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Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer
European Journal of Mechanics A/Solids 24 (2005) 169–182
Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer Fahmi Zaïri a,∗ , Moussa Naït-Abdelaziz a , Krzysztof Woznica b , Jean-Michel Gloaguen a a Université des sciences et technologies de Lille, Polytech’Lille, laboratoire de mécanique de Lille, avenue P. Langevin,
59655 Villeneuve d’Ascq cedex, France b ENSI de Bourges, 10, boulevard Lahitolle, 18020 Bourges cedex, France
Received 23 April 2004; accepted 8 November 2004 Available online 21 December 2004
Abstract In this work, experimental tests in tension on a RT-PMMA material have been achieved under various constant true strain rates and at room temperature. The volumetric dilatation was determined in real time during the deformation by videomeasurements. The experimental results have revealed the presence of both the nucleation and growth deformation mechanisms. Modified viscoplastic constitutive equations for homogeneous glassy polymers at isothermal loading, including strain softening and strain hardening, are proposed. Such a model is based upon an approach originally developed for metals at high temperature. Next, this modified model is coupled with a micromechanics formulation, using the Gurson–Tvergaard model, to investigate the macroscopic mechanical response of the RT-PMMA. The constitutive relation includes strain softening, strain hardening, strain rate sensitivity and void evolution. In the test conditions, the modified viscoplastic model coupled with the original Gurson–Tvergaard model produce quantitative agreement with experimental observations. 2004 Elsevier SAS. All rights reserved. Keywords: Viscoplasticity; Void growth; Rubber-modified polymer
1. Introduction In contrast to extensive qualitative studies on the toughening mechanisms in the literature, constitutive modelling of both homogeneous or blended glassy polymers has received little attention. A quantitative analysis which accounts for both the nonlinear behaviour, the strain rate dependency and the damage of the glassy polymers is still a real challenge. This paper aims to present an experimental and theoretical study of the viscoplastic deformation and damage behaviour of a RT-PMMA. The behaviour of ductile glassy polymers, tested under constant strain rate, is generally highly nonlinear and characterized by an intrinsic strain softening immediately after yield, followed by a progressive strain hardening. Furthermore, glassy polymers behaviour is sensitive to many parameters like strain rate and temperature. The development of models describing the nonlinear post-yield behaviour of glassy polymers has constituted a great interest in the last fifteen years. To date, some physical models based on characteristics of an idealized microstructure have been early developed (Eyring, 1936; Argon, 1973) and further enriched to describe the viscoplastic behaviour of amorphous glassy polymers including strain softening and subsequent strain * Corresponding author. Tel.: +33 3 20 43 46 14; fax: +33 3 28 76 73 01.
E-mail address: [email protected] (F. Zaïri). 0997-7538/$ – see front matter 2004 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2004.11.003
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hardening (Boyce et al., 1988; Arruda et al., 1955; Wu and Van der Giessen, 1995; Van der Aa et al., 2000). It is well established that the physical origin of the strain hardening is entropic and is due to the orientation of the macromolecular chains. However, the physical origin of the strain softening is less known and is commonly modelled using a phenomenological relation. In the referred physical models, the strain hardening is always described using the rubber-elasticity theory, which finds its justification in the almost complete reversibility of the plastic deformation when the glassy polymeric specimen is brought above its glass transition temperature. Although these time dependent physical models were successfully applied to some glassy polymers, it is potentially difficult to obtain material parameters. Furthermore, the viscoplastic deformation mechanisms are not very well understood yet. That is why another approach, based upon a phenomenological point of view, can be used to model the nonlinear and strain rate dependent behaviour of polymers (Frank and Brockman, 2001; Van der Sluis et al., 2001; Ho and Krempl, 2002). The addition of low modulus rubber particles into brittle glassy polymers is often employed to increase their toughness. The toughening of these materials is now generally understood in a qualitative manner. However, very few quantitative data have been published and especially at large strains. In this work, a rubber-modified material was investigated which consists of a PMMA matrix blended with 30% rubber core-shell particles in volume fraction and with overall diameters of 270 nm. The mechanisms responsible of this RT-PMMA toughening during tensile deformation have been investigated. Particularly, the cavitation in the rubber particles and shear banding play an important role in the deformation process. Tests have been achieved to locally measure the plastic flow and volume dilatation response simultaneously for a range of constant true strain rates at room temperature. The competition between cavitation, volume growth and shear deformation is discussed. For a more realistic description of the rubber-toughened glassy polymer behaviour, the void volume variation occurring in the blend must be taken into account in the analysis. In the literature, some authors applied the Gurson (1977) model in a modified form (Lazzeri and Bucknall, 1995; Jeong and Pan, 1995; Steenbrink et al., 1997; Jeong, 2001) or in its original form (Guo and Cheng, 2002; Imanaka et al., 2003) to study the damage behaviour of polymers. Lazzeri and Bucknall (1995), Jeong and Pan (1995) and Jeong (2001) proposed modified Gurson models to account for the mean stress of the matrix introducing a pressure dependence parameter. Steenbrink et al. (1997) proposed a modification of the Gurson expression to account for the elastic effects. As a first approximation, the unmodified model (Gurson, 1977; Tvergaard, 1981) has been used in a uncoupled form and a modification as in Needleman and Tvergaard (1984) has been adopted to capture the plastic volume variation in RT-PMMA. In a first step, the hardening of the matrix has been neglected. After the generation of voids inside the second phase, the rubber particles have been considered equivalent to voids. The nucleation of a void inside a rubber particle in a glassy polymer can be analysed by using analytical models (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). Such models argue that a void is initiated inside the particle when a certain hydrostatic stress limit is reached. In this paper, a statistical model controlled by the hydrostatic stress (Jeong, 2001) has been introduced to take into account for the cavitation of voids. In this work, a model for the nonlinear viscoplastic behaviour of homogeneous glassy polymers is presented. Originally developed for metallic alloys at high temperature, the state variable constitutive equations of the Bodner and Partom (1975) model have been here modified. This model is suitable for multiaxial isotropic deformation. Consequently the development of anisotropy at high elongation due to molecular orientation is ignored. The PMMA matrix is assumed incompressible and described by the above mentioned viscoplastic modified model. In order to introduce new pertinent parameters in the analysis, this model has been next coupled with internal damage including both cavitation and growth of voids using original Gurson’s constitutive relation with the modification introduced by Tvergaard (1981). So, the model we have built by coupling the Bodner–Partom viscoplastic modified constitutive equations with a damage description based upon the Gurson–Tvergaard model, is expected to describe the effect of strain rate sensitivity, strain softening, strain hardening and the void volume fraction evolution. From a set of uniaxial tests, the isochoric parameters have been determined using an original procedure while the micromechanical parameters have been calibrated. The predictions of this model have been generated and compared with experimental data at each strain rate. The results show the ability of the model to correctly predict, as well as qualitatively than quantitatively, both the nonlinearity, strain rate dependence and growth of voids.
2. Experimental study In this part, the deformation mechanisms in a rubber-modified material under different strain rates and at room temperature are investigated. The deformation mechanisms are correlated with the macroscopical viscoplastic behaviour. It is well known that the cavitation in the rubber particles and the matrix yielding around the particles contribute to the deformation. However, the sequence of these toughening deformation mechanisms is not very well clarified and still constitutes one of the remaining
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questions. Furthermore, since very few quantitative data exist, there is a real need to build database to characterise plastic instability, particularly at large strains. 2.1. Material The rubber-toughened material selected in this work consists of a polybutadiene core-shell particles, with a hard shell and a soft rubber core, dispersed in polymethylmethacrylate (PMMA). The refractive index of the rubber particles was adjusted in order to obtain a transparent blend. The glass transition temperature of the blend is about 110 ◦ C and the matrix is characterized by a weight-average molar mass of about 60000 g/mol. The dispersed rubber phase represents 30% in volume fraction of the blend and particles have overall diameters of about 270 nm. This material has been selected because of its simple microstructure and its amorphous matrix. This blend is also known to exhibit ductile properties and significant volumetric growth (Gloaguen et al., 1992; 1993). Moreover, it was shown (Gloaguen et al., 1992; 1993) that shearing in RT-PMMA is promoted for an interparticle distance below approximately 65 nm, which is the case with this modified material since the mean spacings of the rubber particles, estimated from the analysis of Wu (1985), are approximately 55 nm. It can be noticed that this key role of the inter-particle distance in toughening has been extensively studied in the literature for different polymers (Wu, 1985; Borggreve et al., 1987; Muratoglu et al., 1995; Bartczak et al., 1999). The tensile specimens have been cut from rectangular plates which thickness is about 4 mm. These plates have been made from a compression moulded operation. The specimens have been next polished. The geometry of these uniaxial tension specimens, used for the VideoTraction© acquisition and control system, is specific (Fig. 1). In this work, the effect of stress triaxiality on strain localization is not examined. By adopting a large notch radius the stress triaxiality is expected to be low and thus the axial strain is assumed to be uniform across the specimen. The unmodified PMMA homopolymer tensile behaviour, tested under constant true strain rates from 10−5 up to 10−2 s−1 and at room temperature, shows that brittle fracture occurs, and that elastic and crazing deformations dominate. A typical tensile curve for the unmodified material is shown in Fig. 2, where true stress is plotted versus true axial strain. The toughness of the PMMA can be significantly improved by the addition of second phase rubber particles. The rubber toughening effect in RT-PMMA is mainly based on the initiation and growth of plasticity in the PMMA matrix. Furthermore, the macroscopic stress is lower than in the unfilled PMMA (see Figs. 2 and 3). Fig. 2 reveals the extremely brittle behaviour of the PMMA homopolymer at room temperature, where the strain and the stress at break is about 3% and 70 MPa respectively.
Fig. 1. Tensile specimen.
Fig. 2. Experimental tensile true stress-true strain curve for PMMA.
Fig. 3. Experimental tensile true stress-true strain curves at different strain rates and at room temperature for RT-PMMA.
