New bounds and estimates for porous media with rigid perfectly plastic matrixNouvelles bornes et estimations pour les milieux poreux à matrice rigide parfaitement plastique

We derive new rigorous bounds and self-consistent estimates for the effective yield surface of porous media with a rigid perfectly plastic matrix and a microstructure similar to Hashin's composite spheres assemblage. These results arise from a homogenisation technique that combines a pattern-based modelling for linear composite materials and a variational formulation for nonlinear media. To cite this article: N. Bilger et al., C. R. Mecanique 330 (2002) 127–132.     

Porous materials with two populations of voids under internal pressure: I. Instantaneous constitutive relations

This study is devoted to the mechanical behavior of uranium dioxide (UO₂) which is a porous material with two populations of voids of very different size subjected to internal pressure. The smallest voids are intragranular and spherical in shape whereas the largest pores located at the grain boundary are ellipsoidal and randomly oriented. In this first part of the study, attention is focused on the effective properties of these materials with fixed microstructure. In a first step, the poro-elastic properties of these doubly voided materials are studied. Then two rigorous upper bounds are derived for the effective poro-plastic constitutive relations of these materials. The first bound, obtained by generalizing the approach of Gologanu et al. (Gologanu, M., Leblond, J., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids-case of axisymmetric oblate ellipsoidal cavities. ASME J. Eng. Mater. Technol. 116, 290–297) to compressible materials, is accurate at high stress-triaxiality. The second one, which derives from the variational method of Ponte Castañeda (Ponte Castañeda, P., 1991. The effective mechanical properties of non-linear isotropic composites. J. Mech. Phys. Solids 39, 45–71), is accurate when the stress triaxiality is low. A N-phase model, inspired 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. Mécanique 330, 127–132), is proposed which matches the best of the two bounds at low and high triaxiality. The effect of internal pressures is discussed. In particular it is shown that when the two internal pressures coincide, the effective flow surface of the saturated biporous material is obtained from that of the drained material by a shift along the hydrostatic axis. However, when the two pressures are different, the modifications brought to the effective flow surface in the drained case involve not only a shift along the hydrostatic axis but also a change in shape and size of the surface.     

A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields

In this study, we propose a micromechanics-based modification of the Gurson criterion for porous media subjected to arbitrary loadings. The proposed formulation, derived in the framework of limit analysis, consists in the consideration of Eshelby-like exterior point trial velocity fields for the determination of the macroscopic dissipation. This approach is implemented for perfectly plastic rigid von Mises matrix containing spherical voids. After the minimization procedure required by the use of the Eshelby-like trial velocity fields, we derive a two-field estimate of the macroscopic yield function. It is shown that the obtained closed-form estimate provides a significant modification of the Gurson criterion, particularly in the domain of low stress triaxialities. This estimate is first compared with existing criteria. Moreover, its accuracy is assessed through comparison with results derived from numerical exact two-field criterion and with recently available numerical bounds.     

Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis

This study investigates the overall and local response of porous media composed of a perfectly plastic matrix weakened by stress-free voids. Attention is focused on the specific role played by porosity fluctuations inside a representative volume element. To this end, numerical simulations using the Fast Fourier Transform (FFT) are performed on different classes of microstructure corresponding to different spatial distributions of voids. Three types of microstructures are investigated: random microstructures with no void clustering, microstructures with a connected cluster of voids and microstructures with disconnected void clusters. These numerical simulations show that the porosity fluctuations can have a strong effect on the overall yield surface of porous materials. Random microstructures without clusters and microstructures with a connected cluster are the hardest and the softest configurations, respectively, whereas microstructures with disconnected clusters lead to intermediate responses. At a more local scale, the salient feature of the fields is the tendency for the strain fields to concentrate in specific bands. Finally, an image analysis tool is proposed for the statistical characterization of the porosity distribution. It relies on the distribution of the ‘distance function’, the width of which increases when clusters are present. An additional connectedness analysis allows us to discriminate between clustered microstructures.     

Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.     

Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials

Finite element (FE) calculations of a cylindrical cell containing a spherical hole have been performed under large strain conditions for varying triaxiality with three different constitutive models for the matrix material, i.e. rate independent plastic material with isotropic hardening, visco-plastic material under both isothermal and adiabatic conditions, and porous plastic material with a second population of voids nucleating strain controlled. The “mesoscopic” stress-strain and void growth responses of the cell are compared with predictions of the modified Gurson model in order to study the effects of varying triaxiality and strain rate on the critical void volume fraction. The interaction of two different sizes of voids was modelled by changing the strain level for nucleation and the stress triaxiality. The study confirms that the void volume fraction at void coalescence does not depend significantly on the triaxiality if the initial volume fraction of the primary voids is small and if there are no secondary voids. The strain rate does not affect f_c either. The results also indicate that a single internal variable, f, is not sufficient to characterize the fracture processes in materials containing two different size-scales of void nucleating particles.     

