Static deformation due to a long buried dip-slip fault in an isotropic half-space welded with an orthotropic half-space

Closed-form analytical expressions for the displacements and the stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space due to a dip-slip fault of finite width located at an arbitrary distance from the interface in the isotropic half-space are obtained. The Airy stress function approach is used to obtain the expressions for the stresses and the displacements. The case of a vertical dip-slip fault is considered in detail. The variations of the displacements with the distance from the fault and with depth have been shown graphically.     

Inverse crack problem in elasticity

Summary Exact solution is given to the problem of a penny-shaped crack embedded in a transversely isotropic elastic half-space when arbitrary normal displacements are prescribed at its faces. A new integral representation of the kernel of the governing integral equation allowed to obtain closed form expressions for all the quantities of interest like, stresses inside and outside the crack, stress intensity factor, work done to open the crack, directly through the given displacements. Several illustrative examples are considered.     

Periodic array of staggered planar cracks in an anisotropic elastic medium

The problem of determining the elastic displacements and stresses in an infinite anisotropic medium containing a periodic array of staggered planar cracks is considered. It is reduced to a system of Hadamard finite-part singular (hypersingular) integral equations with the crack-opening displacements as unknown functions. The integral equations may be solved numerically by a collocation technique. Numerical results for specific cases involving isotropic and transversely isotropic materials are obtained.     

The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

Summary An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.With 5 Figures     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

The arbitrarily loaded single-edge cracked circular disc; accurate weight function solutions

A general solution for the stresses and displacements of collinear cracks in an infinite homogeneous anisotropic medium subjected to uniform loading at infinity has been given in this paper by using the Stroh's formulation. The solutions are valid not only for plane problems but also for antiplane problems and the problems whose inplane and antiplane deformations couple each other. Two special collinear crack problems are solved explicitly: (1) two collinear cracks, (2) an infinite row of evenly spaced collinear cracks. A closed form solution of the stresses and displacements in the entire domain is obtained. Through the use of identities developed in the literature, the stress intensity factors, crack opening displacements and energy release rate are expressed in real form, which are valid for any kind of anisotropic materials including the degenerate materials such as isotropic materials. The simple explicit form solutions for the crack opening displacements and energy release rate reveal that the effect of anisotropy is totally determined by the fundamental elasticity matrix L. The relation between the stress intensity factors and energy release rate is obtained in quadratic form and related to L.     

Flat crack under shear loading

Summary A solution is called complete when the explicit expressions are derived for the field of displacements as well as stresses in an elastic body. A new method is proposed here which allows us to obtain exact and complete solutions to various crack problems in elementary functions; no integral transforms or special function expansions are involved. The method is based on the new results in potential theory obtained earlier by the author. The method is applied to the case of a concentrated tangential loading of a penny-shaped crack. The main potential function and the relevant Green's functions are derived. An approximate analytical solution is obtained for a flat crack of general shape. A new set of asymptotic expressions is presented for the field of stresses and displacements near the crack tip in a transversely isotropic space. The use of the method is illustrated by examples.     

Stresses and displacements of a transversely isotropic elastic halfspace due to rectangular loadings

Analytical solutions in exact closed-forms are obtained for stresses and displacements in an solid due to rectangular loading. The stresses and displacements are induced in the solid due to the vertical and horizontal loadings uniformly distributed on a rectangular area. The rectangular area is horizontally embedded in or on the solid. The solid occupies a space of semi-infinite extent and has a linear elastic property with transverse isotropy. The classical integral transforms are used in the solution formulation. The solutions are systematically presented in matrix forms and in terms of elementary harmonic functions. The solutions are easily implemented for numerical calculations and applied to problems encountered in engineering. Comparisons of the present solution with existing similar solutions are presented for the stresses and displacements induced by the vertical load. In addition, the numerical results of the stresses and displacements in the solid induced by the horizontal and vertical loads are also presented. These results illustrate the effect of different elastic constants of transversely isotropic solids on the stress and displacement fields.     

Asymmetric displacements of a rigid disc inclusion embedded in a transversely isotropic elastic medium of infinite extent

Asymmetric problems related to a penny-shaped rigid inclusion embedded in bonded contact with a transversely isotropic elastic medium are investigated. The asymmetric displacements of the rigid circular inclusion correspond to a rotation about a diametral axis and an in-plane lateral translation. These problems are formulated in terms of Hankel integral transforms and reduced to systems of dual integral equations. The rotational and translational stiffnesses for the embedded rigid circular disc inclusion are obtained in exact closed forms.     

Exact solution for axisymmetric deformation of laminated transversely isotropic annular plates

Summary Based on the three-dimensional theory of elasticity, this paper presents the state space equation for axisymmetric deformation of a laminated transversely isotropic annular plate. The finite Hankel transform is then introduced and applied to the state space equation. Four exact solutions corresponding to four specified boundary conditions are obtained and expressions for displacements and stresses are presented. Numerical results are finally compared with those obtained by the classical plate theory, the Reissner plate theory and the finite element method.     

