В статье рассмотрены уравнения анизотропной теории пластического течения в пространственном случае. На основе группы непрерывных преобразований, допускаемой системой, построено инвариантное решение. В случае однородного напряженного состояния найдено новое поле скоростей. Это поле имеет функциональный произвол. We considered the equations of the anisotropic theory of plastic flow in the spatial case in this study. We have constructed an invariant solution based on the group of continuous transformations allowed by the system. In the case of a homogeneous stress state, a new velocity field is found. This field has a functional arbitrary value.

При решении задачи групповой классификации уравнений, описывающих движение чисто механического континуума, появились некоторые новые системы дифференциальных уравнений, которые можно использовать для описания реальных физических процессов. Одна из таких новых систем: асимметричная теория упругости. Эта система может быть использована для материалов имеющих малый модуль Юнга, а также для материалов, которые работают при нагрузках близких к критическим. В данной работе изучаются уравнения асимметричной теории упругости на основе их группового расслоения: разложения системы на автоморфную и разрешающую системы, которые являются системами дифференциальных уравнений первого порядка. Построены бесконечные серии законов сохранения для разрешающей системы уравнений и автоморфной системы. Эти законы позволили решить краевую задачу Дирихле для асимметричной теории упругости в двумерном случае. Решения построены в виде квадратур, которые вычисляются по контуру исследуемой области. When solving the problem of group classification of equations describing the motion of a purely mechanical continuum, some new systems of differential equations have appeared that can be used to describe real physical processes. One of such new systems is the asymmetric theory of elasticity. This system can be used for materials with a small Young’s modulus, as well as for materials that operate at loads close to critical. In this paper, the equations of the asymmetric theory of elasticity are studied on the basis of their group bundle: decomposition of the system into automorphic and resolving systems, which are systems of differential equations of the first order. Infinite series of conservation laws are constructed for a resolving system of equations and an automorphic system. These laws made it possible to solve the Dirichlet boundary value problem for the asymmetric theory of elasticity in the two-dimensional case. The solutions are constructed in the form of quadratures, which are calculated along the contour of the studied area.

In the article an elastoplastic boundary for the console being bent with transverse force when the point of force is not situated in the centroid of transverse section was built with the use of the conservation laws. In this case bending moments and torques appear of the console. The case when the point of force is situated in the centroid of transverse section is considered in the previous works of the authors. In the work an infinite system of conservation laws has been built that allows us to reduce the problem of calculating elastoplastic boundary to a few quadratures, at the outer contour of transverse section. At that the contour can be random piecewise smooth. It is assumed that the lateral surface of the console is free from strains and is in its plastic condition

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

In this work we solve an anisotropic antiplane elastoplastic problem about stress state in a body weakened by a hole bounded by a piecewise-smooth contour. We give the conservation laws which allowed us to reduce calculations of stress components to a contour integral over the contour of the hole. The conservation laws allowed us to find the boundary between the elastic and plastic areas