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At the same strain rate, the breaking stress is smaller of about 20 MPa in amplitude for the RT-PMMA. The introduction of the rubber phase in the matrix completely changes the deformation mechanisms of the PMMA, the blend showing a significant ductility. The main mechanisms actived in this blend are the cavitation in the rubber particles and the shear banding in the matrix. Specimens are strained under uniaxial tension, and constant true strain rate during the test, for a range of strain rates and at room temperature (Fig. 3). The modified material which is initially clear, whitens during deformation suggesting the development of the damage mechanisms. As the whitening of this material can disturb the videomeasurement technique, a thin white coat of paint is applied on the specimen. 2.2. Tests and results The experimental setup is mainly constituted by a hydraulic tensile test machine and a videocamera. True strain rate controlled loading of the specimen is performed to remain constant during the test, by image analysis regulating the cross-head speed (G’Sell et al., 2002). The load, the position and the strains are simultaneously recorded by the system. The videomeasurement technique allows to simultaneously record the axial and transverse strains of the specimen. In the region of the specimens where the deformation is concentrated and which is generally characterized by a plastic instability, four black round markers are made on the front surface. Two are aligned along the tensile axis and two others along the transversal axis. The gravity centers of these markers are analysed during the tensile test by a data acquisition system connected to the CCD camera. A more detailed description can be found in G’Sell et al. (2002). From such measurements, volumetric and shear deformations can be therefore determined. Indeed, assuming isotropy of transverse strains, the total true volume strain εvol can be determined from the following equation: V (1) εvol = ln = ε11 + 2ε22 , V0 where V and V0 are the instantaneous and initial volume, respectively, ε11 and ε22 are the axial and transverse true strains, respectively. In the deformation process, because of the cavitation in the particles, the plastic volume strain leads to a volume increase whereas no volume change is attributed to the shear strain. In the elastic stage, a small dilatation occurs and the plastic volume strain (cavitation) is defined as follows: εcav = εvol − εvol(elast) ,
(2)
where εvol(elast) is the elastic contribution to the overall volume strain: εvol(elast) = (1 − 2ν)
σ , E
(3)
ν and E being the Poisson’s ration and Young’s modulus, respectively, and σ is the true stress expressed as: σ=
F F exp(−2ε22 ) , = S S0
(4)
S and S0 are the instantaneous and initial section, respectively, and F is the applied force. Tests have revealed that, above 5 × 10−2 s−1 , this blend is quite brittle and remains transparent; below the behaviour is ductile (Fig. 3). Typical tensile true stress-true strain curves and volume variation evolution are shown in Fig. 4 for two values of the strain rate. The ductile nature and the highly nonlinear behaviour of this RT-PMMA are clearly pointed out on this figure. The cavitation of the rubber particles inducing plastic volume strain starts approximately at the yield. The deformation process seems to be quite complex since, among a certain strain, the volume strain evolution is no longer linear. The curvature which is highlighted probably corresponds to the shearing and distortion of particles. Indeed, towards the final stage of deformation, the rubber particles seems to exhibit a weak strength against distortion compared to that against their expansion. Consequently, an increasing of the rate of shear yielding in the material is expected. To improve the understanding of the macroscopic behaviour, the deformation mechanisms can be highlighted by studying the evolution of the different components of the true axial strain. Assuming the respective contributions of plastic volume (cavitation), shear and elastic strains are linearly additive in the total strain, the shear strain is given as (Heikens et al., 1981): εshear = ε11 − εvol − 2νεelast with εelast = σ/E the elastic strain.
(5)
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Fig. 4. Experimental tensile true stress-true strain curves and plastic volume strain-true strain curves for RT-PMMA.
Fig. 5. True stress-true strain curve and deformation mechanisms in RT-PMMA at 10−3 s−1 and at room temperature.
The contributions of deformation mechanisms can be then quantified for each tensile test. A typical example is given in Fig. 5 which illustrates the general trends observed in the results. This figure shows that, whatever the deformation level is, shear deformation is the dominant dissipative mechanism. After the yield stress, the elastic contribution is globally weak. The onset of shearing occurs before the yield while the cavitation of the first voids begins approximately at the yield. When the strain rate increases, the shear strain and volume strain contributions to the total strain, respectively decreases and increases. In a general way, the strain softening is due to both the void growth and shear bands growth while the strain hardening is almost completely due to the plasticity in the matrix among a given strain value.
3. Stress distributions around a particle Although the qualitative study of the shear band propagation around particules is not the objective of this work, see for this Steenbrink et al. (1997), Smit et al. (1999), Pijnenburg et al. (1999), Socrate and Boyce (2000), Pijnenburg and Van der Giessen (2001), Seelig and Van der Giessen (2002), it has seemed us interesting to evaluate the stress distribution around a particle in order to analyse the influence of the cavitation. Experimental results have confirmed that irreversible strains developed in RT-PMMA in tensile tests may be attributed to shear strain and cavitation strain. A preliminary analysis of the stress field around a dilute spherical rubber particle is given in this section. The rubber particles in the glassy matrix act as stress concentrators when applying a remote stress. This is because of the differences in Young’s modulus and Poisson’s ratio between the particle and the matrix. Some elas-
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(a)
(b) Fig. 6. Radial stress and tangential stress distribution in uniaxial tension around (a) a rubber particle and (b) a void (r and θ are the radial and the tangential coordinate respectively and a is the radius of the particle/void).
tic analytical analyses can give the stress concentrations around a particle (Goodier, 1933; Eshelby, 1957; Tirosh et al., 1995). Fig. 6(a) shows, from the analysis introduced in Tirosh et al. (1995), the radial and tangential stress distribution for two critical directions, i.e. θ = 0 and θ = π/2. θ = 0 is the equator direction and θ = π/2 is the direction of the remote stress. The two phases of our material are considered linear elastic, isotropic, homogeneous and incompressible. Furthermore, the interface between the particle and the matrix is assumed perfect. The elastic properties of the respective phases are: νm = 0.35 and Em = 3000 MPa for the matrix material and νp = 0.499 and Ep = 1 MPa for the rubber particles. The cavitation in the particle is known to be driven by the hydrostatic part of the remote stress (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). Fig. 6(b) gives the stress distribution around the particle after the cavitation, which is mechanically considered as a void because of its low modulus. When the cavitation occurs in the particle, the tangential stress is relatively increased in the surrounding matrix, whereas the radial stress becomes null. That implies an increase of the equivalent stress which may then favor initiation of shear bands at the particle equator. Theoretically, plastic flow of the matrix is easier to be obtained around a void than around an intact rubber particle. But the great difference in the elastic properties values between the two phases implies that the shear bands do not necessary initiate around cavitated particles. In fact, and as we have underlined it in the previous section, the inter-particle distance seems to play a key role in the toughening of the studied material.
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4. Constitutive equations 4.1. Damage evaluation under strain We suppose, in a first approximation, as already done by several authors (Steenbrink et al., 1997; Smit et al., 1999; Pijnenburg et al., 1999; Socrate and Boyce, 2000; Pijnenburg and Van der Giessen, 2001; Seelig and Van der Giessen, 2002) that when cavitation occurs the soft cavitated rubber particles mechanically behave as voids in the matrix. Furthermore, in this part, the matrix behaviour is assumed strain rate independent. The void volume fraction evolution can be obtained using different models. The most popular is the Gurson (1977) potential written as: 2 σe 3 σh − 1 − f 2 = 0, + 2f cosh (6) Φ(σ , σ0 , f ) = σ0 2 σ0 where σ0 is the yield stress, σe is the Mises effective stress and σh is the macroscopic hydrostatic stress: 3 1/2 tr(σ ) and σh = σe = σ :σ . 2 3
(7)
The double dot represents double contraction of tensors and σ = σ − tr(σ )I/3 is the deviatoric stress tensor. In formula (6), f denotes the current void volume fraction: f = Vv /V , where Vv is the current void volume and V is the current total volume. When f = 0, (6) is reduced to the classical von Mises criterion. To account for the interaction between voids, three parameters (q1 , q2 , q3 ) have been empirically introduced by Tvergaard (1981) in the original Gurson form (6), as follows: 2 σe 3 σ + 2f q1 cosh q2 h − 1 − q3 f 2 = 0. (8) Φ(σ , σ0 , f ) = σ0 2 σ0 The void growth is governed by the plastic volumetric strain rate and the nucleation part is neglected since the particles are assumed equivalent to voids. The stress triaxiality is supposed constant during the process. Assuming the normality rule plays for Φ (Gurson, 1977), we easily obtain the dilatational plastic strain rate and from the mass conservation principle we can write, in uniaxial tensile case: q 3 (9) df = q1 q2 sinh 2 f (1 − f ) dεp . 2 2 Integration of Eq. (9) with the initial conditions gives: f (1 − f0 ) q 3 ln = q1 q2 sinh 2 εp , f0 (1 − f ) 2 2
(10)
where f0 is the initial volume fraction of the voids and this initial value represents here the rubber content in PMMA. Because of the plastic incompressibility of the matrix, the relative volume change of the spherical void growth is given by: V 1 − f0 . = 1−f V0 The formula (10) becomes: 3 q V f = f0 exp q1 q2 sinh 2 εp 0 . 2 2 V
(11)
(12)
This expression can be compared with the numerous analytical cavity growth models developed in the literature (McClintock, 1968; Rice and Tracey, 1969). To account for the void distortion and the mode of failure by shear bands growth, we can adapt the modification suggested by Needleman and Tvergaard (1984) by replacing f by f ∗ which is written as: p f, ε p εc , (13) f ∗ = f ∗ (f ) = p fc + k(f − fc ), εp εc , p
where εc and fc represent here the critical values from which the distortion begins, and k is a constant. Fig. 7 shows a quantitative comparison between the experimental results and the model (12). The correction (13) improves the model prediction for this material. The calibration gives q1 = 2.2 and q2 = 0.91 (with q3 = q12 ), values which are close to those suggested by Tvergaard (1981). In order to introduce new pertinent parameters in the micromechanical Gurson–Tvergaard model, this one is coupled with a phenomenological formulation which accounts for viscoplastic constitutive properties and the hardening of the material.
