Пространственная задача математической теории пластичности. Радаев В.Н. - 142 стр.

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THREE-DIMENSIONAL PROBLEM OF THE

MATHEMATICAL

THEORY OF PLASTICITY

Y.N. Radayev, Department of Continuum Mechanics, Samara State University, Samara

443011, Russia

Three-dimensional static and kinematic equations of the theory of perfect elastoplastic

ity are considered in an attempt to ﬁnd approaches to analytical study of three-dimensional

elastic-plastic problems. The Tresca yielding criterion and associated ﬂow rule are employed

to formulate the closed system of equations. If an actual stress state corresponds to an edge

of the Tresca prism then the stress tensor are determined by the maximal (or minimal) prin

cipal stress and the unit vector ﬁeld directed along the principal stress axis associated with

that principal stress, thus allowing the static equilibrium equations to be formally considered

independently of kinematic. As it is shown, these equations are of hyperbolic type that pro

vides signiﬁcant mathematical advantages for the present theory. An important geometrical

property of that unit vector ﬁeld can be derived by analyzing the static equilibrium equa

tions: necessarily it is of complex-lamelar type. It is then found that the complex-lamelar

unit vector ﬁeld, co-oriented to the principal stress axis, determines a special curvilinear co

ordinate system (the Lame isostatic co-ordinate net). The latter is the most appropriate for

further considerations as transformed static equations can be readily integrated providing

a remarkable development of the theory of three-dimensional plasticity. Since the curvilin

ear co-ordinate system generates a canonical transformation of spatial domains then the

canonical transformation technique applicable to three-dimensional, plane strain and axi

ally-symmetric problems can be developed. This is fully implemented in the present study.

The closed system of static and kinematic incremental equations formulated in the local

principal stress frame is obtained and analyzed. A number of self-similar solutions of the

axially-symmetric problem is given by introducing a self-similar variable as the products of

powers of the isostatic co-ordinates. For special values of parameters involved in the self-sim

ilar solution the problem is reduced to obtaining solution of a non-linear non-autonomous

ordinary diﬀerential equation. Then this equation is numerically analyzed. The computation

of principal stresses distributions within the self-similar solutions zone is implemented.

Пространственная задача математической теории пластичности