On One Class of Dual Problems of Mechanics of Deformable Solids

Inelastic body’s plane deformation is described by two vector fields: vector stress potential (gradient of Airy stress function) and vector displacement field. Conditions for possibility of proceeding to the dual problem, when variables change the roles, are described: stress potential is interpreted as displacement field and vice versa. Both a perfectly plastic body model and its dual model of perfectly solidifying matter are considered.


Introduction
Mathematical models of solids' deformation are based on two general concepts: displacement vector and stress tensor. Vector displacement field yields kinematic characteristics of stress; and stress tensor yields power and dynamic ones. In other words, kinematics and dynamics of the matter are described qualitatively by different mathematical objects. In the first case we have to use a first-rank tensor (vector), in the second case -a second-rank tensor (tensor itself). Accordingly there is an asymmetry in the description. It is possible, however, to find a class of problems, when both power and kinematic descriptions are symmetrical and can be described with the aid of two vector fields. Moreover this class of problems is quite extended. The main idea of this work is that, since we are dealing with objects of identical rank, then after solving the equation, their roles can be interchanged. That is, the stress vector field can be regarded as the kinematic field, and vice versa. Thus we get the solution of a new (dual or conjugate) problem.Mathematical models of solids' deformation are based on two general concepts: displacement vector and stress tensor. Vector displacement field yields kinematic characteristics of stress; and stress tensor yields power and dynamic ones. In other words, kinematics and dynamics of the matter are described qualitatively by different mathematical objects. In the first case we have to use a first-rank tensor (vector), in the second case -a second-rank tensor (tensor itself). Accordingly there is an asymmetry in the description. It is possible, however, to find a class of problems, when both power and kinematic descriptions are symmetrical and can be described with the aid of two vector fields. Moreover this class of problems is quite extended. The main idea of this work is that, since we are dealing with objects of identical rank, then after solving the equation, their roles can be interchanged. That is, the stress vector field can be regarded as the kinematic field, and vice versa. Thus we get the solution of a new (dual or conjugate) problem.

Formulation of primal and dual problems.
Let us assume that deformation is plane, and inertial forces and weight can be neglected. Then we have equations for a wide range of deformable continua: 11 12 12 22 where 0x1x2 -Cartesian coordinate system, σ ij and u i -components of stress, displacement or their velocities, i,j=1,2, coefficients a 11 ,… are set. Notations of models (1), (2) are classical. However, in many respects they are "abnormal". [1] For example, five first-order differential equations are reduced to one fourth-order equation only. A natural notation can be obtained by inserting the vector field instead of the stress field {p 1 ,p 2 }-vector potential of the stress field: 1  2  1  2  1  1  11  12  13  2  1  1  2  1  2   2  2  1  1  21  22  23  2  2  1  2   1  2  2  1  1  31  32  33  2  1  2  1 The system is noted in two vector fields   . Suppose that some its solution is built up. As mentioned, the main idea is to interchange roles of vectors p and u : vector u correlates with the stress field, and vector p -with the displacement field of the dual problem. Clearly, this can only be done when p and u are included in the system in some symmetrical manner.
Thus, it is necessary that one of the displacement equations would have the structure of the system's first equation (4). Therefore it is necessary to consider the case of incompressible continuum: where coefficients c 1 ,…,d 3 are set. Let us examine the boundary conditions. Suppose n is an external boundary normal, s is its natural parameter, α is an angle between the axis 0x1 and a normal n . Stress vector components on the area with the normal n are equal to: 1  Let us now turn to the dual problem. Relevant variables will be denoted by ̃ "tilde".
System of equations (5) does not change by such a substitution and takes the following form: .
Here one new factor arises. The original system (1), (2) is invariant with reference to rotation of the body as a rigid solid. This is a necessary correctness condition of any problem on solid deformation. Therefore, system (5) will be invariant. However, after the change of roles, invariant variables 2 The limitation, obtained above, is acceptable. It means that in the original problem principal stresses' maximum displacement and orientation should not depend on hydrostatic compression.

Model of ideal plasticity.
This model and its generalizations nave been studied completely and are widely used for engineering problems [2,3]. The ideal plasticity's closed equation system can be written down as follows: