Yield criteria for porous media in plane strain: second-order estimates versus numerical resultsCritère de plasticité des matériaux poreux en déformation plane : Estimations du second ordre et résultats numériques

This Note presents a comparison of some recently developed “second-order” homogenization estimates for two-dimensional, ideally plastic porous media subjected to plane strain conditions with corresponding yield analysis results using a new linearization technique and systematically optimized finite elements meshes. Good qualitative agreement is found between the second-order theory and the yield analysis results for the shape of the yield surfaces, which exhibit a corner on the hydrostatic axis, as well as for the dependence of the effective flow stress in shear on the porosity, which is found to be non-analytic in the dilute limit. Both of these features are inconsistent with the predictions of the standard Gurson model. To cite this article: J. Pastor, P. Ponte Castañeda, C. R. Mecanique 330 (2002) 741–747.     

Modeling of ductile fracture: Application of the mechanism-based concepts

This paper summarizes our recent studies on modeling ductile fracture in structural materials using the mechanism-based concepts. We describe two numerical approaches to model the material failure process by void growth and coalescence. In the first approach, voids are considered explicitly and modeled using refined finite elements. In order to predict crack initiation and propagation, a void coalescence criterion is established by conducting a series of systematic finite element analyses of the void-containing, representative material volume (RMV) subjected to different macroscopic stress states and expressed as a function of the stress triaxiality ratio and the Lode angle. The discrete void approach provides a straightforward way for studying the effects of microstructure on fracture toughness. In the second approach, the void-containing material is considered as a homogenized continuum governed by porous plasticity models. This makes it possible to simulate large amount of crack extension because only one element is needed for a representative material volume. As an example, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a modified Gologanu–Leblond–Devaux model [Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Tech. 116, 290–297; Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1995. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.) Continuum Micromechanics. Springer-Verlag, pp. 61–130] is used to describe the evolution of void shape and void volume fraction and the associated material softening, and the material failure criterion is calibrated using experimental data. The calibrated computational model successfully predicts crack extension in various fracture specimens, including the compact tension specimen, middle crack tension specimens, multi-site damage specimens and the pressurized cylindrical shell specimen.     

The effect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites

The magnetoelastic homogenization framework and the partial decoupling approximation proposed by Ponte Castañeda and Galipeau (2011) are used to estimate material properties for a class of magnetically susceptible elastomers. Specifically, we consider composites consisting of aligned, ellipsoidal magnetic particles distributed randomly with “ellipsoidal” symmetry under combined magnetic and mechanical loading. The model captures the coupling between the magnetic and mechanical fields, including the effects of magnetic saturation. The results help elucidate the effects of particle shape, distribution, and concentration on properties such as the magnetostriction, actuation stress, magnetic modulus, and magnetization behavior of a magnetorheological composite.     

RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales

The present paper extends the Gurson and GLD models [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. Mech. Phys. Solids 99, 2–15; Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297] to involve the coupled effects of void size and void shape on the macroscopic yield behavior of non-linear porous materials and on the void growth. A spheroidal representative volume element (RVE) under a remote axisymmetric homogenous strain boundary condition is carefully analyzed. A wide range of void aspect ratios covering the oblate spheroidal, spherical and prolate spheroidal void are taken into account to reflect the shape effect. The size effect is captured by the Fleck–Hutchinson phenomenological strain gradient plasticity theory [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, pp. 295–361]. A new size-dependent damage model like the Gurson and GLD models is developed based on the traditional minimum plasticity potential principle. Consequently, the coupled effects of void size and void shape on yield behavior of porous materials and void growth are discussed in detail. The results indicate that the void shape effect on the yield behavior of porous materials and on the void growth can be modified dramatically by the void size effect and vice versa. The applied stress triaxiality plays an important role in these coupled effects. Moreover, there exists a cut-off void radius r_c, which depends only on the intrinsic length l₁ associated with the stretch strain gradient. Voids of effective radius smaller than the critical radius r_c are less susceptible to grow. These findings are helpful to our further understanding to some impenetrable micrographs of the ductile fracture surfaces.     