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.     

Influences of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture

Local mechanical properties in aluminum cast components are inhomogeneous as a consequence of spatial distribution of microstructure,e.g.,porosity,inclusions,grain size and arm spacing of secondary dendrites.In this work,the effect of porosity is investigated.Cast components contain voids with different sizes,forms,orientations and distributions.This is approximated by a porosity distribution in the following.The aim of this paper is to investigate the influence of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture.A micromechanical model with a spherical void located at the center of the matrix material,called the representative volume element(RVE),is developed.Fully periodic boundary conditions are applied to the RVE and the values of stress triaxiality and Lode parameter are kept constant during the entire course of loading.For this purpose,a multi-point constraint(MPC)user subroutine is developed to prescribe the loading.The results of the RVE model are used to establish the constitutive equations and to further investigate the influences of initial porosity,stress triaxiality and Lode parameter on elastic constant,plastic deformation and ductile fracture of an aluminum die casting alloy.     

A finite-strain model for anisotropic viscoplastic porous media: II – Applications

In Part I of this work, we have proposed a new model based on the “second-order” nonlinear homogenization method for determining the effective response and microstructure evolution in viscoplastic porous media with aligned ellipsoidal voids subjected to general loading conditions. In this second part, the new model is used to analyze the instantaneous effective behavior and microstructure evolution in porous media for several representative loading conditions and microstructural configurations. First, we study the effect of the shape and orientation of the voids on the overall instantaneous response of a porous medium that is subjected to principal loading conditions. Secondly, we study the problem of microstructure evolution under axisymmetric and simple shear loading conditions for initially spherical voids in an attempt to validate the present model by comparison with existing numerical and approximate results in the literature. Finally, we study the possible development of macroscopic instabilities for the special case of ideally-plastic solids subjected to plane-strain loading conditions. The results, reported in this paper, suggest that the present model improves dramatically on the earlier “variational” estimates, in particular, because it generates much more accurate results for high triaxiality loading conditions.     

A selfconsistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids

In this work we consider the presence of ellipsoidal voids inside polycrystals subjected to large strain deformation. For this purpose, the originally incompressible viscoplastic selfconsistent (VPSC) formulation of Lebensohn and Tomé (Acta Metall. Mater. 41 (1993) 2611) has been extended to deal with compressible polycrystals. In doing this, both the deviatoric and the spherical components of strain-rate and stress are accounted for. Such an extended model allows us to account for the void and for porosity evolution, while preserving the anisotropy and crystallographic capabilities of the VPSC model. The formulation can be adjusted to match the Gurson model, in the limit of rate-independent isotropic media and spherical voids. We present several applications of this extended VPSC model, which address the coupling between texture, plastic anisotropy, void shape, triaxiality, and porosity evolution.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

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.     

Approximate yield criteria for anisotropic metals with prolate or oblate voidsCritères macroscopiques pour des métaux plastiques anisotropes contenant des cavités non sphériques

Following the study of Gologanu et al. (1997) which has extended the well-known approach of Gurson (1975), we propose approximate yield criteria for anisotropic plastic voided metals containing non spherical cavities. The plastic anisotropy of the matrix is described by means of Hill's quadratic criterion. The procedure to establish the closed form expression of approximate macroscopic criteria, in which void shape and plastic anisotropic effects are included, is detailed. The new criteria allow us to recover existing results in the cases of spherical and cylindrical voids in an Hill type plastic matrix. Moreover, they agree with previous criteria for non spherical voids in an isotropic plastic matrix. Finally, for validation purposes, we provide, in the general case of non spherical cavities in the anisotropic matrix, a comparison with the numerical exact two field criteria. To cite this article: V. Monchiet et al., C. R. Mecanique 334 (2006).     

Void growth and coalescence in anisotropic plastic solids

Large strain finite element calculations of unit cells subjected to triaxial axisymmetric loadings are presented for plastically orthotropic materials containing a periodic distribution of aligned spheroidal voids. The spatial distribution of voids and the plastic flow properties of the matrix are assumed to respect transverse isotropy about the axis of symmetry of the imposed loading so that a two-dimensional axisymmetric analysis is adequate. The parameters varied pertain to load triaxiality, matrix anisotropy, initial porosity and initial void shape so as to include the limiting case of penny-shaped cracks. Attention is focussed on comparing the individual and coupled effects of void shape and material anisotropy on the effective stress–strain response and on the evolution of microstructural variables. In addition, the effect of matrix anisotropy on the mode of plastic flow localization is discussed. From the results, two distinct regimes of behavior are identified: (i) at high triaxialities, the effect of material anisotropy is found to be persistent, unlike that of initial void shape and (ii) at moderate triaxialities the influence of void shape is found to depend strongly on matrix anisotropy. The findings are interpreted in light of recent, microscopically informed models of porous metal plasticity. Conversely, observations are made in relation to the relevance of these results in the development and calibration of a broader set of continuum damage mechanics models.     

