The study of heat transfer and stress-strain state of a material, taking into account its fractal structure

In the work, on the basis of the apparatus of fractional integro-differentiation, the mathematical models of heat-and-moisture transfer and of deformation-relaxation processes in the medium with "memory" effects and self-organization are constructed.  Numerical implementation of the mathematical models of heat-transfer and moisture-transfer is based on the adaptation of the predictor-corrector method.  That is why the mathematical models obtained in this work are in a finite-difference form.  For the explicit difference scheme, the stability conditions are determined on the basis of the method of conditionally defined known functions as well as by means of the Fourier integral method.  An integral representation of the deformation and stress of the fractional-differential rheological model is obtained using the Laplace transform method.  Including the numerical and analytical methods of implementation of the constructed models, in this paper, the main results are presented, in particular, identification of fractal parameters for the creep function according to the experimental data.

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Mathematical Modeling and Computing, Vol. 7, No. 2, pp. 400–409 (2020)