Classical Rate-Independent Plasticity and Viscoplasticity

Abstract

In this chapter we summarize the equations of classical rate-independent plasticity and its viscoplastic regularization. Our presentation is restricted to an outline of the mathematical structure of the governing equations relevant to the numerical solution of boundary-value problems and the analysis of numerical algorithms. First, for the convenience of the reader, we summarize some basic notation of continuum mechanics with attention restricted to the linearized theory. For further details we refer to standard textbooks, e.g., Sokolnikoff [1956] or Gurtin [1972]. Next, we proceed to outline the basic structure of rate-independent plasticity within the classical framework of response functions formulated in stress space, as in Hill [1950] or Koiter [1960]. Special attention is given to the proper (and unique) formulation of loading/unloading conditions in the so-called Kuhn–Tucker form. These are the standard complementarity conditions for problems, such as plasticity, subjected to unilateral constraints. This form of loading/unloading conditions is in fact classical and has been used by several authors, Koiter [1960] and Maier [1970]. Because the algorithmic elastoplastic problem is typically regarded as a strain-driven problem, throughout our discussion we adopt the strain tensor as the primary (driving) variable. Accordingly, although the response functions are formulated in stress space, the theory is essentially equivalent to a strain-space formulation. This is the standard point of view adopted in the numerical analysis literature, starting from the pioneering work of Wilkins [1964]. Alternative stress-space frameworks have been explored by several authors, e.g., Johnson [1977] and Simo, Kennedy, and Taylor [1988]. We consider the thermodynamic basis of the theory within the context of internal variables. As shown subsequently, this structure is important to understand the algorithmic structure of the discrete problem. Finally, we examine the case of associative plasticity which is intimately connected to the principle of maximum plastic dissipation. Because of the important role played by the principle of maximum dissipation in formulating finite-element approximations, a discussion of this and its equivalence with normality, loading/unloading conditions in Kuhn–Tucker form, and convexity of the yield surface is included. We conclude the chapter with an outline of the so-called viscoplastic regularization leading to the classical viscoplastic constitutive 71 72 2. Classical Rate-Independent Plasticity and Viscoplasticity models. As an illustration, we consider in some detail the classical J 2 flow theory. The algorithmic treatment of this important example is considered in subsequent chapters. 2.1 Review of Some Standard Notation Let B ⊂ R n dim be the reference configuration of the body of interest, where 1 ≤ n dim ≤ 3 is the space dimension. We assume that B is open and bounded with smooth boundary ∂B and closure ¯ B : B ∪ ∂B. Let [0, T] ⊂ R + be the time interval of interest, and let u : ¯ B × [0, T] → R