A new approach to solving the solid mechanics problems with matter supply

Abstract

We discuss some aspects of using the spatial description and the finite volume method as applied to the solid mechanics problems. The main objective is to study the differential equation relating the strain measure to the velocity gradient. We show that this equation can be reduced to an integral form and the obtained equation has the structure of the balance equation without a source term. On the one hand, such form of equation for the strain measure is convenient when dynamical problems of solid mechanics are solved by using the finite volume method. On the other hand, the balance equation allows us to look at the strain measure as a parameter of state in a broader sense, not as the purely geometrical characteristic. We believe that this interpretation of the strain measure may open up new prospects for describing the processes connected with the substance supply into solids. In order to solve such problems, we suggest to add a source term into the balance equation for the strain measure. We discuss this idea from different points of view. We also solve some problems where the stress–strain state of solids is caused by the source terms in the balance equation for the strain measure.