Jean-Baptiste Leblond

Approximate models for ductile metals containing non-spherical voids-case of axisymmetric prolate ellipsoidal cavities

Abstract T he aim of this paper is to extend the classical Gurson analysis of a hollow rigid ideal-plastic sphere loaded axisymmetrically to an ellipsoidal volume containing a confocal ellipsoidal cavity, in order to define approximate models for ductile metals containing non-spherical voids. Only axisymmetric prolate cavities are considered here. The analysis makes an essential use of an “expansion” velocity field satisfying conditions of homogeneous boundary strain rate on every ellipsoid confocal with the cavity. A two-field estimate of the overall yield criterion is presented and shown to be reducible, with a few approximations, to a Gurson-like criterion depending on the “shape parameter” of the cavity. The accuracy of this estimate is assessed through comparison with some results derived from a numerical minimization procedure. The two-field approach is also used to derive an approximate evolution equation for the shape parameter ; comparison with some finite element simulations reveals a reasonable qualitative agreement, and suggests a slight modification of the theoretical formula which leads to acceptable quantitative agreement. The application of these results to materials containing axisymmetric prolate ellipsoidal cavities with parallel or random orientations is finally discussed.

Recent extensions of Gurson's model for porous ductile metals

This paper is devoted to two distinct extensions of Gurson’s (1977) famous model for plastic voided metals. Gurson’s work was based on an approximate limit-analysis of a typical elementary volume in a porous material, namely a hollow sphere subjected to conditions of arbitrary homogeneous boundary strain rate. The first extension envisaged consists in considering a more general geometry, namely a spheroidal volume containing some spheroidal confocal cavity. The aim here is to incorporate void shape effects into Gurson’s model. The second extension again considers a hollow sphere, but now subjected to conditions of inhomogeneous boundary strain rate. The goal is to account for possible strong variations of the macroscopic mechanical fields at the scale of the representative cell (i.e. of the void spacing), as encountered near crack tips.

Ductile Fracture by Void Growth to Coalescence

Publisher Summary An important failure mechanism in ductile metals and their alloys is by growth and coalescence of microscopic voids. In structural materials, the voids nucleate at inclusions and second-phase particles by decohesion of the particle–matrix interface or by particle cracking. Void growth is driven by plastic deformation of the surrounding matrix. Early micromechanical treatments of this phenomenon considered the growth of isolated voids. Later, constitutive equations for porous ductile solids were developed based on homogenization theory. Among these, the most widely known model was developed by Gurson for spherical and cylindrical voids.

Approximate Models for Ductile Metals Containing Nonspherical Voids—Case of Axisymmetric Oblate Ellipsoidal Cavities

The aim of this paper is to extend the classical Gurson (1977) analysis of a hollow rigid ideal-plastic sphere loaded axisymmetrically to an ellipsoidal volume containing a confocal oblate ellipsoidal cavity. An “expansion” velocity field satisfying conditions of homogeneous boundary strain rate is used to derive a two-field estimate of the overall yield criterion. The latter is shown to be reducible, with a few approximations, to a Gurson-like criterion depending on the “shape parameter” of the cavity. The accuracy of this estimate is assessed through comparison with some results derived from a numerical minimization procedure. An approximate evolution equation for the shape parameter is also presented; comparison with some finite element simulations suggests a slight modification of the theoretical formula leading to considerably enhanced agreement.

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.

Bifurcation Effects in Ductile Metals With Nonlocal Damage

The purpose of this paper is to investigate some bifurcation phenomena in a porous ductile material described by the classical Gurson (1977) model, but with a modified, nonlocal evolution equation for the porosity. Two distinct problems are analyzed theoretically: appearance of a discontinuous velocity gradient in a finite, inhomogeneous body, and arbitrary loss of uniqueness of the velocity field in an infinite, homogeneous medium. It is shown that no bifurcation of the first type can occur provided that the hardening slope of the sound (void-free) matrix is positive. In contrast, bifurcations of the second type are possible; nonlocality does not modify the conditions of first occurrence of bifurcation but does change the corresponding bifurcation mode, the wavelength of the latter being no longer arbitrary but necessarily infinite. A FE study of shear banding in a rectangular mesh deformed in plane strain tension is finally presented in order to qualitatively illustrate the effect of finiteness of the body; numerical results do evidence notable differences with respect to the case of an infinite, homogeneous medium envisaged theoretically.

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

Abstract 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.

Effect of void locking by inclusions upon the plastic behavior of porous ductile solids—I: theoretical modeling and numerical study of void growth

Abstract The aim of this work is to model the effect of inclusions upon void growth in porous ductile metals. This effect arises from the fact that when such a material is subjected to low tensile, or compressive mean stresses, voids can undergo compressive stresses in some directions; then, if they are still in contact in these directions with the inclusions they originated from, void shrinkage is hindered by the inclusions. A numerical yield surface is first derived through limit-analysis of some RVE, accounting for presence of some rigid inclusion, and compared to the (also numerical) yield surface without inclusion. Using then, as a basis, the Gologanu–Leblond–Devaux (GLD) model accounting for void shape effects but not for influence of inclusions, an analytical approximate model is developed, taking into account both void shape effects and influence of inclusions. This model consists of a macroscopic yield criterion analogous to that of the GLD model, depending upon the porosity and the void shape parameter, a flow rule obeying normality, and evolution equations for the internal parameters. It contains four adjustable coefficients. Two of these are determined through consideration that the yield surface with effect of inclusions should be tangent to the GLD yield surface for a certain “critical” stress state. The other two are adjusted so as to get the best possible fit between the analytical yield surface (with effect of inclusions) and the supposedly exact numerical one. The model is finally critically assessed through comparison of its predictions with results of FE simulations performed in ideal-plasticity (in order for the comparison with the theory developed with this hypothesis to be fully meaningful).

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

Abstract 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 Gursons 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.

Crack front rotation and segmentation in mixed mode I + III or I + II + III. Part II: Comparison with experiments

Abstract Using the results presented in Part I, together with various criteria, we try to explain qualitatively and quantitatively crack front rotation and segmentation under mixed mode I+III or I+II+III conditions. We first give a qualitative explanation of segmentation in mode I+III, based on the energetic theory of fracture. However, a satisfactory quantitative interpretation would require knowledge of a parameter which theory is unable to predict and which is not provided in experimental reports. We then concentrate on crack front rotation. It is shown that a suitable energetic criterion is able to reproduce quantitatively crack front “rotation rates” observed in brittle fracture in mode I+III. In fatigue, a criterion considering the sole mode I stress intensity factor is found to be more appropriate. Finally, for fatigue in general mode I+II+III conditions, we show that the crack front rotation rate is again satisfactorily reproduced by the same criterion as before, and also that a suitable extension of the well-known principle of local symmetry allows satisfactory prediction of the mean kink angle.