Rigorous solution of the problem of the state of a linearly elastic isotropic body under the action of polynomial bulk forces

When solving boundary value problems about the construction of the stress-strain state of an linearly elastic, isotropic body, an important step is finding the internal state generated by the forces, distributed over the area occupied by the body. In the classical version, there is a numerical method for estimating the state at any point of the body based on the singular-integral representation of Cesaro. In the variant of conservative bulk forces, it is possible to construct solutions in an analytical form. With arbitrary regular effects of mechanical and other physical nature the force is not potential and the approaches of Papkovich–Neiber and Arzhanykh–Slobodyansky are powerless. In addition, the solution of nonlinear problems of elastostatics by means of the perturbation method, as well as the use of the Schwarz algorithm in solving problems for the study of multi-cavity solids, lead to the need to solve a sequence of linear problems. At the same time, fictitious bulk forces are necessarily generated, which as a rule have a polynomial nature.

The method proposed by the authors earlier for estimating the stress-strain state of a solid caused by the action of polynomial bulk forces represented in Cartesian coordinates has been improved. The internal state is restored in strict accordance with the forces statically acting on a simply connected bounded linear-elastic body. An effective method for constructing a solution and an algorithm for its computer implementation are proposed and described. Test calculations are demonstrated. The analysis of the state of the ball under the action of a superposition of bulk forces of different nature at different ratios of parameters that emphasize the level of influence of these factors is performed. The results are presented graphically. Conclusions are drawn:

a) the procedure for writing out the stress-strain state on the volume forces represented by polynomials from Cartesian coordinates is justified;
b) the algorithm is implemented in the Mathematica computing system and tested on high-order polynomials;
c) the analysis of the quasi-static state of a linear-elastic isotropic ball exposed to the forces of gravity and inertia at various combinations of parameters corresponding to the variants of slow, fast, compensatory (inertial forces are proportional to the gravitational) rotations is carried out.

The prospects for the development of a new approach to the class of bounded and unbounded bodies containing an arbitrary number of cavities are noted.