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Fig. 7. Void volume fraction versus plastic true axial strain at 10−3 s−1 (fc = 0.35; k = 0.4).
4.2. Viscoplastic constitutive equations Since the mechanical response of the RT-PMMA is characterized by a peak at the yield stress followed by intrinsic strain softening and strain hardening, a model to represent this highly nonlinear behaviour is required. In this section, modified constitutive equations for homogeneous glassy polymers are presented. We assume that the matrix material is isotropic, homogeneous and incompressible. The increase of temperature associated to the deformation is neglected. On a phenomenological point of view, although the deformation mechanisms of polymers and metals are different, viscoplastic constitutive equations originally developed for metals can be used to model the highly nonlinear and strain rate dependent behaviour of polymers. Since the elastic response is found to be linear, the nonlinearity is only included in viscoplastic terms. In a general way, the viscoplastic part can be formulated from the following constitutive equations: Dp = g1 (σ , Zk ),
(14)
Z˙ k = g2 (Dp , Zk ),
(15)
where g1 and g2 are two functions of the internal variables Zk , and respectively of the stress tensor σ and of the viscoplastic strain rate Dp . The variables Zk are taken to represent the internal state of the material. In this part, since the flow is assumed isochoric the viscoplastic strain rate tensor is equivalent to its deviatoric part. The functions g1 and g2 are explicited by using the Bodner–Partom model. Such a model is a unified approach since plasticity and creep are coupled. Furthermore, the viscoplastic contribution exists at all nonzero stress levels. The model gives, as most of others formulations, a relationship between the viscoplastic strain rate, the deviatoric part of the stress tensor, the Mises effective stress and the internal state variables. The original Bodner and Partom (1975) model is expressed as: 3 σ Dp = p˙ , 2 σe
(16)
where σe is the Mises effective stress, σ is the deviatoric part of the stress and p˙ is the accumulated viscoplastic strain rate: 2 p p 1/2 p˙ = . (17) D :D 3 There are many different forms for the accumulated viscoplastic strain rate which have been presented in the literature. To take into account the effect of the nonlinear plastic deformability of the matrix phase, we have modified the expression originally proposed by Frank and Brockman (2001). So, we propose to write the accumulated viscoplastic strain rate as follows: 2n σe 2 , (18) p˙ = √ D0 Z1 + Z2 3 where Z1 and Z2 are the internal state variables which are used to simulate the molecular flow strength.
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The internal state variable Z1 corresponding to the hardening effect arising from development of network alignment, has been also modified and simplified as follows: Z1 − (1 − α)Z10 W˙ p . Z˙ 1 = m (19) Z10 To account for the effect of strain softening, the new internal state variable Z2 we have introduced in the model, is expressed as: Z (20) Z˙ 2 = h 1 − 2 W˙ p . Z2s In Eq. (19) and (20), W˙ p is the viscoplastic work rate given as: ˙ W˙ p = σe p.
(21)
In the model, D0 represents the limiting value of the shear strain rate, n is the strain rate sensitivity parameter, Z10 is the initial value of Z1 which is phenomenologically related to the onset of plasticity at yield stress, Z2s is the saturation value of Z2 , m is the hardening rate parameter, h is the softening rate parameter and α is a parameter which controls the beginning of the strain hardening. The initial condition for the softening parameter is assumed null and only Z10 is taken to represent the internal state of the matrix material at the onset of the shear band nucleation. By coupling the above described viscoplastic model with the Gurson–Tvergaard void growth relationship, an enriched and complete formulation is therefore built. 4.3. Viscoplastic model including damage Some models for porous viscoplastic metallic materials have been proposed in the literature (Pan et al., 1983; Perzyna, 1986). Particularly, the Gurson model for rate independent porous plastic behaviour was used in a rate dependent form by Pan et al. (1983). Most of the models developed for porous glassy polymers are modified (Lazzeri and Bucknall, 1995; Jeong and Pan, 1995; Steenbrink et al., 1997; Pijnenburg and Van der Giessen, 2001; Jeong, 2001) or unmodified (Guo and Cheng, 2002; Imanaka et al., 2003) versions of the Gurson yield surface. The association with the viscoplasticity, in the spirit of the works of Pan et al. (1983), can be found in Steenbrink et al. (1997), Pijnenburg and Van der Giessen (2001). In this work, neither the pressure-sensibility of the matrix nor the elasticity are taken into account in the yield surface. However, the effect of the pressure on plastic flow in the matrix can be introduced directly in the expression (18) of the modified viscoplastic constitutive model. 4.3.1. Tridimensional formulation The viscoplastic constitutive equation describing the mechanical behaviour of the matrix is combined with the micromechanical model of Gurson–Tvergaard to account for both the influence of damage mechanisms in the RT-PMMA and subsequent yielding of the matrix around the particles. It is important to note (see Section 4.1) that in the Gurson–Tvergaard model, the distortional contribution is neglected compared to the dilatation since the void growth is considered to be entirely spherical. We must note that this assumption is clearly a limit of our model when describing such a material behaviour. The macroscopic rate of deformation tensor D is decomposed into an elastic part De and a viscoplastic part Dp : D = De + Dp .
(22)
The elastic constants are assumed to be independent of the strain rate and the elastic part of the rate of deformation is given by: ˙ − ν/E tr(Σ)I, ˙ De = (1 + ν)/E Σ
(23)
where Σ is the macroscopic Cauchy stress tensor and I is the identity tensor. The macroscopic viscoplastic part of the rate of deformation is derived from the normality rule and given by: ∂Φ , ∂Σ where Λ is a parameter which can be determined from the condition of equivalence of plastic work rate: Dp = Λ
(24)
Σ : Dp = (1 − f )σe p. ˙
(25)
In this last equation, σe and p˙ are the Mises effective stress and the accumulated viscoplastic strain rate, respectively, of the matrix material (see Section 4.2).
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Combining Eqs. (24) and (25) leads to: ∂Φ −1 , Λ = (1 − f )σe p˙ Σ : ∂Σ where Φ is the Gurson–Tvergaard’s flow potential (Gurson, 1977; Tvergaard, 1981): 2 Σe 3 Σh + 2f q1 cosh q2 Φ(Σ, σe , f ) = − 1 − q3 f 2 = 0, σe 2 σe
(26)
(27)
qi , 1 i 3, are the three adjustment parameters introduced by Tvergaard (1981) to improve the model predictions, f is the void volume fraction, Σe is the macroscopic Mises effective stress and Σh is the macroscopic hydrostatic stress: 1/2 3 tr(Σ) and Σh = Σe = Σ : Σ , (28) 2 3 where Σ = Σ − tr(Σ)I/3 is the macroscopic deviatoric stress tensor. Introducing the Gurson’s yielding function, the flow rule can be expressed as: f Σ 3 Σ Dp = Λ 3 2 + q1 q2 sinh q2 h I . σe 2 σe σe
(29)
Because of the plastic dilatation by void growth, the viscoplastic strain rate (29) is now both deviatoric and dilatational. We assume that the increase of void volume arises both from the growth of existing voids and from the nucleation of voids. The void volume fraction rate f˙ is decomposed into a void growth f˙grow and a nucleation part f˙nuc , as follows: f˙ = f˙grow + f˙nuc .
(30)
Void growth rate is given by the mass conservation principle: f˙grow = (1 − f ) tr(Dp ), where the trace of viscoplastic strain rate is given by: 3 Σ f tr(Dp ) = 3Λq1 q2 sinh q2 h . σe 2 σe
(31)
(32)
Experimental data suggest that it exists an incubation strain below which the cavitation is prohibited. Recently, three criteria for cavitation in an isolated particle in an infinite glassy polymer matrix, have been presented in the literature (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). According to them, when a critical hydrostatic stress is reached the cavitation simultaneously occurs in the center of all rubber particles. However, in the RT-PMMA, or in others rubber-modified polymers, the rubber particules do not all cavitate at the same time. Therefore, the cavitation of voids can be seen as a continuous process. To capture the accelerated damage due to cavitation of voids, according to the phenomenological hydrostatic stress controlled nucleation law proposed by Jeong (2001), the cavitation part can be given by: fN 1 Σ h − σN 2 (33) exp − Σ˙ h . f˙nuc = √ 2 sZ10 2π sZ10 In the relation (33) three added constants must be determined: the standard deviation s, the mean value of the hydrostatic stress σN and the volume fraction of rubber particles which cavitate fN . 4.3.2. Uniaxial formulation In this part, the equations given earlier are reduced to the uniaxial case. The macroscopic Mises effective stress and the macroscopic hydrostatic stress are defined respectively as follows: Σ11 , 3 where Σ11 is the only component of the macroscopic stress tensor. The components of the macroscopic viscoplastic part of the rate of deformation are: P = Λ 2 Σ11 + q q f sinh q Σ11 D11 1 2 2 σ 2σ σ2 Σe = Σ11
and
Σh =
(34)
(35)
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and:
P = D P = Λ − Σ11 + q q f sinh q Σ11 D22 1 2 2 33 σ 2σ σ2
179
(36)
P + 2D P . with the plastic dilatation: tr(Dp ) = D11 22 The parameter Λ becomes: Σ11 2 Σ11 −1 Σ11 p + q1 q2 f sinh q2 Λ = (1 − f )σ ε˙ 2 σ σ 2σ
(37)
p
with p˙ = ε˙ 11 = ε˙ p and σe = σ11 = σ . ε˙ p is a function of σ as in (18). The void growth rate is governed by: 3q1 q2 f (1 − f ) sinh(q2 Σ11 /(2σ )) p f˙grow = D , 2Σ11 /σ + q1 q2 f sinh(q2 Σ11 /(2σ )) 11
(38)
where the initial void volume fraction in the material is defined by f (t0 ) = f0 . The void volume fraction evolution has an effect on Young’s modulus of the RT-PMMA. In this study, a linear relation between E and f is found the most suitable and is expressed as: E = (1 − 1.2f )E0 , where E0 is now the initial Young’s modulus of the blend. The material constants that need to be determined include: E0 , D0 , n, Z10 , m, h, Z2s , α, f0 , q1 , q2 , q3 , s, σN and fN . 4.4. Identification and validation 4.4.1. Identification of the model parameters A single strain rate independent set of parameters for capturing the nonlinear behaviour, the strain rate effect and the damage evolution is the purchased goal of such a model. The identification has been performed in two stages. In a first step, an approximate estimation of the viscoplastic parameters is achieved using the uniaxial tensile true stress-true strain curves. In a second step, the damage model parameters are calibrated with experimental data and optimized to capture both the nonlinear behaviour and void growth. A first set of isochoric parameters values is determined using an analytical method described in Chan et al. (1988), Rowley and Thornton (1996), Woznica and Klosowski (2000) from the tensile simple tests across a variety of strain rates. The identification procedure was originally developed by Chan et al. (1988) for the unmodified viscoplastic model and is adapted here for the highly nonlinear behaviour of the studied material. The seven independent isochoric strain rate parameters are: D0 , n, Z10 , m, h, Z2s and α. In the original formulation of the Bodner–Partom model, D0 is fixed arbitrarily and in quasi-static problems, we generally assume that D0 = 104 s−1 (Chan et al., 1988). The yield stress σ0 , defined as the peak stress (i.e. just before softening), allows the estimation of n and Z10 from a least squares regression (Fig. 8).
