Porous Plastic Solids Research Articles

Void growth and coalescence in sigmoidal hardening porous plastic solids under tensile and shear loading

Void growth and coalescence in sigmoidal hardening porous plastic solids under tensile and shear loading

Ductile void growing in micromorphic GLPD porous plastic solids containing two populations of cavities with different sizes

Ductile void growing in micromorphic GLPD porous plastic solids containing two populations of cavities with different sizes

An exact consistent tangent stiffness matrix for a second-gradient model for porous plastic solids: Derivation and assessment

It is well known that the use of a consistent tangent stiffness matrix is critical to obtain quadratic convergence of the global Newton iterations in the finite-element simulations of problems involving elasto-plastic deformation of metals, especially for large-scale metallic structure problems. In this article, we derive an exact consistent stiffness matrix for a porous material model, the GLPD model developed by Gologanu, Leblond, Perrin, and Devaux for ductile fracture of porous solids based on generalized continuum mechanics assumptions. Full expressions for the derivatives of the Cauchy stress tensor and the generalized moments stress tensor the model involved are provided and implemented into a finite-element code. The effectiveness and robustness of the proposed tangent stiffness moduli are assessed by applying the formulation in the finite-element simulations of typical ductile fracture problems. Comparisons between the performance of our stiffness matrix and the standard ones are also provided.

Ductile failure as a constitutive instability in porous plastic solids

A criterion for ductile rupture is derived using Rice’s theory for macroscopic strain localization, and a constitutive relation for porous plastic solids that accounts for inhomogeneous yielding at the mesoscale. At the microscopic scale, it is assumed that failure occurs by void coalescence along a band of voids. An approximate, parameter-free closed form expression for the failure criterion is derived as a function of a single scalar damage variable –the porosity– and the macroscopic stress state, characterized by the stress triaxiality and Lode parameters. For practical applications, an uncoupled approach is developed whereby the failure criterion is supplemented with a damage-free plasticity model and a loading path dependent damage evolution law. The predictive capabilities of the approach are illustrated by comparisons with finite element cell model studies. In particular, the influence of strain hardening is investigated in some detail.

Void Growth and Coalescence in Porous Plastic Solids With Sigmoidal Hardening

Abstract This paper presents an analysis of void growth and coalescence in isotropic, elastoplastic materials exhibiting sigmoidal hardening using unit cell calculations and micromechanics-based damage modeling. Axisymmetric finite element unit cell calculations are carried out under tensile loading with constant nominal stress triaxiality conditions. These calculations reveal the characteristic role of material hardening in the evolution of the effective response of the porous solid. The local heterogeneous flow hardening around the void plays an important role, which manifests in the stress–strain response, porosity evolution, void aspect ratio evolution, and the coalescence characteristics that are qualitatively different from those of a conventional power-law hardening porous solid. A homogenization-based damage model based on the micromechanics of void growth and coalescence is presented with two simple, heuristic modifications that account for this effect. The model is calibrated to a small number of unit cell results with initially spherical voids, and its efficacy is demonstrated for a range of porosity fractions, hardening characteristics, and void aspect ratios.

On the role of secondary voids and their distribution in the mechanism of void growth and coalescence in porous plastic solids

Detailed three-dimensional finite element based cell model studies are carried out to understand the role of secondary voids and their distribution in the mechanism of void growth and coalescence in a porous plastic solid. In contrast to most of the previous studies, the two populations of voids (primary and secondary) are modeled explicitly, with more realistic arrangements of the later. Numerical studies revealed that the spacing of the secondary voids sets the length scale at which localization of plastic flow occur. For a given volume fraction of the secondary voids, the mesoscopic ductility depends on the void distribution. In particular, the void arrangements leading to closely spaced secondary voids result in an early onset of plastic flow localization. The mesoscopic ductility, for a fixed arrangement of secondary voids, is observed to depend on the initial volume fraction and also on the location of the secondary voids in the ligament between the primary voids. For the stress triaxiality regime (1 ≤ T ≤ 3), and for the various distributions of the secondary voids analyzed in the present study, some variation in the total void volume fraction corresponding to the onset of flow localization is observed. The variation, however, is not so large as to forbid associating the critical void volume fraction with the void coalescence criterion. Thus, as a practical approach, an average value of this critical porosity may serve as the criterion for simulating accelerated void growth accompanying coalescence in a porous plastic solid with two populations of voids, using some phenomenological constitutive framework.

