Micromechanics of toughened polycarbonate

Micromechanics of toughened polycarbonate

Journal of the Mechanics and Physics of Solids 48 (2000) 233±273 Micromechanics of toughened polycarbonate S. Socrate, M.C. Boyce* Department of Mech...

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Journal of the Mechanics and Physics of Solids 48 (2000) 233±273

Micromechanics of toughened polycarbonate S. Socrate, M.C. Boyce* Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA Received 28 January 1999; accepted 3 June 1999

Abstract Numerical studies are presented on micromechanical and macromechanical aspects of deformation mechanisms in toughened polycarbonate. The dependence of the macroscopic stress±strain behavior, and of the underlying patterns of matrix deformation, on void distribution and triaxiality of the loading conditions are discussed. The presence of voids is shown to create stress ®elds which favor shear yielding over brittle failure mechanisms and thus provide toughness even in the case of highly triaxial stress states. Additionally, we compare predictions obtained using a micromechanical model based on a traditional axisymmetric unit cell, with predictions obtained with an alternative model based on a staggered array of voids. The new model is an axisymmetric equivalent to the Voronoi tessellation of a Body Centered Cubic array of voids (V-BCC model). The V-BCC model appears to be able to better capture essential features of the mechanical behavior of the blends, and provides a more realistic cell-based representation of particle-®lled materials in general. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: C. Polymeric material; B. Voids and inclusions; B. Inhomogeneous material; C. Finite elements; A. Fracture toughness

1. Introduction The toughness of amorphous polymers varies considerably. In the glassy state, polymers such as polystyrene (PS) and polymethylmethacrylate (PMMA) fail in a

* Corresponding author. Tel.: +1-617-253-2342; fax: +1-617-258-8742. E-mail address: [email protected] (M.C. Boyce). 0022-5096/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 9 9 ) 0 0 0 3 7 - X

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brittle manner by the uncontrolled initiation and propagation of crazes. These polymers have been successfully `toughened' by the blending and/or dispersion of second-phase particles, most often some form of elastomer, which act to provide a profusion of craze initiation sites. This controlled crazing behavior is the primary mechanism governing the enhanced toughness of these polymers. Under special circumstances, crazing can be arrested altogether in favor of shear yielding, which provides even more dramatic increases in toughness. In addition to brittle PS and PMMA type polymers, there are glassy polymers, such as polycarbonate (PC), which are considered to be intrinsically tough. During most loading conditions, PC will favor shear yielding over brittle failure mechanisms such as crazing. In the event of shear yielding, a polymer will typically undergo extensive plastic strains and therefore will display a higher toughness level due to the work associated with the large strain plastic ¯ow of the material. Although considered a tough material, PC will craze and fail in a brittle manner under plane strain notch conditions (Yee, 1977; Nimmer and Woods, 1992) where the high triaxiality state favors crazing over shear yielding. In order to suppress such brittle behavior, PC is toughened by incorporation of second-phase particles (Yee 1977; van der Sanden et al., 1994; Cheng et al., 1995), following approaches used for brittle polymers such as PS and PMMA. The particles are thought to promote shear yielding of the matrix material rather than brittle failure even in the presence of extreme geometry and loading conditions. In this paper, we explore the micromechanics of particle-enhanced polycarbonate under uniaxial and triaxial loading. Typically, PC is modi®ed by blending with elastomeric particles which cavitate early-on during loading due to the combination of thermal residual stresses (Boyce et al., 1987) and applied tension. This study therefore focuses on the case of PC toughened by spherical voids (i.e. cavitated elastomeric particles). While the analyses to be presented are speci®c to PC, qualitative aspects of the results are also applicable to other amorphous polymers that shear yield, including thermoplastic and thermoset polymers. The emphasis of the paper is on how the presence of heterogeneities promote shear yielding of the matrix material even in the presence of high triaxiality. In particular, we introduce a simpli®ed three-dimensional micromechanical model which is able to capture some essential patterns of void interaction which are overlooked in traditional models. The ability of the new micromechanical model to more accurately account for the in¯uence of neighboring heterogeneities can be valuable in the study of other two-phase material systems of similar morphology. The dependence of the macroscopic stress±strain behavior and of the underlying matrix deformation patterns on void volume fraction, void distribution, and triaxiality of the loading condition is addressed. 2. Description of micromechanical models Micromechanical models are frequently invoked in order to understand the

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local mechanics and mechanisms governing the macroscopic elastic±plastic deformation of heterogeneous solids. There are four basic features to a micromechanical model for a generic multi-phase composite: (1) the geometric de®nition of a representative volume element (RVE) which embodies the essence of the microstructure under consideration; (2) the constitutive description of the mechanical behavior of each phase; (3) the constitutive description of the interphase boundaries (interfaces); and (4) a homogenization strategy for predicting the macroscopic mechanical behavior of the aggregate based on the mechanical response of the RVE. As previously mentioned, this study is focused on voided PC, where the voids represent non-adhering (cavitated) rubbery particles. Therefore, only the constitutive model for the matrix material (PC) needs to be de®ned, and the particle/matrix interface can be modeled as a traction-free boundary for the matrix phase. In the following sections we introduce the RVEs selected for our study; describe the essential features of a constitutive model for PC; and discuss the boundary conditions imposed on the RVEs and the averaging scheme used to obtain macroscopic quantities from local (microscopic) ®elds. 2.1. The representative volume element (RVE) The material microstructure of a matrix ®lled with spheroidal inhomogeneities (whether voids or second-phase particles) is frequently encountered in several classes of materials, including metals, ceramics, and polymers. Typically, the second-phase particles are not uniform in size, and are dispersed in an irregular pattern throughout the matrix. A well established approach for predicting the macroscopic mechanical behavior of these morphologically complex two-phase systems relies on the introduction of a spatially periodic Representative Volume Element (RVE). It is assumed that each RVE deforms in a repetitive way, identical to its neighbors. Periodic boundary conditions are imposed on each RVE in order to ensure compatibility of the deformation ®eld (i.e. no separation or overlapping of material at the boundary between two adjacent RVEs). While analytical approaches can be used to model RVEs characterized by simple geometry and elastic constituent materials, it is necessary to introduce numerical techniques, such as the ®nite element method, in order to study increasingly complex aspects of behavior. The accuracy of predicted macroscopic composite behavior based on the averaged RVE stress±strain ®elds clearly depends on how closely the RVE captures the morphological features of the actual microstructure. 2.1.1. E€ects of RVE morphology for plane strain RVEs The degree of complexity of the system is often reduced by considering a twodimensional approximation of the three-dimensional continuum, assuming plane strain conditions for a two-dimensional model of the material de®ned in a plane parallel to the loading direction. Thus, a 3-D distribution of spherical particles

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becomes a 2-D distribution of cylindrical particles with the axes of the cylinders normal to the plane of the model. In its simplest implementation, the 2-D RVE approach is based on a Simple Square Array (SSA) of particles, with the axes of the array parallel to the loading direction(s) (see Fig. 1(a)). Using symmetry and periodic boundary conditions, the RVE is reduced to the shaded area in Fig. 1(a), often referred to as the `unit cell'. Compatibility of the deformation ®eld is enforced by constraining the boundaries of the cell to remain straight and parallel to the principal axes of the model throughout the loading history. This somewhat crude idealization of the complex morphology of the two-phase system still retains some important features of the actual material, allowing investigations of the e€ects of microstructural parameters such as volume fraction, constitutive behavior of the constituent phases, and interface properties. As long as the second phase volume fraction, f, is low, the particles are essentially isolated, and their interaction does not dominate the composite response. Under these conditions, the SSA model can yield suciently accurate predictions for the macroscopic behavior of the two-phase aggregate. On the other hand, high levels of f are associated with a signi®cant overlap between the stress ®elds of neighboring particles. Typically, particle interactions become signi®cant and strongly a€ect macroscopic composite behavior when the interparticle distance approaches the average particle diameter. For a regular particle spacing, this condition corresponds to a volume fraction of about 8%. Commercial toughened polymer blends are characterized by a wide range of rubber volume fractions, but typically f is between 5 and 30%. Due to processing, the spatial distribution of rubber particles often exhibits a degree of clustering. Experimental observations by Yee (1977) indicate that in PC blends the rubber phase tends to aggregate in two-to-three particle clusters. Therefore, it can be expected that particle interactions will play a substantial role in the aggregate behavior, even in blends with lower rubber content. A successful model for this

Fig. 1. Geometries of two-dimensional (plane strain) RVEs. Only the shaded area is modeled by virtue of symmetry and periodicity of the void array. (a) Simple square array; (b) staggered square array; (c) hexagonal array.