Fig. 8. Yield stress versus true strain rate.
Fig. 9. Work hardening rate versus true stress.
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The others isochoric parameters have been determined from the work hardening rate defined as: γ=
dσ 1 dσ (εp ) = . p dW σ (εp ) dεp
(39)
Using polynomial curve fitting to correlate the stress versus the plastic strain at each strain rate, the analytical work hardening rate expression can be then determined. The hardening parameters h and m are computed from the negative and positive slope, respectively, of the work hardening rate plotted versus the stress (Fig. 9). Z2s and α are determined from a work hardening rate value (noted γ1 in the work hardening rate-stress curve) and the minimum stress σa after yield (σa also corresponds to zero value in γ (σ ) curve). Z2s and α are provided by the following equations: γ (Z 0 )2 Z2s = 1 1 − (1 − α)Z10 , m σ0 p γ1 Z10 mW σ0 , α = σa − exp − 0a m Z
(40) (41)
1
p p p ε p where Wa is the viscoplastic work defined by Wa = 0 a σ (εp ) dεp , εa being the plastic strain corresponding to σa . Thus, each stress-strain experimental record leads to a set of parameters. The retained values are issued from an averaging on all the sets. Correction and readjustment of some parameters are then required to account for damage in the blend. The values of the isochoric parameters obtained are: E0 = 1780 MPa, D0 = 104 s−1 , n = 10.85, Z10 = 107.25 MPa, Z2s = −22.5 MPa, m = 6.9, h = −40, α = 0.14. The parameters q1 , q2 and q3 are chosen by simultaneously fitting true stress-true strain and volume variation curves. The parameter f0 is chosen such that the model gives the best fit of dilatational curves. Since the adjustment of f0 is arbitrary this one suffers of a lack of physical basis. Taking as smallest as possible the value of f0 , the selection of the three others parameters of the normal distribution function (33) completes the model identification. The standard deviation s is arbitrary fixed, the mean value σN is equal to the initial value of Z1 and the volume fraction fN is equal to the volume fraction of rubber particles. The values of the micromechanics parameters we have obtained are: f0 = 0.01, q1 = 2.6, q2 = 2.2, q3 = q12 , s = 0.7, σN = Z10 , fN = 0.27.
4.4.2. Numerical integration and validation of the model An explicit integration scheme in a Fortran program is used to numerically integrate the system of nonlinear constitutive differential equations and to generate data for the monotonic uniaxial test at each strain rate. From the constitutive equations, the values of the unknown variables at the current time step are evaluated from their values at the previous time step using the trapezoidal method. The numerical integration stability is preserved by assigning that the rate of deformation must be always greater than the rate of the viscoplastic part. As an example, in Fig. 10, a numerical response is simulated and compared with the experimental data for two strain rates. A quite nice agreement is observed between nonlinear hardening experimental results and the predicted behaviour given by the model. The comparison with the dilatation curves is also reasonably satisfactory but the model under-estimates the void growth.
Fig. 10. Experimental and modeled true stress-true strain curves and plastic volume strain-true strain curves for RT-PMMA.
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Although the model is in agreement with the experimental data, it is important to note that validation was made in a limited range of test conditions. So, a complete validation of the model requires to enrich the data basis.
5. Conclusion The deformation mechanisms of a typical rubber-modified glassy polymer, a RT-PMMA, has been investigated over a range of strain rates and at room temperature. The links between these mechanisms and macroscopical behaviour was discussed. The toughening effect is mainly based on the initiation and growth of shear bands in the matrix. Plastic flow of the surrounding matrix leads to an arising of the void growth rate in a first step, and when a certain strain is reached, this rate decreases. Furthermore, experimental results have shown that shear bands occur before rubber particles cavitation. The cavitation of the rubber particles reduces hydrostatic tension and accelerate the rate of shear yielding. In order to describe the mechanical behaviour of the glassy polymeric matrix, we have modified the viscoplastic Bodner– Partom model. Such a model is based upon a thermodynamics formalism of irreversible processes. To account for both the dependence on shear and mean stresses, the modified viscoplastic model was then coupled with the Gurson–Tvergaard’s flow potential. The parameters are determined from the experiments. An original analytical method has been developed to estimate the modified viscoplastic model. Next, the micromechanical parameters have been optimized using an iterative process in order to obtain the best agreement between the experimental data and the predicted curves. A numerical procedure has been developed to solve the system of nonlinear constitutive equations using a trapezoidal scheme. The predictions have shown the ability of the modified viscoplastic model coupled with damage to correctly describe the experimental observations: nonlinearity, strain rate sensitivity and damage evolution are well captured. To our knowledge, it is the first time that a unified set of constitutive equations that combine viscoplasticity and damage is used to represent a quantitative polymer response. However, several aspects need to be detailed and completed in the future. The viscoplastic modified model has to be implemented in a finite element program. Since model predictions sensitively depend on micromechanics parameters, a robust parameter identification using a proper scheme is required. Furthermore, to increase the understanding of the interplay between shear and mean stress and the role of stress triaxiality, tests must be made over a range of notched specimen types with different radius. Moreover, in order to improve the model prediction, the effect of anisotropy due to molecular orientation has to be taken into account.
References Argon, A.S., 1973. A theory for the low temperature plastic deformation of glassy polymers. Philos. Mag. 28, 839–865. Arruda, E.M., Boyce, M.C., Jayachandran, R., 1955. Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mech. Mater. 19, 193–212. Bartczak, Z., Argon, A.S., Cohen, R.E., Weinberg, M., 1999. Toughness mechanism in semicrystalline polymer blends. Part I: High-density polyethylene toughened with rubbers. Pol. 40, 2331–2346. Bodner, S.R., Partom, Y., 1975. Constitutive equations for elastic-viscoplastic strain-hardening materials. J. Appl. Mech. 42, 385–389. Borggreve, R.J.M., Gaymans, R.J., Schuijer, J., Housz, J.F.I., 1987. Brittle-tough transition in nylon-rubber blends: effect of rubber concentration and particle size. Pol. 28, 1489–1496. Boyce, M.C., Parks, D.M., Argon, A.S., 1988. Large inelastic deformation of glassy polymers. Part I: rate dependent constitutive model. Mech. Mater. 7, 15–33. Chan, K.S., Bodner, S.R., Lindholm, U.S., 1988. Phenomenological modelling of hardening and thermal recovery in metals. J. Engrg. Mater. Technol. 110, 1–8. Dompas, D., Groeninckx, G., 1994. Toughening behaviour of rubber-modified thermoplastic polymers involving very small rubber particles: 1. A criterion for internal rubber cavitation. Pol. 35, 4743–4749. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. London Ser. A 241, 376–396. Eyring, H., 1936. Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J. Chem. Phys. 4, 283–291. Fond, C., Lobbrecht, A., Schirrer, R., 1996. Polymers toughened with rubber microspheres: an analytical solution for stresses and strains in the rubber particles at equilibrium and rupture. Int. J. Frac. 77, 141–159. Frank, G.J., Brockman, R.A., 2001. A viscoelastic-viscoplastic constitutive model for glassy polymers. Int. J. Solids Struct. 38, 5149–5164. G’Sell, C., Hiver, J.M., Dahoun, A., 2002. Experimental characterization of deformation damage in solid polymers under tension, and its interrelation with necking. Int. J. Solids Struct. 39, 3857–3872. Gloaguen, J.M., Heim, P., Gaillard, P., Lefebvre, J.M., 1992. Plasticity of rubber-toughened PMMA: effect of particle size. Pol. 33, 4741. Gloaguen, J.M., Steer, P., Gaillard, P., Wrotecki, C., Lefebvre, J.M., 1993. Plasticity and fracture initiation in rubber-toughened PMMA. Pol. Engrg. Sci. 33, 748.
