Nonstationary 1D Dynamics Problems for Heteromodular Elasticity with Piecewise-Linear Approximation of Boundary Conditions

Abstract

The paper provides the investigation of a heteromodular elastic medium under dynamic loading. The heteromodularity (when the stress - strain relation depends on the deformation direction) is a distinctive feature of many natural and structural materials: rocks, porous and cohesive bulk media, fibrous and granular composites, some metal alloys, etc. The fact that the listed materials show the heteromodular property at the stage of elastic deformation should be especially taken into account when solving problems of their shock dynamics. To describe the heteromodular behavior of an elastic medium in terms of small strains we use the physically nonlinear model of V.P. Myasnikov. The accepted assumption about the one-dimensional straining reduces the nonlinear relationship of stresses and small strains to piecewise linear equations. In the case of dynamic shock deformation, the initial nonlinearity of the model is concentrated in the equations which define the velocity of the shock wave abruptly transforming the heteromodular medium from a stretched to a compressed state. In this paper we investigate the processes of generation, motion, and possible interactions of plane one-dimensional deformation waves (including shock ones) in a heteromodular elastic half-space. The points of the half-space boundary undergo one-dimensional motions according to a given non-linear law corresponding to the “stretching-compression” mode. We suggest replacing the nonstationary boundary condition of the problem by its piecewise linear approximation and constructing a connected sequence of analytical solutions with a linear boundary condition at each local time interval. The proposed approach is the basis of the numerical solving algorithm for a boundary value problem with a given nonlinear condition. It is shown that the general solution behind the shock wave consists of several local layers, which number is related to the quantity of nodes in the piecewise linear decomposition of the boundary condition. In these layers, the compression deformation is defined by the relevant part of the boundary condition and simultaneously “stores” information on the preliminary tension, which should be considered an important feature of the heteromodular medium dynamics.

On the Convexity of Potential of Nonlinear Elastic Media Model With Tensor Damage Parameter

“…Для описания деформационного поведения нелинейных разномодульных сред используются различные подходы, которые можно разделить на два класса: модели с упругими модулями, зависящими от типа напряженно-деформированного состояния [19-26], модели с постулированием упругого потенциала особой формы [27][28][29][30][31][32][33][34]. В независимости от способа описания нелинейного деформационного отклика при решении краевых задач определяющее значение имеет вопрос устойчивости материала, которая определяется, в свою очередь, выпуклостью используемого потенциала, обеспечивающую единственность решения, и эллиптичностью уравнения движения.…”