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class of material must be able to accurately capture the actual mechanisms of interaction between rubber particles. Numerous investigators have demonstrated, through experimental and modeling studies (Magnuson et al., 1988; Spitzig et al., 1988; Christman et al., 1989; Bao et al., 1991; Povirk et al., 1992; Becker and Smelser, 1994), that the spatial distribution of second phase inhomogeneities and the RVE geometry can strongly a€ect macroscopic material response, due to the di€erent patterns of deformation and localization which arise in the matrix. Of particular relevance for the class of materials examined in this study is a recent investigation by Smit et al. (1998a) which explores microstructural e€ects on plastic deformation of voided polycarbonate under uniaxial tension. By considering plane strain RVEs containing random distributions of multiple particles of di€erent sizes, the nature of matrix plastic straining is revealed to be one of shear banding at angles between particles. The prevailing direction of shear localization appears to be at an angle of approximately 2458 to the direction of the applied stress. These results are in good agreement with experimental studies (Kinloch et al., 1983; Pearson and Yee, 1986; Sue and Yee, 1988; Kinloch, 1989), where localization of plastic ¯ow in the matrix at angles of approximately 458 to the direction of maximum tensile stress is observed. Micrographs by van der Sanden et al. (1994) also substantiate these ®ndings. This pattern of deformation cannot be obtained by modeling the material through the SSA model of Fig. 1(a). In the SSA model, the enforced alignment of voids on planes normal to the loading direction brings about a highly nonuniform distribution of void area fraction along the loading axis: layers of pure matrix alternate with layers with high concentrations of voids. For the range of void volume fraction of relevance for this study, matrix ¯ow localizes in these weaker layers, with shear patterns which propagate across the ligaments, transversely to the principal loading direction. Predictions in better agreement with the observed diagonal shear banding pattern between particles can be obtained with RVEs based on a staggered square array, the two-dimensional analog to a three dimensional BCC lattice (see Fig. 1(b)). Several investigators (e.g. Fukui et al., 1991; Huang and Kinloch, 1992; Smit et al., 1998b) considered RVEs based on this spatial arrangement. This particle distribution fundamentally alters the pattern of matrix deformation. While plastic deformation still initiates at the equator, the plastic deformation propagates in the form of diagonal bands at 458 between the particles. Steenbrink and van der Giessen (1998) considered an alternative staggered array with particles arranged on a hexagonal array (see Fig. 1(c)). This particle arrangement also promotes the formation of diagonal shear bands. While unit cell RVEs based on staggered lattices can e€ectively capture some important features of the deformation patterns, they cannot truly represent the complexity of the two-phase material. In particular, the regularity of the model forces each deformation event to occur simultaneously at each particle in the composite. Multi-particle models, such as the RVEs introduced in Smit et al. (1998a), can capture the e€ects of an irregular microstructure. The results of Smit

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et al. reveal the important percolation of plastic ¯ow through a distributed heterogeneous structure (somewhat analogous to the simulations of Boyce and Chui (1998) on the e€ect of the distributed nature of polymer structure on plastic ¯ow). Smit et al. discuss at length how the distributed nature of the structure fundamentally alters the macroscopic stress±strain behavior of the composite, eliminating any macroscopic strain softening that is present in the homo-polymer (and is predicted to occur when a SSA model is used to model the ®lled polymer). It should be noted that, due to clustering e€ects associated with the random distribution of voids, di€erences in macroscopic behavior predicted using the multiparticle RVE and the SSA model can be observed at fairly low void volume fractions ( f = 2.5%), con®rming the need for accurate modeling of particle interactions even for blends with lower rubber content. These results illustrate the important role of particle distribution on the progression of plastic strain in heterogeneous material systems. However, the twodimensional plane strain nature of these models misrepresents the e€ects of the actual three-dimensional nature of the structure. Therefore, three-dimensional loading conditions, such as the highly triaxial stress state at the root of a notch, cannot be realistically simulated. In many instances, PC being one such case, it is

Fig. 2. The SHA model. (a) A three dimensional array of stacked hexagonal cylinders, each containing a spherical particle; (b) the SHA axisymmetric RVE in its reference con®guration; (c) compatible deformation of the SHA cell under applied axial and radial macroscopic stresses. A neighboring cell is outlined as a graphical aid to visualize the RVE compatibility constraints.

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only in the presence of high triaxiality that toughness is lost. Therefore, it is as important, if not more important, to understand the e€ects of particles on matrix plasticity under conditions of high triaxiality. Thus, the objective of this study is to construct a simpli®ed RVE which is able to capture the essential features of particle interactions and can be used to simulate triaxial loading conditions.

2.1.2. Stacked hexagonal array RVE Since its introduction (Tvergaard, 1982) to aid in the study of porous plasticity in metals, the axisymmetric RVE based on a Stacked Hexagonal Array (SHA) of particles has become one of the staples in micromechanical analyses. A random distribution of particles is idealized by considering a regular three-dimensional array of hexagonal cylinders of matrix material, each containing a spherical particle (see Fig. 2(a)). A further simpli®cation is introduced by modeling the problem in axisymmetric geometry, as discussed in Tvergaard (1982), where it is argued that an in®nite series of stacked circular cylinders containing spherical particles is an acceptable approximation for the idealized three-dimensional con®guration (see Fig. 2(b)). Symmetry arguments are then used to limit the RVE to 1/4 of the axisymmetric cell. The geometry of the SHA unit cell is fully de®ned through three parameters: R0, the initial radius of the unit cell; H0, the initial height of the cell; and r0, the initial radius of the particle. The most common  determined by the choice for these parameters is R0=H0=1, so that r0 is uniquely 1 3 3 initial void volume fraction, f0 through the relation r0 ˆ … 2 f0 † . This representation obviously requires the loading conditions to be axisymmetric as well, with a macroscopic stress state which can be fully characterized by specifying the levels of the applied axial and radial stresses, Sz, and Sr (see Fig. 2(c)). The e€ects of triaxial loading conditions on the RVE can then be easily investigated by varying the triaxiality TS of the applied macroscopic stress: TS ˆ

Sm , Se

…1†

where Se and Sm are, respectively, the macroscopic Mises stress and mean stress: Se ˆ kSz ÿ Sr k,

1 Sm ˆ …Sz ‡ 2Sr †: 3

…2†

This traditional RVE is used in this paper as a point of reference for comparison purposes and will be referred to as the SHA model. In line with similar studies in the literature, we have selected a geometry of the cell with R0=H0=1. The in¯uence of neighboring particles is captured by enforcing the symmetry of the particle distribution through proper speci®cation of the boundary conditions, i.e. by constraining the RVE to remain a circular cylinder throughout the analysis (see Fig. 2(c)). The macroscopic deformation of the RVE can then be described

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through the axial displacement of the top plane (z = 1.0) of the cell, Uzv1.0, and the radial displacement at the outer cell radius, Ur(z )=constant=Urv0.5. It should be noted that, conceptually, the SHA model is the 3-D analog to the 2-D SSA model of Fig. 1(a), and in fact it su€ers from the same shortcomings, as further illustrated later in sections 3 and 4. While the SHA model has been used extensively in the study of deformation mechanisms in a variety of materials, the number of studies speci®cally addressing elastic±plastic deformation of ®lled polymers is somewhat limited. Huang and Kinloch (1992) utilized the SHA model to study the plastic deformation of an epoxy ®lled with either rubber particles or voids subjected to uniaxial tension. As expected, they found that plastic deformation initiates at the equator of the particle due to the stress concentration e€ect, and then proceeds to spread radially across the remaining ligament and then axially up the ligament. The authors found that the SHA model was unable to capture the experimentally observed nature of matrix deformation between particles. They therefore abandoned the SHA model in favor of the two-dimensional plane strain model based on a staggered particle array, as discussed in the previous section. In more recent studies, Steenbrink et al. (1997, 1998) utilized the SHA model together with a more physically representative matrix constitutive model to study void growth in polycarbonate under various levels of triaxial loading. They considered blends with two levels of initial void volume fraction. At the lower level of volume fraction considered in their study, f0=0.5%, the voids are essentially isolated, so that the particular choice of RVE should not a€ect the results of the analyses. At the higher level of volume fraction, f0=8.3%, e€ects of void interactions are not negligible. Patterns of matrix deformation identi®ed in

Fig. 3. Construction of the three-dimensional V-BCC cell. (a) Particles arranged on a regular BCC array; (b) the truncated octahedron is obtained through Voronoi tessellation of the BCC lattice.

