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).     

Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu–Leblond–Devaux’s (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill’s [Hill, R., 1948. A theory of yielding and plastic flow of anisotropic solids. Proc. Roy. Soc. London A 193, 281–297] anisotropic yield criterion) and the representative volume element is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.     

Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry

This paper is devoted to modeling the effects of the tension-compression asymmetry of the matrix on yielding of the void-matrix aggregate. The matrix plastic behavior is described by the Cazacu et al. [2006. Orthotropic yield criterion for hexagonal closed packed metals. Int. J. Plasticity 22, 1171-1194] isotropic yield criterion, which captures strength differential effects. Using an upper-bound approach, a new analytic isotropic plastic potential for a random distribution of spherical voids is obtained. The derived analytic potential is sensitive to the third invariant of the stress deviator and displays tension-compression asymmetry. In the case when the matrix material has the same yield in tension and compression, it reduces to Gurson's [1977. Continuum theory of ductile rupture by void nucleation and growth: Part I: Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. Trans. ASME Ser. H 99, 2-15.] criterion. Furthermore, the proposed criterion predicts the exact solution of a hollow sphere loaded in hydrostatic tension or compression. The accuracy of the proposed analytical criterion is assessed through comparisons with finite-element cell calculations.     

The mechanical behaviour of ductile materials damaged by two generations of voids

Large strain finite element method is employed to investigate the damaging effect of two generations of voids in ductile materials. An axisymmetric cylinder embedding an initially spherical void is chosen as the model cell. Secondary voids will initiate around the initial void when the local stress/strain in the matrix increases to certain critical conditions. This event is numerically simulated through an empty element technique. The interaction between these two generations of voids has been proved to be favourable to the voiding condition, thus accelerating the material damage, characterized by the value of the overall elastic modulus which may undergo drastic drop when nearing final fracture.     

Effects of stress invariants and reverse loading on ductile fracture initiation

Recent research studies on ductile fracture of metals have shown that the ductile fracture initiation is significantly affected by the stress state. In this study, the effects of the stress invariants as well as the effect of the reverse loading on ductile fracture are considered. To estimate the reduction of load carrying capacity and ductile fracture initiation, a scalar damage expression is proposed. This scalar damage is a function of the accumulated plastic strain, the first stress invariant and the Lode angle. To incorporate the effect of the reverse loading, the accumulated plastic strain is divided into the tension and compression components and each component has a different weight coefficient. For evaluating the plastic deformation until fracture initiation, the proposed damage function is coupled with the cyclic plasticity model which is affected by all of the stress invariants and pervious plastic deformation history.For verification and evaluation of this damage-plasticity constitutive equation a series of experimental tests are conducted on high-strength steel, DIN 1.6959. In addition finite element simulations are carried out including the integration of the constitutive equations using the modified return mapping algorithm. The modeling results show good agreement with experimental results.     

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.     

Modeling of crack growth in ductile solids: a three-dimensional analysis

A population of several spherical voids is included in a three-dimensional, small scale yielding model. Two distinct void growth mechanisms, put forth by [Int. J. Solids Struct. 39 (2002) 3581] for the case of a two-dimensional model containing cylindrical voids, are well contained in the model developed in this study for spherical voids. A material failure criterion, based on the occurrence of void coalescence in the unit cell model, is established. The critical ligament reduction ratio, which varies with stress triaxiality and initial porosity, is used to determine ligament failure between the crack tip and the nearest void. A comparison of crack initiation toughness of the model containing cylindrical voids with the model containing spherical voids reveals that the material having a sizeable fraction of spherical voids is tougher than the material having cylindrical voids. The proposed material failure determination method is then used to establish the fracture resistance curve (J–R curve) of the material. For a ductile material containing a small volume fraction of microscopic voids initially, the void by void growth mechanism prevails, which results in a J–R curve having steep slope. On the other hand, for a ductile material containing a large volume fraction of initial voids, the multiple voids interaction mechanism prevails, which results in a flat J–R curve. Next, the effect of T-stress on fracture resistance is examined. Finally, nucleation and growth of secondary microvoids and their effects on void coalescence are briefly discussed.     