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Goodier, J.N., 1933. Concentration of stress around spherical and cylindrical inclusions and flaws. J. Appl. Mech. 55, 39–44. Guo, T.F., Cheng, L., 2002. Modeling vapor pressure effects on void rupture and crack growth resistance. Acta Mater. 50, 3487–3500. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I – yield criteria and flow rules for porous ductile media. J. Engrg. Mater. Technol. 99, 2–15. Heikens, D., Sjoerdsma, S.D., Coumans, W.J., 1981. A mathematical relation between volume strain, elongation strain and stress in homogeneous deformation. J. Mater. Sci. 16, 429. Ho, K., Krempl, E., 2002. Extension of the viscoplasticity theory based on overstress (VBO) to capture non-standard rate dependence in solids. Int. J. Plasticity 18, 851–872. Imanaka, M., Nakamura, Y., Nishimura, A., Iida, T., 2003. Fracture toughness of rubber-modified epoxy adhesives: effect of plastic deformability of the matrix phase. Comput. Sci. Technol. 63, 41–51. Jeong, H.Y., 2001. A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices. Int. J. Solids Struct. 39, 1385–1403. Jeong, H.Y., Pan, J., 1995. A macroscopic constitutive law for porous solid with pressure-sensitive matrices and it’s implications to plastic flow localization. Int. J. Solids Struct. 32, 3669–3691. Lazzeri, A., Bucknall, C.B., 1995. Applications of a dilatational yielding model to rubber-toughened polymers. Pol. 36, 2895–2902. McClintock, F.A., 1968. A criterion for ductile for ductile by growth of holes. J. Appl. Mech. 35, 363–371. Muratoglu, O.K., Argon, A.S., Cohen, R.E., 1995. Toughening mechanism of rubber-modified polyamides. Pol. 36, 921–930. Needleman, A., Tvergaard, V., 1984. An analysis of ductile rupture in notched bars. J. Mech. Phys. Solids 32, 461–490. Pan, J., Saje, M., Needleman, A., 1983. Localization of deformation in rate sensitive porous plastic solids. Int. J. Fract. 21, 261–278. Perzyna, P., 1986. Internal state variable description of dynamic fracture of ductile solids. Int. J. Solids Struct. 22, 797–818. Pijnenburg, K.G.W., Steenbrink, A.C., Van der Giessen, E., 1999. Shearing of particles during crack growth in polymer blends. Pol. 40, 5761– 5771. Pijnenburg, K.G.W., Van der Giessen, E., 2001. Macroscopic yield in cavitated polymer blends. Inter. J. Solids Struct. 38, 3575–3598. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201–217. Rowley, M.A., Thornton, E.A., 1996. Constitutive modeling of the visco-plastic reponse of hastelloy-x and aluminium alloy 8009. J. Engrg. Mater. Technol. 118, 19–27. Seelig, Th., Van der Giessen, E., 2002. Localized plastic deformation in ternary polymer blends. Inter. J. Solids Struct. 39, 3505–3522. Smit, R.J.M., Brekelmans, W.A.M., Meijer, H.E.H., 1999. Prediction of the large-strain mechanical response of heterogeneous polymer systems: local and global deformation behavior of a representative volume element of voided polycarbonate. J. Mech. Phys. Solids 47, 201–221. Socrate, S., Boyce, M.C., 2000. Micromechanics of toughened polycarbonate. J. Mech. Phys. Solids 48, 233–273. Steenbrink, A.C., Van der Giessen, E., Wu, P.D., 1997. Void growth in glassy polymers. J. Mech. Phys. Solids 45, 405–437. Tirosh, J., Nachlis, W., Hunston, D., 1995. Strength behavior of toughened polymers by fibrous (or particulate) elastomers. Mech. Mater. 19, 329–342. Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17, 389–407. Van der Aa, H.C.E., Van der Aa, M.A.H., Schreurs, P.J.G., Baaijens, F.P.T., Van Veenen, W.J., 2000. An experimental and numerical study of the wall ironing process of polymer coated sheet metal. Mech. Mater. 32, 423–443. Van der Sluis, O., Schreurs, P.J.G., Meijer, H.E.H., 2001. Homogenisation of structured elastoviscoplastic solids at finite strains. Mech. Mater. 33, 499–522. Woznica, K., Klosowski, P., 2000. Evaluation of viscoplastic parameters and its application for dynamic behavior of plates. Arch. Appl. Mech. 70, 561–570. Wu, S., 1985. Phase structure and adhesion in polymer blends: a criterion for rubber toughening. Pol. 26, 1855–1863. Wu, P.D., Van der Giessen, E., 1995. On neck propagation in amorphous glassy polymers under plane strain tension. Int. J. Plasticity 11, 211–235.
Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer Fahmi Zaïri a,∗ , Moussa Naït-Abdelaziz a , Krzysztof Woznica b , Jean-Michel Gloaguen a a Université des sciences et technologies de Lille, Polytech’Lille, laboratoire de mécanique de Lille, avenue P. Langevin,
59655 Villeneuve d’Ascq cedex, France b ENSI de Bourges, 10, boulevard Lahitolle, 18020 Bourges cedex, France
Received 23 April 2004; accepted 8 November 2004 Available online 21 December 2004
Abstract In this work, experimental tests in tension on a RT-PMMA material have been achieved under various constant true strain rates and at room temperature. The volumetric dilatation was determined in real time during the deformation by videomeasurements. The experimental results have revealed the presence of both the nucleation and growth deformation mechanisms. Modified viscoplastic constitutive equations for homogeneous glassy polymers at isothermal loading, including strain softening and strain hardening, are proposed. Such a model is based upon an approach originally developed for metals at high temperature. Next, this modified model is coupled with a micromechanics formulation, using the Gurson–Tvergaard model, to investigate the macroscopic mechanical response of the RT-PMMA. The constitutive relation includes strain softening, strain hardening, strain rate sensitivity and void evolution. In the test conditions, the modified viscoplastic model coupled with the original Gurson–Tvergaard model produce quantitative agreement with experimental observations. 2004 Elsevier SAS. All rights reserved. Keywords: Viscoplasticity; Void growth; Rubber-modified polymer
1. Introduction In contrast to extensive qualitative studies on the toughening mechanisms in the literature, constitutive modelling of both homogeneous or blended glassy polymers has received little attention. A quantitative analysis which accounts for both the nonlinear behaviour, the strain rate dependency and the damage of the glassy polymers is still a real challenge. This paper aims to present an experimental and theoretical study of the viscoplastic deformation and damage behaviour of a RT-PMMA. The behaviour of ductile glassy polymers, tested under constant strain rate, is generally highly nonlinear and characterized by an intrinsic strain softening immediately after yield, followed by a progressive strain hardening. Furthermore, glassy polymers behaviour is sensitive to many parameters like strain rate and temperature. The development of models describing the nonlinear post-yield behaviour of glassy polymers has constituted a great interest in the last fifteen years. To date, some physical models based on characteristics of an idealized microstructure have been early developed (Eyring, 1936; Argon, 1973) and further enriched to describe the viscoplastic behaviour of amorphous glassy polymers including strain softening and subsequent strain * Corresponding author. Tel.: +33 3 20 43 46 14; fax: +33 3 28 76 73 01.
E-mail address: [email protected] (F. Zaïri). 0997-7538/$ – see front matter 2004 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2004.11.003
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hardening (Boyce et al., 1988; Arruda et al., 1955; Wu and Van der Giessen, 1995; Van der Aa et al., 2000). It is well established that the physical origin of the strain hardening is entropic and is due to the orientation of the macromolecular chains. However, the physical origin of the strain softening is less known and is commonly modelled using a phenomenological relation. In the referred physical models, the strain hardening is always described using the rubber-elasticity theory, which finds its justification in the almost complete reversibility of the plastic deformation when the glassy polymeric specimen is brought above its glass transition temperature. Although these time dependent physical models were successfully applied to some glassy polymers, it is potentially difficult to obtain material parameters. Furthermore, the viscoplastic deformation mechanisms are not very well understood yet. That is why another approach, based upon a phenomenological point of view, can be used to model the nonlinear and strain rate dependent behaviour of polymers (Frank and Brockman, 2001; Van der Sluis et al., 2001; Ho and Krempl, 2002). The addition of low modulus rubber particles into brittle glassy polymers is often employed to increase their toughness. The toughening of these materials is now generally understood in a qualitative manner. However, very few quantitative data have been published and especially at large strains. In this work, a rubber-modified material was investigated which consists of a PMMA matrix blended with 30% rubber core-shell particles in volume fraction and with overall diameters of 270 nm. The mechanisms responsible of this RT-PMMA toughening during tensile deformation have been investigated. Particularly, the cavitation in the rubber particles and shear banding play an important role in the deformation process. Tests have been achieved to locally measure the plastic flow and volume dilatation response simultaneously for a range of constant true strain rates at room temperature. The competition between cavitation, volume growth and shear deformation is discussed. For a more realistic description of the rubber-toughened glassy polymer behaviour, the void volume variation occurring in the blend must be taken into account in the analysis. In the literature, some authors applied the Gurson (1977) model in a modified form (Lazzeri and Bucknall, 1995; Jeong and Pan, 1995; Steenbrink et al., 1997; Jeong, 2001) or in its original form (Guo and Cheng, 2002; Imanaka et al., 2003) to study the damage behaviour of polymers. Lazzeri and Bucknall (1995), Jeong and Pan (1995) and Jeong (2001) proposed modified Gurson models to account for the mean stress of the matrix introducing a pressure dependence parameter. Steenbrink et al. (1997) proposed a modification of the Gurson expression to account for the elastic effects. As a first approximation, the unmodified model (Gurson, 1977; Tvergaard, 1981) has been used in a uncoupled form and a modification as in Needleman and Tvergaard (1984) has been adopted to capture the plastic volume variation in RT-PMMA. In a first step, the hardening of the matrix has been neglected. After the generation of voids inside the second phase, the rubber particles have been considered equivalent to voids. The nucleation of a void inside a rubber particle in a glassy polymer can be analysed by using analytical models (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). Such models argue that a void is initiated inside the particle when a certain hydrostatic stress limit is reached. In this paper, a statistical model controlled by the hydrostatic stress (Jeong, 2001) has been introduced to take into account for the cavitation of voids. In this work, a model for the nonlinear viscoplastic behaviour of homogeneous glassy polymers is presented. Originally developed for metallic alloys at high temperature, the state variable constitutive equations of the Bodner and Partom (1975) model have been here modified. This model is suitable for multiaxial isotropic deformation. Consequently the development of anisotropy at high elongation due to molecular orientation is ignored. The PMMA matrix is assumed incompressible and described by the above mentioned viscoplastic modified model. In order to introduce new pertinent parameters in the analysis, this model has been next coupled with internal damage including both cavitation and growth of voids using original Gurson’s constitutive relation with the modification introduced by Tvergaard (1981). So, the model we have built by coupling the Bodner–Partom viscoplastic modified constitutive equations with a damage description based upon the Gurson–Tvergaard model, is expected to describe the effect of strain rate sensitivity, strain softening, strain hardening and the void volume fraction evolution. From a set of uniaxial tests, the isochoric parameters have been determined using an original procedure while the micromechanical parameters have been calibrated. The predictions of this model have been generated and compared with experimental data at each strain rate. The results show the ability of the model to correctly predict, as well as qualitatively than quantitatively, both the nonlinearity, strain rate dependence and growth of voids.
