Approaches to formulation and methods of solving boundary value problems of solid mechanics

Abstract

The article presents approaches to the formulation and methods of solving initial boundary value problems of solid mechanics. Classical formulations of problems are considered. Principal scheme of the generalized formulation of problems in the form of operator equations in function spaces is presented and illustrated on the example of boundary value problems in the theory of small elastic deformations. The mathematical structure of iterative methods (method of elastic solutions and its modifications) and incremental approaches are stated in the article. Theorems on the existence and uniqueness of solutions, the convergence of the methods are given. The author discusses specific issues of formulating initial boundary value problems for finite deformations. The difficulties of the Lagrangian and finiteness of Eulerian descriptions are noticed. The conditions of possibility of effective application of the Euler formulation of problems that lead to substantial restrictions on the mechanical properties of material are displayed. The examples of the lack of the solutions at finite strains are given and unreasonableness of a requirement of uniqueness of solutions of problems of statics is shown. For evolutionary problems the author proposes a hypothesis on uniqueness of solutions as continuous-time field processes.