Prediction of ductile fracture in tension by bifurcation,localization, and imperfection analyses

In this investigation, it is shown that the onset of ductile fracture in tension can be interpreted as the result of a supercritical bifurcation of homogeneous deformation and that this fact can be applied to predict ductile fracture initiation of materials with general imperfections or flaws. We focus on one dimensional quasi-static simple tension for rate-independent isotropic plastic materials. For deformation beyond the bifurcation point, multiple equilibrium paths appear. The homogeneous deformation, as one of the equilibrium paths, loses stability while the inhomogeneous paths are stable, thus indicating the occurrence of strain localization. This investigation also provides a physical example for the application of the Lambert W function in material localization analyses. Material instability is treated as the instability of a static system with dynamic perturbation. We also address the presence of microvoids in a power law plastic material as an unfolding of the supercritical pitchfork bifurcation. The imperfect system, idealized as spherical voids within the plastic matrix, is analyzed using the familiar Gurson model which is based on the presumption of a randomly voided material and characterized by the volume fraction of voids. If, in addition, the sizes of the microvoids are known, this then provides a length scale for the imperfection zone. In this manner, relevance to the sample size effects of strain-to-failure for ductile fracture initiation is addressed by considering separate zones with variations in void volume fractions. Fracture initiation predictions are presented and compare very well to existing experimental results.     

A constitutive model for elastoplastic solids containing primary and secondary voids

In many ductile metallic alloys, the damage process controlled by the growth and coalescence of primary voids nucleated on particles with a size varying typically between 1 and 100 μm, is affected by the growth of much smaller secondary voids nucleated on inclusions with a size varying typically between 0.1 and 3 μm. The goal of this work is first to quantify the potential effect of the growth of these secondary voids on the coalescence of primary voids using finite element (FE) unit cell calculations and second to formulate a new constitutive model incorporating this effect. The nucleation and growth of secondary voids do essentially not affect the growth of the primary voids but mainly accelerate the void coalescence process. The drop of the ductility caused by the presence of secondary voids increases if the nucleation strain decreases and/or if their volume fraction increases and/or if the primary voids are flat. A strong coupling is indeed observed between the shape of the primary voids and the growth of the second population enhancing the anisotropy of the ductility induced by void shape effects. The new micromechanics-based coalescence condition for internal necking introduces the softening induced by secondary voids growing in the ligament between two primary voids. The FE cell calculations were used to guide and assess the development of this model. The use of the coalescence condition relies on a closed-form model for estimating the evolution of the secondary voids in the vicinity of a primary cavity. This coalescence criterion is connected to an extended Gurson model for the first population including the effect of the void aspect ratio. With respect to classical models for single void population, this new constitutive model improves the predictive potential of damage constitutive models devoted to ductile metal while requiring only two new parameters, i.e. the initial porosity of second population and a void nucleation stress, without any additional adjustment.     

受有两级空洞损伤时韧性材料的力学行为

本文利用大应变有限元方法研究了两级空洞对韧性材料的损伤作用.模型是以轴对称圆柱基体作为胞元,内含一初始的球型空洞.基体内的应力/应变随胞元外载的增大而达到临界状态,从而在围绕初级空洞的基体内将萌生次级空洞.后者是由空单元实现的.两级空洞的交互作用被证明将促进材料中的空洞化现象从而加速损伤并导至材料的总体弹性模量值在临近破断时急剧下降.     

Accelerated void growth in porous ductile solids containing two populations of cavities

This paper presents an analytical and numerical study of accelerated void growth in porous ductile solids arising from the presence of two populations of cavities very different in size. It is based on the model problem of some hollow sphere made of porous plastic material and subjected to hydrostatic tension. The central hole plays the role of a typical big cavity of the first population while those dispersed in the matrix stand for the small cavities of the second one. The behavior of the matrix is supposed to obey Gurson's famous “homogenized” model for porous ductile solids (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. ASME J Engng Materials Technol 99, 2–15). The analytic solution of this model problem shows that the small voids located near the big one grow twice as fast as the latter void. This suggests that in a subsequent step, these small cavities may reach coalescence prior to the big ones, thus creating spherical shells of ruined matter around the cavities of the first population and leading to accelerated growth of the latter cavities; this scenario is in agreement with experimental evidence. Since this subsequent step is not amenable to a complete analytic solution, it is studied numerically. Finally, a simplified model reproducing the two steps of void growth (prior to coalescence of the small voids and after it has started) is developed on the basis of the analytical solution for the first step and some elements of a similar solution for the second one. The results derived from this simplified model are in good quantitative agreement with those obtained through the complete numerical simulations.     

