An internal variable finite-strain theory of plasticity within the framework of convex analysis

Abstract

Abstract. An internal variable constitutive theory for elastic-plastic materials undergoing finite strains is presented. The theory is based on a corresponding study in the context of small strains [6], and has the following features: first, with a view to embracing the classical notions of convex yield surfaces and the normality law, the evolution law is developed within the framework of nonsmooth convex analysis, which proves to be a powerful unifying tool; secondly, the special case of elastic materials is recovered from the theory in a natural manner. After presentation of the theory a concrete example is discussed in detail.1. Introduction. The theory of plasticity in its classical small-strain form is wellestablished, especially that form which the theory takes in its application to metals. The finite-strain theory, on the other hand, while showing indications that it is on the way to becoming an established branch of mechanics, is still nevertheless the subject of considerable effort and debate, and certain of its aspects remain unsettled. The literature on the subject has in the meantime acquired voluminous proportions; without attempting a comprehensive survey we mention as important contributions the early work of Green and Naghdi [7] and of Lee [17] in which were addressed, inter alia, the question of the decomposition of deformation into elastic and plastic parts, the former proposing an additive decomposition of strain and the latter a multiplicative decomposition of deformation gradient. The multiplicative decomposition has proved more popular, and forms the basis for a number of alternative theories; as examples we mention the works of Mandel [18], Halphen and Nguyen [8], Simo and

Plasticity

“…For example, (III) is the most often used, and in fact has been the goal of the theory developed in Chapter 3. Formulation (II) has been used in [26,48,116,118]; we will see later that it leads to a variational formulation of the initial-boundary value problem, which can be regarded as the natural extension of the corresponding displacement problem for linearized elasticity. Formulation (I) has limitations in that it is not simple nor natural to formulate evolution equations in this form, except perhaps for problems posed in one dimension.…”

Plasticity

“…Formulation (II) has been used in [19,36,84,86]; we will see later that it leads to a variational formulation of the initial-boundary value problem, which can be regarded as the natural extension of the corresponding displacement problem for linear elasticity. For example, (III) is the most often used, and in fact has been the goal of the theory developed in Chapter 3.…”

The variational formulation and solution of problems of finite‐strain elastoplasticity based on the use of a dissipation function

“…We use as a basis for the description of elastic-plastic deformations the multiplicative decomposition of deformation gradient F proposed by Lee.5 The deformation gradient is written as F = F'FP (1) where F' and F P are respectively its elastic and plastic parts. We define an elastic counterpart C' = FcTFC to the right Cauchy-Green deformation tensor C = FTF; where no plastic deformation has taken place C' = C. The velocity gradient L is related to the rate of change of deformation gradient and by using (1) can be decomposed in a natural way into elastic and plastic parts, in the form Each of these tensors may in turn be decomposed into symmetric and antisymmetric parts: The symmetric parts D, D' and DP are called the total, elastic and plastic deformation rates while the antisymmetric parts W, We and Wp are called the total, elastic and plastic spins, respectively.…”