Sadovskiy V.M.  

Mathematical models of structurally inhomogeneous media and their numerical implementation on multiprocessor systems

       Rheological method for the construction of constitutive equations is the basis of phenomenological approach to the description of a stressed-strained state of structurally inhomogeneous materials with complex mechanical properties. This method enables one to construct mathematical models which describe quantitative characteristics of materials with a satisfactory accuracy and are of a good mathematical structure. To take into account different resistance of materials to tension and compression, rheological method is supplemented by a new element, a rigid contact, which serves for imitation of perfectly granular material with rigid particles. By using a rigid contact in combination with conventional rheological elements (an elastic spring, a viscous damper, and a plastic hinge) one can construct constitutive equations of materials with complex structure.
       Parallel computational algorithm is worked out for the analysis of obtained mathematical models. This algorithm is based on the splitting methods with respect to physical processes and spatial variables. One-dimensional hyperbolic systems of equations in spatial directions are solved numerically by means of the explicit monotone ENO schemes. The algorithm is implemented as a complex of parallel programs for multiprocessor computer systems of the cluster type.
       A series of computations was performed, which demonstrate the characteristic features of processes of wave propagation in structurally inhomogeneous elastic-plastic materials (granular media, rocks and porous metals). The results of numerical analysis of the propagation of signotons (compression waves in a loosened granular medium) with the formation of a cumulative splash are presented. The process of passage of stress and strain waves through a block medium with thin viscoelastic interlayers is analyzed. Computations of the dissipation of mechanical energy in the aluminum foam due to plastic collapse of pores during the intensive impulse action were performed.

This work was supported by the Russian Foundation for Basic Research (project no. 11-01-00053), by Complex Fundamental Research Program no. 2 of the Presidium of RAS (Intelligent Information Technologies, Mathematical Modeling, System Analysis and Automation) and by the Interdisciplinary Integration Project no. 40 of the Siberian Branch of RAS.