Boundary – value static problem and modeling of elastic deformations of structures

The problem of determining the undeformed (natural) configuration of a body for a given deformed state is solved. The mathematical model for this problem is based on the solution of the boundary value problem of the theory of elasticity in the non-classical formulation, obtained in [1]. The final state is the domain for determining the equations of equilibrium and compatibility of deformations, as well as boundary conditions. This state of equilibrium is considered to be given, not sought. Otherwise, it is impossible to mathematically correctly indicate the positions of the forces distributed in the volume and on the surface of the body. It turned out that with this approach, the final deformations can be described by linear tensors of displacement gradients at any values of the latter. The displacements determined by the Cesaro’s formulas represent those displacements that bring the body into the region of the final (deformed) state. They are used to find the coordinates of the points of the initial state of the body and build its configuration.