Analytical Solution of Coupled Thermoelastic Axisymmetric Transient Waves in a Transversely Isotropic Half-Space

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
M. Raoofian Naeeni ◽  
M. Eskandari-Ghadi ◽  
Alireza A. Ardalan ◽  
M. Rahimian ◽  
Y. Hayati
Keyword(s):  
Potential Function ◽  
Half Space ◽  
Equations Of Motion ◽  
Scalar Potential ◽  
Transversely Isotropic ◽  
Integral Transforms ◽  
Heaviside Step Function ◽  
Isotropic Materials ◽  
Coupled Equations ◽  
Degree Of Anisotropy

A half-space containing transversely isotropic thermoelastic material with a depth-wise axis of material symmetry is considered to be under the effects of axisymmetric transient surface thermal and forced excitations. With the use of a new scalar potential function, the coupled equations of motion and energy equation are uncoupled, and the governing equation for the potential function, is solved with the use of Hankel and Laplace integral transforms. As a result, the displacements and temperature are represented in the form of improper double integrals. The solutions are also investigated in detail for surface traction and thermal pulses varying with time as Heaviside step function. It is also shown that the derived solutions degenerate to the results given in the literature for isotropic materials. Some numerical evaluations for displacement and temperature functions for two different transversely isotropic materials with different degree of anisotropy are presented to portray the dependency of response on the thermal properties as well as the degree of anisotropy of the medium.

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.

Thermoelastodynamics in Transversely Isotropic Media With Scalar Potential Functions1

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Morteza Eskandari-Ghadi ◽  
Mohammad Rahimian ◽  
Stein Sture ◽  
Maysam Forati
Keyword(s):  
Heat Equation ◽  
Potential Function ◽  
Convex Domain ◽  
Scalar Potential ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Newtonian Potential ◽  
Existence Of A Solution ◽  
Axis Of Symmetry ◽  
Special Case

A complete set of potential functions consisting of three scalar functions is presented for coupled displacement-temperature equations of motion and heat equation for an arbitrary x3-convex domain containing a linear thermoelastic transversely isotropic material, where the x3-axis is parallel to the axis of symmetry of the material. The proof of the completeness theorem is based on a retarded logarithmic potential function, retarded Newtonian potential function, repeated wave equation, the extended Boggio's theorem for the transversely isotropic axially convex domain, and the existence of a solution for the heat equation. It is shown that the solution degenerates to a set of complete potential functions for elastodynamics and elastostatics under certain conditions. In a special case, the number of potential functions is reduced to one, and the required conditions are discussed. Another special case involves the rotationally symmetric configuration with respect to the axis of symmetry of the material.

Solution of contact problems for a transversely isotropic elastic layer bonded to an elastic half-space

2009 ◽  
Vol 223 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
V I Fabrikant
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Integral Transform ◽  
Transversely Isotropic ◽  
Contact Problems ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Electrostatic Problem ◽  
Crack Problems ◽  
New Interpretation

The idea, first used by the author for the case of crack problems, is applied here to solve a contact problem for a transversely isotropic elastic layer bonded to an elastic halfspace, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the layer's free surface. The governing integral equation is derived; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged discs in the shape of the domain of contact. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.

An Efficient Numerical Model of Elastohydrodynamic Lubrication for Transversely Isotropic Materials

2019 ◽  
Vol 141 (9) ◽  
Author(s):  
Zhanjiang Wang ◽  
Yinxian Zhang
Keyword(s):  
Film Thickness ◽  
Half Space ◽  
Fluid Pressure ◽  
Elastohydrodynamic Lubrication ◽  
Transversely Isotropic ◽  
Vertical Displacement ◽  
Constant Load ◽  
Von Mises Stress ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials

An elastohydrodynamic lubrication model for a rigid ball in contact with a transversely isotropic half-space is constructed. Reynolds equation, film thickness equation, and load balance equation are solved using the finite difference method, where the surface vertical displacement or deformation of transversely isotropic half-space is considered through the film thickness equation. The numerical methods are verified by comparing the displacements and stresses with those from Hertzian analytical solutions. Furthermore, the effects of elastic moduli, entertainment velocities, and lubricants on fluid pressure, film thickness, and von Mises stress are analyzed and discussed under a constant load. Finally, the modified Hamrock–Dowson equations for transversely isotropic materials to calculate central film thickness and minimum film thickness are proposed and validated.

Impedance Functions for Surface Rigid Rectangular Foundations on Transversely Isotropic Multilayer Half-Spaces

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Amirhossein Amiri-Hezaveh ◽  
Morteza Eskandari-Ghadi ◽  
Mohammad Rahimian ◽  
Amir K. Ghorbani-Tanha
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Mixed Boundary ◽  
Surface Displacement ◽  
Transversely Isotropic ◽  
Mixed Boundary Conditions ◽  
Integral Transforms ◽  
Material Symmetry ◽  
Time Harmonic ◽  
Impedance Functions

A horizontally multilayered Green elastic transversely isotropic half-space is considered as the domain of the boundary value problem involved in this paper, such that the axes of material symmetry of different layers are parallel to the axis of material symmetry of the lowest half-space, which is depthwise. The domain is assumed to be affected by an arbitrary time-harmonic forced vibration due to a rigid rectangular surface foundation. With the use of a potential function method and the Hankel integral transforms, the displacements and stresses Green's functions are determined in each layer. The unknown functions due to integrations in each layer are transformed to the unknown functions of the surface layer with the use of the concept of propagator matrix and the continuity conditions. The mixed boundary conditions at the surface of the whole domain are numerically satisfied with the assumption of piecewise constant distribution of tractions in the contact area. It is numerically shown that the surface displacement and stress boundary conditions are satisfied very well. The vertical and horizontal impedance functions of the rectangular foundation are determined, which may be used as lumped parameters in time-harmonic soil-structure interaction with transversely isotropic horizontally layered domain as the soil. It is shown that the impedance functions determined in this paper coincide with the same functions for the simpler case of isotropic homogeneous half-space as degenerations of this study.