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Fig. 4. The three-dimensional V-BCC cell. Each layer of material is constituted by two antisymmetric families of cells. The reference RVE, centered on particle P0, is shaded, while a neighboring (antisymmetric) RVE, centered on particle P1, is outlined.

their study, and qualitative characteristics of the macroscopic response of the SHA model, are consistent with our ®ndings for a similar level of second phase volume fraction, f0=10%, as further discussed in the section 3. 2.1.3. Body centered cubic and body centered tetragonal RVEs In order to overcome the inherent limitations of the SHA model, we have developed an alternative formulation for an axisymmetric cell based on a staggered array of second phase particles. In our formulation, the random twophase composite is idealized by arranging the particles on a regular Body Centered Cubic (BCC) lattice (Fig. 3(a)), and building the RVE cell using a Voronoi tessellation procedure. This procedure is based on three elementary steps (Dib and Rodin, 1993). First, the particle at the center of a reference BCC cube (particle P0 in Fig. 3(b)) is connected by straight line segments to the eight particles (P1) at the vertices of the cube, and to the particles (P2) at the center of the six adjacent BCC cubes. Second, each segment is bisected by a plane. Third, a truncated octahedron is formed as a body bounded by these planes (Fig. 3(b)). Square facets bisect the `center-center' segments, while hexagonal faces bisect the `center-vertex' segments. The truncated octahedron, also known as the Wigner± Seitz cell, is a highly symmetric polyhedron (it possesses nine symmetry planes), and it completely ®lls space1. In order to mathematically describe the cell, we introduce a cartesian coordinate system with origin at the center of the particle, P0, located at the centroid of the reference truncated octahedron, and with axes aligned with the principal directions

1 These properties can be fully exploited to model creep deformation in polycrystals, considering octahedral grains arranged on a periodic three-dimensional array (Anderson and Rice, 1985; Dib and Rodin, 1993).

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of the lattice (Fig. 3(b)). We can normalize the cell dimensions by assuming unit distance between (staggered) lattice planes, so that the positions of the particles P1 at the vertices of the reference BCC cube are given by permutations of the coordinates {21, 21, 21}. If we consider a layer of material between z = 0 and z = 1, we see (Fig. 4) that the layer is constituted by two kinds of cells: cells centered at particles on the z = 0 plane, and an equal number of cells centered at particles on the z = 1 plane. In Fig. 4 we consider two cells representative of these two families: cell C0 centered on particle P0, and cell C1 centered on one of the P1 particles at z = 1. Let us consider a cross-section of the cells with the plane z=x, and denote the areas of the cross-sections of cells C0 and C1 as A0(x ) and A1(x ), respectively. Symmetry conditions, together with the space-®lling properties of the cells, provide the following constraints: A0 …x† ‡ A1 …x† ˆ constant,

…3†

A0 …x† ˆ A1 …1 ÿ x†,

…4†

which yield:

Fig. 5. The axisymmetric V-BCC model. (a) the V-BCC axisymmetric RVE in its reference con®guration; (b) compatible deformation of the V-BCC cell under applied axial and radial macroscopic stresses. A neighboring cell is outlined as a graphical aid to visualize the RVE compatibility constraints.

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A0 …x† ‡ A0 …1 ÿ x† ˆ 2A0 j0:5 ,

243

…5†

where A0v0.5 is the cross-sectional area at the midplane of the cell (z = 0.5). The actual pro®le of cell cross-sectional area can be easily obtained from the truncated octahedron geometry:  2…2:0 ÿ …0:5 ‡ z†2 † if 0:0RzR0:5 : …6† A0 …z† ˆ 2…1:5 ÿ z†2 if 0:5RzR1:0 For this geometry, as well as for any cell geometry based on tessellation of a normalized BCC lattice, A0v0.5=2. If we limit our model to loading conditions which are axisymmetric about the zaxis, using arguments similar to those invoked in constructing the SHA model, we can substitute the three-dimensional truncated octahedron cell with an equivalent axisymmetric RVE (Fig. 5). We will refer to this axisymmetric cell as the V-BCC model, as it is based on the Voronoi tessellation of the BCC lattice. As in the SHA model, the height of the V-BCC cell in its reference con®guration is H0=1; however, the external radius of the cell, R0, varies with z so as to match the A0 pro®le as de®ned in Eq. (6): p …7† R0 …z† ˆ A0 …z†=p: The compatibility constraint (5) can then be recast in terms of R0(z ): …R0 …x††2 ‡ …R0 …1 ÿ x††2 ˆ 2…R0 j0:5 †2 ,

…8†

where R0v0.5 is the radius of the cell at the midplane (see Fig. 5(a)). For a cell based on a regular BCC lattice of second phase particles, such as the V-BCC cell, p …9† R0 j0:5 ˆ 2=p ˆ 0:798: When axisymmetric loading is applied to the composite (see Fig. 5(b)), geometric compatibility of the deformation in the two families of antisymmetric cells requires that the pro®le of the deformed cell outer radius, R(z )=R0(z )+Ur (z ), satis®es a relationship analogous to (8), so that the following constraint must be applied to the pro®le of radial displacement at the outer boundary of the cell, Ur (z ): …R0 …x† ‡ Ur …x††2 ‡ …R0 …1 ÿ x† ‡ Ur …1 ÿ x††2 ˆ 2…R0 j0:5 ‡Ur j0:5 †2 ,

…10†

where Ur(x ) is the radial displacement for a point at the outer radius of the cell which is at z=x in the undeformed cell con®guration, and Urv0.5 is the radial displacement of the point with initial coordinates (r=R0v0.5, z = 0.5). Symmetry of the axial deformation for the two families of cells about the cell midplane (z = 0.5) introduces an additional constraint on the pro®le of axial displacement at the outer cell radius: Uz …x† ‡ Uz …1 ÿ x† ˆ 2Uz j0:5 ,

…11†

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Fig. 6. Construction of cylindrical RVEs based on staggered particle arrays. (a) Tessellation of the BCC lattice through rectangular cylinders; (b) cylindrical BCC RVE; (c) cylindrical BCT RVE.

where Uz(x ) is the axial displacement for a point at the outer radius of the cell which is at z=x in the undeformed cell con®guration, and Uzv0.5 is the axial displacement of the point with initial coordinates (r=R0v0.5, z = 0.5). Eqs. (10) and (11) de®ne the constraints which must be imposed on the V-BCC cell to account for the presence of the anti-symmetric family of cells, which, together with the family of the reference cell, forms the BCC lattice. At this point we should note that there is nothing special or unique about the particular pro®le of cell outer radius associated with the V-BCC model. This is simply the pro®le associated with one particular choice of cut planes introduced to apportion matrix material to each of the two families of cells centered on the staggered BCC lattice. We can choose an in®nite number of di€erent pro®les of the cell, and, as long as the compatibility condition (8) is satis®ed, and the cell midplane-radius is given by (9), each of these choices is a valid representation of a two-phase composite, with second phase particles arranged on a regular BCC lattice. In particular, we can consider a cylindrical cell (Fig. p6(b)) with a uniform value of the undeformed cell outer radius, R0 …z† ˆ 2=p. This axisymmetric cell corresponds to the three-dimensional RVE con®guration depicted in Fig. 6(a). Notice that the initial geometry of the cylindrical BCC cell in Fig. 6(b) di€ers from the initial geometry of the SHA cell only in the value of the parameter R0. In order to directly compare the results of the two models, isolating the e€ects of the di€erent boundary conditions, we also considered a slight variation of the

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Fig. 7. Model predictions for stress±strain behavior of polycarbonate in uniaxial tension and uniaxial compression. Di€erences between the two curves re¯ect the pressure-dependence of yield, and molecular network orientation e€ects.

cylindrical BCC cell, and introduced a BCT model2 with R0(z )=H0=1 (Fig. 6(c)). Results of studies conducted using the BCT model are presented in Appendix A. Here, we want to stress that the key element of the BCC and BCT models which di€erentiates them from the SHA model are the boundary conditions expressed through Eqs. (10) and (11). As illustrated in Appendix A, di€erences in predictions obtained through the V-BCC and the BCT models, which are related to the slightly di€erent lattice con®gurations, are negligible when compared to the more signi®cant discrepancies between the V-BCC and SHA model behaviors. There is, however, one aspect in which the V-BCC model is more versatile than the cylindrical BCC and BCT models. Due to the curved cell boundary, it can model composites with very high levels of f0, and can therefore be used to study particle interactions in particle clusters. As a ®nal remark for this section, we want to point out that the idea of an axisymmetric model with the anti-symmetric boundary conditions (10) and (11) is not new to the ®eld of micromechanical studies. Tvergaard (1996, 1998) introduced analogous models to study the e€ects of void size di€erences on the mechanisms of void interaction in porous elastic±plastic materials. The Tvergaard studies focused on low levels of void volume fractions, and thus the distinctive

2 This model corresponds to an idealized two-phase composite in which the second phase particles are p p arranged on a Body Centered Tetragonal (BCT) lattice, with lattice spacings f p=2, p=2, 1g along the {x, y, z } lattice axes.

