Exact closed-form solutions for Lamb’s problem—III: the case for buried source and receiver

SUMMARY In this paper, we derive the exact closed-form solution for the displacement in the interior of an elastic half-space due to a buried point force with Heaviside step function time history. It is referred to as the tensor Green’s function for the elastic wave equation in a uniform half-space, also a natural generalization of the classical 3-D Lamb’s problem, for which previous solutions have been restricted to the cases of either the source or the receiver or both are located on the free surface. Starting from the complex integral solutions of Johnson, we follow the similar procedures presented by Feng and Zhang to obtain the closed-form expressions in terms of elementary functions as well as elliptic integrals. Numerical results obtained from our closed-form expressions agree perfectly with those of Johnson, which validates our explicit formulae conclusively.

Exact closed-form solutions for Lamb’s problem—II: a moving point load

SUMMARY In this paper, we report on an exact closed-form solution for the displacement in an elastic homogeneous half-space elicited by a downward vertical point source moving with constant velocity over the surface of the medium. The problem considered here is an extension to Lamb’s problem. Starting with the integral solutions of Bakker et al., we followed the method developed by Feng and Zhang, which focuses on the displacement triggered by a fixed point source observed on the free surface, to obtain the final solution in terms of elementary algebraic functions as well as elliptic integrals of the first, second and third kind. Our closed-form results agree perfectly with the numerical results of Bakker et al., which confirms the correctness of our formulae. The solution obtained in this paper may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.

NUMERICAL ANALYSIS OF EARTH DAM STRESS-STRAIN STATE UNDER SEISMIC IMPACT CONSIDERING THE WAVE DYNAMICS

the design, construction and operation of high-rise earth dams in seismic regions, such as the territory of the Republic of Uzbekistan, requires constant improvement of the methods to calculate them under various loads, both of a static nature (gravitational forces, hydrostatic, etc.), and of a dynamic nature, including seismic effects. Emergency situations at such facilities or their partial destruction under any impact can lead to disastrous aftermath. The aim of this study is to develop a mathematical statement and an algorithm for numerical solution to an unsteady-state problem for an earth dam in a plane elastic statement. To verify the proposed methodology and the corresponding complex of applied programs, a solution to the test problem was given (the Lamb’s problem). According to the developed methodology and algorithm based on numeri-cal method of finite differences, the problem of studying the stress-strain state was solved under shear stress on the foundation (in the form of a seismogram) on the example of the high-rise Charvak earth dam located near Tashkent city. The solution is presented in the form of distribution lines of equal displacements, stress-es along the dam body, depending on time. The most vulnerable zones of the earth dam under consideration were identified.

Lamb’s problem: a brief history

In this review note, a historical scientific investigation is presented for Lamb’s problem in the mathematical theory of elasticity. This problem first appeared in 1904 in the pioneering paper of Professor Sir Horace Lamb (Lamb, H. On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc Lon 1904; 203: 1–42). Of special interest here are the analytical studies of the three-dimensional version of Lamb’s problem, which consists of a semi-infinite, homogeneous, isotropic elastic solid that is set in motion by the exertion of a dynamical point force applied suddenly on the surface of the domain. The objective of this paper is to offer a comprehensive introduction to Lamb’s problem for the reader, along with discussing its mathematical complexities. An account is given of the history of this ever-significant problem from its earlier stages to the more recent investigations via outlining and discussing different rigorous approaches and methods of solution that have been hitherto suggested. The limitations of different methods, if they exist, are also discussed. Eventually, various solution methods are compared considering their nature, advantages, and restrictions.

Transient interior analytical solutions of Lamb’s problem

The classical three-dimensional Lamb’s problem is considered for an inclined surface point load of Heaviside time dependence. Attention is focused upon the acquisition of the transient elastodynamic analytical solutions for interior points through a unified method of analysis that is valid for arbitrary Lamé constants. The method of elastodynamic potentials is employed jointly with integral transforms to treat the corresponding initial boundary value problem. To derive the time-domain solutions, some integral equations are encountered, the solutions of which are found via a modified version of the Cagniard–Pekeris method. The final solutions are obtained as finite integrals that are amenable to numerical calculations. They are also expressed in the form of Green’s functions. The limit case of infinite time is investigated analytically to derive the closed-form expressions for the limits of the solutions as the temporal variable tends to infinity. As expected, the results are found to be equivalent to Boussinesq–Cerruti solutions in elastostatics. The elastodynamic solutions are also evaluated numerically to plot several time-history diagrams, depicting the transient motions of the interior points, especially of the points close to the boundary so as to illustrate the formation of forced Rayleigh waves at shallow depths within the elastic half-space.