Porous Viscoplastic Materials Research Articles

On the optimality of the variational linear comparison bounds for porous viscoplastic materials

This paper is concerned with the optimality of the variational bounds of the Hashin-Shtrikman type (VHS) for nonlinear composites, first obtained by Ponte Castañeda (1991a) by means of the corresponding HS bounds for suitably optimized linear comparison composites (LCCs). For simplicity, this problem is addressed in the context of porous viscoplastic materials with incompressible, isotropic matrix phase and two-dimensional microstructures, subjected to plane-strain loading conditions. Although special, this case—exhibiting infinite heterogeneity contrast and compressible macroscopic response—is expected to be fully representative of more general three-dimensional porous materials, as well as more general two-phase, well-ordered composites. Thus, it is shown that the VHS bounds, which were originally derived for the class of nonlinear composites with statistically isotropic microstructures, are in fact attained over the larger class of microstructures with anisotropic nonlinear response but isotropic linear response. By appealing to an exact variational representation for the effective potential of finite-rank nonlinear laminates, it is shown that there exist certain values of the applied macroscopic stress for which the finite-rank laminate (closed-cell porous) microstructures attaining the linear HS bounds also attain the nonlinear VHS bound. Explicit results are obtained in the ideally plastic limit for the yield surface of the finite-rank laminates attaining the VHS bound. In particular, the results of the paper highlight the fact that bounds for nonlinear composites are much more sensitive to microstructural details than bounds for linear composites.

Anisotropic finite-strain models for porous viscoplastic materials with microstructure evolution

In this work, we propose a constitutive model for the finite-strain, macroscopic response of porous viscoplastic solids, accounting for deformation-induced changes in the size, shape and distribution of the voids. The model makes use of consistent homogenization estimates obtained by the “iterated variational linear comparison” procedure of Agoras and Ponte Castañeda (2013) to characterize both the instantaneous effective response of the porous material and the evolution of the underlying microstructure. The proposed model applies for general, three-dimensional loading conditions and can be implemented numerically for use in standard FEM codes. We also investigate the interplay between the evolution of the microstructure and the macroscopic stress–strain response, in the context of displacement-controlled, plane strain loading (bi-axial straining) of initially isotropic, porous, rigid-plastic materials with power-law hardening. We focus on the effect of strain triaxiality, and consider both extensional and contractile loading conditions leading to porosity growth and collapse, respectively. For both types of loadings, it is found that the macroscopic behavior of the material exhibits an initial hardening regime followed by a softening regime at sufficiently large strains. Consistent with earlier models and experimental results, the softening regime for extensional loadings is a consequence of porosity growth. On the other hand, the softening behavior predicted for contractile loadings is found to be a consequence of void collapse, i.e., of rapid changes in the average shape of the pores leading to crack-like shapes. For both types of loadings, the transition from hardening to softening in the macroscopic response can be identified with the onset of macroscopic strain localization. The associated critical conditions at the onset of localization are determined as a function of the strain triaxiality. The type of localization band ranges from dilatational to compaction bands as the bi-axial straining varies from uniaxial extension to uniaxial contraction.

Iterated linear comparison bounds for viscoplastic porous materials with “ellipsoidal” microstructures

Analytical estimates are obtained for the effective constitutive response of porous viscoplastic materials consisting of aligned ellipsoidal voids that are distributed randomly with “ellipsoidal” symmetry in the matrix material. These estimates are obtained by means of a novel iterative homogenization strategy recently proposed by Ponte Castañeda (2012), and can be shown to be bounds for certain classes of multi-scale microstructures. By design, the resulting constitutive model agrees exactly with the earlier “variational linear comparison” model at the first iteration (N=1), and provides estimates that get progressively more accurate as the number of iterations increases (N→∞), especially for high-triaxiality loading conditions, and low porosity and strain-rate sensitivity. However, in practice, a small number of iterations (N≈10) is sufficient to get very accurate results. It is important to emphasize that, unlike other models that have been proposed in the literature, the new model requires no fitting parameters, solely depending on the properties of the matrix phase and microstructural information, such as the porosity, the average void shape and orientation, as well as the generally different shape and orientation of their distribution. Results are given for the yield and gauge surfaces of ideally plastic and power-law viscoplastic porous materials for the special cases of aligned spheroidal and ellipsoidal voids, and the results are compared with available numerical results and with the results of other models. Compared to available numerical results, the new estimates are found to be quite accurate, while they also provide more flexibility than competing models in terms of the characterization of the microstructure. In particular, it was found that the effect of different shapes for the average pore shape and distribution on the yield surfaces of the porous materials can be significant at high triaxialities, even for very small porosities. In addition, consistent estimates are obtained for the average strain rate in the voids, and explicit results have been computed for initially spherical voids, which exhibit a strong effect of the triaxiality, especially at low porosities and strain-rate sensitivities.

