Functional kinetic equations in mathematical modeling of coupled processes in solids

摘要

In this paper, we consider the role of functional kinetic equations in models of solid mechanics. It is shown that the choice of time-non-locality kernels allows both the finite speed of signal propagation and the inhomogeneity of material to be taken into account. Using analysis of the rheological kinetic equation for the entropy flux, we propose the motion energy, defined in the space of the momenta of considered motions. We also propose a method for constructing mathematical models that take into account inertia of various physical processes. By interposing a component, proportional to the Dirac's delta function, into the kernel of the rheological kinetic equation for mass flux, we can introduce the energy of an inhomogeneous medium and describe the near-surface inhomogeneity and its associated size effects. The latter is illustrated by the example of an elastic layer and its strength.