Constitutive potential for ductile damaged material and void growth rate

In this paper a general formula for plastic potential including isotropic ductile damage has been presented on the basis of the thermodynamics for irreversible process and intemal variable theory. With this formula the mass conservation law is satisfied and it also contains a series of unknow coefficients which are the function of macro equivalent stress and the average micro equivalent stress and an unknown function which is the function of two generalized forces. The approximate yield surface equation for isotropically damaged materials is developed. Using this equation the void growth rate is calculated for nonlinear material containing voids. The present results are in good agreement with the numerical results given by the cell model.     

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 physical model for nucleation and early growth of voids in ductile materials under dynamic loading

Spall fracture and other rapid tensile failures in ductile materials are often dominated by the rapid growth of voids. Recent research on the mechanics of void growth clearly shows that void nucleation may be represented as a bifurcation phenomenon, wherein a void forms spontaneously followed by highly localized plastic flow around the new void. Although thermal, viscoplastic, and work hardening effects all play an essential role in the earliest stages of nucleation and growth, the flow becomes dominated by spherical radial inertia, which soon causes all voids to grow asymptotically at the same rate, regardless of differences in initial conditions or constitutive details, provided only that there is the same density of matrix material and the same excess loading history beyond the cavitation stress.These two facts, initiation by bifurcation at a cavitation stress, at which a void first appears, and rapid domination by inertia, are used to postulate a simple, but physically realistic, model for nucleation and early growth of voids in a ductile material under rapid tensile loading. A reasonable statistical distribution for the cavitation stress at various nucleation sites and a simple similarity solution for inertially dominated void growth permit a simple calculation of the initiation and early growth of porosity in the material.Parametric analyses are presented to show the effect that loading rate, peak loading stress, density of nucleation sites, physical properties of the material, etc. have on the applied pressure and distribution of void sizes when a critical porosity is reached.     

EVOLUTION OF VOIDS IN DUCTILE POROUS MATERIALS AT HIGH STRAIN RATE

In the present work,a dynamic damage model in ductile materials underthe application of dynamic general stresses loading is presented.The evolution equationof ductile voids has the closed form,in which work-hardening,the change of surfaceenergy of voids,rate-dependent,inertial effects are taken into account.Theexpressions of critical stresses for the growth and compaction of voids are directlyobtained from the evolution equations of voids.From the expressions,the resultobtained by Carroll and Holt,as a special example,is given.Numerical analysis ofthe model indicates that the growth of voids is sensitive to the strain rates.The voidsgrow quickly as the increase of strain rates.It is also shown that the influence of theinertial effects on the void growth is great at high loading rates.It appears to resist thegrowth of voids.In addition,a dynamic collapse model of ductile voids is alsoproposed,which can be applied to study the problems of compaction in powder andother materials.     

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.     

高应变率下延性多孔介质中孔洞的动态演化

本文提出了一个新的材料延性动态损伤模型，模型中不但包括了率效应，同时还考虑了惯性效应，孔洞表面能变化和材料硬化对孔洞演化的影响。此外，在模型中同时考虑了体应力和偏应力对孔洞演化的作用，从孔洞演化方程地接到了孔洞增长和压缩应力临界表达式，Ｃａｒｒｏｌｌ和Ｈｏｌｔ结果作为该表达式的一个特例而得出，模型的数值分析得出以下结论：①延性孔洞的动太增长对率效应十分敏感，应变率越高，孔洞增长越快；②惯性效应在主     

A model for dynamic ductile behavior applicable for arbitrary triaxialities

Previous models for ductile behavior incorporating dynamic effects were derived assuming the triaxiality to be very high. Such is indeed the case in experiments of impact of plates, but not in those of expansion of rings or shells. Here we propose a model for dynamic ductile behavior applicable for arbitrary triaxialities. The material is schematized as a porous, viscoplastic Norton medium. The essential approximation made consists in accounting for this acceleration arising from growth of the voids, but not for that arising from their change of shape. Within the framework of this approximation, the special case of a hollow cylinder loaded axisymmetrically in generalized plane strain is treated exactly. This special, analytic solution is used as a guide to propose a model for the practically more significant case of spherical cavities. This model is finally extended in a heuristic way to incorporate elasticity effects.