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Plasticity - Mathematical Theory and Numerical Analysis

by Weimin Han B. Daya Reddy

Springer 1999

The basis for the modern theory of elastoplasticity was laid in the nineteenthcentury,
by Tresca, St. Venant, L´evy, 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 constitutive
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 outstanding
examples being the articles by Moreau, and the monographs
by Duvaut and J.-L. Lions, and Temam. In this manner the stage was
set for comprehensive investigations of the well-posedness of problems in
elastoplasticity, while the simultaneous rapid growth in interest in numer-
ical methods ensured that equal attention was given to issues such as the
development of solution algorithms, and their convergences.
The interaction between elastoplasticity and mathematics has spawned
among many engineering scientists an interest in gaining a better understanding
of the modern mathematical developments in the subject. In the
same way, given the richness of plasticity in interesting and important
mathematical problems, many mathematicians, either students or mature
researchers, have developed an interest in understanding the mechanical
and engineering basis of the subject, and its connections with the mathematical
theory. While there are many textbooks and monographs on plasticity
that deal with the mechanics of the subject, they are written mainly
for a readership in the engineering sciences; there does not appear to us
to have existed an extended account of elastoplasticity which would serve
these dual needs of both engineering scientists and mathematicians. It is
our hope that this monograph will go some way towards filling that gap.
We present in this work three logically connected aspects of the theory of
elastic-plastic solids: the constitutive theory, the variational formulations of
the related initial boundary value problems, and the numerical analysis of
these problems. These three aspects determine the three parts into which
the monograph is divided.
The constitutive theory, which is the subject of Part I, begins with a
motivation grounded in physical experience, whereafter the constitutive
theory of classical elastoplastic media is developed. This theory is then cast
in a convex analytic setting, after some salient results from convex analysis
have been reviewed. The term “classical” refers in this work to that theory
of elastic-plastic material behavior which is based on the notion of convex
yield surfaces, and the normality law. Furthermore, only the small strain,
quasi-static theory is treated. Much of what is covered in Part I will be
familiar to those working on plasticity, though the greater insights offered
by exploiting the tools of convex analysis may be new to some researchers.
On the other hand, mathematicians unfamiliar with plasticity theory will
find in this first part an introduction that is self-contained and accessible.
Part II of the monograph is concerned with the variational problems in
elastoplasticity. Two major problems are identified and treated: the primal
problem, of which the displacement and internal variables are the primary
unknowns; and the dual problem, of which the main unknowns are the
generalized stresses.
Finally, Part III is devoted to a treatment of the approximation of the
variational problems presented in the previous part. We focus on finite element
approximations in space, and both semi- and fully discrete problems.
In addition to deriving error estimates for these approximations, attention
is given to the behavior of those solution algorithms that are in common
use.
Wherever possible we provide background materials of sufficient depth
to make this work as self-contained as possible. Thus, Part I contains a
Preface ix
review of topics in continuum mechanics, thermodynamics, linear elasticity,
and convex analytic setting of elastoplasticity. In Part II we include a
treatment of those topics from functional analysis and function spaces that
are relevant to a discussion of the well-posedness of vatriational problems.
And Part III begins with an overview of the mathematics of finite elements.
In writing this work we have drawn heavily on the results of our joint collaboration
in the past few years. We have also consulted, and made liberal
use of the works of many: we mention in particular the major contributions
of G. Duvaut and J.-L. Lions, C. Johnson, J.B. Martin, H.
Matthies, and J.C. Simo. While we acknowledge this debt with gratitude,
the responsibility for any inaccuracies or erroneous interpretations
that might exist in this work, rests with its authors.
We thank our many friends, colleagues and family members whose interest,
guidance, and encouragement made this work possible.
W.H.
Iowa City
B.D.R.
Cape Town

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