Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space

The idea of generalized images, first used by the author for the case of crack problems, is applied here to solve a contact problem for n transversely isotropic elastic layers, with smooth interfaces, resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the top layer’s free surface. The governing integral equation is derived for the case of two layers; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. This result is then generalized for an arbitrary number of layers. 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.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Dynamic interaction of a pile with a transversely isotropic elastic half-space under transverse excitations

This investigation is concerned with a mathematical analysis of an elastic circular cylindrical pile embedded in a transversely isotropic half-space under lateral dynamic excitations. A combination of time-harmonic horizontal shear force and moment are applied at the top end of the pile. The boundary value problem is formulated by decomposing the pile-medium system into a fictitious pile and an extended transversely isotropic half-space. A Fredholm integral equation of the second kind governs the interaction problem, whose solution is then computed numerically. Selected results for dynamic compliance bending moment, displacement and slope profiles are presented for different transversely isotropic half-spaces to portray the influence of degree of anisotropy of the medium on various aspects of the solution.     

横观各向同性材料三维裂纹问题的数值分析

严格从三维横观各向同性材料弹性空间问题的Green函数出发,采用Hadamard有限部积分概念,导出了三维状态下单位位移间断(位错)集度的基本解.在此基础上,将三维任意形状的片状裂纹问题归结为求解-组以未知位移间断表示的超奇异积分方程;并给出了边界元离散形式.对方程中出现的超奇异积分,采用了Had-alnard定义的有限部积分来处理.论文最后给出了若干典型片状裂纹问题的数值算例,数值结果表明了本文方法是非常有效的.     

Stress State of a Transversely Isotropic Medium with an Arbitrarily Oriented Spheroidal Inclusion

The stress-concentration problem for an elastic transversely isotropic medium containing an arbitrarily oriented spheroidal inclusion (inhomogeneity) is solved. The stress state in the elastic space is represented as the superposition of the principal state and the perturbed state due to the inhomogeneity. The problem is solved using the equivalent-inclusion method, the triple Fourier transform in space variables, and the Fourier-transformed Green function for an infinite anisotropic medium. Double integrals over a finite domain are evaluated using the Gaussian quadrature formulas. In special cases, the results are compared with those obtained by other authors. The influence of the geometry and orientation of the inclusion and the elastic properties of the medium and inclusion on the stress concentration is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 33–40, February 2005.     

Exact analytic solutions of three-dimensional boundary-value problems of the statics of a transversely isotropic body of canonical form (Survey)

Conclusion Thus, we have described some of the approaches used to solve three-dimensional problems of the statics of isotropic and transversely isotropic bodies, and we have systematized results from the construction of analytic general solutions to problems of the theory of elasticity of transversely isotropic bodies of canonical form. These solutions have in turn been used to solve numerous specific problems on stress concentration near notches, cavities, inclusions, and cracks, as well as problems concerning the contact interaction of an elastic half-space with elliptical and parabolic dies. Translated from Prikladnaya Mekhanika, Vol. 33, No. 10, pp. 3–30, October, 1997.     

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.     

Exact solution of external tangential contact problem for a transversely isotropic elastic half-space

Summary  The following mixed boundary-value problem for a transversely isotropic elastic half-space is considered. Arbitrary tangential displacements are prescribed at the exterior of a circle, while the interior of the circle is free of tangential stress, and the normal stress vanishes all over the boundary. The governing integral equation is solved exactly, in closed form, and in terms of elementary functions. The method of continuation of solutions previously published by the author has been used here. Several examples are considered. No similar results has been reported before, even in the case of an isotropic body. Received 8 May 2000; accepted for publication 20 July 2000     

Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch

This paper presents the fundamental contact solutions of a magneto-electro-elastic half-space indented by a smooth and rigid half-infinite punch. The material is assumed to be transversely isotropic with the symmetric axis perpendicular to the surface of the half-space. Based on the general solutions, the generalized method of potential theory is adopted to solve the boundary value problems. The involved potentials are properly assumed and the corresponding boundary integral equations are solved by using the results in literature. Complete and exact fundamental solutions are derived case by case, in terms of elementary functions for the first time. The obtained solutions are of significance to boundary element analysis, and an important role in determining the physical properties of materials by indentation technique can be expected to play.     

