Varygina M.  

Numerical Algorithm for the Solution of Dynamic Problems in Micropolar Theory of Rods and Thin Plates

The micropolar (Cosserat) model is used to describe materials with microstructure like composite, granulate, powdery and porous media [1, 2]. In this model independent small rotations of particles described by the angular velocity vector are taken into consideration. In addition to asymmetric stress tensor an asymmetric couple stress tensor is introduced. The issues of numerical solution of dynamic problems in the framework of three-dimensional Cosserat elasticity theory are presented in [3, 4]. In this paper the numerical algorithms for the solution of dynamic problems in micropolar rods and thin plates are proposed.

The systems of equations describing rods and thin plates are constructed with the help of reduction of three-dimensional equations of the micropolar media by integration over the thickness. These systems can be written in matrix form convenient for numerical realisation. Under some restrictions on material parameters the systems of equations are hyperbolic in the sense of Friedrichs. For these systems the energy balance equation is fulfilled.

The algorithms of the solution of dynamic problems in micropolar thin-walled structures are based on the two-cyclic decomposition method with respect to the spatial variables and time. For two-dimensional case the two-cyclic decomposition method on time interval (t; t + τ) consists of five stages: the solution of one-dimensional problem in the x1 direction on the interval (t; t + τ/2); similar stage in the x2 direction, the stage of solution of a system of linear ordinary differential equations; and two stages of repeated recalculation of the problem in the x2 and x1 directions on time interval (t + τ /2; t + τ). At all stages except the third one for the solution of one-dimensional systems in spatial directions the explicit monotone finite-difference scheme of the “predictor-corrector” type is used. At third stage Crank-Nicholson finite-difference scheme with full time step is used. This scheme has good computational properties: it is conservative in the sense of consistency with the corresponding energy balance equation.

A series of numerical computations of elastic wave propagation induced by the action of instant concentrated loads and periodic distributed loads is performed. A verification of a problem is made by comparing the results of computations with the exact solutions describing elastic wave propagation in special cases.

This work was supported by the Russian Foundation for Basic Research (project no. 16-31-00078).

References

[1] E. Cosserat and F. Cosserat, “Theorie des Corps Deformables”, Librairie Scientifique A. Hermann et Fils, Paris (1909).

[2] V.A. Palmov, “Constitutive equations of asymmetric elasticity theory”, Prikl. Mat. Mekh. 28, No. 3, P. 401-408 (1964). (in Russian)

[3] M.P. Varygina, I.V. Kireev, O.V. Sadovskaya, V.M. Sadovskii, “Software for analysis of wave motions in Cosserat media on multiprocessor computer systems”, Vestnik of Siberian State Aerospace University, 2 (23), P 104-108 (2009). (in Russian)

[4] V. Sadovskii, O. Sadovskaya and M. Varygina, “Numerical solution of dynamic problems in couple-stressed continuum on multiprocessor computer systems”, International Journal of Numerical Analysis and Modeling, Series B. 2, No. 2-3. P. 215-230 (2011).