture the inelastic behaviour of polytetrafluoroethylene. More recently, similar equations were developed by Khan et al. (2006) to describe the inelastic response of adiprene-L100. Based on the concept of thixotropic behaviour proposed in the framework of non-newtonian fluid mechanics (Leonov, 1976), a few models for polymers that combine viscoelasticity and viscoplasticity were also derived (Sobhanie et al., 1997; Saramito, 2007). This review is not complete and many other phenomenological inelastic constitutive models for polymers were developed in the literature (e.g. Bardenhagen et al., 1997; Al-Haik et al., 2004).

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Polymer behaviour is generally well described by using these inelasticity theories when deformation results from the development and propagation of shear bands. However, when damage occurs in the material, assumption of constant inelastic volume is no more acceptable. In particular, multiphase polymers consisting of a rubbery component dispersed in a glassy matrix, deformation proceeds with an increase in energy dissipation by comparison to the case of homogeneous glassy polymers. This behaviour corresponds to an elasto-viscoplastic response accompanied by damage in the form of nucleation and growth of voids. In a qualitative manner these deformation processes are generally wellknown. However, quantitative modelling has not been well established to date. During the last 40 years, there was considerable effort to develop predictive tools for the growth of voids in metallic materials (e.g. McClintock, 1968; Rice and Tracey, 1969; Gurson, 1977; Tvergaard, 1981; Pan et al., 1983; Sun and Wang, 1989; Gologanu et al., 1993; Leblond et al., 1994; Pardoen and Hutchinson, 2000; Wen et al., 2005). Recent studies show that existing classical micromechanical models accounting for voiding processes in metals can be used to investigate the growth of voids in rubber-modified glassy polymers. Most of these inelastic damage models, based on the well-known Gurson potential, have been used under the assumption of time independent rigid-plasticity. Furthermore, these studies remain qualitative in nature. In order to take into account the pressure dependence of polymer matrix yielding, Lazzeri and Bucknall (1995), Jeong and Pan (1995) and Jeong (2002) proposed modified versions of the Gurson potential. To investigate the damage behaviour of polymers, the original version of the Gurson model was also used by Guo and Cheng (2002) and Imanaka et al. (2003). Considering that the elastic effect can be significant in voids growth, it has been introduced in the Gurson potential by Steenbrink et al. (1997) and Pijnenburg and Van der Giessen (2001). Their yield surface was coupled with a physically based viscoplastic model (Wu and Van der Giessen, 1995). The complexity of the latter yield surface led Seelig and Van der Giessen (2002) and Pijnenburg et al. (2005) to propose a more convenient yield surface based on phenomenological considerations for the correlation with their numerical results on unit cell model. Inelastic constitutive models taking into account the strain induced chain scission of polymers were developed by Wineman and Rajagopal (1990), Rajagopal and Wineman (1992), and Rajagopal et al. (2007). In a rubber-modified glassy polymer, it is commonly recognized that as the strain is increased, both shear band nucleation and damage mechanisms may take place. Commonly observed nucleation damages in such a system are crazing in the matrix and internal cavitation in the rubber particles. Extensive studies on internal cavitation in rubber particles have been conducted by many researchers. Using the energy balance principles, analytical investigations for cavitation in isolated particle in an infinite glassy polymer matrix were presented in the literature (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). In these studies the matrix material is assumed to be purely elastic; the effects of strain rate and nonlinearity are not considered. Moreover, cavitation is a continuous process influenced by the interactions between the rubber particles, a fact not taken into consideration in isolated particle models. Since the last 40 years, the unstable character of crazing phenomenon has been intensively examined (Sternstein et al., 1968; Kambour, 1973) through its competition with shear yielding in homogeneous glassy polymers. The stable inelastic response involving multiple crazing in rubber-toughened glassy polymers is much less reported in the literature. One can however quote the numerical work of Socrate et al. (2001).

