Decomposition of systems of equations for continuum mechanics 2. General results and applications

In the second part of work presents the general decomposition methods for systems of linear partial differential equations that arise in continuum mechanics, in particular, in the theory of elasticity and thermoelasticity and poroelasticity. A systematic approach to the decomposition of the equations of continuum mechanics is proposed. Asymmetrical and symmetrical decomposition methods for various classes of three-dimensional linear (and model nonlinear) systems of equations arising in the theory of elasticity, thermoelasticity, and thermoviscoelasticity, the mechanics of viscous and viscoelastic incompressible and compressible barotropic gas are described. These methods are based on the decomposition of systems of coupled equations into several simpler independent equations and the use of two stream functions. It is shown that in the absence of body forces any solution of considered steady and unsteady three-dimensional systems is expressed in terms of solutions of two independent equations. The methods of direct decomposition that do not require expansion of the right hand side of the equations into the components are proposed. A generalization of the considered methods to the decomposition of higher orders systems of equations, as well as to special classes of model nonlinear equations are obtained. The examples of the decomposition of specific systems are given. Formulas and split equations given in the work significantly simplify the qualitative study and the interpretation of the most important physical properties of a wide class of coupled systems of equations for continuum mechanics and allow studying their wave and dissipative properties. These results can be used for the exact integration of linear systems of mechanics, as well as for testing of numerical methods for nonlinear equations of continuum mechanics.