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features of this class of models, and their ability to capture di€erent patterns of matrix deformation, were not fully explored. 2.2. Matrix constitutive behavior The large strain elastic±viscoplastic stress±strain behavior of glassy polymers has been found to be well-modeled using the constitutive model of Boyce et al. (1988) as modi®ed by Arruda and Boyce (1993a). Variations of this model have been utilized in several micromechanical studies (Smit et al., 1998a; Steenbrink et al., 1997, 1998). Here, the Arruda and Boyce model will be used to represent the behavior of the polycarbonate matrix and is brie¯y reviewed in Appendix B. The constitutive model has been shown to be predictive of the three-dimensional nature of the mechanical behavior of glassy polymers. The model predictions for the stress±strain behavior of polycarbonate in uniaxial tension and uniaxial compression are shown in Fig. 7. 2.3. Problem de®nition and homogenization method As previously mentioned, the objective of this study is to determine the e€ects of void volume fraction and void distribution on the mechanical behavior of voided PC. In particular, we compare predictions obtained using two di€erent RVEs, the traditional SHA model and the proposed V-BCC model, in order to illustrate the implications of the corresponding modeling assumptions. We use a commercially available ®nite element program, ABAQUS, to solve the boundary value problems posed on the RVEs. We ®rst examine the behavior of voided PC under uniaxial tension (TS=0.33), for six levels of initial void volume fraction: f0={0.10, 0.15, 0.20, 0.25, 0.30, 0.35}. We then investigate the e€ects of an increase in triaxiality for a blend with an

Fig. 8. Finite element meshes used for the analysis of blends with an initial void volume fraction f0=0.25. (a) V-BCC mesh; (b) SHA mesh.

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initial volume fraction f0=0.25. In particular, we consider two levels of triaxiality: TS=1.3, typical of the root of a mild notch, and TS=2.3, typical of the region ahead of a crack tip3. The eight boundary value problems which correspond to the various combinations of porosity and triaxiality levels are investigated using both the SHA and the V-BCC models. An example of the ®nite element discretization for the two RVEs is illustrated in Fig. 8 for the case of an initial void volume fraction f0=0.25. The RVEs are discretized using axisymmetric 8-node biquadratic elements, and the meshes are suciently re®ned to capture the localized nature of matrix deformation. As discussed in section 2.1, di€erences in the geometry of the void arrays on which the two models are based are re¯ected in di€erent boundary conditions on the RVEs. In the SHA model, the cell surrounding the particle is required to remain cylindrical throughout the deformation history, resulting in the following boundary conditions on the ®nite element models of the RVE (see Fig. 2(c)): 1. nodes along the z-axis are, by de®nition, constrained to have zero radial displacement; 2. nodes along the bottom surface of the cell are constrained to have zero axial displacement, as required by symmetry about the void midplane (z = 0); 3. nodes along the top plane (z = 1.0) of the cell are required to have equal axial displacement, Uzv1.0; 4. (a) SHA nodes along the outer radius of the cell are required to have equal radial displacement, Ur(z )=constant=Urv0.5. For V-BCC ®nite element models, boundary conditions (1), (2), and (3) remain unchanged, while condition (4(a)) is replaced by the following condition (see Fig. 5(b)); (4) (b) V-BCC nodes along the outer radius of the cell are required to have radial and axial displacements which satisfy the conditions expressed through Eqs. (10) and (11). Note that, due to the symmetry of Uz(x ) about the midplane and the constraint (2) of zero axial displacement for nodes along the bottom plane (z = 0), for the V-BCC model Uz j0:5 ˆ 12 Uz j1:0 . For both the SHA and the V-BCC cells, the axial and radial components of the macroscopic logarithmic strain can be expressed as: 

H0 ‡ Uz j1:0 Ez ˆ ln H0

 ˆ ln…1 ‡ Uz j1:0 †,

…12†



 R0 j0:5 ‡Ur j0:5 , Er ˆ ln R0 j0:5 3

p The Prandtl slip-line ®eld gives a peak value TS ˆ …1 ‡ p†= 3 ˆ 2:39 ahead of the crack.

…13†

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from which the e€ective strain, Ee, and volumetric strain, EV, can be evaluated: 2 Ee ˆ kEz ÿ Er k, 3

…14†

EV ˆ ln…V=V0 † ˆ Ez ‡ 2Er ,

…15†

where V and V0 represent, respectively, the deformed and undeformed volume. Note that (15) holds for the V-BCC cell in virtue of relations (10) (11). The macroscopic stress components, Sr and Sz, are computed as appropriate volume averages of the microscopic stress components: … 1 sr …x† dV, Sr ˆ V x2V Sz ˆ

1 V

cell and the …16†

… x2V

sz …x† dV:

…17†

In the ®nite element implementation, the above expressions are evaluated in terms of the equivalent surface integrals along the outer boundary of the cells (Smit et al., 1998b; Tzika et al., 1999). The corresponding macroscopic measures for the Mises stress, Se, and mean stress, Sm, are evaluated according to Eq. (2). The loading conditions on the . cells are prescribed by specifying a constant macroscopic axial strain rate, Ez=0.01 (sÿ1)4, while the desired triaxiality of the macroscopic stress state is enforced, and kept constant throughout the deformation history, by applying appropriate levels of radial traction along the RVE outer radius (Tzika et al., 1999). The traction boundary conditions are imposed by introducing a user-element which monitors the magnitude of the axial reaction force at the displaced top boundary of the RVE, and computes the required level of radial force, which is then applied at the middle node of the outer cell boundary. The displacement boundary conditions (4(b)) are imposed via a `Multi Point Constraint (MPC)' user subroutine.

3. Results Di€erences in the geometry of the void arrays on which the SHA and V-BCC cells are based result in distinct patterns of localization of plastic deformation in the matrix, which in turn give rise to di€erent macroscopic cell behaviors. In the following sections we detail the results of our studies both in terms of homogenized RVE responses and of local deformation mechanisms. 4

In this study we do not examine the e€ects of temperature increase due to dissipation mechanisms.

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3.1. Predicted uniaxial response The RVEs are deformed up to a ®nal macroscopic axial strain Ez=40%. The macroscopic RVE responses are characterized by plotting the evolution of the macroscopic equivalent stress, Se, and of the macroscopic volumetric strain, EV, as a function of the applied level of axial deformation, Ez (Fig. 9).

Fig. 9. Macroscopic responses of the RVEs under uniaxial loading. (a) Evolution of the macroscopic Mises stress, Se, with macroscopic axial strain, Ez, predicted by the V-BCC model; (b) evolution of the macroscopic volumetric strain, EV, with macroscopic axial strain, Ez, predicted by the V-BCC model; (c) evolution of the macroscopic Mises stress, Se, with macroscopic axial strain, Ez, predicted by the SHA model; (d) evolution of the macroscopic volumetric strain, EV, with macroscopic axial strain, Ez, predicted by the SHA model.

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Fig. 10. Contour plots of g_ p , predicted by the V-BCC model, under uniaxial loading, for f0=0.10 and f0=0.25. Patterns of localizations are shown for increasing levels of axial deformation: (a) Ez 25% (macroscopic yield); (b) Ez 210%; (c) Ez 225%; (d) Ez 240%. The modeled RVE has been mirrored in order to better show the shear band patterns.

