Extremal paths and holonomic constitutive laws in elastoplasticity

Abstract

Abstract. Extremal stress paths between any two stresses are investigated for elasticplastic materials, extending existing results which hold for the case when one of the stress points is at the origin. Assumptions about the differentiability of the various work and complementary work functions are relaxed, and it is shown that the maximum complementary work U is a potential for the strain in the sense that the strain lies in the subdifferential of U. In the same way the minimum work W is a potential for stress. Parallel investigations with respect to maximum complementary plastic work and plastic work show that these quantities are potentials for stress and plastic strain increment, respectively. A holonomic constitutive law based on the relations between stress and strain, obtained when the stress history follows an extremal path, is constructed.

A convergent adaptive finite element method for the primal problem of elastoplasticity

“…The primal problem of elastoplasticity of Reference [14] assumes (u, p, ) as primary variables. Choosing Q := {q ∈ L 2 ( ; R d×d sym ) : tr q = 0 a.e.…”

“…With the energy J defined from (14) and := J (w ) − J (w), Algorithm 5.1 enforces a fixed rate reduction of the energy (up to control of data oscillations). This means that for the family of triangulations T generated by the Algorithm 5.1 there exist positive constants E , C with E <1, depending on the regularity of the initial triangulation T 0 and on the material parameters, such that…”

Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem

“…Suppose that at t=t~ At the body force on f2 is f. Suppose further that the material obeys the holonomic constitutive law proposed by Reddy et al [18], which is based on extremal stress paths from a~ to a(x, t). Then we seek the diaplacement field u(x, t) and the plastic strain field p(x, t) which satisfy We shall refer to the above statement as Problem (S).…”

“…In a recent paper (Reddy et al [18]) a set of holonomic constitutive equations approximating the behaviour of an elastic-plastic material have been proposed. These equations, which relate stress to elastic and plastic strain, have been derived on an assumption that stress paths followed by material points are extremal, in the sense that complementary work is maximised.…”

Finite element analysis of a holonomic elastic-plastic problem

“…In [13], a holonomic model is proposed to approximate the behavior of an elastic-plastic material. The holonomic constitutive law is derived based on the assumption that stress paths followed by material points are extremal, in the sense that complementary work is maximised.…”