Application of a model of plastic porous materials including void shape effects to the prediction of ductile failure under shear-dominated loadings

An extension of Gurson's famous model (Gurson, 1977) of porous plastic solids, incorporating void shape effects, has recently been proposed by Madou and Leblond (Madou and Leblond, 2012a, 2012b, 2013; Madou et al., 2013). In this extension the voids are no longer modelled as spherical but ellipsoidal with three different axes, and changes of the magnitude and orientation of these axes are accounted for. The aim of this paper is to show that the new model is able to predict softening due essentially to such changes, in the absence of significant void growth. This is done in two steps. First, a numerical implementation of the model is proposed and incorporated into the SYSTUS® and ABAQUS® finite element programmes (through some freely available UMAT (Leblond, 2015) in the second case). Second, the implementation in SYSTUS® is used to simulate previous “numerical experiments” of Tvergaard and coworkers (Tvergaard, 2008, 2009, 2012, 2015a; Dahl et al., 2012; Nielsen et al., 2012) involving the shear loading of elementary porous cells, where softening due to changes of the void shape and orientation was very apparent. It is found that with a simple, heuristic modelling of the phenomenon of mesoscopic strain localization, the model is indeed able to reproduce the results of these numerical experiments, in contrast to Gurson's model disregarding void shape effects.

Steady shock waves in porous plastic solids

The affect of material porosity on propagation of shock waves in solids is examined in the context of finite strain, associated plasticity, with porosity incorporated via the Gurson model and accounting for material hardening. Setting is analogous to the fluid dynamics piston shock model so that deformation of the semi-infinite medium is permitted only in the longitudinal direction. The steady response, which develops by imposing constant piston velocity in either tension or compression, is examined by sectors mapping of the characteristic velocity as determined by the constitutive model. It is shown that even the slightest levels of initial porosity can have an appreciable effect on field response, inducing destructive unsteady behavior accompanied by increased shock dissipation. Numerical illustration of limit velocities at appearance of a plastic shock and at onset of that unsteady behavior are presented, showing that material porosity delays initiation of plastic shock waves and promotes higher energy consumption which may, in turn, enhance protective capabilities.

Coalescence of voids by internal necking: Theoretical estimates and numerical results

Upper-bound estimates and supposedly exact numerical results are obtained for the limit loads associated with cylindrical cells containing voids and subjected to boundary conditions that are consistent with post-localization kinematics in porous plastic solids. When supplemented with evolution equations for the microstructural variables, the results can be used in the modeling of void coalescence by internal necking in ductile materials.

On the effect of the matrix tension–compression asymmetry on damage evolution in porous plastic solids

Strength differential effects (different behavior in tension versus compression) are observed at the polycrystal level, if either twinning or non-Schmid type slip are contributors to plastic deformation at the single crystal level. Despite recent progress in modeling the effects of this asymmetry on the plastic flow of the fully dense polycrystal, modeling the response of such polycrystals in the presence of voids remains a challenge. The aim of this paper is to numerically assess the influence of the tension–compression asymmetry of the plastic flow in the matrix (solid material) on void evolution and the location of the zone corresponding to maximum damage in round tensile specimens subject to uniaxial tension. It is shown that if the matrix tensile strength is higher than its compressive strength, void growth and damage distribution are similar to that in classical materials obeying Gurson's criterion. On the other hand, for certain porous polycrystals in which the matrix tensile strength is lower than its compressive strength, the rate of void growth rate is much slower. Damage distribution is significantly different, the location of the zone of maximum porosity shifts from the center of the specimen outwards. Furthermore, void growth is significantly affected by the rate of change of the tension–compression asymmetry of the matrix with accumulated plastic strain.

Effect of Microstructure Evolution on the Overall Response of Porous-Plastic Solids

Ductile fracture is the macroscopic result of a micromechanical process consisting in void nucleation and growth to coalescence. While growing in size, voids also evolve in shape because of the non-uniform deformation field in the surrounding material; this shape evolution is either disregarded or approximately accounted for by constitutive laws for porous-plastic solids. To assess the effect of void distortion on the overall properties of a porous-plastic material prior to any coalescence-dominated event, we here present a micromechanical study in which the void-containing material is treated as a two-phase (matrix and inclusion) composite. A cylindrical representative volume element (RVE), featuring elliptic cross-section and containing a coaxial and confocal elliptic cylindrical cavity, is considered. In case of a matrix obeying flow theory of plasticity, the overall yield domain and the evolution laws for the volume fraction and aspect ratio of the void are obtained. Under assigned strain histories, these theoretical findings are then compared to finite element unit-cell simulations, in order to assess the capability of the proposed results to track microstructure evolution. The improvements with respect to the customarily adopted Gurson’s model are also discussed.