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Quantitative inelastic modelling development of rubber-modified glassy polymers (or more generally, regarding polymers in the solid state) is obviously linked to theoretical and numerical methods but also to experimental techniques. If very few studies deal with inelastic behaviour at large strains, it is partly because of the inefficiency of conventional experimental methods to provide reliable data. Since the inelastic behaviour of rigid polymers is highly nonlinear, strong instability takes place at yield. In order to determine a pertinent behaviour the viscoplastic response must be measured in the instability. Recent developments in experimental techniques (G’Sell et al., 2002) allow to challenge this problem. Indeed, a modern testing method, using a video-controlled device, allows both to record simultaneously the principal strains, axial stress and volumetric strain in the plastic instability region, and to regulate the local strain rate at a constant value during the test. In the present work, the macromechanical response and damage micromechanism by void growth in rubber-toughened glassy polymer systems are theoretically and experimentally investigated. Firstly, a modified Bodner–Partom (BP) model extended to include both the macroscopic nonlinear behaviour and the damage evolution is presented. This modelling is mainly based on internal variable inelastic theory supported by phenomenological and micromechanical considerations. Secondly, a brief sensitivity analysis is performed in order to give a better understanding of the respective role of material parameters in the plastic behaviour. Thirdly, in order to illustrate the performance of the model we present some numerical simulations compared to experimental results on two representative rubber-modified glassy polymers: RTPMMA and HIPS. From the experimental point of view, the true stress–strain and volumetric strain behaviour is determined by using an optical technique to measure locally the strains and to control the true axial strain rate. Finally, conclusions and future works are discussed. 2. Macroscopic constitutive equations with void evolution In this section, we introduce constitutive equations for nonlinear damaged polymers. A macroscopic constitutive relation is built according to both micromechanical (Gurson potential) and phenomenological (modified BP model) considerations. According to the BP model, which is written in the framework of infinitesimal strain, the inelastic behaviour of the solid ligaments between voids is governed only by the second invariant of the deviatoric stress. Both the elastic and inelastic strain rates are non-zero at all stages of loading since the BP formulation is without yield surface. Moreover, no kinematic hardening mechanism is included; the material is assumed isotropic. Although the scheme used in this paper needs to be improved in this way, we attempt to extend this kind of modelling to polymers exhibiting large strains. However, in order to improve the modelling the development of a thermodynamically consistent scheme accounting for finite strains is required. A brief description of such a development is given at the end of this section. The inelastic flow rule for an isotropic material can be given by (Bodner and Partom, 1975) dp ¼ kr0 ¼ kðr  rh IÞ;

ð1Þ

where r0 is the deviatoric stress tensor, rh is the hydrostatic stress, I is the second order unit tensor and k is a scalar parameter defined by

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

k¼

3 p_ ; 2 re

949

ð2Þ

in which p_ is the effective inelastic strain rate and re is the effective stress  1  1 2 p p 2 3 0 0 2 and re ¼ p_ ¼ d :d r :r ; 3 2

ð3Þ

where the double dotes (:) denotes the standard inner product of second order tensor. In the BP model, we assume that p_ ¼ nðre Þ:

ð4Þ

Various expressions for the scalar function n have been presented in the literature. In the plastic regime, glassy polymers are known to exhibit softening after yield and hardening. Here, we retain a formulation suitable to describe the characteristic behaviour of glassy polymers (Zaı¨ri et al., 2005)  2n 2 re nðre Þ ¼ pffiffiffi D0 : ð5Þ Z1 þ Z2 3 The quantities Z1 and Z2 are internal variables depending on the inelastic work Wp and given by   _Z 1 ¼ m1 Z 1  ð1  aÞZ 10 W_ p and Z_ 2 ¼ m2 ðZ 2s  Z 2 ÞW_ p ; ð6Þ Z 10 where Z1 and Z2 are related to the micro-structural arrangements during the strain hardening and the strain softening. Z1 corresponds to the hardening effect arising from development of network alignment and Z2 is introduced to account for the effect of strain softening. By definition Z t Z t p p W ¼ r : d dt ¼ re nðre Þdt: ð7Þ t0

t0

The initial conditions on the internal variables Z1 and Z2 are Z 1 ð0Þ ¼ Z 10

and

Z 2 ð0Þ ¼ 0:

ð8Þ

In Eqs. (5) and (6), D0 is the limiting shear strain rate, n controls the rate sensitivity (from a physical viewpoint, it controls the viscosity of viscoplastic flow), Z10 is the initial value of Z1, Z2s is the saturation value of Z2, a is a hardening parameter controlling the transition between softening and hardening, m1 and m2 are the hardening and softening rate parameters. The macroscopic strain rate tensor D is the sum of the elastic De and inelastic Dp rates D ¼ De þ Dp

ð9Þ

in which the elastic strain rate tensor has the form _ De ¼ C1 R;

ð10Þ

where R_ is the time rate of the macroscopic stress tensor R and C is the fourth-order isotropic elastic modulus tensor:

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C ijkl

  E 2m ðdik djl þ dil djk Þ þ dij dkl : ¼ 2ð1 þ mÞ 1  2m

ð11Þ

In relation (11), E is the Young’s modulus, m is the Poisson’s ratio and d is the Kroneckerdelta symbol. Damage evolution is assumed isotropic and is represented by a scalar: the current void volume fraction f defined by f ¼

Vv ; V

ð12Þ

f 0 6 f < 1;

where Vv is the void volume and V is the elementary apparent volume of the material. The initial void volume fraction is noted f0. To include damage into the model, we assume additive decomposition of the macro0 scopic inelastic strain rate tensor Dp into a deviatoric part Dp and a hydrostatic part Dph : 0

Dp ¼ Dp þ Dph I:

