Explicit Analytical Solutions for the Complete Elastic Field Produced by an Ellipsoidal Thermal Inclusion in a Semi-Infinite Space

2018 ◽  
Vol 85 (5) ◽  
Author(s):  
Ding Lyu ◽  
Xiangning Zhang ◽  
Pu Li ◽  
Dahui Luo ◽  
Yumei Hu ◽  
...  
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Micromechanical Model ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Elastic Field ◽  
Elastic Half Space ◽  
Infinite Space ◽  
Space Case ◽  
Thermal Inclusion

Thermal inclusion in an elastic half-space is a classical micromechanical model for describing localized heating near a surface. This paper presents explicit analytical solutions for the complete elastic fields, including displacements, strains, and stresses, produced by an ellipsoidal thermal inclusion in a three-dimensional semi-infinite space. Unlike the famous Eshelby solution corresponding to the infinite space case, the present work demonstrates that the interior strain and stress components are no longer uniform and appear to be much more complex. Nevertheless, the results can be represented in a more compact and geometrically meaningful form by constructing auxiliary confocal ellipsoids. The derived explicit solution indicates that the shear components of the stress and strain may be represented in closed-form. The jump conditions are examined and proven to be exactly identical to the infinite space case. A purposely selected benchmark example is studied to illustrate the free boundary surface effects. The degenerate case of a spherical thermal inclusion may be derived in a closed form, and is verified by the well-known Mindlin solution.

Transient interior analytical solutions of Lamb’s problem

2019 ◽  
Vol 24 (11) ◽  
pp. 3485-3513 ◽  
Author(s):  
Mohamad Emami ◽  
Morteza Eskandari-Ghadi
Keyword(s):  
Analytical Solutions ◽  
Three Dimensional ◽  
Time History ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Point Load ◽  
Lamb’S Problem ◽  
Interior Points

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.

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.

Elastic Fields Resulting From Concentrated Loading on a Three-Dimensional Incompressible Wedge

1995 ◽  
Vol 62 (3) ◽  
pp. 557-565 ◽  
Author(s):  
M. T. Hanson
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Elastic Field ◽  
Point Force ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Elementary Functions ◽  
Normal Loading ◽  
Direction Parallel ◽  
Elastic Fields

This paper considers point force or point moment loading applied to the surface of a three-dimensional wedge. The wedge is two-dimensional in geometry but the loading may vary in a direction parallel to the wedge apex, thus creating a three-dimensional problem within the realm of linear elasticity. The wedge is homogeneous, isotropic, and the assumption of incompressibility is taken in order for solutions to be obtained. The loading cases considered presently are as follows: point normal loading on the wedge face, point moment loading on the wedge face, and an arbitrarily directed force or moment applied at a point on the apex of the wedge. The solutions given here are closed-form expressions. For point force or point moment loading on the wedge face, the elastic field is given in terms of a single integral containing associated Legendre functions. When the point force or moment is at the wedge tip, closed-form (nonintegral) expressions are obtained in terms of elementary functions. An interesting result of the present research indicates that the wedge paradox in two-dimensional elasticity also exists in the three-dimensional case for a concentrated moment at the wedge apex applied in one direction, but that it does not exist for a moment applied in the other two directions.

Complete Solution of the Linear Magnetoelasticity and the Magnetic Fields in a Magnetized Elastic Half-Space

1995 ◽  
Vol 62 (4) ◽  
pp. 930-934 ◽  
Author(s):  
Huang Ke-Fu ◽  
Wang Min-Zhong
Keyword(s):  
Magnetic Fields ◽  
General Solution ◽  
Closed Form ◽  
Half Space ◽  
Complete Solution ◽  
Three Dimensional ◽  
Phenomenological Theory ◽  
Elastic Half Space ◽  
Linearized Theory ◽  
Single Force

In this paper, a general solution of the equations in the linearized theory of magnetoe-lasticity, which was developed by Pao and Yeh (1973) on the basis of Brown’s phenomenological theory of magnetoelasticity (1966), is obtained. As in some applications, the magnetic fields caused by the mechanical singularities in a magnetized elastic half-space are considered. Using the general solution and the Mindlin state of the elastic half-space (1936), the exact three-dimensional solutions for the generated magnetic fields due to various mechanical singularities, such as a single force and a doublet, are obtained in closed form.

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