Léo Morin

A new technique for finite element limit-analysis of Hill materials, with an application to the assessment of criteria for anisotropic plastic porous solids

Abstract The present work is devoted to the numerical limit-analysis of Hill materials with particular emphasis on anisotropically plastic porous solids. Its aim is to provide an efficient method of limit-analysis based on the standard finite element method including elasticity, and present a few applications. We first present the numerical implementation of Hill’s criterion. We then describe the procedure used for the numerical limit-analysis, which basically consists of using a single large load step ensuring that the limit-load is reached, without updating the geometry. Also, the convergence of the elasto–plastic iterations is accelerated by suitably adjusting the elastic properties of the material. The method is applied to assess Gurson-like criteria for orthotropically plastic materials containing spheroidal voids. This is done by performing numerical limit-analyses of elementary cells made of a Hill material and containing confocal spheroidal voids, subjected to classical conditions of homogeneous boundary strain rate. The numerical results are compared to the model predictions for both the yield surface and the flow rule, and this permits to discuss the accuracy of the theoretical models considered.

Gurson's Criterion and Its Derivation Revisited

This paper revisits Gurson’s [1,2] classical limit-analysis of a hollow sphere made of some ideal-plastic von Mises material and subjected to conditions of homogeneous boundary strain rate (Mandel [3], Hill [4]). Special emphasis is placed on successive approximations of the overall dissipation, based on a Taylor expansion of one term appearing in the integral defining it. Gurson considered only the approximation based on the first-order expansion, leading to his well-known homogenized criterion; higher-order approximations are considered here. The most important result is that the correction brought by the second-order approximation to the first-order one is significant for the porosity rate, if not for the overall yield criterion. This bears notable consequences upon the prediction of ductile damage under certain conditions.

A damage model for ductile porous materials with a spherically anisotropic matrix

In the present study, we investigate the macroscopic strength of ductile porous materials having a Hill-type radial anisotropic matrix. The procedure is based on a limit analysis (LA)-based kinematic approach of a rigid plastic hollow sphere. We first established the exact solution (stress and velocity fields) to the problem of the hollow sphere subjected to an external hydrostatic loading. Then, we propose, for general loadings, an appropriate trial velocity field which allows to implement the kinematic LA procedure. The resulting macroscopic criterion, whose closed-form expression is provided, extends the well-known Gurson criterion to materials with radial anisotropy. Numerical limit analyses are provided by performing standard finite elements computations which validate the new criterion. Finally, the yield criterion is supplemented by a plastic flow rule and evolution equations of the internal parameters, allowing to study the predictions of the complete model for axisymmetric proportional loadings at fixed stress triaxiality. A strong influence of the radial anisotropy is observed on the stress softening and the growth of the porosity.