ð13Þ

Different yield surfaces proposed in the literature are presented in Fig. 1 in terms of the normalized macroscopic effective and hydrostatic stresses. The phenomenological potential proposed by Seelig and Van der Giessen (2002) is very close to the lower-bound derived by Sun and Wang (1989). The original version of the Gurson (1977) potential looses its upper-bound character around the normalized hydrostatic stress axis but recovers it around the normalized effective stress axis. Because of its simplicity, the scheme of Pan et al. (1983) for a rate-dependent form of the Gurson (1977) model is adopted in the present study. Regarding the associated plasticity, the macroscopic inelastic strain rate tensor is governed by the normality rule and the following flow rule is proposed oUp ¼ KðR0 þ Xre IÞ: ð14Þ oR The deviatoric and volumetric terms of the macroscopic inelastic strain rate tensor are then given by Dp ¼ K

1 G. 1977 S-W.1989 L-B.1995 J-P.1995 Ste et al. 1997 J. 2002 Se-VdG. 2002

0.8

Σ e /σo

0.6

0.4

0.2

0 0

0.3

0.6

0.9

1.2

1.5

1.8

Σ h /σo

Fig. 1. Initial yield surfaces of Gurson (1977), Sun and Wang (1989), Lazzeri and Bucknall (1995), Jeong and Pan (1995), Steenbrink et al. (1997), Jeong (2002), Seelig and Van der Giessen (2002) for f = 0.2 and qi = 1 (for L–B, J–P and J the pressure-sensitivity coefficient is 0.2 and for Ste et al. the initial yield strain is 0.05).

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965 0

Dp ¼ KR0

and

1 oUp : I ¼ KXre ; Dph ¼ K 3 oR

951

ð15Þ

where K is a scalar parameter in the dilatance regime computed from the equivalence between the plastic power dissipated into the porous material (macroscopic state) and into the corresponding solid ligaments between the voids (microscopic state), and X is a parameter depending on the void volume fraction f:    1 oUp f 3 Rh q2 and X ¼ q1 q2 sinh K ¼ ð1  f Þre p_ R : : ð16Þ 3 2 re oR For an undamaged material (i.e. X = f = 0, K = k and R0 = r0 ), the inelastic flow rule (1) is recovered. The inelastic potential of the porous material, which is also used as a yield function, is given by (Gurson, 1977; Tvergaard, 1981)   3 Rh q2 Up ðR; re ; f Þ ¼ R2e þ 2fq1 r2e cosh ð17Þ  r2e ð1 þ q21 f 2 Þ ¼ 0: 2 re The extended inelastic flow rule now takes the following form:    1 2 f 3 Rh q2 Dp ¼ ð1  f Þkr2e R0 : R0 þ q1 q2 re sinh R:I 3 3 2 re    f 3 Rh q  R0 þ q1 q2 re sinh I : 3 2 2 re

ð18Þ

In relations (17) and (18), Re is the macroscopic effective stress, Rh is the macroscopic hydrostatic stress, re is the microscopic effective stress of the solid ligaments through which the material hardening and rate sensitivity are described and q1, q2 are material parameters that were introduced to take into account the interaction between voids/particles (Tvergaard, 1981). The inelastic volumetric strain evolution as function of the macroscopic strain which is predicted in this case (i.e. q1 and q2 constant) is linear. To take into account the distortion phenomena, q1 and q2 can be seen as internal variables. In order to fit the expected nonlinear evolution of the damage, the following empirical power laws can be assumed to express these variables: q1 ¼ q10 ð1 þ cpÞN

and

q2 ¼ q20 ð1 þ cpÞN ;

ð19Þ

where q10 and q20 are respectively the initial values of q1 and q2, c and N are constants. The void volume fraction rate is decomposed as follows: f_ ¼ f_ n þ f_ g

ð20Þ

in which f_ g and f_ n are the growth rate of existing voids and the nucleation rate of new voids, respectively. The plastic incompressibility assumption leads to the following expression of the void volume fraction growth rate: f_ g ¼ 3ð1  f ÞDph ¼ 3ð1  f ÞKXre :

ð21Þ

In relation (21), the inertia effect on the voids growth rate is neglected and we suppose that the matrix viscosity is dominant. This relation does not capture the accelerated damage due to nucleation of voids which seems to be significant. The kinetics of the nucleation process of voids (internal cavitation/crazing) is complex and is not known very well yet.

