## 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.

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.

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.

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