2. Experimental study In this part, the deformation mechanisms in a rubber-modified material under different strain rates and at room temperature are investigated. The deformation mechanisms are correlated with the macroscopical viscoplastic behaviour. It is well known that the cavitation in the rubber particles and the matrix yielding around the particles contribute to the deformation. However, the sequence of these toughening deformation mechanisms is not very well clarified and still constitutes one of the remaining
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questions. Furthermore, since very few quantitative data exist, there is a real need to build database to characterise plastic instability, particularly at large strains. 2.1. Material The rubber-toughened material selected in this work consists of a polybutadiene core-shell particles, with a hard shell and a soft rubber core, dispersed in polymethylmethacrylate (PMMA). The refractive index of the rubber particles was adjusted in order to obtain a transparent blend. The glass transition temperature of the blend is about 110 ◦ C and the matrix is characterized by a weight-average molar mass of about 60000 g/mol. The dispersed rubber phase represents 30% in volume fraction of the blend and particles have overall diameters of about 270 nm. This material has been selected because of its simple microstructure and its amorphous matrix. This blend is also known to exhibit ductile properties and significant volumetric growth (Gloaguen et al., 1992; 1993). Moreover, it was shown (Gloaguen et al., 1992; 1993) that shearing in RT-PMMA is promoted for an interparticle distance below approximately 65 nm, which is the case with this modified material since the mean spacings of the rubber particles, estimated from the analysis of Wu (1985), are approximately 55 nm. It can be noticed that this key role of the inter-particle distance in toughening has been extensively studied in the literature for different polymers (Wu, 1985; Borggreve et al., 1987; Muratoglu et al., 1995; Bartczak et al., 1999). The tensile specimens have been cut from rectangular plates which thickness is about 4 mm. These plates have been made from a compression moulded operation. The specimens have been next polished. The geometry of these uniaxial tension specimens, used for the VideoTraction© acquisition and control system, is specific (Fig. 1). In this work, the effect of stress triaxiality on strain localization is not examined. By adopting a large notch radius the stress triaxiality is expected to be low and thus the axial strain is assumed to be uniform across the specimen. The unmodified PMMA homopolymer tensile behaviour, tested under constant true strain rates from 10−5 up to 10−2 s−1 and at room temperature, shows that brittle fracture occurs, and that elastic and crazing deformations dominate. A typical tensile curve for the unmodified material is shown in Fig. 2, where true stress is plotted versus true axial strain. The toughness of the PMMA can be significantly improved by the addition of second phase rubber particles. The rubber toughening effect in RT-PMMA is mainly based on the initiation and growth of plasticity in the PMMA matrix. Furthermore, the macroscopic stress is lower than in the unfilled PMMA (see Figs. 2 and 3). Fig. 2 reveals the extremely brittle behaviour of the PMMA homopolymer at room temperature, where the strain and the stress at break is about 3% and 70 MPa respectively.
Fig. 1. Tensile specimen.
Fig. 2. Experimental tensile true stress-true strain curve for PMMA.
Fig. 3. Experimental tensile true stress-true strain curves at different strain rates and at room temperature for RT-PMMA.
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At the same strain rate, the breaking stress is smaller of about 20 MPa in amplitude for the RT-PMMA. The introduction of the rubber phase in the matrix completely changes the deformation mechanisms of the PMMA, the blend showing a significant ductility. The main mechanisms actived in this blend are the cavitation in the rubber particles and the shear banding in the matrix. Specimens are strained under uniaxial tension, and constant true strain rate during the test, for a range of strain rates and at room temperature (Fig. 3). The modified material which is initially clear, whitens during deformation suggesting the development of the damage mechanisms. As the whitening of this material can disturb the videomeasurement technique, a thin white coat of paint is applied on the specimen. 2.2. Tests and results The experimental setup is mainly constituted by a hydraulic tensile test machine and a videocamera. True strain rate controlled loading of the specimen is performed to remain constant during the test, by image analysis regulating the cross-head speed (G’Sell et al., 2002). The load, the position and the strains are simultaneously recorded by the system. The videomeasurement technique allows to simultaneously record the axial and transverse strains of the specimen. In the region of the specimens where the deformation is concentrated and which is generally characterized by a plastic instability, four black round markers are made on the front surface. Two are aligned along the tensile axis and two others along the transversal axis. The gravity centers of these markers are analysed during the tensile test by a data acquisition system connected to the CCD camera. A more detailed description can be found in G’Sell et al. (2002). From such measurements, volumetric and shear deformations can be therefore determined. Indeed, assuming isotropy of transverse strains, the total true volume strain εvol can be determined from the following equation: V (1) εvol = ln = ε11 + 2ε22 , V0 where V and V0 are the instantaneous and initial volume, respectively, ε11 and ε22 are the axial and transverse true strains, respectively. In the deformation process, because of the cavitation in the particles, the plastic volume strain leads to a volume increase whereas no volume change is attributed to the shear strain. In the elastic stage, a small dilatation occurs and the plastic volume strain (cavitation) is defined as follows: εcav = εvol − εvol(elast) ,
(2)
where εvol(elast) is the elastic contribution to the overall volume strain: εvol(elast) = (1 − 2ν)
σ , E
(3)
ν and E being the Poisson’s ration and Young’s modulus, respectively, and σ is the true stress expressed as: σ=
F F exp(−2ε22 ) , = S S0
(4)
S and S0 are the instantaneous and initial section, respectively, and F is the applied force. Tests have revealed that, above 5 × 10−2 s−1 , this blend is quite brittle and remains transparent; below the behaviour is ductile (Fig. 3). Typical tensile true stress-true strain curves and volume variation evolution are shown in Fig. 4 for two values of the strain rate. The ductile nature and the highly nonlinear behaviour of this RT-PMMA are clearly pointed out on this figure. The cavitation of the rubber particles inducing plastic volume strain starts approximately at the yield. The deformation process seems to be quite complex since, among a certain strain, the volume strain evolution is no longer linear. The curvature which is highlighted probably corresponds to the shearing and distortion of particles. Indeed, towards the final stage of deformation, the rubber particles seems to exhibit a weak strength against distortion compared to that against their expansion. Consequently, an increasing of the rate of shear yielding in the material is expected. To improve the understanding of the macroscopic behaviour, the deformation mechanisms can be highlighted by studying the evolution of the different components of the true axial strain. Assuming the respective contributions of plastic volume (cavitation), shear and elastic strains are linearly additive in the total strain, the shear strain is given as (Heikens et al., 1981): εshear = ε11 − εvol − 2νεelast with εelast = σ/E the elastic strain.
(5)
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Fig. 4. Experimental tensile true stress-true strain curves and plastic volume strain-true strain curves for RT-PMMA.
Fig. 5. True stress-true strain curve and deformation mechanisms in RT-PMMA at 10−3 s−1 and at room temperature.
The contributions of deformation mechanisms can be then quantified for each tensile test. A typical example is given in Fig. 5 which illustrates the general trends observed in the results. This figure shows that, whatever the deformation level is, shear deformation is the dominant dissipative mechanism. After the yield stress, the elastic contribution is globally weak. The onset of shearing occurs before the yield while the cavitation of the first voids begins approximately at the yield. When the strain rate increases, the shear strain and volume strain contributions to the total strain, respectively decreases and increases. In a general way, the strain softening is due to both the void growth and shear bands growth while the strain hardening is almost completely due to the plasticity in the matrix among a given strain value.
3. Stress distributions around a particle Although the qualitative study of the shear band propagation around particules is not the objective of this work, see for this Steenbrink et al. (1997), Smit et al. (1999), Pijnenburg et al. (1999), Socrate and Boyce (2000), Pijnenburg and Van der Giessen (2001), Seelig and Van der Giessen (2002), it has seemed us interesting to evaluate the stress distribution around a particle in order to analyse the influence of the cavitation. Experimental results have confirmed that irreversible strains developed in RT-PMMA in tensile tests may be attributed to shear strain and cavitation strain. A preliminary analysis of the stress field around a dilute spherical rubber particle is given in this section. The rubber particles in the glassy matrix act as stress concentrators when applying a remote stress. This is because of the differences in Young’s modulus and Poisson’s ratio between the particle and the matrix. Some elas-
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(a)
(b) Fig. 6. Radial stress and tangential stress distribution in uniaxial tension around (a) a rubber particle and (b) a void (r and θ are the radial and the tangential coordinate respectively and a is the radius of the particle/void).
tic analytical analyses can give the stress concentrations around a particle (Goodier, 1933; Eshelby, 1957; Tirosh et al., 1995). Fig. 6(a) shows, from the analysis introduced in Tirosh et al. (1995), the radial and tangential stress distribution for two critical directions, i.e. θ = 0 and θ = π/2. θ = 0 is the equator direction and θ = π/2 is the direction of the remote stress. The two phases of our material are considered linear elastic, isotropic, homogeneous and incompressible. Furthermore, the interface between the particle and the matrix is assumed perfect. The elastic properties of the respective phases are: νm = 0.35 and Em = 3000 MPa for the matrix material and νp = 0.499 and Ep = 1 MPa for the rubber particles. The cavitation in the particle is known to be driven by the hydrostatic part of the remote stress (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). Fig. 6(b) gives the stress distribution around the particle after the cavitation, which is mechanically considered as a void because of its low modulus. When the cavitation occurs in the particle, the tangential stress is relatively increased in the surrounding matrix, whereas the radial stress becomes null. That implies an increase of the equivalent stress which may then favor initiation of shear bands at the particle equator. Theoretically, plastic flow of the matrix is easier to be obtained around a void than around an intact rubber particle. But the great difference in the elastic properties values between the two phases implies that the shear bands do not necessary initiate around cavitated particles. In fact, and as we have underlined it in the previous section, the inter-particle distance seems to play a key role in the toughening of the studied material.
