scholarly journals Discussion: “Exact Transient Response of an Elastic Half Space Loaded Over a Rectangular Region of Its Surface” (Norwood, F. R., 1969, ASME J. Appl. Mech., 36, pp. 516–522)

1970 ◽  
Vol 37 (1) ◽  
pp. 236-236
Author(s):  
D. C. Gakenheimer
Keyword(s):  
Half Space ◽  
Transient Response ◽  
Elastic Half Space ◽  
Rectangular Region

Exact Transient Response of an Elastic Half Space Loaded Over a Rectangular Region of Its Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 516-522 ◽  
Author(s):  
F. R. Norwood
Keyword(s):  
Half Space ◽  
Real Axis ◽  
Transient Response ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Impulsive Loading ◽  
Rectangular Region ◽  
The Real ◽  
Algebraic Expressions ◽  
Straight Lines

The response of an elastic half space to a normal impulsive loading over one half and also over one quarter of its bounding surface is considered. By a simple superposition the solution is obtained for a half space loaded on a finite rectangular region. In each case the solution was found to be a superposition of plane waves directly under the load, plus waves emanating from bounding straight lines and the corners of the loaded region. The solution was found by Cagniard’s technique and by extending the real transformation of de Hoop to double Fourier integrals with singularities on the real axis of the transform variables. Velocities in the interior of the half space are given for arbitrary values of Poisson’s ratio in terms of single integrals and algebraic expressions.

Transient fundamental solutions for a transversely isotropic elastic half space

1993 ◽  
Vol 442 (1916) ◽  
pp. 505-531 ◽  
Keyword(s):  
Half Space ◽  
Transient Response ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Integral Transforms ◽  
Fundamental Solutions ◽  
Elastic Half Space ◽  
Kernel Functions ◽  
Two Dimensional

This paper is concerned with the study of transient response of a transversely isotropic elastic half-space under internal loadings and displacement discontinuities. Governing equations corresponding to two-dimensional and three-dimensional transient wave propagation problems are solved by using Laplace–Fourier integral transforms and Laplace−Hankel integral transforms, respectively. Explicit general solutions for displacements and stresses are presented. Thereafter boundary-value problems corresponding to internal transient loadings and transient displacement discontinuities are solved for both two-dimensional and three-dimensional problems. Explicit analytical solutions for displacements and stresses corresponding to internal loadings and displacement discontinuities are presented. Solutions corresponding to arbitrary loadings and displacement discontinuities can be obtained through the application of standard analytical procedures such as integration and Fourier expansion to the fundamental solutions presented in this article. It is shown that the transient response of a medium can be accurately computed by using a combination of numerical quadrature and a numerical Laplace inversion technique for the evaluation of integrals appearing in the analytical solutions. Comparisons with existing transient solutions for isotropic materials are presented to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses due to a buried circular patch load are presented to portray some features of the response of a transversely isotropic elastic half-space. The fundamental solutions presented in this paper can be used in the analysis of a variety of transient problems encountered in disciplines such as seismology, earthquake engineering, etc. In addition these fundamental solutions appear as the kernel functions in the boundary integral equation method and in the displacement discontinuity method.

Response of an Elastic Half Space to Impulsive Stationary Finite Line Sources

1971 ◽  
Vol 38 (2) ◽  
pp. 549-550 ◽  
Author(s):  
F. R. Norwood
Keyword(s):  
Half Space ◽  
Transient Response ◽  
Elastic Half Space ◽  
Line Load ◽  
Finite Line ◽  
Infinite Line

This Note considers the transient response in the interior of a half space acted upon by a normal impulsive stationary semi-infinite line load. The solution for the corresponding infinite line load problem is contained in the solution for the case of a semi-infinite line load. By a simple superposition, the solution is obtained for a half space acted upon by a finite line load.

scholarly journals To Solution of Contact Problem for Rectangular Plate on Elastic Half-Space

2020 ◽  
Vol 19 (3) ◽  
pp. 224-229
Author(s):  
S. V. Bosakov
Keyword(s):  
Contact Problem ◽  
Rectangular Plate ◽  
Half Space ◽  
Chebyshev Polynomials ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Rectangular Region ◽  
Double Row ◽  
Linear Algebraic Equations ◽  
Boussinesq Solution

Until the present time there is no exact solution to the contact problem for a rectangular plate on an elastic base with distribution properties. Practical analogues of this design are slab foundations widely used in construction. A lot of scientists have solved this problem in various ways. The methods of finite differences, B. N. Zhemochkin and power series do not distinguish a specific feature in contact stresses at the edges of the plate. The author of the paper has obtained an expansion of the Boussinesq solution for determining displacements of the elastic half-space surface in the form of a double series according to the Chebyshev polynomials of the first kind in a rectangular region. For the first time, such a representation for the symmetric part of the Boussinesq solution was obtained by V. I. Seimov and it has been applied to study symmetric vibrations of a rectangular stamp, taking into account inertial properties of the half-space. Using this expansion, the author gives a solution to the problem for a rectangular plate lying on an elastic half-space under the action of an arbitrarily applied concentrated force. In this case, the required displacements are specified in the form of a double row in the Chebyshev polynomials of the first kind. Contact stresses are also specified in the form of a double row according to the Chebyshev polynomials of the first kind with weight. In the integral equation of the contact problem integration over a rectangular region is performed while taking into account the orthogonality of the Chebyshev polynomials. In the resulting expression the coefficients are equal for the same products of the Chebyshev polynomials. The result is an infinite system of linear algebraic equations, which is solved by the amplification method. Thus the sought coefficients are found in the expansion for contact stresses.

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