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Therefore, the nucleation rate is empirically expressed as a function of the plastic strain. Then, the probability of nucleating a given number of voids is assumed to be proportional to effective inelastic strain rate and given by the following nucleation rate:   fN 1  p  eN  2 _f n ¼ p ffiffiffiffiffiffi exp  _ p: ð22Þ 2 s s 2p In Eq. (22), fN is a constant denoting the volume fraction of voids resulting from void nucleation, eN is the mean value of the normal distribution function and s is its standard deviation. The scheme used here is written under small strains assumption but can be extended to finite strains. The scheme can be developed from theories where thermodynamically consistent framework was well defined (Boyce et al., 1988; Arruda et al., 1995; Wu and Van der Giessen, 1995; Anand and Gurtin, 2003; Anand and Ames, 2006). More precisely, the theory can be formulated in terms of the basic assumption of the multiplicative decomposition of the total deformation gradient F into elastic and plastic parts according to Lee (1969): F = FeFp. The polar decomposition of Fe is given by: Fe = VeRe where Ve is the left elastic stretch tensor and Re is the elastic rotation. The velocity gradient tensor L in the _ 1 ¼ D þ W where D and W are the rate current configuration can be expressed as: L ¼ FF of deformation (symmetric part) and material spin tensors (skewsymmetric part), respec1 tively. The tensor L can be re-written as: L = Le + Lp where Le ¼ F_ e Fe ¼ De þ We and 1 1 Lp ¼ Fe F_ p Fp Fe ¼ Dp þ Wp are the elastic and plastic velocity gradient tensors, respectively. By assuming the plastic flow irrotational, i.e. Wp = 0 and Lp = Dp, the evolution of 1 Fp is then given by: F_ p ¼ Fe Dp Fe Fp in which Dp is the rate of macroscopic plastic deformation tensor developed above. The hypoelastic rate form (Eq. (10)) can be used to give the Cauchy stress tensor by using its objective rate. 3. Numerical examples The stress–strain curves were computed based on the assumption of uniform strain across the section. Equations described above have differential form. An appropriate numerical time integration method is required to include them into a computer algorithm. Because of its simplicity, the trapezoidal method was found the most suitable:

Dt Dt _ DX ¼ ½hðRtDt ; Z 1tDt ; Z 2tDt ; ftDt Þ þ hðRt ; Z 1t ; Z 2t ; ft Þ ¼ X tDt þ X_ i1 ; ð23Þ t 2 2 where X denotes the updated variables computed at each time increment. Considering a given time in the iteration process (i denotes the number of the iteration), the program calculates the updated quantities from the known quantities. The integrated current values are given by: p p Ept ¼ Epi t ¼ EtDt þ DE ;

Z 1t ¼ Z i1t ¼ Z 1tDt þ DZ 1 ;

f t ¼ fti ¼ ftDt þ Df Z 2t ¼ Z i2t ¼ Z 2tDt þ DZ 2

ð24Þ

where Ep is the macroscopic inelastic strain tensor. Evaluation of the model requires the knowledge of stresses. The macroscopic stress is computed from Hooke’s generalized law with macroscopic elastic strain (relations (9) and (10)). The microscopic stress, defining the stress state of the solid ligaments between the voids, is determined from the condition Up = 0 (relation (17)).

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

953

In order to test the model, some numerical calculations, which can be considered as numerical experiments, are performed with the hypothetical material data listed in Table 1. Numerical results for one-dimensional behaviour of the material are shown in Fig. 2. The influence of damage and initial porosity on the behaviour can be clearly seen. The increase Table 1 Material data used in computations Parameter

E

D0

n

Z10

m1

m2

Z2s

a

f0

eN

s

fN

q10

q20

c

N

Unit Value

MPa 2000

s1 104

– 10

MPa 100

– 10

MPa1 2

MPa 15

– 0.15

– 0

– 0.03

– 0.1

– 0.1

– 1.5

– 1.5

– 0

– 1

50 undamaged material

stress (MPa)

40

30

20

f0=0 f0=0.05 f0=0.10 f0=0.20

damaged material

10

0 0

0.05

0.1

0.15

0.2

strain Fig. 2. Strain controlled tensile response for an undamaged and damaged material with different initial void volume fractions.

0.1

40

0.08

0.06

20

0.04

10

0.02

0

0 0

0.05

0.1

strain

0.15

0.2

50

0.12 0.1

40

void volume fraction

α=0.25 α=0.20 α=0.15

30

b

stress (MPa)

50

void volume fraction

stress (MPa)

a

0.08

(1.0;1.5) (1.5;1.0) (1.5;1.5) (q10;q20) (1.5;2.0) (2.0;1.5)

30

20

0.06 0.04

10

0.02

0

0 0

0.05

0.1

0.15

0.2

strain

Fig. 3. Influence of (a) the hardening parameter a and (b) the micromechanics parameters q10 and q20 on the strain controlled tensile response.

