Introduction to the Variational Formulation in Mechanics

6Materials Exhibiting Plasticity

6.1 Introduction

This chapter tackles the mechanics of bodies featuring elasto‐plastic behavior. The term plasticity refers to time‐independent inelastic processes, excluding viscous phenomena such as creep and relaxation, among others.

The presentation is divided into two parts. First, we describe the mathematical models which characterize the stress–strain constitutive relation for elasto‐plastic materials. From that point, the variational principles specified to plasticity are developed. Such variational problems provide the roots for the posterior computational analysis in plasticity, leading to limit analyses and the study of strain and stress evolution.

6.2 Elasto‐Plastic Materials

First, let us introduce some characteristic elements from tensile tests in order to identify the physical phenomena which must be embedded in a mathematical model that represents the material elasto‐plastic behavior.

Figure 6.1 illustrates typical results obtained from the tensile testing of a metal specimen (think of a bar). The axial load is monotonically increasing, and with a controlled strain rate . The different curves in the figure contain a linear part, that is a region of proportionality between the applied load and the strain in the specimen, and a nonlinear region featuring a reduced slope. The slope in this second part is always positive in a strain ...