Numerical modeling of one-dimensional wave processes in a bimodular medium: software implementation and convergence analysis

Modern mechanics of deformable solids is actively advancing towards modeling bimodular materials – media with different resistance to tension and compression. This class includes both natural and artificial materials widely used in engineering practice. The difference in elastic moduli under tension and compression leads to a strong nonlinearity of the mechanical response, which under dynamic loading generates qualitatively complex wave patterns with the formation and interaction of strain discontinuities. As a result, the analytical solution of the corresponding boundary value problems becomes quite challenging due to numerous wavefront interactions. This paper presents the development, software implementation, and verification of a specialized system for the automated solution of boundary value problems of one-dimensional dynamics of bimodular elastic media under arbitrary boundary loading regimes. The system is based on a formalized iterative method for constructing the wave pattern, which uses piecewise-linear approximation of both the boundary conditions and the constitutive relations of the Myasnikov-Oleynikov bimodular medium. The software fully automates the construction of the wave pattern: it tracks the evolution of the wave system, classifies events, automatically generates and solves systems of nonlinear equations to determine the parameters of new local domains between wavefronts, and controls the evolutionarity of strain discontinuities and introduction of rigid layers. Verification was performed on test problems involving the formation of shock waves and rigid layers. A convergence analysis of the results against known analytical solutions confirms the high accuracy of the implemented method with an increasing number of boundary condition approximation nodes. Thus, the developed software tool is an effective instrument for automating the study of wave processes, providing physically correct modeling of the one-dimensional dynamic deformation of bimodular materials. The results obtained are of practical value to researchers in solid mechanics, specialists in numerical modeling, and engineers involved in the design of structures made of bimodular materials under dynamic loads.

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