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in void volume fraction leads to a reduced yield stress and strain hardening rate. Since it is not of particular interest in this paper, the dependence of the initial void volume fraction on the macroscopic elastic properties is not taken into consideration. Some material parameters have a strong effect on the mechanical response while other parameters play only a secondary role. Sensitivity analysis was performed to identify the material parameters that have the most significant effect on the mechanical response and the evolution of porosity. As an illustrative example, Fig. 3 presents a sensitivity analysis showing the influence of the parameters a, q10 and q20. The parameter a has an important effect on the stress–strain behaviour while it has no significant influence on damage evolution. The parameters qi have a significant effect on both the stress–strain and the damage curves. Furthermore, a competition between damage and viscoplastic hardening is clearly highlighted. 4. Quantitative investigations of local damage in typical rubber-modified polymers The numerical results show that macroscopic strain rate, isochoric (viscosity and hardening) parameters, volumetric (micromechanics) parameters all play key roles in the overall mechanical behaviour of the material. An important aspect is now the ability of the constitutive equations to account for the real behaviour of materials, allowing therefore to open the way towards quantitative damage predictions. Experimental tests were performed in order to analyse and quantify the kinetics and mechanisms of damage in representative rubber-modified polymers. 4.1. Materials under investigation Two rubber-modified polymers were experimentally investigated: a rubber-toughened polymethylmethacrylate (RTPMMA) and a high impact polystyrene (HIPS). These materials display different internal microstructures for the reinforcing particles: core–shell type for RTPMMA and salami type for HIPS. Fig. 4 presents transmission electron microscopy observations in the undeformed state.

Fig. 4. Transmission electron microscopy of (a) HIPS and (b) RTPMMA showing respectively the morphology of salami and core–shell particles (the rubber, stained in OSO4, appears in dark gray and the matrix in lighter gray).

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

955

In RTPMMA, the particles (core–shell type), with inner and outer diameters of 230 nm and 270 nm, respectively, have a PMMA shell and a styrene-co-butyl acrylate core. This core–shell structure promotes the cohesion of the rubber–matrix interphase and favors cavitation in the rubbery core at the expense of interfacial decohesion. The material contains a 30% volume fraction of rubber phase. The glass transition temperature of the rubbery phase and matrix are 21 °C and 110 °C, respectively. The PMMA matrix has a weight-average molar mass of about 130.000 g/mol. In HIPS, the particles (salami type), with diameters distributed in the range 2–4 lm, consist of a polybutadiene continuous phase with multiple PS inclusions. The rubbery phase represents a volume fraction of 12.5%. The glass transition temperature of the PS matrix is 105 °C. Its weight-average molar mass is about 200.000 g/mol. Contrary to the situation in HIPS, the optical index of the rubber phase in RTPMMA is identical to that of the matrix so that the material is transparent before damage. 4.2. Tests and experimental method Tensile tests were carried out using an electromechanical InstronÓ universal testing machine connected with an optical device (G’Sell et al., 2002) Video-TractionÓ (CCD video–camera interfaced with a computer) used to estimate the local damage occurring in the material at a constant true strain rate. The materials were processed by compression moulding at 180 °C in the shape of 4 mm thick plates and slowly cooled down to room temperature. Polymer samples optimized for tensile testing were designed to more accurately measure the plastic flow parameters (Fig. 5). Seven marks made on the surface of the sample were used to evaluate the true axial and transverse strains at a given location (G’Sell et al., 2002) defined by a representative volume element (RVE) containing the two transverse marks.

4

15

120

RV E

6

R=120

Fig. 5. Schematic illustration of the sample (dimensions in mm) and disposition of markers defining the representative volume element (RVE) in which the strains are measured.

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In order to avoid all perturbations (due to whitening) during videomeasurements, a thin black paint coating, able to follow the imposed deformation without breaking, is firstly applied on the sample lateral surface. Then, five white round dots are positioned along the tensile axis. Their diameter and relative distance is about 0.5 mm and 1 mm, respectively. Two other dots are also positioned on both sides of the first series of aligned dots and normal to them in the minimum cross-section. An important advantage of the videomeasurement technique is to avoid a potential deformation initiation in the sample due to perturbation by contact extensometry. Additionally, because of the necking, the local true strain rate was under control during the test by regulating the cross-head speed. Moreover, the strains are measured in a RVE with a thin layer (about 0.2 mm) rigorously included in the plastic instability. Assuming transversal isotropy, the true stress–strain and volumetric strain behaviour of RTPMMA and HIPS were calculated. The macroscopic true stress is calculated as R11 ¼