Estimate of the homogenized hyperelastic behavior of periodic fiber-reinforced composites using the second-order procedure

We use the second-order procedure of P. Ponte Castañeda to estimate the hyperelastic behavior of periodic composites. The results obtained for a fiber-reinforced composite show that the accuracy of the homogenized energy estimated using this procedure is excellent in the case of compressible materials and poor in the case of quasi-incompressible materials. Then we develop a new formulation for incompressible materials using a Lagrange multiplier which improves the quality of the estimate given by the procedure.     

On reflected interactions in elastic solids containing inhomogeneities

Interactions in linear elastic solids containing inhomogeneities are examined using integral equations. Direct and reflected interactions are identified. Direct interactions occur simply because elastic fields emitted by inhomogeneities affect each other. Reflected interactions occur because elastic fields emitted by inhomogeneities are reflected by the specimen boundary back to the individual inhomogeneities. It is shown that the reflected interactions are of critical importance to analysis of representative volume elements. Further, the reflected interactions are expressed in simple terms, so that one can obtain explicit approximate expressions for the effective stiffness tensor for linear elastic solids containing ellipsoidal and non-ellipsoidal inhomogeneities. For ellipsoidal inhomogeneities, the new approximation is closely related to that of Mori and Tanaka. In general, the new approximation can be used to recover Ponte Castañeda–Willis׳ and Kanaun–Levin׳s approximations. Connections with Maxwell׳s approximation are established.     

Macroscopic strain potentials in nonlinear porous materials

By taking a hollow sphere as a representative volume element (RVE), the macroscopic strain potentials of porous materials with power-law incompressible matrix are studied in this paper. According to the principles of the minimum potential energy in nonlinear elasticity and the variational procedure, static admissible stress fields and kinematic admissible displacement fields are constructed, and hence the upper and the lower bounds of the macroscopic strain potential are obtained. The bounds given in the present paper differ so slightly that they both provide perfect approximations of the exact strain potential of the studied porous materials. It is also found that the upper bound proposed by previous authors is much higher than the present one, and the lower bounds given by Cocks is much lower. Moreover, the present calculation is also compared with the variational lower bound of Ponte Castañeda for statistically isotropic porous materials. Finally, the validity of the hollow spherical RVE for the studied nonlinear porous material is discussed by the difference between the present numerical results and the Cocks bound.     

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.     

On Hashin–Shtrikman-type bounds for nonlinear conductors

For linear composite conductors, it is known that the celebrated Hashin–Shtrikman bounds can be recovered by the translation method. We investigate whether the same conclusion extends to nonlinear composites in two dimensions. To that purpose, we consider two-phase composites with perfectly conducting inclusions. In that case, explicit expressions of the various bounds considered can be obtained. The bounds provided by the translation method are compared with the nonlinear Hashin–Shtrikman-type bounds delivered by the Talbot–Willis (1985) [2] and the Ponte Castañeda (1991) [3] procedures.     

Homogenization estimates for multi-scale nonlinear composites

This work is concerned with the generalization of the “variational linear comparison” method of Ponte Castañeda (J. Mech. Phys. Solids 39 (1991) 45)) to multi-scale, random, heterogeneous material systems with nonlinear isotropic constituents. This method has the distinguishing feature of allowing the conversion of bounds or estimates that might be available for linear systems into corresponding bounds or estimates for the nonlinear composites of interest. Furthermore, the method is fairly simple to implement and quite general. General estimates are developed for two-scale systems and applied to various model composites with “particulate” and “granular” micro- and meso-structures, and compared with the corresponding results for their single-scale counterparts. It is found that the way that the material heterogeneity is distributed at the two separate scales can in most cases have a significant effect on the macroscopic behavior of the composite system.     

