A new approach is proposed for obtaining the dynamic elastic response of a multilayered elastic solid caused by axisymmetric, time-harmonic elastic singularities. The method for obtaining the elastodynamic Green’s functions of the point force, double forces and center of dilatation is presented. For this purpose, the boundary conditions in an infinite solid at the plane passing through the singularity are derived first by using Helmholtz potentials. Then the Green’s function solution for layered solids is obtained by solving a set of simultaneous linear algebraic equations using the boundary conditions for both the singularities and for the layer interfaces. The application of the point force solution for the oscillating normal indentation problem is also given. The solution of the forced normal oscillation is formulated by integrating the point force Green’s function over the contact area with unknown surface traction. The dual integral equations of the unknown surface traction are established by considering the boundary conditions on the contact surface of the multilayered solid, which can be converted into a Fredholm integral equation of the second kind and solved numerically.

中文翻译：

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时谐弹性奇异性和层状固体的振动压痕

提出了一种新的方法来获得由轴对称，时谐弹性奇异性引起的多层弹性固体的动态弹性响应。提出了获得点力，双力和扩张中心弹力格林函数的方法。为此，首先通过使用亥姆霍兹势来推导通过奇异性的平面上的无限实体中的边界条件。然后，通过使用奇点和层界面的边界条件求解一组联立的线性代数方程组，获得层状固体的格林函数解。还给出了点力解在振动法向压痕问题中的应用。强制法向振动的解决方案是通过积分接触力未知的表面上的点力格林函数来确定的。通过考虑多层固体接触表面上的边界条件，建立了未知表面牵引力的对偶积分方程，可以将其转换为第二类Fredholm积分方程并进行数值求解。