On some features of the effective behaviour of porous solids with J2- and J3-dependent yielding matrix behaviourSur quelques effets de l'angle de Lode sur les caractéristiques macroscopiques de matériaux en rupture ductile

Some features od the constitutive behaviour of voided materials taking into account possible effects of the Lode angle in the yielding behaviour of the matrix are discussed. The Gurson approach is used to this end. After providing a parametric representation of the effective behaviour of such materials, some closed-form results are given for pure shear stress states and also at very high stress triaxialities. In the former case corresponding to a zero macroscopic mean stress, the contour of the yield domain in the π-plane has exactly the shape of the yield surface of the matrix in the deviatoric plane, but a size reduced by a factor $1?f$, with f the porosity of the voided material. In the latter, effective yield stresses for the voided material are slightly different from the Gurson result and found to be set by the yield stress at a microscopic stress Lode angle $\frac{\pi }{3}$ for very high positive triaxiality and by the yield stress at a microscopic stress Lode angle 0 for very high negative triaxiality. This last result is extended for porous materials with yielding depending further on the hydrostatic stress, fully exhibiting the interaction between volumetric and shear interactions on the yielding behaviour of isotropic porous materials. Applications to many usual yielding criteria for the matrix are also provided.     

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation

The present two-part study aims at investigating the specific effects of Mohr–Coulomb matrix on the strength of ductile porous materials by using a kinematic limit analysis approach. While in the Part II, static and kinematic bounds are numerically derived and used for validation purpose, the present Part I focuses on the theoretical formulation of a macroscopic strength criterion for porous Mohr–Coulomb materials. To this end, we consider a hollow sphere model with a rigid perfectly plastic Mohr–Coulomb matrix, subjected to axisymmetric uniform strain rate boundary conditions. Taking advantage of an appropriate family of three-parameter trial velocity fields accounting for the specific plastic deformation mechanisms of the Mohr–Coulomb matrix, we then provide a solution of the constrained minimization problem required for the determination of the macroscopic dissipation function. The macroscopic strength criterion is then obtained by means of the Lagrangian method combined with Karush–Kuhn–Tucker conditions. After a careful analysis and discussion of the plastic admissibility condition associated to the Mohr–Coulomb criterion, the above procedure leads to a parametric closed-form expression of the macroscopic strength criterion. The latter explicitly shows a dependence on the three stress invariants. In the special case of a friction angle equal to zero, the established criterion reduced to recently available results for porous Tresca materials. Finally, both effects of matrix friction angle and porosity are briefly illustrated and, for completeness, the macroscopic plastic flow rule and the voids evolution law are fully furnished.     

A yield function for single crystals containing voids

A yield function for single crystals containing voids has been developed based on a variational approach. This first yield function is phenomenologically extended by modifying the dependence on the mean stress and introducing three adjustable parameters. Unit cell finite element calculations are performed for various stress triaxiality ratios, main loading directions and porosity levels in the case of a perfectly plastic FCC single crystal. The three model parameters are adjusted on the unit cell calculations so that a very good agreement between simulation results and the proposed model is obtained.     

Yield criterion for porous media with spherical voids

The yield criterion of a porous material satisfying Gurson criterion conditions is studied here. We use the twofold limit analysis approach applied to a representative volume element of the porous media and recent optimization codes. Both upper and lower bounds are very close, and they give quasi-exact solutions. As a result, the Gurson approach is slightly improved and, for the first time, validated by a rigorous, full 3D static approach.     

Analytical study of a hollow sphere made of plastic porous material and subjected to hydrostatic tension-application to some problems in ductile fracture of metals

A solution to the problem stated in the title is given by treating the porous matrix as a homogeneous material obeying the Gurson-Tvergaard model. As a first application, it is shown that the flow stress under hydrostatic loading of a material containing two populations of voids with very different sizes is almost the same as that of a material with only one population of voids and the same total porosity. As a second application, a self-consistent method is used to derive a value for the parameter introduced by Tvergaard in the original Gurson model to account for the interactions between cavities. Other models are also considered and shown to fail to satisfy the self-consistency requirement, whatever the value chosen for the parameter characterizing interactions between voids.