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3.1.1. Voronoi-body centered cubic RVEs In Figs. 9(a) and (b) we compare the behavior of homogeneous PC under uniaxial loading to the macroscopic behavior of the V-BCC RVEs for the six levels of initial void volume fraction considered in this study. The evolutions of macroscopic Mises stress for the six blends, illustrated in Fig. 9(a), display identical qualitative features, with gradual variations in the macroscopic cell properties. For increasing levels of void volume fraction, we note progressive reductions both in the sti€ness of the small-strain elastic response and in the macroscopic yield stress; also, the stress drop associated with intrinsic strain softening decreases in magnitude and becomes less abrupt, while the macroscopic hardening response is evident at lower strain levels. The uniformity of the macroscopic response mirrors the consistency in the local mechanisms of deformation for the six blends. To illustrate this point, contour plots of plastic shear strain rate in the matrix, g_ p , at four levels (a, b, c, d ) of macroscopic axial strain, are shown in Fig. 10 for two levels of initial void volume fraction ( f0=10%, f0=25%). These contour plots provide snapshots of the active regions of matrix shearing at typical stages of the deformation process. As an aid to visualize deformation patterns in the matrix, the unit cell contour plot is mirrored to represent the neighbouring antisymmetric family of cells5. Viscoplastic shearing initiates early-on at the equator of the void, however, macroscopic yield conditions are reached only when the plastic zone propagates in a shear band across the diagonal ligament (a ) between the two antisymmetric families of voids. In the subsequent stage, active deformation localizes in the shear band, where the material undergoes intrinsic softening. Due to the di€use nature of the shear band, the resulting drop in the macroscopic stress level is not quite as abrupt and severe as the corresponding drop in the stress strain behavior of the PC homopolymer. With increasing strain, the core of the band starts to harden, so that the rate of deformation at the center of the band decreases, and the active region of deformation spreads along the ligament (b ). The fronts of the sheared region advance as two bands of high strain rate which propagate towards the pole and towards the equator (c and d ). The macroscopic deformation of the RVE is accomplished both through ligament stretching (hence the necking-drawing characteristic of the deformation), and through ligament rotation, which brings the ligament in better alignment with the axis of loading. This additional mechanism increases the axial compliance of the cell, which can thus accommodate higher levels of macroscopic strain with a relatively moderate increase in matrix stretching. Note that the gradual rotation of the ligament is associated with a continuous variation in the relative orientation of the shear bands with respect to the loading direction, promoting alterations in the location and angle of the bands across the ligament, thus heightening the di€use nature of the deformation pattern.

5 Note that the compatibility condition expressed by Eq. (3) applies to the cell cross sections and not to the cell radii, hence the apparent mismatch at the cell outer boundaries.

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Fig. 11. Contour plots of g_ p , predicted by the SHA model, under uniaxial loading, for f0=0.10 and f0=0.25. Patterns of localizations are shown for increasing levels of axial deformation: (a) Ez 25% (macroscopic yield); (b) Ez 210%; (c) Ez 225%; (d) Ez 240%. The modeled RVE has been mirrored in order to better show the shear band patterns.

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The ligament rotation, which is made possible by the antisymmetric radial boundary conditions on the V-BCC cells, is associated with high levels of negative radial strain at the equator of the RVEs. The voids become very elongated along the loading direction without undergoing a large volume change, so that the VBCC RVEs do not display substantial levels of volumetric strain (see Fig. 9(b)). Higher levels of initial void volume fraction, f0, are naturally associated with higher levels of volumetric strain, EV; however, for increasing levels of f0, the relative increases of EV taper o€, as slender ligaments rotate more freely. 3.1.2. Stacked hexagonal array RVEs Fig. 9(c) and (d) display the macroscopic behavior of the six blends as predicted by the SHA RVEs under uniaxial loading. It is apparent that the SHA predictions and the V-BCC predictions present substantial di€erences, both in the axial stress±strain behavior as well as in the evolution of volumetric strain with axial strain. These di€erences can be interpreted in terms of the underlying patterns of plastic ¯ow in the matrix. While for the V-BCC cells the rate of internal dissipation is always minimized when the deformation localizes across the diagonal ligaments between the two antisymmetric families of voids, for the SHA cells two di€erent mechanisms of deformation can be activated, depending on the initial level of void volume fraction and on the triaxiality of the stress state6. Plastic ¯ow localization patterns characteristic of low ( f0=0.10) and high ( f0=0.25) levels of void volume fraction are shown in Fig. 11. As for the V-BCC results, the progression of local deformation is illustrated by contour plots of plastic shear strain rate in the matrix, g_ p , at four levels (a, b, c, d ) of macroscopic axial strain. At low level of initial void volume fraction, plastic ¯ow localizes in shear bands which originate at the equator of the void, and are oriented, approximately, at 458 relative to the loading direction (a ). In Steenbrink et al. (1997, 1998) this type of shear band is referred to as a `wing-like' shear band. As the plastic ¯ow in the band stabilizes (b ), further macroscopic elongation of the cell is accomplished through di€use deformation over the vertical ligament (c ), and, at higher levels of macroscopic strain (d ), over the conical region around the pole of the RVE. As for the V-BCC cell, the di€use deformation patterns are associated with a fairly stable macroscopic behavior. The rigid constraint applied at the outer radius of the SHA cell limits the negative radial strain at the equator of the void, thus necessitating void expansion in order to accommodate matrix stretching; therefore, the SHA model predicts a higher level of volumetric strain, as compared to the VBCC model. At higher levels of initial void volume fraction, the vertical ligament between adjacent voids becomes much more slender, so that the rate of internal dissipation

6 These ®ndings for the SHA RVEs are in agreement with the results of studies by Steenbrink et al. (1997, 1998), where traditional SHA unit cells were used to model the growth of voids in glassy polymers.

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is minimized when the deformation localizes at the equator of the cell, between adjacent voids. For the initial stages of the deformation history, the macroscopic axial deformation of the RVE is accomplished through a process of necking and drawing of the vertical ligament. In a manner analogous to the deformation

Fig. 12. Macroscopic responses of the RVEs under triaxial loading for a blend with initial void volume fraction f0=0.25. (a) Evolution of the macroscopic Mises stress, Se, with macroscopic axial strain, Ez, predicted by the V-BCC model; (b) evolution of the macroscopic volumetric strain, EV, with macroscopic axial strain, Ez, predicted by the V-BCC model; (c) evolution of the macroscopic Mises stress, Se, with macroscopic axial strain, Ez, predicted by the SHA model; (d) evolution of the macroscopic volumetric strain, EV, with macroscopic axial strain, Ez, predicted by the SHA model.

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mechanisms in an hourglass tensile specimen (see Fig. 11), macroscopic yield is associated with the formation of a narrow, declining shear band, at an angle of approximately 608 relative to the loading direction (a ). In the SHA studies of Steenbrink et al. (1997, 1998) this type of shear band is referred to as a `dog-ear' shear band. Note the substantially higher levels of shear strain rate associated with this pattern of plastic ¯ow localization. The intrinsic softening in the narrow band is associated with a substantial drop in the macroscopic stress. As stability is recovered at the core of the band, the deformation propagates axially under increasing levels of macroscopic stress (b ). When the root of the dog-ear shear band at the void surface reaches, approximately, the 458 location (c ), a secondary localization patterns emerges, with diagonal shear bands along the 458 direction. This secondary localization (d ) is associated with a new macroscopic softening in the RVE response, until the new band stabilizes and begins to propagate towards the pole of the RVE. The di€erent deformation patterns associated with the formation of dog-ear shear bands have signi®cant consequences on the macroscopic dilatancy of the RVE (see Fig. 9(d)). As the vertical ligament stretches, it undergoes a substantial reduction in cross-section so that the void bulges out at the equator. The rate of increase in void volume with axial strain is very large throughout the ligament drawing stage, and the values for the macroscopic volumetric strain predicted by the SHA model are much higher than the corresponding values predicted by the V-BCC model. For each of the other levels of f0 considered in this study, it is easy to infer the pattern of plastic ¯ow localization from the macroscopic behaviors plotted in Fig. 9. The three SHA RVEs with the higher values of volume fractions, f0={0.25, 0.30,0.35}, display similar deformation mechanisms, with the formation of dog-ear shear bands. With increasing levels of void volume fraction, the sharpness of the localization patterns is enhanced, so that the macroscopic behavior displays a higher degree of instability. The three SHA RVEs with the lower values of volume fraction, f0={0.10, 0.15, 0.20}, display similar deformation mechanisms, with the formation of wing-like shear bands.

3.2. Predicted response under triaxial loading As noted earlier, it is under high levels of triaxiality that toughness is lost in the homopolymer. The axisymmetric nature of both the V-BCC and SHA RVEs permits the study of the in¯uence of triaxiality on the behavior of the blended polymer as described in section 2. The RVEs are now deformed up to a ®nal macroscopic axial strain Ez=80%, while maintaining ®xed triaxiality levels TS={0.33, 1.3, 2.3}, characteristic of uniaxial tension, mild notch, and crack environment. The macroscopic RVE responses are illustrated by plotting the evolutions of the macroscopic equivalent stress, Se, and of the volumetric strain, EV, with increasing levels of applied axial strain, Ez (Fig. 12).

Fig. 13. Contour plots of g_ p , predicted by the V-BCC model, under triaxial loading, for f0=0.25. Patterns of localizations are shown for increasing levels of axial deformation: (a) Ez 24% (macroscopic yield); (b) Ez 2 15%; (c) Ez 240%; (d) Ez 250%. The modeled RVE has been mirrored in order to better show the shear band patterns.

