The Reissner-Sagoci Problem for the Transversely Isotropic Half-Space

1997 ◽  
Vol 64 (3) ◽  
pp. 692-694 ◽  
Author(s):  
M. T. Hanson ◽  
I. W. Puja
Keyword(s):  
Half Space ◽  
Previous Analysis ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Point Force ◽  
Force Potential

This paper evaluates the elastic field in a transversely isotropic half-space caused by a circular flat bonded punch under torsion loading. The elastic field is found by integrating the point force potential functions. For the case of isotropy the present results agree with previous analysis.

The Elastic Field Resulting From Elliptical Hertzian Contact of Transversely Isotropic Bodies: Closed-Form Solutions for Normal and Shear Loading

1997 ◽  
Vol 64 (3) ◽  
pp. 457-465 ◽  
Author(s):  
M. T. Hanson ◽  
I. W. Puja
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Normal Pressure ◽  
Shear Loading ◽  
Potential Functions ◽  
Hertzian Contact ◽  
Elementary Functions ◽  
Shear Traction

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

Elastic inclusions and inhomogeneities in transversely isotropic solids

1994 ◽  
Vol 444 (1920) ◽  
pp. 239-252 ◽  
Keyword(s):  
Closed Form ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Closed Form Expression ◽  
Inclusion Problem ◽  
Stress Vector ◽  
Equivalent Inclusion Method ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

A method that introduces a new stress vector function ( the hexagonal stress vector ) is applied to obtain, in closed form, the elastic fields due to an inclusion in transversely isotropic solids. The solution is an extension of Eshelby’s solution for an ellipsoidal inclusion in isotropic solids. The Green’s functions for double forces and double forces with moment are derived and these are then used to solve the inclusion problem. The elastic field inside the inclusion is expressed in terms of the newtonian and biharmonic potential functions, which are similar to those needed for the solution in isotropic solids. Two more harmonic potential functions are introduced to express the solution outside the inclusion. The constrained strain inside the inclusion is uniform and the relation between the constrained strain and the misfit strain has the same characteristics as those of the Eshelby tensor for isotropic solids, namely, the coefficients coupling an extension to a shear or one shear to another are zero. The explicit closed form expression of this tensor is given. The inhomogeneity problem is then solved by using Eshelby’s equivalent inclusion method. The solution for the thermoelastic displacements due to thermal inhomogeneities is also presented.

Wave motion in multi-layered transversely isotropic porous media by the method of potential functions

2019 ◽  
Vol 25 (3) ◽  
pp. 547-572 ◽  
Author(s):  
Hamid Teymouri ◽  
Ali Khojasteh ◽  
Mohammad Rahimian ◽  
Ronald Y S Pak
Keyword(s):  
Differential Equations ◽  
Half Space ◽  
Riemann Surfaces ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Transformation Method ◽  
Harmonic Excitation ◽  
Form Solution ◽  
Potential Functions ◽  
Special Cases

Wave propagation in a multi-layered transversely isotropic porous medium has been considered in this paper, which consists of n parallel layers overlying on a half-space. Potential functions are used to solve elastodynamic differential equations of the poroelastic medium. Time-harmonic excitation is assumed and the procedure of solution is performed in the frequency domain. Generalized reflection and transmission matrices are generated for compressional and shear waves separately. By means of the Hankel transformation method, coupled differential equations are altered to ordinary ones and Riemann surfaces are used to establish the path of integrations. A closed-form solution is described to reach Green’s functions of displacements and stresses. Some special cases of excitations are discussed and verification of the solution is presented. The numerical results of a three-layered medium on a porous half-space are determined and discussed.

Solution of the Contact Problem of a Rigid Conical Frustum Indenting a Transversely Isotropic Elastic Half-Space

2013 ◽  
Vol 81 (4) ◽  
Author(s):  
X.-L. Gao ◽  
C. L. Mao
Keyword(s):  
Contact Problem ◽  
Closed Form ◽  
Half Space ◽  
Cone Angle ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Elastic Half Space ◽  
Form Solution ◽  
Potential Functions ◽  
Displacement Method

The contact problem of a rigid conical frustum indenting a transversely isotropic elastic half-space is analytically solved using a displacement method and a stress method, respectively. The displacement method makes use of two potential functions, while the stress method employs one potential function. In both the methods, Hankel's transforms are applied to construct potential functions, and the associated dual integral equations of Titchmarsh's type are analytically solved. The solution obtained using each method gives analytical expressions of the stress and displacement components on the surface of the half-space. These two sets of expressions are seen to be equivalent, thereby confirming the uniqueness of the elasticity solution. The newly derived solution is reduced to the closed-form solution for the contact problem of a conical punch indenting a transversely isotropic elastic half-space. In addition, the closed-form solution for the problem of a flat-end cylindrical indenter punching a transversely isotropic elastic half-space is obtained as a special case. To illustrate the new solution, numerical results are provided for different half-space materials and punch parameters and are compared to those based on the two specific solutions for the conical and cylindrical indentation problems. It is found that the indentation deformation increases with the decrease of the cone angle of the frustum indenter. Moreover, the largest deformation in the half-space is seen to be induced by a conical indenter, followed by a cylindrical indenter and then by a frustum indenter. In addition, the axial force–indentation depth relation is shown to be linear for the frustum indentation, which is similar to that exhibited by both the conical and cylindrical indentations—two limiting cases of the former.

Point Force Solution for an Infinite Transversely Isotropic Solid

1976 ◽  
Vol 43 (4) ◽  
pp. 608-612 ◽  
Author(s):  
Y.-C. Pan ◽  
T.-W. Chou
Keyword(s):  
Elastic Constants ◽  
Numerical Evaluation ◽  
Systematic Approach ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Point Force ◽  
Isotropic Materials ◽  
Isotropic Solid ◽  
Individual Term ◽  
Transversely Isotropic Materials

The solution for a point force applied at the interior of an infinite transversely isotropic solid is obtained by introducing three potential functions which govern the displacements. Unlike previous publications where the solutions are expressed in different forms depending on the conditions satisfied by the elastic constants, the present paper provides a systematic approach to obtain a unified solution which is applicable for all stable transversely isotropic materials. The expression obtained does not have the deficiency suffered by previous solutions, namely, each individual term in the present expression does not tend to infinity on the z-axis. Thus accurate numerical evaluation of the Green’s function can be directly performed without the need to resolve the singularity algebraically.

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