横观各向同性饱和地基上中厚圆板的非轴对称振动

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题。基于横观各向同性饱和介质Biot波动方程的一般解,按混合边值问题建立了饱和地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程转化为易于计算的第二类Fredholm积分方程;采用数值方法求解该积分方程。数值算例结果表明,当h/a>0.05时,饱和半空间体上中厚度圆板在不同频率下的振动特性与相应频率下的刚性板的振动特性基本相同,当h/a<0.05时,板中心的位移将随h/a的减小而增大。     

Rolling without Slip on a Transversely Isotropic Thermoelastic Half-space

Rolling without slip by a rigid cylinder on a transversely isotropic, coupled thermoelastic half-space at constant subcritical speed is studied. The cylinder is of infinite length, surface heat convection is neglected, and a dynamic steady state of plane strain is treated. The unmixed problem of traction applied to a translating surface strip is addressed first. A robust asymptotic form of the exact transform solution, valid when Fourier heat conduction dominates any thermal relaxation effect, is extracted, and inverted analytically. Use of material characterization and identification of parameters that vanish in the isotropic limit or are invariant under an isothermal–thermoelastic transformation result in compact full-field solutions. These expressions are used to construct analytical solutions that satisfy the mixed boundary value problem and auxiliary conditions of rolling contact. For the hexagonal material zinc, calculations are made for contact zone width and temperature increases near onset of zone yield. Mathematics Subject Classifications (2000) 73B30, 73C25, 73C30, 73C35.     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the axisymmetric problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and is subjected over a part of its top surface to normal tractions while the rest of it is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using Hankel transform, the axisymmetric elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact radius. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using orthogonal Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

The Boussinesq problem, that is, determining the deformation in a hyperelastic half-space due to a point force normal to the boundary, is an important problem of engineering, geomechanics, and other fields to which elasticity theory is often applied. While linear solutions produce useful Greens functions, they also predict infinite displacements and other physically inconsistent results nearby and at the point of application of the load where the most critical and interesting material behavior occurs. To illuminate the deformation due to such a load in the region of interest, asymptotic analysis of the nonlinear Boussinesq problem has been considered in the context of isotropic hyperelasticity. Studies considering transversely isotropic materials have also been broadly used in the linear theory, but have not been treated within the nonlinear framework. In this paper we extend the nonlinearly elastic isotropic analysis to transverse isotropy, producing a more general theory which also better encompasses applications involving layered media. The governing equations for nonlinearly elastic, transversely isotropic solids are derived, conservation laws of elastostatics are invoked, asymptotic forms of the deformation solutions are hypothesized, and the differential equations governing deformation near the point load are determined. The analysis also develops sequences of simple tests to determine if a transversely isotropic material can possibly sustain a finite deflection under the point load. The results are applied to a variety of transversely isotropic materials, and the effects of the anisotropy considered is demonstrated by comparison of the resulting deformation with similar asymptotic solutions in the isotropic theory. Mathematics Subject Classifications (2000) 74B20, 74E10, 74G10, 74G15, 74G70.     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

A crack of arbitrary shape inside an infinite transversely isotropic cylinder under arbitrary load

Summary  In the first part of the article an infinite circular cylinder is considered, made of transversely isotropic elastic material and weakened by a plane crack perpendicular to its axis O z. The crack is opened by an arbitrary normal stress. The second part is devoted to the same crack loaded by an arbitrary tangential stress. The complete solution in both cases is presented as a sum of the solution of a similar problem of a crack in an infinite space and an integral transform term, the parameters of which are determined from a set of linear algebraic equations derived from the boundary conditions. Governing integral equations with respect to the yet unknown crack displacement discontinuities are obtained. In the case of a circular crack, these equations can be inverted and solved by the method of consecutive interations. Received 30 November 2000; accepted for publication 3 May 2001     

Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.     

A dynamic problem for a prestressed compressible layered half-space

The linearized theory of elasticity for prestressed bodies is used to solve a stationary plane problem for a prestressed two-layer half-space under a surface load moving with constant velocity. The half-space is assumed to be compressible and to have an arbitrary elastic potential. The Fourier transform is used to obtain the fundamental solution of the problem for different contact conditions and load velocities. A compressible material with a harmonic elastic potential is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 35–55, April 2008.     

SECOND ORDER EFFECTS IN AN ELASTIC HALF－SPACE ACTED UPON BY A NON－UNIFORM NORMAL LOAD

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On the relationship between the solutions of static contact problems of elasticity and electroelasticity for a half-space

The paper establishes the relationship between the static contact problems of elasticity and electroelasticity (in the absence of friction) for a transversely isotropic half-space whose surface is the isotropy plane. This makes it possible to avoid solving the electroelastic problem by finding all the characteristics of electroelastic contact from known cases of purely elastic interaction. Moreover, the electroelastic state of the half-space can be fully described using a known harmonic function, which is a solution of the purely elastic problem. The approach is exemplified by solving contact problems of electroelasticity for flat, elliptic, two circular, conical, and paraboloidal (circular and elliptic in plan) punches __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 69–84, November 2006.