F F ¼ : S S 0 k222

ð25Þ

The true inelastic volumetric strain which measures the degree of damage is defined as   V V R11 R11 ln ¼ ln k11 þ 2 ln k22  ð1  2mÞ : ð26Þ ¼ ln  ð1  2mÞ V0 n V0 E E The link between volume strain and current void volume fraction may be expressed as   V0 R11 f ¼ 1  ð1  f0 Þ exp ð1  2mÞ : ð27Þ V E In Eqs. (25)–(27), F is the total load, S and S0 are the current and initial cross-sectional area of the sample, V and V0 are the current and initial volume, k11 and k22 are the axial and transverse stretching ratios. PS and PMMA are amorphous polymers which exhibit brittle behaviour at room temperature. Introduction of the rubber phase in the brittle glassy polymer matrix has a dramatic impact on the deformation behaviour. Reinforcing particles both act as nucleation promoters for elementary plasticity events in the matrix, as well as potential cavitational sites. Indeed, the mechanical responses of HIPS and RTPMMA are very different from those of their parent homopolymers PS and PMMA. Typical experimental true stress– strain curves at room temperature are displayed in Fig. 6. HIPS and RTPMMA present the characteristic behaviour of ductile glassy polymers. However, HIPS exhibits a higher strain softening following by a moderate strain hardening. It is worth pointing out that the marked yield drop is not an artifact since the local strain rate is kept constant, owing to the video-controlled procedure. In the region of the RVE (Fig. 5), RTPMMA and HIPS samples have developed dense whitening without a significant cross-section reduction for HIPS. The whitening is generally attributed to damage processes occurring in the material. The nucleation and growth levels of voids are also revealed in Fig. 6 for HIPS and RTPMMA. The incubation strain, below which void nucleation is prohibited, is higher in RTPMMA (about 3%) than in HIPS (1%). Furthermore, this strain corresponds approximately to the initial yield strain. The kinetics of damage mechanism are completely different for these two materials. In HIPS, the inelastic volumetric strain is nearly equal to the true axial strain, a confirmation of the unique activation of crazing/cavitation phenomena. RTPMMA presents a lower

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

9 0.1 6 0.05 3

0

0 0

0.05

0.1

0.15

0.2

0.3

true inelastic volumetric strain

0.15

true inelastic volumetric strain

12

true stress (MPa)

b 50

0.2 HIPS 10-3s-1 and 22˚ C

40

true stress (MPa)

a 15

957

0.2

RTPMMA 10-3s-1 and 22˚ C

30

20

0.1 10

0

0 0

true strain

0.1

0.2

0.3

true strain

Fig. 6. Experimental stress–strain and inelastic volumetric strain–strain curves for (a) HIPS and (b) RTPMMA.

damage due to the enhancement of shear yielding. Damage kinetics exhibit a curved behaviour thus indicating that particle destructuration leads to an acceleration in the rate of shear yielding. In this material, the interparticle spacing, which depends on volume fraction and average diameter of the particles, plays a key role (Bucknall et al., 1984). Indeed, according to the interparticle spacing the response can change from brittle to viscoplastic behaviour (Gloaguen et al., 1992, 1993). Regarding HIPS, the diameter of the particles is known to be of prime importance in initiation and stabilisation of crazing. The whitening zones in HIPS were characterized by means of a field emission scanning electron microscope (FESEM) using secondary electrons. Prior to observation, samples were coated with thin conductive layer (few nanometers) in order to restrict surface charges, to minimize radiation damage and to increase electron emission. Fig. 7 shows an example of FESEM images in HIPS. Two types of morphological characterisations were performed by FESEM. Firstly, the break surfaces, obtained after tensile test, were observed (Fig. 7a). Secondly, after a metallographic polish on one side of sample, deformation mechanisms thus have been examined during in situ deformation directly in chamber microscope operated at 1 kV (Fig. 7b). The observations confirm that particle cavitation with crazing is the common deformation mode. Salami particles show prominent internal fibrillation and this step of rubber cavitation is known to play a key role in multiple crazing development (Bucknall and Smith, 1965). The present observations confirm the greater ability of salami particles to achieve a stabilised craze morphology in HIPS. 4.3. Correlation between experimentation and modelling In order to determine the parameters appearing in the model described in Section 2, the system of equations is reduced to the uniaxial tensile loading case. The material parameters to be determined from experiments are: E, m (from linear elastic range), D0, n, Z10 (from yield stress at various strain rates), m1, m2 , Z2s, a (from work hardening rate function) and f0, eN, s, fN, q10, q20, c and N (from inelastic volumetric strain). The number of parameters is very high and their estimation must be done from a robust identification

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F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

Fig. 7. Scanning electron microscopy of HIPS after deformation: (a) break surface and (b) during in situ deformation.

procedure eliminating all ambiguous results. In order to determine the isochoric material parameters, the experimental stress–strain curves on damaged RTPMMA and HIPS samples have to be converted to the response of undamaged materials. Furthermore, in the case of the uniaxial stress state: d p11 ¼ Dp11 ¼ p_ ¼ e_ p

and

r11 ¼ R11 ¼ r:

ð28Þ

The BP model written in Eq. (1) can be expressed in the following form: r ¼ ðZ 1 þ Z 2 Þgð_ep Þ;

ð29Þ

p

where gð_e Þ is a functional given by pffiffiffi !2n1 3 p p gð_e Þ ¼ : e_ 2D0

ð30Þ

In the yield stress r0 region (defined as the peak stress in the stress–strain curve), Z1 = Z10, Z2 = 0 and e_ ¼ e_ p . According to Eq. (29), the yield stress is then given by r0 ¼ Z 10 gð_eÞ:

