International Conference
"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

Varygina M.P.   Sadovskaya O.V.   Sadovskii V.M.  

Models of materials with microstructure and their numerical realization on supercomputers

Reporter: Sadovskii V.M.

Important factors, which it is necessary to consider when modeling the deformation of structurally inhomogeneous media (granular and porous materials, metal foams, fiber composites), are the different resistance to tension and compression, and the relative rotation of particles of the materials microstructure. For example, the allowable tensile stresses in cohesive soils and rocks are much smaller than compressive stresses. Conversely, fiber composites have higher stiffness in tension in the direction of fiber orientation than in compression in the same direction. Under certain special types of mechanical effects in granular and porous media is observed the rotational motion of particles relative to the binder material, which causes an elastic response from the neighboring particles.
To construct constitutive equations of granular media with elastic and plastic particles and porous media, the stiffness of which increases significantly with the collapse of pores, we apply a generalized rheological method, supplemented by a new element, taking into account the different resistance of materials. The rotational degrees of freedom of the particles motion are considered.
Parallel computational algorithm is worked out for the analysis of mathematical models. This algorithm is based on the method of splitting with respect to the physical processes and the spatial variables, where for the numerical solution of one-dimensional hyperbolic systems the explicit monotone ENO schemes (modifications of the scheme by S.K. Godunov) are used. The algorithm is realized as a software package for the cluster-type supercomputers.
The results of numerical analysis of the processes of propagation of signotons (the compression waves in a loosened granular medium) are presented, as well as the results of computation of the dissipation of mechanical energy in a foam aluminum caused by plastic collapse of pores under a concentrated impulsive action. In the framework of the Cosserat elasticity theory the computations of resonant excitation of a medium at the frequency of natural oscillations of rotational motion of the particles were performed.
This work was supported by RFBR (project 11–01–00053), the Complex Fundamental Research Program of the Presidium of RAS № 2, and the Interdisciplinary Integration Project of SB RAS № 40.