Decomposition of systems of equations for continuum mechanics 1. Elasticity, thermoelasticity, poroelasticity

The work is devoted to the development of decomposition methods for systems of linear partial differential equations that arise in continuum mechanics, in particular, in the theory of elasticity and thermoelasticity and poro-elasticity. These methods are based on the decomposition (splitting) of systems of coupled equations into several independent equations. The decomposition significantly simplifies the qualitative study and interpretation of the most important physical properties related to three-dimensional equations and allows an effective study of their wave and dissipative properties. Moreover in certain cases the decomposition makes it possible to find exact analytical solutions of the corresponding boundary and initial-boundary value problems, and greatly simplifies the application of numerical methods, allowing us to use the appropriate routines for simpler equations and independent subsystems. In the first part of the work various systems of equations, including equations of elasticity theory in the form of Tedone and in the form of Beltrami-Donati-Michell are given, their dynamic generalizations are proposed, and various forms of the equations of classical and hyperbolic thermoelasticity as well as the equations poroelasticity are described. A number of historical facts, which are directly related with the considered questions and weakly reflected in Russian literature,are presented. Various types of decomposition and their generalizations are described. The representation of solutions of dynamical systems of equations resulting from the toroidal-poloidal decomposition, as well as the decompositions of Green-Lamé, Cauchy-Kovalevski-Somigliana, Naghdi-Hsu-Chandrasekharaiah, and Teodorescu types are discussed in details. Special attention is given to their static analogues. A generalization of the representation of Savin for the dynamic equations of elasticity is obtained. The representations in curvilinear coordinates, in particular, the representations of Boussinesq, Timpe, Love, Michell, and Muki types are given. The bibliographical references to the original papers are listed.