ð31Þ

According to the work of Chan et al. (1988) and Rowley and Thornton (1996) for the original BP model, the parameter D0 is arbitrarily selected as D0 = 104 s1. Fixing the parameter D0, identification of parameters n and Z10 is done by the least-squares method from the yield stress versus strain rate data. Once these parameters are determined, the hardening parameters are estimated from the work hardening rate function expressed as

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

959

dr 1 dr ¼ : ð32Þ dW p r dep The stress-viscoplastic strain curve is approximated by a polynomial function which is used to plot the work hardening rate as a function of stress. Fig. 8 shows that the work hardening rate exhibits a bilinear behaviour which leads to the estimation of the two hardening parameters m1 and m2. In order to explicitly express the work hardening rate, Z1 and Z2 can be integrated   m1 p p Z 2 ¼ Z 2s ½1  expðm2 W Þ and Z 1 ¼ ð1  aÞZ 10 þ aZ 10 exp W : ð33Þ Z 10 c¼

According to (29,32) and (33), the work hardening rate function can be written as    Z1 þa1 : ð34Þ c ¼ gð_eÞ m2 ðZ 2s  Z 2 Þ þ m1 Z 10 In the yield stress region, Z1 = Z10 and the work hardening rate takes the following form: c0 ¼ gð_eÞ½m2 ðZ 2s þ Z 10 Þ þ m1 a  m2 r:

ð35Þ

Relation (35) indicates that for small viscoplastic strains, the work hardening rate versus stress curve is linear with a slope m2. In the plateau stress region (defined as the minimum stress in the stress–strain curve and as zero value in c(r) curve), Z2 = Z2s and expression (34) becomes   Z 2s m1 cp ¼ gð_eÞm1 a  1  r: ð36Þ þ Z 10 Z 10 Formula (36) shows that in the plateau stress region, the c(r) function becomes linear with a slope m1/Z10. According to relations (33) and (34), the parameter a is estimated in the plateau stress region by the condition c = 0 which leads to     m2 m1 a ¼  Z 2s exp  m2 þ W pp ; ð37Þ m1 Z 10 where W pp is the viscoplastic work at the plateau stress.

a

b

30

5 m2

m2

γ1 -10 0

σp m1/Z10

5

10

15

-30 -50

σp

0

HIPS 10-3s-1 and 22˚C

work hardening rate

work hardening rate

10

30

40

45

m1/Z10 -10

γ1 RTPMMA 10-3s-1 and 22˚C

σ0

-90

-20

stress (MPa)

50

-5

-15

-70

35

stress (MPa)

Fig. 8. Work hardening rate versus stress for (a) HIPS and (b) RTPMMA.

σ0

960

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

Taking into account Eq. (36), the parameter Z2s is calculated from a c-value c1 (Fig. 8) by the following formula: Z 2s ¼ 

m2 Z 1 10 þ m1

1 1 þ m1cgð_   eÞ  : exp  m2 þ Zm101 W pp

ð38Þ

Table 2 Parameter values for RTPMMA and HIPS Parameter

E

m

D0

n

Z10

m1

m2

Z2s

a

f0

eN

s

fN

q10

q20

c

N

Unit RTPMMA HIPS

MPa 1800 1700

– 0.4 0.38

s1 104 104

– 9.8 9.2

MPa 111.3 34.9

– 8.5 20

MPa1 1.5 12.1

MPa 26 12.8

– 0.2 0.28

– 0 0

– 0.03 0.01

– 0.15 0.1

– 0.15 0.22

– 0.9 1.9

– 1.2 1.9

– 0.2 0

– 1.5 1

0.2 HIPS 10-3s-1 and 22˚C

12

0.15

0.15

9 0.1 6

9 0.1 6

0.05 3

experimental model

0

0 0.05

0.1

true strain

0.15

15

true stress (MPa)

9 0.1 6 0.05 3

experimental model

0

0 0

0.05

0.1

true strain

0.15

0.2

0.1

true strain

0.15

0.2

0.2 HIPS 10-5s-1 and 22˚C

12

0.15 9 0.1 6 0.05 3

experimental model

0

true inelastic volumetric strain

0.15

0.05

d 15 true inelastic volumetric strain

12

0 0

0.2 HIPS 10-4s-1 and 22˚C

experimental model

0

0.2

true stress (MPa)

0

c

0.05 3

true inelastic volumetric strain

true inelastic volumetric strain

HIPS 5.10-3s-1 and 22˚C

12

true stress (MPa)

b 15

0.2 15

true stress (MPa)

a

0 0

0.05

0.1

true strain

0.15

0.2

Fig. 9. Numerical and experimental stress–strain and inelastic volumetric strain–strain curves for HIPS: (a) 5  103 s1, (b) 103 s1, (c) 104 s1, (d) 105 s1.

