Investigation on the Processes of Deformation, Heat- and-Moisture Transfer in Media with the Properties of the Effects of "Memory" and Self-Similarity

On the basis of the laws of the mechanics of hereditary media and the properties of the mathematical apparatus of fractional integro-differentiation, the paper investigates the one- and two-dimensional Kelvin rheological model in media with fractal structure. In the two-dimensional Kelvin mathematical model in fractional-differential form, the deformation components in different directions of anisotropy are caused by changes in temperature and moisture content of the material. Thus, one of the objectives of the study in this work is to develop a model of heat-and-moisture transfer, taking into account the inhomogeneity of the material and its effects of "memory". For the mathematical model of heat or moisture transfer, taking into account the fractal structure of the medium, a numerical implementation method, which is based on explicit and implicit finite-difference schemes, has been devised. The difference of implicit finite-difference scheme is shown when using integer differentiation operators and fractional operators. The stability condition for an explicit difference scheme containing fractional derivatives is given. Application software has been developed and the results of its application for the study of deformation processes, heat transfer taking into account the self-similarity and material effects of "memory" have been analyzed.