A theoretical approach of strain localization within thin planar bands in porous ductile materials

Propagation of cracks in ductile materials is well known to occur through two possible mechanisms: coalescence of cavities and formation of shear bands (‘void sheet mechanism’). The classical Gurson–Tvergaard–Needleman (GTN) homogenized model for such materials incorporates some phenomenological modelling of coalescence, but not of formation of shear bands assisted by the presence of microvoids, and this generates a number of shortcomings. In order to solve these difficulties, this paper presents a unified model of both coalescence and formation of shear bands in porous plastic solids, including the possible couplings between the two. Both phenomena are viewed as expressions of the same basic effect, namely strain localization within thin planar bands, the only difference being the mode of deformation. The model is first developed assuming a periodic distribution of cavities, then critically assessed through comparison with some micromechanical numerical simulations based on the same assumption, and finally extended to the case of a random distribution of voids. To cite this article: J.-B. Leblond, G. Mottet, C. R. Mecanique 336 (2008).

Numerical implementation and assessment of a phenomenological nonlocal model of ductile rupture

Just like all constitutive models involving softening, Gurson’s [A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteria and flow rules for porous ductile media, ASME J. Engrg. Mater. Technol. 99 (1977) 2–15] model of ductile rupture predicts unlimited localization of strain and damage. Leblond et al.’s [J.B. Leblond, G. Perrin, J. Devaux, Bifurcation effects in ductile metals with nonlocal damage, ASME J. Appl. Mech. 61 (1994) 236–242] have proposed to solve this problem in a heuristic way by using a nonlocal evolution equation for the porosity, which expresses its time-derivative as a spatial convolution integral of some “local porosity rate”. This paper is devoted to the numerical implementation and assessment of this phenomenological variant of Gurson’s model. The numerical implementation proposed bears some resemblance to Aravas’s [N. Aravas, On the numerical integration of a class of pressure-dependent plasticity models, Int. J. Numer. Methods Engrg. 24 (1987) 1395–1416] classical one but departs from it through use of an explicit algorithm with respect to the porosity, if not with respect to other parameters. The main reason for this choice lies in the proof of existence and uniqueness, for such an algorithm, of the solution of the “projection problem” onto Gurson’s yield locus. The assessment of the model is based on two criteria, absence of mesh size effects in finite element computations and agreement of experimental and numerical results for some typical ductile fracture tests. Since the first topic has already been investigated by Tvergaard and Needleman’s [V. Tvergaard, A. Needleman, Effects of nonlocal damage in porous plastic solids, Int. J. Solids Struct. 32 (1995) 1063–1077], we concentrate on the second one. Unfortunately, numerical experience reveals that the modification of Gurson’s model envisaged degrades the agreement between experimental results and model predictions. A theoretical analysis of this phenomenon is presented. Although based on crude simplifying assumptions, this analysis suffices to qualitatively explain the failure of the model and suggest a simple remedy. Numerical experience confirms that this remedy considerably improves numerical predictions. Thus, with this modification, Leblond et al. (1994)’s proposal appears as a viable solution to the problem of unlimited localization of strain and damage in Gurson (1997)’s model.

A new model for porous nonlinear viscous solids incorporating void shape effects – II: Numerical validation

Abstract The aim of this work is to critically assess the new model for porous, nonlinear viscous solids incorporating void shape effects proposed in Part I, by comparing its predictions with the results of some numerical micromechanical simulations. Two kinds of simulations are performed. First, the gauge surface of spheroidal representative volume elements, as considered in Part I, is determined for various values of the porosity, the aspect ratio of the void and the Norton exponent. This is done through minimization of the macroscopic viscous potential over a family of trial velocity fields especially adapted to the spheroidal geometry, which was proposed by Lee and Mear. Such simulations allow not only for satisfactory validation of the approximate analytical gauge surface proposed, but also for adjustment of the heuristic coefficients involved in the evolution equation for the void shape parameter. Second, the evolution in time of cylindrical cells subjected to various mechanical loads is determined by the finite element method. The quasi-periodicity of this new geometry is intended to approximately represent interactions between neighbouring voids. These simulations also reveal very good agreement between model predictions and numerical calculations, provided that the effect of the new geometry considered is accounted for by using a non-unity value for the analog of Tvergaard's famous “ q 1 ” parameter for porous plastic solids.