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Fig. 14. Contour plots of g_ p , predicted by the SHA model, under triaxial loading, for f0=0.25. Patterns of localizations are shown for increasing levels of axial deformation: (a) Ez 24% (macroscopic yield); (b) Ez 2 15%; (c) Ez 240%; (d) Ez 250%. The modeled RVE has been mirrored in order to better show the shear band patterns.

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3.2.1. Voronoi-body centered cubic RVEs Fig. 12(a) and (b) compare the V-BCC predictions for the macroscopic behavior of an RVE with initial void volume fraction f0=0.25 (25/75 blend) under triaxial loading (TS=1.3, TS=2.3), to the V-BCC predictions for the RVE under uniaxial loading (TS=0.33). The macroscopic Mises stress curves, in Fig. 12(a), display the inverse relationship between macroscopic yield stress and triaxiality typical of porous materials, as well as a slight increase in the macroscopic elastic compliance with increasing levels of triaxiality. The more abrupt post-yield softening behavior in the high triaxiality cases is associated with sharper patterns of plastic ¯ow localization, as illustrated in Fig. 13, which display contour plots of g_ p in the matrix at four levels (a, b, c, d ) of macroscopic axial strain for the highest triaxiality case (TS=2.3). Following initial yielding at the equator of the void, the deformation localizes in a narrow shear band which spans the diagonal ligament between antisymmetric RVEs (a ). The next stage of RVE deformation is accomplished through necking/drawing of the ligament (b ). The macroscopic Mises stress remains essentially constant throughout the drawing stage, and increases again only after the entire ligament has been stretched (c ). In the ®nal stage of the deformation history, the plastic zone propagates to the regions around the pole of the RVE (d ), where the triaxial stress state generates substantial radial straining. The gradual void growth associated with these deformation patterns is mirrored in a gradual increase in volumetric strain with axial strain. A slight rise in volumetric strain rate accompanies the onset of yielding and radial stretching at the pole region. Di€erent levels of triaxiality for the macroscopic loading conditions are re¯ected in substantial variations in cell dilatancy, which are very apparent even at low levels of macroscopic deformation. 3.2.2. Stacked hexagonal array RVEs The macroscopic response of the SHA cell for three levels of triaxiality (TS=0.33, TS=1.3, TS=2.3) is shown in Fig. 12(c) and (d) for the 25/75 blend. We note a marked e€ect of triaxiality on the macroscopic yield stress (Fig. 12(c)), analogously to the V-BCC predictions, while the volumetric strain curves in Fig. 12(d) display an initial regime where the triaxiality level has no e€ect on the macroscopic cell dilatancy. Again, we can interpret these results in terms of local mechanisms of deformation. At moderate and high levels of triaxiality, the SHA model predicts dog-ear patterns of plastic ¯ow localization even for very low levels (less than 1%) of initial void volume fraction, as extensively discussed in Steenbrink et al. (1998). The progression of local deformation mechanisms shown in Fig. 14 for the 25/75 blend under high triaxiality (TS=2.3) is representative of the cell behavior for a wide range of triaxiality and porosity levels. In the initial regime of the deformation history (a and b ), visco-plastic stretching of the PC matrix is limited to the vertical interparticle ligament, which undergoes a necking-drawing process. Throughout this stage the upper portion of the cell remains in the elastic regime, accommodating extremely low levels of radial strain. The void growth is constrained by the rigid outer perimeter of the

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cell, so that the macroscopic volumetric strain rate is essentially independent of the applied level of macroscopic radial stress. It is only after drawing of the vertical ligament is complete, that active plastic deformation localizes at the pole of the RVE, and propagates along the horizontal ligament (c ). As the horizontal ligament yields and stretches (d ) the RVE radial compliance increases dramatically, resulting in a sharp sudden rise in the macroscopic volumetric strain rate (Fig. 12(d)). 4. Discussion The micromechanical studies presented in this paper provide insight into two areas. Comparisons between predictions for the blend behavior provided by the VBCC model and the SHA model illustrate the implications of modeling idealizations. Comparisons of macroscopic and local ®elds for the V-BCC cell and for the PC homopolymer yield some practical understanding in relation to toughening mechanisms in ®lled polymers. 4.1. Modeling implications Under uniaxial loading, for the three blends at low porosity levels (i.e. f0 R 20%), the V-BCC and SHA models appear to provide similar macroscopic predictions. However, upon closer examination of the results in Fig. 9(a) and (c), it is apparent that the V-BCC macroscopic stress±strain curves are characterized by a lower macroscopic yield strength, reduced strain softening, and an earlier onset of macroscopic strain hardening. Moreover, the SHA model shows strain softening to increase with increasing levels of void volume fraction, whereas the V-BCC model displays the opposite trend. It should be noted that even these arguably minor di€erences in the macroscopic stress±strain behavior will result in substantially di€erent predictions for the macroscopic deformation patterns in a tensile specimen; a material characterized by the SHA stress±strain behavior will undergo pronounced localization of deformation in shear bands, while a material characterized by the V-BCC behavior will display more di€use patterns of deformation. Predictions for the evolution of volumetric strain (Fig. 9(b) and (d)) also strongly depend on the modeling assumptions7 . Consistent with experimental results reported by van der Sanden et al. (1994), the V-BCC volumetric strain begins to saturate at macroscopic axial strains above 10%; conversely, the SHA curves exhibit a linearly increasing volumetric strain. These di€erences would clearly impact the characteristics of any Gurson-like homogenized constitutive model for the porous polymer.

7 We should note that a characterization of the blend dilatancy cannot be obtained through 2D plane strain models. Even the elaborate multiparticle model of Smit et al. (1998a) provides extremely high predictions for the evolution of volumetric strain under `uniaxial tension'.

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Fig. 15. A comparison of pro®les of transverse void area fraction, Af , as a function of axial position for the SHA and the V-BCC models. A random array of void would display a constant level of Af .

Much more substantial di€erences between the predictions of the two models arise under uniaxial loading at higher levels of porosity (i.e. f0 r 25%), and under triaxial loading. Under these conditions8, while the V-BCC model predicts diagonal patterns of localization which are in good agreement with experimental evidence and multi-particle plane strain model predictions (as discussed in section 2.1), the SHA deformation mechanisms involve the formation of dog-ear shear bands and axial ligament stretching, for which we are not aware of any substantiating experimental evidence. Under uniaxial loading conditions, for high levels of void volume fractions, the SHA predictions for the macroscopic stress state (Fig. 9(c)) are characterized by dramatic softening, and a secondary instability associated with the de®nition of a secondary pattern of localization, as discussed in section 3.1. Conversely, the VBCC model presents consistent predictions at all levels of second-phase volume fraction with gradual variation of the macroscopic cell properties. Predictions for the evolution of volumetric macroscopic strain display an even higher degree of inconsistency; while the SHA model predicts dramatic increases in volumetric strain (up to 35%), V-BCC model predictions saturate at a volumetric strain

8

It is important to point out that these are the most relevant conditions in understanding toughening mechanisms, as the crucial initial stages of macroscopic deformation of the blend are always controlled by local patterns of stress and matrix straining within particle clusters, and/or within regions of high triaxiality.

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around 5% for all values of initial porosity. Again, these di€erences would have a signi®cant impact in the construction of a homogenized constitutive model for the blend. Di€erences in model predictions for high levels of triaxiality are signi®cant both in terms of the evolution of the macroscopic equivalent stress, where the SHA model displays two abrupt softening events, as well as in the predicted cell dilatancy. The latter is particularly signi®cant at lower levels of macroscopic deformation (Ez < 40%), while di€erences are less dramatic at the latest stages of deformation. The task of ®tting SHA model predictions with a Gurson-type model appears particularly daunting when considering the apparent lack of correlation between triaxiality and volumetric strain, up to a signi®cant level of macroscopic deformation (Ez 2 20%). As shown in Fig. 12, the V-BCC model predicts a more realistic and stable macroscopic response in terms of both stress± strain behavior and cell dilatancy. The shortcomings and inherent limitations of the traditional SHA model arise, to a large extent, from the highly non-uniform distribution of porosity along the principal loading axis (z-axis), which characterizes the void-array on which the model is based. In Fig. 15, we compare pro®les of void area fraction, along the zaxis, in the SHA and in the V-BCC models9, for a blend with 25% void volume fraction. For a random void distribution, over large volumes of the blend, the void area fraction, Af , should be approximately constant at 25%. For the V-BCC pro®le, there is only a small range of variation in Af with axial position. In contrast, for the SHA model, layers of pure PC alternate with layers of high porosity. This nonuniformity results in the localization of matrix deformation in the weak porous layer, while the pure PC layer rigidly translates, still remaining in the elastic regime. Due to the constraint of uniform radial displacement at the outer boundary of the cell, the radial compliance of the SHA RVE is controlled by the compliance of the sti€ polymer layer, so that the radial macroscopic cell strain remains negligible, under all triaxiality levels, up to when plastic deformation spreads to the pure PC layer (Fig. 16(b)). Due to these e€ects, predictions for the blend dilatancy obtained through the SHA model seem, at best, questionable. Conversely, the macroscopic ligament rotation permitted by the more ¯exible boundary conditions on the V-BCC RVE allows the cell to respond to the triaxiality of the loading condition with substantially di€erent trends of evolution for the macroscopic radial strain (Fig. 16(a)), which are accompanied by more realistic predictions for the blend dilatancy. As discussed in Smit et al. (1998a), predictions of post-yield softening in toughened polymeric blends are an artifact introduced by unit cell models, due to the underlying assumption that yield occurs simultaneously at each (identical) particle in the blend. It is only through the introduction of a random spatial distribution of second-phase particles/voids that a model can capture the stable macroscopic blend behavior, resulting from the arbitrary order of subsequent yield 9

Averaged over two neighboring antisymmetric cells.

