Пространственная задача математической теории пластичности. Радаев Ю.Н. - 458 стр.

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458
THREE-DIMENSIONAL PROBLEM OF THE
MATHEMATICAL
THEORY OF THE PERFECT 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 plasticity are
considered in an attempt to find new 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)
principal 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 provides significant mathematical advantages for the present theory. An important
geometrical property of that unit vector field can be derived by analyzing the static equilibrium
equations: 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 curvilinear co
ordinate system generates a canonical transformation of spatial domains then the canonical
transformation technique applicable to three-dimensional, plane strain and axially-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.
Slip kinematic on a surface of maximum shear strain rate in perfectly plastic continuous
media is studied. Sliding on the surface is shown can be realized only along asymptotic
directions and only within hyperbolic zones of the surface (wherein the Gaussian curvature
of the surface is negative). Integrable equations along asymptotic lines of the maximum shear
strain rate surface for the jumps of tangent velocities are obtained. Kinematic equations
corresponding to elliptic zones on a maximum shear strain rate surface (i.e. if the Gaussian
curvature of the surface is positive) are derived and analysed.
The problem of transformation of three-dimensional equations of the perfect plasticity
into their simplest Cauchy’s normal form by changing independent variables is considered.
The original essentially non-linear system of partial differential equations is represented by
the isostatic co-ordinates. The maximum simplicity principle for Cauchy’s normal forms is
formulated whereby the independent variables which afford one of the simplest normal forms
for 3-dimensional equations of the perfect plasticity are found. The principle is shown be
stronger than the t-hyperbolic condition. The original system is found belong to t-hyperbolic
type with respect to the third canonical isostatic co-ordinate thus allowing to correctly
formulate the Cauchy problems on layers normal to the third principal directions (i.e.
directions of the maximum (or minimum) principal stresses) ensuring the existence, uniqueness
and stability of the Cauchy problem solutions.
Group analysis of the system of partial differential equations of three-dimensional plastic
equilibrium is given. The system of static equilibrium equations is represented in the stress
Пространственная задача математической теории пластичности, 3-е издание

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