Effects of nonlocal damage in porous plastic solids

A nonlocal constitutive formulation for a porous ductile material is investigated, in which delocalization relates to the damage mechanism. An elastic-viscoplastic material model is used, with delocalization incorporated in terms of an integral condition on the rate of increase of the void volume fraction. Two model problems are analysed to study the effect of including this material length, one relates to localization of plastic flow in shear bands, while the other considers a metal matrix composite. Both model problems involve final failure in strongly nonhomogeneous strain fields, and it is shown that the inherent mesh sensitivity of the numerical failure predictions can be removed by using the nonlocal material model considered here.

The influence of nucleation criterion on shear localization in rate-sensitive porous plastic solids

The influence of the void nucleation criterion on localization in strain rate-sensitive porous plastic solids is analyzed under plane strain and axisymmetric tensile loading conditions using an infinite band model. The void nucleation process is described in terms of a void nucleation stress that depends linearly on the hydrostatic tension with a coefficient that is less than unity. Although the nucleation strain depends monotonically on the void nucleation stress coefficient, the dependence of the localization strain on this quantity is not necessarily monotonic. With the void nucleation stress coefficient fixed, there can also be a nonmonotonic dependence of the critical strain for localization on imposed strain rate.

Void growth and coalescence in porous plastic solids

A boundary value problem simulating a periodic array of spherical voids in an isotropically hardening elastic-viscoplastic matrix is analyzed. The calculations show a shift from a general axisymmetric deformation state to a mode of uniaxial straining at which point the plastic deformation localizes to the ligament between neighboring voids. This event is associated with the accelerated void growth accompanying coalescence. The numerical results are related to the description of void growth and coalescence within a phenomenological constitutive framework for progressively cavitating solids.

The effect of porosity distribution on ductile failure

T he effect of a nonuniform distribution of porosity on flow localization and failure in a porous material is analyzed numerically. The void density distribution and properties used to characterize the material behavior were obtained from measurements on partially consolidated and sintered iron powder. The calculations were carried out using an elastic viscoplastic constitutive relation for porous plastic solids. Local material failure is incorporated into the model through the dependence of the flow potential on void volume fraction. The region modelled is a small portion of a larger body, subject to various triaxial stress conditions. Both plane strain and axisymmetric deformations are considered with imposed periodic boundary conditions. Interactions between regions with higher void fractions promote plastic flow localization into a band. Local failure occurs by void growth and coalescence within the band. The results suggest a failure criterion based on a critical void volume fraction that is only weakly dependent on stress history. The critical void fraction does. however, depend on the initial void distribution and material hardening characteristics.

Effect of Yield Surface Curvature on Necking and Failure in Porous Plastic Solids

The effect of material path dependent hardening on neck development and the onset of ductile failure is analyzed numerically. The calculations are carried out using an elastic-viscoplastic constitutive relation that has isotropic hardening and kinematic hardening behaviors as limiting cases and that accounts for the weakening due to the growth of micro-voids. Final material failure is incorporated into the constitutive model by the dependence of the plastic potential on void volume fraction. Results are obtained for both axisymmetric and plane strain tension. Failure is found to initiate by void coalescence at the neck center in axisymmetric tension and within a shear band in plane strain tension. The increased curvature of flow potential surfaces associated with the kinematic hardening solid leads to somewhat more rapid diffuse neck development than occurs for the isotropic hardening solid. However, a much greater difference between the predictions of the two constitutive models is found for the onset of ductile failure.

Effects of microvoids on crack blunting and initiation in ductile materials

Effects of microvoid nucleation and growth on the stress and strain fields near a crack tip are studied by employing the Gurson's constitutive model of porous plastic solids and by using a finite element method suitably formulated to admit large geometry change. It is shown that microvoids nucleate and grow rapidly in the region at a distance less than the blunted tip diameter away from the crack tip and have a great influence on the stress fields therein. Effects of a large void near a crack tip on crack initiation are also investigated by taking the shear localization into account. The critical value of the crack tip opening displacement predicted by the present investigation is shown to coincide with the published experimental data.