Vertical Action of a Concentric Multi-Annular Punch on a Transversely Isotropic Elastic Half-Space

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
Morteza Eskandari-Ghadi ◽  
Ronald Y. S. Pak ◽  
Azizollah Ardeshir-Behrestaghi
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Transversely Isotropic ◽  
Elastic Half Space ◽  
Computational Approach ◽  
Mixed Boundary Value Problem ◽  
Contact Conditions ◽  
Degree Of Anisotropy ◽  
Annular Punch ◽  
Vertical Action

In this paper, the response of a transversely isotropic half-space under the punch action of a set of rigid concentric annuli frictionless contacts is considered. By virtue of a compact potential representation and Hankel transforms, a set of ring-load Green’s functions for the axisymmetric equations of equilibrium are derived and shown to be expressible in terms of standard elliptic integrals. With the aid of a rigorous yet highly efficient numerical method, the integral equation is solved for the multi-interval singular mixed boundary value problem. Detailed solutions to illustrate the performance of the computational approach and the influence of the degree of anisotropy and contact conditions on the mechanics problem are presented.

DISTURBANCE DUE TO PERIODIC THERMAL LOAD IN A PIEZOTHERMOELASTIC HALF-SPACE

2009 ◽  
Vol 01 (04) ◽  
pp. 607-629 ◽  
Author(s):  
J. N. SHARMA ◽  
ANITA D. THAKUR ◽  
Y. D. SHARMA
Keyword(s):  
Cadmium Selenide ◽  
Half Space ◽  
Epicentral Distance ◽  
Electric Displacement ◽  
Transversely Isotropic ◽  
Integral Transforms ◽  
Open Circuit ◽  
Fortran Code ◽  
Periodic Strip ◽  
Thermally Conducting

The present paper deals with the disturbances due to periodic strip thermal sources, acting on the rigidly fixed, charge free (open circuit) or electrically shorted (closed circuit) surface of a homogeneous, transversely isotropic, thermally conducting generalized piezothermoelastic half-space. The Laplace and Fourier transform technique have been employed to solve the model consisting of partial differential equations and boundary conditions in the transformed domain. In order to obtain the displacements, temperature change, stresses, electric potential and electric displacement in the physical domain, the integral transforms have been inverted by using a numerical inversion technique. Finally, to illustrate the analytical results, the numerical computations of various field parameters have been carried out from the relevant expressions and relations for cadmium selenide (CdSe) material by developing a FORTRAN code. The computer simulated results are presented graphically in respect of different considered field functions with load frequency and epicentral distance for 6 mm cadmium selenide (CdSe) material half-space.

scholarly journals Dynamic interaction between elastic plate and transversely isotropic poroelastic medium

2019 ◽  
Vol 258 ◽  
pp. 05016
Author(s):  
Suraparb Keawsawasvong ◽  
Teerapong Senjuntichai
Keyword(s):  
Circular Plate ◽  
Half Space ◽  
Contact Surface ◽  
Equations Of Motion ◽  
Admissible Function ◽  
Transversely Isotropic ◽  
Vertical Displacement ◽  
Generalized Coordinates ◽  
Vertical Loading ◽  
Elastic Circular Plate

In this paper, dynamic response of an elastic circular plate, under axisymmetric time-harmonic vertical loading, resting on a transversely isotropic poroelastic half-space is investigated. The plate-half-space contact surface is assumed to be smooth and fully permeable. The discretization techniques are employed to solve the unknown normal traction at the contact surface based on the solution of flexibility equations. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results corresponding to the deflections of a circular plate, with different degrees of flexibility, resting on a transversely isotropic poroelastic half-space are presented.

Rocking forced displacement of a rigid disc embedded in a functionally graded transversely isotropic half-space

2020 ◽  
pp. 108128652097935
Author(s):  
Maziar Kalantari ◽  
Naser Khaji ◽  
Morteza Eskandari-Ghadi
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Half Space ◽  
Soil Structure ◽  
Mixed Boundary ◽  
Transversely Isotropic ◽  
Integral Transforms ◽  
Soil Structure Interaction ◽  
Mixed Boundary Value Problem ◽  
Structure Interaction

Recent studies have confirmed that rockable structures have beneficial effects in earthquakes due to uniform dynamic behavior of the structure. For these kinds of structures, an equivalent static analysis is accurate enough, as the rocking motion is the dominant mode of their interaction with the surrounding soil (i.e. soil–structure interaction problem). In this study, the soil–structure interaction problems are extended to consider the effect of elastic non-homogeneities as well as changing the elasticity constants with depth in an exponential manner for the rocking loads. This paper analytically investigates the mixed boundary value problem regarding the forced rocking interaction of a rigid foundation embedded in a finite depth of an exponentially graded transversely isotropic half-space. The potential function method accompanied by Hankel integral transforms is applied to solve the system of the equations of motion of the media. Due to integral transforms used in the solving procedure, the mixed boundary value problem raised may be apt to be transformed to dual integral equations, which in turn, could be reduced to Fredholm integral equation of the second kind. To evaluate the solution of the integral equations, a robust numerical procedure has been developed for different anisotropic materials. Regarding the complicated integrand functions, the integrals are numerically and graphically presented to cover the effect of degree of change of material properties that plays a key role.

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