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On the Lamb problem: forced vibrations in a homogeneous and isotropic elastic half-space
Archive of Applied Mechanics ( IF 2.8 ) Pub Date : 2020-07-11 , DOI: 10.1007/s00419-020-01724-0
B. F. Apostol

The problem of vibrations generated in a homogeneous and isotropic elastic half-space by spatially concentrated forces, known in Seismology as (part of) the Lamb problem, is formulated here in terms of Helmholtz potentials of the elastic displacement. The method is based on time Fourier transforms, spatial Fourier transforms with respect to the coordinates parallel to the surface (in-plane Fourier transforms) and generalized wave equations, which include the surface values of the functions and their derivatives. This formulation provides a formal general solution to the problem of forced elastic vibrations in the homogeneous and isotropic half-space. Explicit results are given for forces derived from a gradient, localized at an inner point in the half-space, which correspond to a scalar seismic moment of the seismic sources. Similarly, explicit results are given for a surface force perpendicular to the surface and localized at a point on the surface. Both harmonic time dependence and time \(\delta \)-pulses are considered (where \(\delta \) stands for the Dirac delta function). It is shown that a \(\delta \)-like time dependence of the forces generates transient perturbations which are vanishing in time, such that they cannot be viewed properly as vibrations. The particularities of the generation and the propagation of the seismic waves and the effects of the inclusion of the boundary conditions are discussed, as well as the role played by the eigenmodes of the homogeneous and isotropic elastic half-space. Similarly, the distinction is highlighted between the transient regime of wave propagation prior to the establishment of the elastic vibrations and the stationary-wave regime.

中文翻译:

关于羔羊问题:在均质各向同性弹性半空间中的强迫振动

在这里,根据弹性位移的亥姆霍兹势,用空间集中力在同质各向同性的弹性半空间中产生的振动问题(在地震学中称为兰姆问题的一部分)被提出。该方法基于时间傅立叶变换,相对于与曲面平行的坐标的空间傅立叶变换(面内傅立叶变换)和广义波动方程,其中包括函数的表面值及其导数。该公式为均质和各向同性半空间中的强迫弹性振动问题提供了正式的一般解决方案。对于从梯度得出的力给出了明确的结果,该梯度位于半空间的内部点,对应于震源的标量地震矩。同样,对于垂直于表面并位于表面上某个点的表面力,给出了明确的结果。谐波时间相关性和时间考虑\(\ delta \)- pulses(其中\(\ delta \)代表Dirac delta函数)。结果表明,类似力的时间(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\-\“随地消失),从而无法将其适当地视为振动。讨论了地震波的产生和传播的特殊性以及边界条件包括在内的影响,以及均质和各向同性弹性半空间的本征模式所起的作用。同样,在建立弹性振动之前,波传播的瞬态状态与固定波状态之间的区别也得到了强调。
更新日期:2020-07-11
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