Bounds for nonlinear composites via iterated homogenization

Improved estimates of the Hashin–Shtrikman–Willis type are generated for the class of nonlinear composites consisting of two well-ordered, isotropic phases distributed randomly with prescribed two-point correlations, as determined by the H-measure of the microstructure. For this purpose, a novel strategy for generating bounds has been developed utilizing iterated homogenization. The general idea is to make use of bounds that may be available for composite materials in the limit when the concentration of one of the phases (say phase 1) is small. It then follows from the theory of iterated homogenization that it is possible, under certain conditions, to obtain bounds for more general values of the concentration, by gradually adding small amounts of phase 1 in incremental fashion, and sequentially using the available dilute-concentration estimate, up to the final (finite) value of the concentration (of phase 1). Such an approach can also be useful when available bounds are expected to be tighter for certain ranges of the phase volume fractions. This is the case, for example, for the “linear comparison” bounds for porous viscoplastic materials, which are known to be comparatively tighter for large values of the porosity. In this case, the new bounds obtained by the above-mentioned “iterated” procedure can be shown to be much improved relative to the earlier “linear comparison” bounds, especially at low values of the porosity and high triaxialities. Consistent with the way in which they have been derived, the new estimates are, strictly, bounds only for the class of multi-scale, nonlinear composites consisting of two well-ordered, isotropic phases that are distributed with prescribed H-measure at each stage in the incremental process. However, given the facts that the H-measure of the sequential microstructures is conserved (so that the final microstructures can be shown to have the same H-measure), and that H-measures are insensitive to length scales, it is conjectured that the new bounds may hold for more general classes of microstructures with prescribed volume fractions and H-measures (independent of the separation of length scales hypotheses that was made in the derivation of the result using iterated homogenization).

A semi-analytical model for the behavior of saturated viscoplastic materials containing two populations of voids of different sizes

This paper presents a micromechanical model for a porous viscoplastic material containing two populations of pressurized voids of different sizes. Three scales are distinguished: the microscopic scale (corresponding to the size of the small voids), the mesoscopic scale (corresponding to the size of the large voids) and the macroscopic scale. It is assumed that the first homogenization step is performed at the microscopic scale, and, at the mesoscopic scale, the matrix is taken to be homogeneous and compressible. At the mesoscopic scale, the second homogenization step, on which the present study focuses, is based on a simplified representative volume element: a hollow sphere containing a pressurized void surrounded by a nonlinear viscoplastic compressible matrix. The nonlinear behavior of the matrix, which is expressed using the results obtained in the first homogenization step, is approached using a modified secant linearization procedure involving the discretization of the hollow sphere into concentric layers. Each layer has uniform secant moduli. The predictions of the model are compared with the more accurate numerical results obtained using the finite element method. Good agreement is found to exist with all the macroscopic stress triaxialities and all the porosity and nonlinearity values studied.

Three-dimensional analysis of compaction of metal powder

Three-dimensional compaction simulations of geometries with circular and quadratic cross-sections are reported using dilatant viscoplastic material models representing a ductile steel powder. The simulations are compared with experiments using in situ measurements of the wall-tractions within the die. Two different porous viscoplastic material models have been used: the combined Fleck–Kuhn–McMeeking (FKM) and Gurson model and the Shima–Oyane model. The contact between the die and the powder was modelled with Coulomb-type friction. Effects such as the applied traction – density response and average relations for the upper, lower and the normal tractions against the die wall are compared. Comparisons between the numerical results and experimental results show that both material models can well be used to capture the overall density distributions for the geometries studied and that many of the coarse three-dimensional features of the powder-pressing process are well described by the simulations.

Inertial Effects on Void Growth in Porous Viscoplastic Materials

The present work examines the inertial effects on void growth in viscoplastic materials which have been largely neglected in analyses of dynamic crack growth and spallation phenomena using existing continuum porous material models. The dynamic void growth in porous materials is investigated by analyzing the finite deformation of an elastic/viscoplastic spherical shell under intense hydrostatic tensile loading. Under typical dynamic loading conditions, inertia is found to have a strong stabilizing effect on void growth process and consequently to delay coalescence even when the high rate-sensitivity of materials at very high strain rates is taken into account. Effects of strain hardening and thermal softening are found to be relatively small. Approximate relations are suggested to incorporate inertial effects and rate sensitivity of matrix materials into the porous viscoplastic material constitutive models for dynamic ductile fracture analyses for certain loading conditions.

Exact results and approximate models for porous viscoplastic solids

Abstract The aim of this paper is to present new approximate macroscopic models for porous viscoplastic materials, based on partial but exact results applicable to such media. Available results are first supplemented by providing a new inequality (which, in addition to its intrinsic interest, allows one to rederive in a simpler way some previous bounds of Ponte-Castaneda and Talbot and Willis), and by exhibiting the exact form of the overall potential of a typical porous viscoplastic volume element, namely a hollow cylinder loaded in generalized plane strain. Approximate expressions for the macroscopic viscoplastic potentials of materials containing cavities of cylindrical or spherical shape are then proposed, based on these and other results; these expressions satisfy, in particular, the three following natural requirements: (i) reproduce the exact solution of a hollow cylinder or sphere loaded in hydrostatic tension or compression; (ii) be a quadratic form of the overall stress tensor in the extreme case of a Newtonian (linear) behaviour; and (iii) yield the currently accepted Gurson criterion in the other extreme case of an ideal-plastic behavior.