A secant approximation for holonomic elastic—plastic incremental analysis with a von Mises yield condition

Abstract

An algorithm is described for the incremental solution of elastic—plastic finite element analysis using a piecewise holonomic constitutive law based on a von Mises yield condition. The holonomic assumption effectively converts each incremental problem into a non‐linear elastic—plastic problem. The algorithm is iterative, substituting the non‐linear strain potential by a quadratic potential at each iteration, and convergence is proved. The algorithm has been implemented into a finite element program as a series of secant modulus approximations, and results for a variety of problems are given. The rate of convergence is fairly slow, but the algorithm can be very easily programmed as an extension of an elastic program, and may have value as an independent method of determining incremental elastic—plastic solutions.

Plasticity

“…For completeness, we shall refer briefly to the concept of a secant predictor used in Bird and Martin [14]. Once again assuming that members of Λ are arrays of n-tuples, we replace D in the active region by a quadratic function D (i) , defined in such a way that (12.81) and The quadratic function D (i) thus fits inside the cone D, touching at the point λ i−1 (see Figure 12.2).…”

“…Once again assuming that members of Λ are arrays of n-tuples, we replace D in the active region by a quadratic function D (i) , defined in such a way that (12.81) and The quadratic function D (i) thus fits inside the cone D, touching at the point λ i−1 (see Figure 12.2). The exact form of D (i) will depend on the nature of D; the von Mises case is discussed in [14].…”

Plasticity

“…For completeness, we shall refer briefly to the concept of a secant predictor used in Bird and Martin [10]. Here we replace D in the active region (12.53) and…”