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4. Constitutive equations 4.1. Damage evaluation under strain We suppose, in a first approximation, as already done by several authors (Steenbrink et al., 1997; Smit et al., 1999; Pijnenburg et al., 1999; Socrate and Boyce, 2000; Pijnenburg and Van der Giessen, 2001; Seelig and Van der Giessen, 2002) that when cavitation occurs the soft cavitated rubber particles mechanically behave as voids in the matrix. Furthermore, in this part, the matrix behaviour is assumed strain rate independent. The void volume fraction evolution can be obtained using different models. The most popular is the Gurson (1977) potential written as: 2 σe 3 σh − 1 − f 2 = 0, + 2f cosh (6) Φ(σ , σ0 , f ) = σ0 2 σ0 where σ0 is the yield stress, σe is the Mises effective stress and σh is the macroscopic hydrostatic stress: 3 1/2 tr(σ ) and σh = σe = σ :σ . 2 3
(7)
The double dot represents double contraction of tensors and σ = σ − tr(σ )I/3 is the deviatoric stress tensor. In formula (6), f denotes the current void volume fraction: f = Vv /V , where Vv is the current void volume and V is the current total volume. When f = 0, (6) is reduced to the classical von Mises criterion. To account for the interaction between voids, three parameters (q1 , q2 , q3 ) have been empirically introduced by Tvergaard (1981) in the original Gurson form (6), as follows: 2 σe 3 σ + 2f q1 cosh q2 h − 1 − q3 f 2 = 0. (8) Φ(σ , σ0 , f ) = σ0 2 σ0 The void growth is governed by the plastic volumetric strain rate and the nucleation part is neglected since the particles are assumed equivalent to voids. The stress triaxiality is supposed constant during the process. Assuming the normality rule plays for Φ (Gurson, 1977), we easily obtain the dilatational plastic strain rate and from the mass conservation principle we can write, in uniaxial tensile case: q 3 (9) df = q1 q2 sinh 2 f (1 − f ) dεp . 2 2 Integration of Eq. (9) with the initial conditions gives: f (1 − f0 ) q 3 ln = q1 q2 sinh 2 εp , f0 (1 − f ) 2 2
(10)
where f0 is the initial volume fraction of the voids and this initial value represents here the rubber content in PMMA. Because of the plastic incompressibility of the matrix, the relative volume change of the spherical void growth is given by: V 1 − f0 . = 1−f V0 The formula (10) becomes: 3 q V f = f0 exp q1 q2 sinh 2 εp 0 . 2 2 V
(11)
(12)
This expression can be compared with the numerous analytical cavity growth models developed in the literature (McClintock, 1968; Rice and Tracey, 1969). To account for the void distortion and the mode of failure by shear bands growth, we can adapt the modification suggested by Needleman and Tvergaard (1984) by replacing f by f ∗ which is written as: p f, ε p εc , (13) f ∗ = f ∗ (f ) = p fc + k(f − fc ), εp εc , p
where εc and fc represent here the critical values from which the distortion begins, and k is a constant. Fig. 7 shows a quantitative comparison between the experimental results and the model (12). The correction (13) improves the model prediction for this material. The calibration gives q1 = 2.2 and q2 = 0.91 (with q3 = q12 ), values which are close to those suggested by Tvergaard (1981). In order to introduce new pertinent parameters in the micromechanical Gurson–Tvergaard model, this one is coupled with a phenomenological formulation which accounts for viscoplastic constitutive properties and the hardening of the material.
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Fig. 7. Void volume fraction versus plastic true axial strain at 10−3 s−1 (fc = 0.35; k = 0.4).
4.2. Viscoplastic constitutive equations Since the mechanical response of the RT-PMMA is characterized by a peak at the yield stress followed by intrinsic strain softening and strain hardening, a model to represent this highly nonlinear behaviour is required. In this section, modified constitutive equations for homogeneous glassy polymers are presented. We assume that the matrix material is isotropic, homogeneous and incompressible. The increase of temperature associated to the deformation is neglected. On a phenomenological point of view, although the deformation mechanisms of polymers and metals are different, viscoplastic constitutive equations originally developed for metals can be used to model the highly nonlinear and strain rate dependent behaviour of polymers. Since the elastic response is found to be linear, the nonlinearity is only included in viscoplastic terms. In a general way, the viscoplastic part can be formulated from the following constitutive equations: Dp = g1 (σ , Zk ),
(14)
Z˙ k = g2 (Dp , Zk ),
(15)
where g1 and g2 are two functions of the internal variables Zk , and respectively of the stress tensor σ and of the viscoplastic strain rate Dp . The variables Zk are taken to represent the internal state of the material. In this part, since the flow is assumed isochoric the viscoplastic strain rate tensor is equivalent to its deviatoric part. The functions g1 and g2 are explicited by using the Bodner–Partom model. Such a model is a unified approach since plasticity and creep are coupled. Furthermore, the viscoplastic contribution exists at all nonzero stress levels. The model gives, as most of others formulations, a relationship between the viscoplastic strain rate, the deviatoric part of the stress tensor, the Mises effective stress and the internal state variables. The original Bodner and Partom (1975) model is expressed as: 3 σ Dp = p˙ , 2 σe
(16)
where σe is the Mises effective stress, σ is the deviatoric part of the stress and p˙ is the accumulated viscoplastic strain rate: 2 p p 1/2 p˙ = . (17) D :D 3 There are many different forms for the accumulated viscoplastic strain rate which have been presented in the literature. To take into account the effect of the nonlinear plastic deformability of the matrix phase, we have modified the expression originally proposed by Frank and Brockman (2001). So, we propose to write the accumulated viscoplastic strain rate as follows: 2n σe 2 , (18) p˙ = √ D0 Z1 + Z2 3 where Z1 and Z2 are the internal state variables which are used to simulate the molecular flow strength.
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The internal state variable Z1 corresponding to the hardening effect arising from development of network alignment, has been also modified and simplified as follows: Z1 − (1 − α)Z10 W˙ p . Z˙ 1 = m (19) Z10 To account for the effect of strain softening, the new internal state variable Z2 we have introduced in the model, is expressed as: Z (20) Z˙ 2 = h 1 − 2 W˙ p . Z2s In Eq. (19) and (20), W˙ p is the viscoplastic work rate given as: ˙ W˙ p = σe p.
(21)
In the model, D0 represents the limiting value of the shear strain rate, n is the strain rate sensitivity parameter, Z10 is the initial value of Z1 which is phenomenologically related to the onset of plasticity at yield stress, Z2s is the saturation value of Z2 , m is the hardening rate parameter, h is the softening rate parameter and α is a parameter which controls the beginning of the strain hardening. The initial condition for the softening parameter is assumed null and only Z10 is taken to represent the internal state of the matrix material at the onset of the shear band nucleation. By coupling the above described viscoplastic model with the Gurson–Tvergaard void growth relationship, an enriched and complete formulation is therefore built. 4.3. Viscoplastic model including damage Some models for porous viscoplastic metallic materials have been proposed in the literature (Pan et al., 1983; Perzyna, 1986). Particularly, the Gurson model for rate independent porous plastic behaviour was used in a rate dependent form by Pan et al. (1983). Most of the models developed for porous glassy polymers are modified (Lazzeri and Bucknall, 1995; Jeong and Pan, 1995; Steenbrink et al., 1997; Pijnenburg and Van der Giessen, 2001; Jeong, 2001) or unmodified (Guo and Cheng, 2002; Imanaka et al., 2003) versions of the Gurson yield surface. The association with the viscoplasticity, in the spirit of the works of Pan et al. (1983), can be found in Steenbrink et al. (1997), Pijnenburg and Van der Giessen (2001). In this work, neither the pressure-sensibility of the matrix nor the elasticity are taken into account in the yield surface. However, the effect of the pressure on plastic flow in the matrix can be introduced directly in the expression (18) of the modified viscoplastic constitutive model. 4.3.1. Tridimensional formulation The viscoplastic constitutive equation describing the mechanical behaviour of the matrix is combined with the micromechanical model of Gurson–Tvergaard to account for both the influence of damage mechanisms in the RT-PMMA and subsequent yielding of the matrix around the particles. It is important to note (see Section 4.1) that in the Gurson–Tvergaard model, the distortional contribution is neglected compared to the dilatation since the void growth is considered to be entirely spherical. We must note that this assumption is clearly a limit of our model when describing such a material behaviour. The macroscopic rate of deformation tensor D is decomposed into an elastic part De and a viscoplastic part Dp : D = De + Dp .
(22)
The elastic constants are assumed to be independent of the strain rate and the elastic part of the rate of deformation is given by: ˙ − ν/E tr(Σ)I, ˙ De = (1 + ν)/E Σ
(23)
where Σ is the macroscopic Cauchy stress tensor and I is the identity tensor. The macroscopic viscoplastic part of the rate of deformation is derived from the normality rule and given by: ∂Φ , ∂Σ where Λ is a parameter which can be determined from the condition of equivalence of plastic work rate: Dp = Λ
(24)
Σ : Dp = (1 − f )σe p. ˙
(25)
In this last equation, σe and p˙ are the Mises effective stress and the accumulated viscoplastic strain rate, respectively, of the matrix material (see Section 4.2).
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Combining Eqs. (24) and (25) leads to: ∂Φ −1 , Λ = (1 − f )σe p˙ Σ : ∂Σ where Φ is the Gurson–Tvergaard’s flow potential (Gurson, 1977; Tvergaard, 1981): 2 Σe 3 Σh + 2f q1 cosh q2 Φ(Σ, σe , f ) = − 1 − q3 f 2 = 0, σe 2 σe
(26)
(27)
qi , 1 i 3, are the three adjustment parameters introduced by Tvergaard (1981) to improve the model predictions, f is the void volume fraction, Σe is the macroscopic Mises effective stress and Σh is the macroscopic hydrostatic stress: 1/2 3 tr(Σ) and Σh = Σe = Σ : Σ , (28) 2 3 where Σ = Σ − tr(Σ)I/3 is the macroscopic deviatoric stress tensor. Introducing the Gurson’s yielding function, the flow rule can be expressed as: f Σ 3 Σ Dp = Λ 3 2 + q1 q2 sinh q2 h I . σe 2 σe σe
(29)
Because of the plastic dilatation by void growth, the viscoplastic strain rate (29) is now both deviatoric and dilatational. We assume that the increase of void volume arises both from the growth of existing voids and from the nucleation of voids. The void volume fraction rate f˙ is decomposed into a void growth f˙grow and a nucleation part f˙nuc , as follows: f˙ = f˙grow + f˙nuc .
