A Direct Method for the Determination of Effective Strength Domains for Periodic Elastic-Plastic Media

Abstract

This work is devoted to the analysis of the mechanical strength of heterogeneous periodic media submitted to variable loads. The shakedown static approach is coupled with the homogenisation theory to obtain the strength domains: the effective ultimate yield surfaces and shakedown domains. This direct method leads to constrained optimization problems on a three-dimensional unit cell. In order to treat general microstructures, the fem is used and a specific formulation is introduced to rigorously take into account the specificities of the localisation problems. It eventually leads to discretized constrained optimization problems. Using this method, this work is focused on the case of heterogeneous plates, which exhibit periodic microstructures only in the two in-plane directions. The effective model (Love-Kirchhoff) is obtained by solving problems on a 3D unit cell. Membrane-bending couplings and three dimensional effects are naturally taken into account. The effective strength domains are obtained in terms of in-plane stresses and bending moments. The method is first validated for a homogeneous plate. Two other examples are also proposed : a sandwich plate and a periodically perforated plate.

Statistical Analyses of the Strengths of Particulate Reinforced Metal Matrix Composites (PRMMCs) Subjected to Multiple Tensile and Shear Stresses

“…To this end, direct method based on Melan and Koiter's theorem has a remarkable advantage over the conventional incremental method since they allow the strengths to be calculated without considering the specific load history. Using direct methods (DM) constructed from static theorem in conjunction with the finite element (FE) based homogenization techniques, Weichert et al [9], Magoariec et al [10], Bourgeois et al [11], Zhang et al [12], You et al [13], Chen et al [14], Zhang et al [15,16] elucidated in their works how the macroscopic feasible load domains of different heterogeneous materials can be calculated. Similarly, macroscopic strengths were also evaluated from numerical methods developed based on kinematic theorem, Carvelli [17], Chen and Ponter [18], Barrera et al [19], Li and Yu [20,21], Canh et al [22].…”

Statistical Analyses of the Strengths of Particulate Reinforced Metal Matrix Composites (PRMMCs) Subjected to Multiple Tensile and Shear Stresses

“…To this end, direct method based on Melan and Koiter's theorem has a remarkable advantage over the conventional incremental method since they allows the strengths to be calculated without considering the specific load history. Using direct methods (DM) constructed from static theorem in conjunction with the finite element (FE) based homogenization techniques, Weichert et al [11], Magoariec et al [12], Bourgeois et al [13], H.T. Zhang et al [14], J.H.…”