F. Zaı¨ri et al. / International Journal of Plasticity 24 (2008) 945–965

961

Values of the hardening parameters are determined by averaging each set obtained with each experimental true stress–strain data. Once the isochoric parameters are determined, the volumetric parameters must be estimated. The macroscopic change in porosity in RTPMMA and HIPS during inelastic tensile deformation originates from both nucleation and growth of nucleated voids (internal cavitation/crazing). The homogeneous continuum damage f, corresponding to an heterogeneous void distribution at different growth stages, is a coupling function of these two phenomena (Eq. (20)). Then, the identification is performed separately. The initial void volume fraction f0 has a direct physical meaning and in order to be physically correct it was fixed to zero. The parameters appearing in the nucleation rate of voids (22) are next determined. The strain eN is assumed to be equal to the critical strain beyond which the first voids appear. The volume fraction fN and the standard deviation s are arbitrarily fixed. The parameters q10, q20, c and N controlling the damage evolution (Eqs. (19) and (21)) are determined by fitting the true inelastic volumetric strain curves. At the same time, some hardening parameters must be readjusted to obtain a better fit to the experimental

experimental

20

0.1

model

0

40

0.1

true strain

0.2

experimental model 20 0.1 10

0

0 0.1

true strain

0.2

0.3

0.1

0.2

0.3

true strain

d 50

0.3 RTPMMA 10-5s-1 and 22˚C

40

0.2

30 experimental model

20

0.1

10

0

true inelastic volumetric strain

0.2 30

0.1

0 0

true inelastic volumetric strain

40

experimental model

0

0.3 RT PMMA 10 -4 s-1 and 22 ˚C

0

20

0.3

c 50

0.2

RT PMMA 10 -3 s-1 and 22 ˚C

30

10

0 0

0.3

true stress (MPa)

true stress (MPa)

30

50

true inelastic volumetric strain

0.2

RTPMMA 5.10-3s-1 and 22˚C

true inelastic volumetric strain

40

10

true stress (MPa)

b

0.3 50

true stress (MPa)

a

0 0

0.1

true strain

0.2

0.3

Fig. 10. Numerical and experimental stress–strain and inelastic volumetric strain–strain curves for RTPMMA: (a) 5  103 s1, (b) 103 s1, (c) 104 s1, (d) 105 s1.

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observations. The values of the identified parameters are shown in Table 2 for the materials under study. One can notice that for HIPS, the parameters qi are found to be independent of plastic strain (c = 0). Furthermore, a single value is necessary to apprehend the inelastic volume. The lower value of q10 and q20 that we found for RTPMMA indicates weaker interactions between particles/voids than in HIPS. In Figs. 9 and 10, the numerical responses were generated in tension and compared with experimental data at various strain rates for HIPS and RTPMMA respectively. A nice agreement is pointed out between the two kinds of solutions both for the nonlinear behaviour, the strain rate sensitivity and the damage evolution. However, a relatively poor agreement is observed in the softening region of the hardening curves of HIPS material. Although the constitutive equations can quite well capture the elasto-viscoplastic damage response of these materials, some improvements are still necessary. It is necessary to extend the presented mathematical description of the polymer deformation behaviour to finite deformation. Furthermore, an extended formulation incorporating the effect of void shape evolution has to be developed in order to take into account explicitly the induced damage anisotropy. The modified Gurson yield surface derived by Gologanu et al. (1993) for spheroidal voids can be employed instead of the potential (Eq. (17)) used in this study. It is worthnoting that the abilities of the presented constitutive equations to describe the behaviour of the polymers were only examined under monotonic uniaxial tension. The constitutive equations need to be further validated under more complex loading conditions (e.g. loading/unloading, relaxation, creep, recovery and multiaxial state). 5. Conclusion We have presented an extension of the Bodner–Partom model to include the specific behaviour of glassy polymers and account for damage effects. The proposed model was applied to the case of a rubber-toughened polymethylmethacrylate with core–shell particle morphology and to a high impact polystyrene with salami particles. From the experimental point of view, the true stress–strain and volumetric strain behaviour were determined by using an optical technique to measure the strains locally and to control the true axial strain rate. In order to obtain realistic material parameters from test data, a parameter identification scheme was established with examples based upon experimental results of these two typical rubber-modified polymers. Simulated results were confronted to experiments and a good agreement was obtained. The present approach is quite interesting since it can be applied to materials which deform by means of different elementary mechanisms, and identification may be achieved through simple but pertinent tests. Thus, this study opens the way towards quantitative damage predictions through the implementation of the proposed approach into more general finite element codes. However in order to improve the modelling there is a need to develop a rigorous thermodynamic framework. Moreover, further investigations on materials response under various loading paths are still needed (e.g. cyclic loading, multiaxial state. . .). References Al-Haik, M.S., Garmestani, H., Savran, A., 2004. Explicit and implicit viscoplastic models for polymeric composite. International Journal of Plasticity 20, 1875–1907.

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