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Fig. 16. Evolution of the macroscopic radial strain, Er, for a blend with initial void volume fraction f0=0.25, as predicted by the V-BCC model and by the SHA model, under the three levels of triaxiality considered in this study.

events at single particles. However, while a multiparticle model is a feasible proposition for two-dimensional studies, a three-dimensional multiparticle model introduces much higher computational requirements. If the blend behavior is to be modeled via a unit cell RVE, it is arguable that predictions of severe instability in the macroscopic blend response, as predicted by the SHA model, should be considered with extreme caution. In light of these remarks, the sharpness and severity of the post-yield softening behavior associated with the formation of dogear shear bands in SHA cells are particularly questionable, especially since this pattern of deformation is a direct consequence of the arti®cial enhancement of void area fraction at the equator of the RVE cell. The V-BCC cell predictions for the macroscopic stress±strain blend behavior display much more stable and realistic trends. Di€erences between the two model predictions are not limited to macroscopic cell behavior, but are re¯ected in the relative magnitude of local quantities as well. As an illustration of these e€ects, Fig. 17 compares contour plots of mean (hydrostatic) stress in the matrix for the 25/75 blend under uniaxial loading, as predicted by the two models, at a level of axial deformation corresponding to macroscopic yield. The crossed dog-ear shear band pattern in the SHA cell is associated with a tensile mode of deformation, and therefore with high levels of local hydrostatic stress at the center of the shear band intersection, while the single diagonal shear band across the interparticle ligament in the V-BCC cell is associated with a shearing mode of deformation, and therefore a lower level of hydrostatic stress. These di€erences are of relevance when results of the

Fig. 17. Distribution of mean (hydrostatic) stress in the matrix at macroscopic yield (Ez 25%), under uniaxial loading, as predicted by the V-BCC model and by the SHA model, for a blend with initial void volume fraction f0=0.25. The peak level for the SHA model is 80 MPa, while the peak level for the V-BCC model is 46 MPa. (The stress levels in the legend are given in Pa.)

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micromechanical models are to be used to predict the possible onset of crazing phenomena. 4.2. Toughening implications It is widely recognized that the increase in toughness associated with the introduction of second-phase rubber particles relies on the combination of two mechanisms: (1) promotion of matrix yielding by localization of shear deformation across the interparticle regions; and (2) a reduction in the level of hydrostatic stress in the matrix. The latter e€ect is important in relation to the possibility of suppressing crazing events, which are the precursors to brittle failure modes. The craze initiation mechanism is still the subject of ongoing investigations, as some aspects of the process are not yet fully understood. Studies indicate that craze initiation can be related to various possible stress conditions, including a critical mean (hydrostatic) stress (Ishikawa et al., 1977; Nimmer and Woods, 1992), a critical maximum principal stress (Nimmer and Woods, 1992), a combination of principal stresses (Sternstein and Ongchin, 1969), and a combination of hydrostatic and shear stresses (Argon and Hannoosh, 1977). For discussion purposes, in view of the key role of the stress state triaxiality in initiating crazes below the notch tip surface, we adopt a simplistic model which associates craze initiation with a critical level of mean stress. Estimates for a critical mean stress for craze nucleation in the PC homopolymer, scraze, have been obtained in the literature by relating the experimentally observed onset of crazing to the local state of stress at the nucleation site. While earlier investigations relied on slip line solutions to characterize the stress state ahead of sharp notches, (Ishikawa et al, 1977), subsequent three-dimensional studies by Nimmer and Woods (1992) relied on ®nite element analyses to assess the e€ects of changes in geometrical details upon the stress state at the notch. The constitutive model used by Nimmer and Woods was based on a standard J2 ¯ow theory, with a constant ¯ow stress and no pressure dependence. It is then not surprising that estimates for scraze obtained in the two studies are in good agreement, with values between 90 and 100 MPa. Interestingly, a recent study by Lai and van der Giessen (1997), which incorporates a realistic model for the constitutive behavior of the homopolymer, provides stress ®eld solutions ahead of a crack tip which also yield a similar estimate for scraze. Through a ®nite element analysis of the stress ®eld around a blunted mode I crack in PC, Lai and van der Giessen obtain estimates for the peak mean stress values ahead of the crack tip for di€erent levels of the applied stress intensity factor, KI. The critical stress intensity factor for crack initiation in p PC,  as measured by Parvin and Williams (1975), is KC ˆ 2:24 MPa m. If we assume that the onset of crazing and the fracture event occur at close levels of the loading parameter, KI, we can equate pthe peak mean stress obtained by Lai and van der Giessen for KI ˆ 2:24 MPa m to the critical hydrostatic stress for craze nucleation. This provides an estimate scraze 1 92 MPa. The V-BCC predictions for the macroscopic response of the 25/75 blend under

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Fig. 18. Equivalent and mean stress levels at macroscopic yield. (a) In pure PC; (b) macroscopic, homogenized stress in a 25/75 blend; (c) peak local stress in the matrix for a 25/75 blend.

increasing levels of triaxiality (Fig. 12(a)) clearly show how the presence of nonadhering particles acts to substantially lower the macroscopic stress needed to yield the polymer blend under increasing levels of triaxiality. Comparing, in Fig. 18(a) and (b), the levels of equivalent (Mises) stress and mean (hydrostatic) stress at macroscopic yield in the PC homopolymer (a ) and in the blend (b ), we see the dramatic e€ects of the second-phase particles on the macroscopic stress state. While the mean hydrostatic stress in the homopolymer reaches critical crazing levels for triaxialities above 1.3, the macroscopic mean stress in the blend remains below 50 MPa up to the highest level of triaxiality. It is important to notice that these macroscopic measures of stress state in the blend are not of relevance in assessing the potential for crazing. As the typical interparticle spacing is substantially greater than the characteristic length-scale associated with craze initiation, we can apply the craze initiation criterion to the local stress ®eld in the matrix10. Therefore, we take the distribution of the local mean stress in the matrix around the particle and, in particular, the peak level of mean hydrostatic stress in the matrix, to be of relevance in predicting possible brittle failure mechanisms11. As previously discussed, at macroscopic yield, well

10 In cases where the interparticle spacing is of the same order of magnitude as the critical craze initiation length-scale, we can expect crazing to be suppressed altogether. 11 In this regard, we must recognize that while multiparticle models can provide important indications concerning the macroscopic blend response, the necessary coarseness of the mesh around each particle limits the accuracy of any assessment of the magnitude of local ®eld quantities.

