# Plasticity Mathematical Theory and Numerical Analysis

## Plasticity Mathematical Theory and Numerical Analysis

The basis for the modern theory of elastoplasticity was laid in the nineteenth-
century, by Tresca, St. Venant, Levy  ́ , and Bauschinger. Further
major advances followed in the early part of this century, the chief contributors during this period being Prandtl, von Mises, and Reuss. This early phase in the history of elastoplasticity was characterized by the introduction and development of the concepts of irreversible behavior, yield
criteria, hardening and perfect plasticity, and of rate or incremental con-
stitutive equations for the plastic strain.
Greater clarity in the mathematical framework for elastoplasticity theory
came with the contributions of Prager, Drucker, and Hill, during the
period just after the Second World War. Convexity of yield surfaces, and
all its ramifications was a central theme in this phase of the development
of the theory.
The mathematical community, meanwhile, witnessed a burst of progress
in the theory of partial differential equations and variational inequalities
from the early 1960s onwards. The timing of this set of developments was
particularly fortuitous for plasticity, given the fairly mature state of the
subject, and the realization that the natural framework for the study of
initial boundary value problems in elastoplasticity was that of variational
inequalities. This confluence of subjects emanating from mechanics and

mathematics resulted in yet further theoretical developments, the out-
standing examples being the articles by Moreau, and the monographs

by Duvaut and J.-L. Lions, and Temam.

The theory of elastoplastic media is now a mature branch of solid and
structural mechanics, having experienced significant development during
the latter half of this century. In particular, the classical theory, which
deals with small-strain elastoplasticity problems have a firm mathematical-
basis, and from this basis further developments, both mathematical
and computational, have evolved. Small-strain elastoplasticity is well
understood, and the understanding of its governing equations can be said to
be almost complete. Likewise, theoretical, computational, and algorithmic
work on approximations in the spatial and time domains are at a stage at
which approximations of the desired accuracy can be achieved with confidence.
The finite-strain theory has evolved along parallel lines, although it is
considerably more complex and is subject to a number of alternative
treatments. The form taken by the governing equations is reasonably settled,
though there is as yet no mathematical treatment of existence, uniqueness,
and stability analogous to those of the small-strain case. Computationally,
great strides have been made in the last two decades, and it is now possible
to solve highly complex problems with the aid of the computer.

This monograph focuses on theoretical aspects of the small-strain theory
of elastoplasticiy with hardening assumptions. It is intended to provide
a reasonably comprehensive and unified treatment of the mathematical
theory and numerical analysis, exploiting in particular the great advantages

to be gained by placing the theory in a convex-analytic context.