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

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.

2020 ◽  
Vol 224 (1) ◽  
pp. 517-532
Author(s):  
Xi Feng ◽  
Haiming Zhang

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.


2019 ◽  
Vol 25 (3) ◽  
pp. 547-572 ◽  
Author(s):  
Hamid Teymouri ◽  
Ali Khojasteh ◽  
Mohammad Rahimian ◽  
Ronald Y S Pak

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.


1997 ◽  
Vol 64 (3) ◽  
pp. 457-465 ◽  
Author(s):  
M. T. Hanson ◽  
I. W. Puja

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.


2019 ◽  
Vol 24 (10) ◽  
pp. 3231-3253 ◽  
Author(s):  
Marco Salviato ◽  
Sean E Phenisee

The new generation of manufacturing technologies such as additive manufacturing and automated fiber placement has enabled the development of material systems with desired functional and mechanical properties via particular designs of inhomogeneities and their mesostructural arrangement. Among these systems, particularly interesting are materials exhibiting curvilinear transverse isotropy (CTI), in which the inhomogeneities take the form of continuous fibers following curvilinear paths designed to, for example, optimize the electric and thermal conductivity, and the mechanical performance of the system. In this context, the present work proposes a general framework for the exact, closed-form solution of electrostatic problems in materials featuring CTI. First, the general equations for the fiber paths that optimize the electric conductivity are derived, leveraging a proper conformal coordinate system. Then, the continuity equation for the curvilinear transversely isotropic system is derived in terms of electrostatic potential. A general exact, closed-form expression for the electrostatic potential and electric field is derived and validated by finite element analysis. Finally, potential avenues for the development of materials with superior electric conductivity and damage sensing capabilities are discussed.


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