Abstract

Approximate theoretical solutions are presented for the boundary value problem of the shock load on the boundary of a circular cylindrical cavity in the space occupied by an incompressible elastic medium undeformed previously. We assume that the shock load causes the movement of the medium particles along helical trajectories. Data on the types and velocities of shock waves are based on the analysis of the dynamic conditions for the compatibility of discontinuities; and these conditions are supplemented by the relations along the characteristic directions. We show that the leading front of the dynamic process in an undeformed previously medium is a plane-polarized shock wave which is simultaneously included in one of the families of characteristics. This property results in the constancy of shear direction on the plane-polarized shock wave. This condition makes it possible to significantly simplify the obtaining of approximate theoretical solutions for the near-front neighborhood of the shock wave. The two approximate solutions of the problem are presented. One of them bases on solving a system of evolutionary equations that was obtained using the matched asymptotic expansions method; and the second solution, on a version of the ray method.

中文翻译：

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先前未受压的不可压缩弹性介质的轴对称冲击变形动力学问题的一些近似解

摘要

针对由先前未变形的不可压缩弹性介质占据的空间中的圆柱腔边界上的冲击载荷的边值问题，给出了近似的理论解。我们假设冲击载荷引起介质颗粒沿着螺旋形轨迹运动。关于冲击波的类型和速度的数据基于对不连续性相容性的动态条件的分析；这些条件通过沿特征方向的关系得到补充。我们表明，在未变形的先前介质中动态过程的前沿是平面极化的冲击波，该冲击波同时包含在一系列特性中。该特性导致平面极化冲击波上的剪切方向恒定。该条件使得可以显着简化对于冲击波的近前邻域的近似理论解的获得。给出了该问题的两个近似解。其中之一基于求解使用匹配渐近展开法获得的演化方程组。第二种解决方案是关于ray方法的版本。