Bounds and estimates for the effective yield surface of porous media with a uniform or a nonuniform distribution of voids

This paper aims at studying the effects of a nonuniform distribution of voids on the macroscopic yield response of porous media with a rigid-perfectly plastic matrix. For this purpose, a semi-analytical model, recently proposed by Bilger et al. [Bilger, N., Auslender, F., Bornert, M., Masson, R., 2002. New bounds and estimates for porous media with rigid perfectly plastic matrix. C. R. Mecanique 330, 127–132], is extended to more general situations where the local porosity can fluctuate. The microstructure is described by a generalized Hashin-type assemblage of hollow spheres and the distribution of the local porosity is obtained from a three-dimensional simulated microstructure. The matrix layer around the voids is discretized into concentric sub-layers so as to take better into account the plasticity gradient along the radial direction. Classical homogenization techniques then provide new self-consistent estimates and upper bounds for the macroscopic yield surface. These results are compared first to the predictions of the Gurson model and its extensions and then to numerical results derived from three-dimensional Fast Fourier Transform (FFT) calculations carried out with the same material porosity distribution. A good agreement is obtained with the three-dimensional FFT calculations and with Gurson–Tvergaard's predictions even for high triaxiality and without fitting any parameter. Nevertheless, when the heterogeneous distribution of voids tends to form clusters, the proposed model fails to capture the properties of the macroscopic yield surface for large triaxiality factors.     

Onset of Cavitation in Compressible,Isotropic, Hyperelastic Solids

In this work, we derive a closed-form criterion for the onset of cavitation in compressible, isotropic, hyperelastic solids subjected to non-symmetric loading conditions. The criterion is based on the solution of a boundary value problem where a hyperelastic solid, which is infinite in extent and contains a single vacuous inhomogeneity, is subjected to uniform displacement boundary conditions. By making use of the “linear-comparison” variational procedure of Lopez-Pamies and Ponte Castañeda (J. Mech. Phys. Solids 54:807–830, 2006), we solve this problem approximately and generate variational estimates for the critical stretches applied on the boundary at which the cavity suddenly starts growing. The accuracy of the proposed analytical result is assessed by comparisons with exact solutions available from the literature for radially symmetric cavitation, as well as with finite element simulations. In addition, applications are presented for a variety of materials of practical and theoretical interest, including the harmonic, Blatz-Ko, and compressible Neo-Hookean materials.     

Effects of internal pore pressure on closed-cell elastomeric foams

A micromechanics framework for porous elastomers with internal pore pressure (Idiart and Lopez-Pamies, 2012) is used together with an earlier homogenization estimate for elastomers containing vacuous pores (Lopez-Pamies and Ponte Castañeda, 2007a) to investigate the mechanical response and stability of closed-cell foams. Motivated by applications of technological interest, the focus is on isotropic foams made up of a random isotropic distribution of pores embedded in an isotropic matrix material, wherein the initial internal pore pressure is identical to the external pressure exerted by the environment (e.g. atmospheric pressure). It is found that the presence of internal pore pressure significantly stiffens and stabilizes the response of elastomeric foams, and hence that it must be taken into account when modeling this type of materials.     

Nonlinear composites: a linearization procedure,exact to second-order in contrast and for which the strain-energy and affine formulations coincideComposites non linéaires : un schéma exact au second ordre pour lequel la formulation en énergie de déformation coı̈ncide avec la formulation affine

This Note presents a new approximate scheme for nonlinear composites. The approximation which is made preserves certain features of the original second-order scheme of Ponte Castañeda, exactness to second-order in the contrast and existence of an effective energy, but improves on one drawback, which is the gap between the strain-energy formulation and the affine formulation. A numerical example shows the accuracy of the present method. To cite this article: N. Lahellec, P. Suquet, C. R. Mecanique 332 (2004).     

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—theory

This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method (Ponte Castañeda, J. Mech. Phys. Solids 39 (1991) 45) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method (Ponte Castañeda, J. Mech. Phys. Solids 44 (1996) 827) makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work.     