(30)
Void growth rate is given by the mass conservation principle: f˙grow = (1 − f ) tr(Dp ), where the trace of viscoplastic strain rate is given by: 3 Σ f tr(Dp ) = 3Λq1 q2 sinh q2 h . σe 2 σe
(31)
(32)
Experimental data suggest that it exists an incubation strain below which the cavitation is prohibited. Recently, three criteria for cavitation in an isolated particle in an infinite glassy polymer matrix, have been presented in the literature (Dompas and Groeninckx, 1994; Lazzeri and Bucknall, 1995; Fond et al., 1996). According to them, when a critical hydrostatic stress is reached the cavitation simultaneously occurs in the center of all rubber particles. However, in the RT-PMMA, or in others rubber-modified polymers, the rubber particules do not all cavitate at the same time. Therefore, the cavitation of voids can be seen as a continuous process. To capture the accelerated damage due to cavitation of voids, according to the phenomenological hydrostatic stress controlled nucleation law proposed by Jeong (2001), the cavitation part can be given by: fN 1 Σ h − σN 2 (33) exp − Σ˙ h . f˙nuc = √ 2 sZ10 2π sZ10 In the relation (33) three added constants must be determined: the standard deviation s, the mean value of the hydrostatic stress σN and the volume fraction of rubber particles which cavitate fN . 4.3.2. Uniaxial formulation In this part, the equations given earlier are reduced to the uniaxial case. The macroscopic Mises effective stress and the macroscopic hydrostatic stress are defined respectively as follows: Σ11 , 3 where Σ11 is the only component of the macroscopic stress tensor. The components of the macroscopic viscoplastic part of the rate of deformation are: P = Λ 2 Σ11 + q q f sinh q Σ11 D11 1 2 2 σ 2σ σ2 Σe = Σ11
and
Σh =
(34)
(35)
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and:
P = D P = Λ − Σ11 + q q f sinh q Σ11 D22 1 2 2 33 σ 2σ σ2
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(36)
P + 2D P . with the plastic dilatation: tr(Dp ) = D11 22 The parameter Λ becomes: Σ11 2 Σ11 −1 Σ11 p + q1 q2 f sinh q2 Λ = (1 − f )σ ε˙ 2 σ σ 2σ
(37)
p
with p˙ = ε˙ 11 = ε˙ p and σe = σ11 = σ . ε˙ p is a function of σ as in (18). The void growth rate is governed by: 3q1 q2 f (1 − f ) sinh(q2 Σ11 /(2σ )) p f˙grow = D , 2Σ11 /σ + q1 q2 f sinh(q2 Σ11 /(2σ )) 11
(38)
where the initial void volume fraction in the material is defined by f (t0 ) = f0 . The void volume fraction evolution has an effect on Young’s modulus of the RT-PMMA. In this study, a linear relation between E and f is found the most suitable and is expressed as: E = (1 − 1.2f )E0 , where E0 is now the initial Young’s modulus of the blend. The material constants that need to be determined include: E0 , D0 , n, Z10 , m, h, Z2s , α, f0 , q1 , q2 , q3 , s, σN and fN . 4.4. Identification and validation 4.4.1. Identification of the model parameters A single strain rate independent set of parameters for capturing the nonlinear behaviour, the strain rate effect and the damage evolution is the purchased goal of such a model. The identification has been performed in two stages. In a first step, an approximate estimation of the viscoplastic parameters is achieved using the uniaxial tensile true stress-true strain curves. In a second step, the damage model parameters are calibrated with experimental data and optimized to capture both the nonlinear behaviour and void growth. A first set of isochoric parameters values is determined using an analytical method described in Chan et al. (1988), Rowley and Thornton (1996), Woznica and Klosowski (2000) from the tensile simple tests across a variety of strain rates. The identification procedure was originally developed by Chan et al. (1988) for the unmodified viscoplastic model and is adapted here for the highly nonlinear behaviour of the studied material. The seven independent isochoric strain rate parameters are: D0 , n, Z10 , m, h, Z2s and α. In the original formulation of the Bodner–Partom model, D0 is fixed arbitrarily and in quasi-static problems, we generally assume that D0 = 104 s−1 (Chan et al., 1988). The yield stress σ0 , defined as the peak stress (i.e. just before softening), allows the estimation of n and Z10 from a least squares regression (Fig. 8).
Fig. 8. Yield stress versus true strain rate.
Fig. 9. Work hardening rate versus true stress.
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The others isochoric parameters have been determined from the work hardening rate defined as: γ=
dσ 1 dσ (εp ) = . p dW σ (εp ) dεp
(39)
Using polynomial curve fitting to correlate the stress versus the plastic strain at each strain rate, the analytical work hardening rate expression can be then determined. The hardening parameters h and m are computed from the negative and positive slope, respectively, of the work hardening rate plotted versus the stress (Fig. 9). Z2s and α are determined from a work hardening rate value (noted γ1 in the work hardening rate-stress curve) and the minimum stress σa after yield (σa also corresponds to zero value in γ (σ ) curve). Z2s and α are provided by the following equations: γ (Z 0 )2 Z2s = 1 1 − (1 − α)Z10 , m σ0 p γ1 Z10 mW σ0 , α = σa − exp − 0a m Z
(40) (41)
1
p p p ε p where Wa is the viscoplastic work defined by Wa = 0 a σ (εp ) dεp , εa being the plastic strain corresponding to σa . Thus, each stress-strain experimental record leads to a set of parameters. The retained values are issued from an averaging on all the sets. Correction and readjustment of some parameters are then required to account for damage in the blend. The values of the isochoric parameters obtained are: E0 = 1780 MPa, D0 = 104 s−1 , n = 10.85, Z10 = 107.25 MPa, Z2s = −22.5 MPa, m = 6.9, h = −40, α = 0.14. The parameters q1 , q2 and q3 are chosen by simultaneously fitting true stress-true strain and volume variation curves. The parameter f0 is chosen such that the model gives the best fit of dilatational curves. Since the adjustment of f0 is arbitrary this one suffers of a lack of physical basis. Taking as smallest as possible the value of f0 , the selection of the three others parameters of the normal distribution function (33) completes the model identification. The standard deviation s is arbitrary fixed, the mean value σN is equal to the initial value of Z1 and the volume fraction fN is equal to the volume fraction of rubber particles. The values of the micromechanics parameters we have obtained are: f0 = 0.01, q1 = 2.6, q2 = 2.2, q3 = q12 , s = 0.7, σN = Z10 , fN = 0.27.
4.4.2. Numerical integration and validation of the model An explicit integration scheme in a Fortran program is used to numerically integrate the system of nonlinear constitutive differential equations and to generate data for the monotonic uniaxial test at each strain rate. From the constitutive equations, the values of the unknown variables at the current time step are evaluated from their values at the previous time step using the trapezoidal method. The numerical integration stability is preserved by assigning that the rate of deformation must be always greater than the rate of the viscoplastic part. As an example, in Fig. 10, a numerical response is simulated and compared with the experimental data for two strain rates. A quite nice agreement is observed between nonlinear hardening experimental results and the predicted behaviour given by the model. The comparison with the dilatation curves is also reasonably satisfactory but the model under-estimates the void growth.
Fig. 10. Experimental and modeled true stress-true strain curves and plastic volume strain-true strain curves for RT-PMMA.
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Although the model is in agreement with the experimental data, it is important to note that validation was made in a limited range of test conditions. So, a complete validation of the model requires to enrich the data basis.
5. Conclusion The deformation mechanisms of a typical rubber-modified glassy polymer, a RT-PMMA, has been investigated over a range of strain rates and at room temperature. The links between these mechanisms and macroscopical behaviour was discussed. The toughening effect is mainly based on the initiation and growth of shear bands in the matrix. Plastic flow of the surrounding matrix leads to an arising of the void growth rate in a first step, and when a certain strain is reached, this rate decreases. Furthermore, experimental results have shown that shear bands occur before rubber particles cavitation. The cavitation of the rubber particles reduces hydrostatic tension and accelerate the rate of shear yielding. In order to describe the mechanical behaviour of the glassy polymeric matrix, we have modified the viscoplastic Bodner– Partom model. Such a model is based upon a thermodynamics formalism of irreversible processes. To account for both the dependence on shear and mean stresses, the modified viscoplastic model was then coupled with the Gurson–Tvergaard’s flow potential. The parameters are determined from the experiments. An original analytical method has been developed to estimate the modified viscoplastic model. Next, the micromechanical parameters have been optimized using an iterative process in order to obtain the best agreement between the experimental data and the predicted curves. A numerical procedure has been developed to solve the system of nonlinear constitutive equations using a trapezoidal scheme. The predictions have shown the ability of the modified viscoplastic model coupled with damage to correctly describe the experimental observations: nonlinearity, strain rate sensitivity and damage evolution are well captured. To our knowledge, it is the first time that a unified set of constitutive equations that combine viscoplasticity and damage is used to represent a quantitative polymer response. However, several aspects need to be detailed and completed in the future. The viscoplastic modified model has to be implemented in a finite element program. Since model predictions sensitively depend on micromechanics parameters, a robust parameter identification using a proper scheme is required. Furthermore, to increase the understanding of the interplay between shear and mean stress and the role of stress triaxiality, tests must be made over a range of notched specimen types with different radius. Moreover, in order to improve the model prediction, the effect of anisotropy due to molecular orientation has to be taken into account.
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