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de®ned patterns of shear ¯ow localization have already arisen in the matrix. It is the intensity and geometry of these deformation patterns which de®nes the characteristics of the local stress state. It is then of essence that the micromechanical model provide a realistic description of the local deformation mechanisms. The magnitudes of the peak Mises stress and mean stress in the matrix at macroscopic yield predicted by the V-BCC model are plotted in Fig. 18(c). While local levels of the mean stress are somewhat higher than the cell average, they are still within safe bounds, with respect to the estimated values of scraze, up to the highest level of triaxiality. Thus the model, in agreement with experimental evidence, predicts that the presence of particles, at this level of second phase volume fraction, provide local conditions which favor the occurrence of shear yielding over brittle crazing12. As a supporting tool in designing optimized polymer blends, a parametric study can be carried out, varying the model parameter f0, to identify a range of particle volume fractions for which the model would predict the suppression of crazing events under the considered range of triaxiality TS={0.33, 1.3, 2.3}. The results of this study would be, however, of limited relevance, as this range of triaxiality is somewhat arbitrary and based on estimates derived from metal plasticity. The desired condition for an optimized blend is that the rubber content should be high enough to suppress crazing under stress triaxialities typical of crack tip environments for the blend. The Lai and van der Giessen study (1997) clearly indicates that the crack tip ®elds strongly depend on the material constitutive behavior. The level of triaxiality of concern for a speci®c blend should be therefore identi®ed through crack-tip studies in which the constitutive behavior of the blend is obtained by homogenization of the corresponding RVE response. 5. Concluding remarks We introduced a novel axisymmetric unit cell (V-BCC) based on a regular array of voids arranged on a BCC lattice. The proposed model is able to overcome some inherent limitations of the traditional axisymmetric unit cell (SHA). The less restrictive boundary conditions applied to the V-BCC model result in more realistic patterns of matrix deformation, and, therefore, more reliable predictions of microscopic and macroscopic features of the mechanical behavior of toughened polymeric blends. These qualities can be exploited both in micromechanical RVE studies, to optimize the properties of polymer blends, as well as in the de®nition of an

12

As discussed in section 4.1 and illustrated in Fig. 17, the SHA model tends to predict higher levels of local hydrostatic stress. Under the same conditions, while the V-BCC model predicts peak mean stresses between 45 and 75 MPa, as illustrated in Fig. 18(c), the SHA model predicts peak mean stresses between 80 and 90 MPa.

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improved homogenized macroscopic constitutive model for porous glassy polymers. The model is also applicable for the study of deformation mechanisms in other heterogeneous systems of similar morphology, such as semi-crystalline polymer blends (Tzika et al., 1999), porous metals obtained via powder compaction, and metal matrix composites, where high levels of second phase volume fraction compromise the adequacy of the traditional axisymmetric cell. It should also be noted that the three-dimensional Voronoi cell is well suited, due to its high degree of symmetry, for building a fully three-dimensional model

Fig. A1. Comparison of macroscopic predictions of the V-BCC, BCT, and SHA models for a blend with an initial void volume fraction f0=0.25, under uniaxial (TS=0.33) and triaxial (TS=2.3) loading.

Fig. A2. Strain localization patterns predicted by the BCT model at low (a) and high (b) triaxiality for a macroscopic axial strain Ez 215%.

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of the composite, which can then be used to investigate the e€ects of nonaxisymmetric loading conditions, such as pure shear. Acknowledgements This research has been supported in part by the MIT CMSE through NSF grants No. DMR-94-00334 and No. DMR-98-08941. We acknowledge stimulating discussions with D. M. Parks and P. Tzika. Appendix A Body centered tetragonal RVE In order to illustrate the similarities between the V-BCC and the BCT models, we compare the macroscopic and microscopic predictions of the BCT cell for a blend with initial void volume fraction f0=0.25 (25/75 blend) with the results of the V-BCC and SHA models. In Fig. A1 we compare macroscopic predictions of the three RVEs under uniaxial (TS=0.33) and triaxial (TS=2.3) loading. It is apparent that the BCT model is in excellent agreement with the V-BCC model. Similarities between the two models are not limited to macroscopic cell behavior, but are re¯ected in the local pattern of deformation as well. In Fig. A2 we provide two snapshots of the strain localization patterns in the BCT cell, at low and high triaxiality, which are essentially identical to the V-BCC patterns illustrated, respectively, in Figs. 10 and 13. Appendix B Constitutive behavior of polycarbonate The polycarbonate constitutive model requires three elements: a linear spring used to characterize the initial response as elastic; a viscoplastic dashpot representing the rate and temperature-dependent yield that monitors an isotropic resistance to chain segment rotation; and a nonlinear rubber elasticity spring element that accounts for an anisotropic resistance to chain alignment, which develops with plastic strain. Constitutive descriptions for each of these elements are summarized below within the context of a general ®nite strain deformation framework. The representation of the kinematics begin with the multiplicative decomposition of the deformation gradient: F ˆ Fe Fp ,

…B1†

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where Fp is the deformation gradient of the relaxed con®guration obtained by elastically unloading to a stress-free state via Feÿ1. Furthermore, in the model adopted for this study, the elastic component of the deformation gradient, Fe, is restricted to be a stretch only, Fe=Ve=Ue. Note that this results in no loss of generality as shown in Boyce et al. (1989). The velocity gradient, L, can be expressed as the sum of the symmetric tensor D, the rate of deformation, and the skew-symmetric tensor W, the spin, and is given as follows: Ç ÿ1 ˆ D ‡ W ˆ Le ‡ Fe Lp Feÿ1 , L ˆ FF

…B2†

p where the velocity gradient of the relaxed con®guration, Lp ˆ FÇ Fpÿ1 , may be represented in terms of its symmetric and skew-symmetric components:

Lp ˆ D p ‡ W p ,

…B3†

where Wp is the plastic spin, and Dp is the rate of shape change in the relaxed con®guration, which is constitutively prescribed through the characterization of the viscoplastic element. It may be shown, as in Boyce et al. (1989), that Wp is algebraically prescribed without loss of generality as a result of the imposed symmetry on the elastic deformation gradient. The linear elastic spring used to characterize the initial response of the material is constitutively characterized by the fourth order tensor operator of elastic constants, Le. 1 T ˆ Le ‰ ln Ue Š, J

…B4†

where T is the Cauchy stress, (ln Ue) is the Hencky strain and J = det Ue. The nonlinear rubber elasticity spring element introduces a convected network stress, TN, which captures the e€ect of orientation-induced strain hardening. The underlying macromolecular network orients with strain and has been found to be well-modeled using the Arruda and Boyce (1993b) eight-chain model of rubber elasticity. The network stress tensor (taken to be deviatoric in the case of strain hardening in the glassy state) is given by: p   N ÿ1 lchain N p ‰B ÿ l2chain IŠ, L …B5† T ˆ mR lchain N 1

where B=FpFpT, and lchain ˆ ‰ 13 trBŠ 2 is the stretch on each chain in the network. The material properties describing the strain hardening characteristics are mR, the initial hardening modulus, and N, the number of rigid molecular units between entanglements. The Langevin function L is given by:   1 lchain …B6† L…b† ˆ coth…b† ÿ ; b ˆ Lÿ1 p , b N

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and its inverse provides the functionality that as the chain stretch, lchain, p approaches its limiting extensibility … N†, the stress increases dramatically. The viscoplastic element describes the rate and temperature-dependent yield behavior. The e€ective equivalent shear stress on this element is found from the tensorial di€erence between the total stress, or Cauchy stress, T, and the convected network stress from the nonlinear spring element, TN: 1 T ˆ T ÿ Fe TN FeT : J

…B7†

T is the driving stress state, i.e. the portion of the total stress which continues to activate plastic ¯ow. The deviatoric component of the driving stress state is denoted T ', and it can be expressed in terms of its magnitude, t, the e€ective equivalent shear stress, and its tensorial direction, N: 1=2  1   T 0T 0 , …B8† tˆ 2 1 N ˆ p T 0: 2t

…B9†

The rate of shape change Dp in the viscoplastic element is assumed to be aligned with the deviatoric driving stress state: Dp ˆ g_ p N,

…B10†

where g_ p is the plastic shear strain rate, which ensues once isotropic barriers to chain segment rotation are overcome. The magnitude of g_ p is taken to depend on the relative values of the applied shear stress, t, and the athermal shear strength, s, according to the relation: " (  5=6 )# As t p 1ÿ , …B11† g_ ˆ g_ o exp ÿ kY s where g_ o is the pre-exponential factor proportional to the attempt frequency, As is the zero stress level activation energy, k is Boltzmann's constant, and Y is absolute temperature. Table 1 Material parameters for the PC matrix Elastic

Viscoplastic

Softening

Orientation hardening

E (MPa)

n

g_ o (sÿ1)

As (J)

h (MPa)

sss/so

a

mR (MPa)

N

2300

0.33

2  1015

3.3  10ÿ19

500

0.78

0.08

18

2.8

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Strain softening is modeled, after Boyce et al. (1988), by taking the athermal shear strength, s, to evolve from its initial annealed value, so=0.077 m/(1ÿn ) (where m is the elastic shear modulus, and n is Poisson's ratio), to a `preferred' state, sss,   s …B12† g_ p , s_ ˆ h 1 ÿ sss where h is the softening slope. Pressure dependence is also accounted for, as described in Boyce et al. (1988), by adding to the athermal shear resistance the pressure contribution, ap, where p is the pressure, and a is the pressure-dependent coecient of the material. The constitutive description is thus summarized in Eqs. (B4)±(B12). The material properties, as obtained in Arruda and Boyce (1993a), are listed in Table 1.

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