Second-order estimates for the large-deformation response of particle-reinforced rubbersEstimations homogénéisées pour le comportement mécanique des caoutchoucs renforcés en grandes déformations

This paper presents the application of a recently proposed ‘second-order’ homogenization method (J. Mech. Phys. Solids 50 (2002) 737–757) to the estimation of the effective behavior of hyperelastic composites subjected to finite deformations. The main feature of the method is the use of ‘generalized’ secant moduli that depend not only on the phases averages of the fields, but also on the phase covariance tensors. The use of the method is illustrated in the context of particle-, or fiber-reinforced elastomers and estimates analogous to the well-known Hashin–Shtrikman estimates for linear-elastic composites are generated. The new estimates improve on earlier estimates (J. Mech. Phys. Solids 48 (2000) 1389–1411) neglecting the use of fluctuations. In particular, the new estimates, unlike the earlier ones, are capable of recovering the exact incompressibility constraint when the matrix is also taken to be incompressible. To cite this article: O. Lopez-Pamies, P. Ponte Castañeda, C. R. Mecanique 331 (2003).     

A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids—I: Limit-analysis of some representative cell

Gurson (1977)'s famous model of the behavior of porous ductile solids, initially developed for spherical cavities, was extended by Gologanu et al., 1993, Gologanu et al., 1994, Gologanu et al., 1997 to spheroidal, both prolate and oblate voids. The aim of this work is to further extend it to general (non-spheroidal) ellipsoidal cavities, through approximate homogenization of some representative elementary porous cell. As a first step, we perform in the present Part I a limit-analysis of such a cell, namely an ellipsoidal volume made of some rigid-ideal-plastic von Mises material and containing a confocal ellipsoidal void, loaded arbitrarily under conditions of homogeneous boundary strain rate. This analysis provides an estimate of the overall plastic dissipation based on a family of trial incompressible velocity fields recently discovered by Leblond and Gologanu (2008), satisfying conditions of homogeneous strain rate on all ellipsoids confocal with the void and the outer boundary. The asymptotic behavior of the integrand in the expression of the global plastic dissipation is studied both far from and close to the void. The results obtained suggest approximations leading to explicit approximate expressions of the overall dissipation and yield function. These expressions contain parameters the full determination of which will be the object of Part II.     

Infinite-contrast periodic composites with strongly nonlinear behavior: Effective-medium theory versus full-field simulations

This paper presents a combined numerical-theoretical study of the macroscopic behavior and local field distributions in a special class of two-dimensional periodic composites with viscoplastic phases. The emphasis is on strongly nonlinear materials containing pores or rigid inclusions. Full-field numerical simulations are carried out using a fast Fourier transform algorithm [Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linear and nonlinear properties of composites. C. R. Acad. Sci. Paris II 318, 1417–1423.], while the theoretical results are obtained by means of the ‘second-order’ nonlinear homogenization method [Ponte Castañeda, P., 2002. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I. Theory. J. Mech. Phys. Solids 50, 737–757]. The effect of nonlinearity and inclusion concentration is investigated in the context of power-law (with strain-rate sensitivity m) behavior for the matrix phase under in-plane shear loadings. Overall, the ‘second-order’ estimates are found to be in good agreement with the numerical simulations, with the best agreement for the rigidly reinforced materials. For the porous systems, as the nonlinearity increases (m decreases), the strain field is found to localize along shear bands passing through the voids (the strain fluctuations becoming unbounded) and the effective stress exhibits a singular behavior in the dilute limit. More specifically, for small porosities and fixed nonlinearity $m>0$, the effective stress decreases linearly with increasing porosity. However, for ideally plastic behavior $(m=0)$, the dependence on porosity becomes non-analytic. On the other hand, for rigidly-reinforced composites, the strain field adopts a tile pattern with bounded strain fluctuations, and no singular behavior is observed (to leading order) in the dilute limit.