Concentrated force in an infinite space of transversely isotropic material

Summary An exact closed form solution for the displacements and stresses in a transversely isotropic infinite space due to concentrated point forces is presented, which contains the solution for the corresponding granular material as its special case and the well-known three-dimensional Kelvin solution as its limiting case.     

Wave propagations in exponentially graded transversely isotropic half-space with potential function method

Time-harmonic response of a vertically graded transversely isotropic, linearly elastic half-space is analytically determined by introducing a new set of potential functions. The potential functions are set in such a way that the governing equations be simple and with physical meaning as well. In addition, the potential functions introduced in this paper are degenerated to a complete set of potential functions used frequently for wave propagations in homogeneous transversely isotropic media. Utilizing Fourier series and Hankel integral transforms, the governing equations for the potential functions are solved, after which the displacements and stresses are presented in the form of line integrals. Both the displacements and stresses determined here are collapsed on the solution previously reported for the constant profile transversely isotropic material. Because of complicated integrand functions, the integrals are evaluated numerically and presented graphically, where the effect of degree of change of material properties plays a major role, which may be recognized easily.     

裂纹稳态扩展下正交异性材料的动应力强度因子K_(Ⅲ)解答

研究了无限大正交异性材料中半无限长Ⅲ型裂纹的动态扩展问题。裂纹尖端附近的应力和位移被表达为解析复函数的形式,而复函数可以表达为幂级数的形式,幂级数的系数由研究问题的边界条件来确定。这样就给出了裂纹尖端附近的应力分量和位移分量的简单近似表达式,由推导出的动应力分量和动位移分量可以退化为其在各向同性材料静态断裂问题中的情况。最后,裂纹扩展特性由裂纹几何参数和裂纹扩展速度来反映出来,相同的几何参数情况下,裂纹扩展愈快,裂纹尖端附近的最大应力分量和最大位移分量愈大。     

Influence of anisotropy,porosity and initial stresses on crack propagation due to Love‐type wave in a poroelastic medium

An analytical solution has been attained to establish the closed form expression of stress intensity factor at the tip of a semi‐infinite crack, dynamically propagating in an initially stressed transversely isotropic poroelastic strip due to Love‐type wave for the case of concentrated force of constant intensity as well as for the case of constant load. The study presents the sound effect of various affecting parameters viz. speed of the crack, length of the crack, horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, porosity parameter and anisotropy parameter on the stress intensity factor. In order to delineate the effects of these aforementioned parameters on the stress intensity factor graphically, numerical simulations have been accomplished. One of the major highlight of the paper is the comparative study carried out for horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, porosity parameter and anisotropy parameter with the case when the strip is isotropic, non‐porous and free from initial stresses. Wiener–Hopf technique and the Fourier integral transform has been effectuated for the procurement of the closed form expression (exact solution) of stress intensity factor.     

Planar elastic laminates and their homogenization

Summary The state of stress and deformation of a planar elastic-homogeneous transversely isotropic thick layer in the case of plane and axisymmetric strain respectively is determined in a systematic and uniform manner using integral transforms and transfer matrices. Next, a laminate with an arbitrary number of different layers is considered without any simplifying assumptions. Then we analyze a periodic structure consisting of many thin and identical layer groups by means of a suitable homogenization, where a layer group contains two or more different transversely isotropic homogeneous basic layers. As an example exact closed form solutions for a periodically layered half space are evaluated. The well known result, that a medium which is composed of alternating thin layers of two different elastic-isotropic substances is elastostatically equivalent to a homogeneous transversely isotropic medium is extended to the above mentioned more general case. Further, the in-plane normal stresses which are discontinuous for a finite layering are evaluated in addition to the smeared ones which are continuous and correct only in the limit of a vanishing thickness of the individual layers. The explicit knowledge of the resultant elastic constants (called effective ones) turns out to be dispensable; rather, the effective material parameters (defined in this paper) which are the weighted sums of the material parameters of the basic layers are proved to be relevant. Nevertheless, for the purpose of comparison with some published results the effective elastic constants, especially for a layer group consisting of two different elastic-isotropic substances, are evaluated additionally.Dedicated to Prof. Gallus Rehm on the occasion of his 75th birthday     

Axisymmetric time-harmonic response of a transversely isotropic substrate–coating system

By virtue of a method of displacement potentials, an analytical treatment of the response of a transversely isotropic substrate–coating system subjected to axisymmetric time-harmonic excitations is presented. In determination of the corresponding elastic fields, infinite line integrals with singular complex kernels are encountered. Branch points, cuts, and poles along the path of integration are accounted for exactly, and the physical phenomena pertinent to wave propagation in the medium are also highlighted. For evaluation of the integrals at the singular points, an accurate analytical residual theory is presented. Comparisons with the existing numerical solutions for a two-layered transversely isotropic medium under static surface load, and a transversely isotropic half-space subjected to buried time-harmonic load are made to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses are presented to portray the dependence of the response of the substrate–coating system on the frequency of excitation and the role of coating layer.     

Hygrothermal stresses for a plane crack in a generally anisotropic plate

The solutions are presented for the hygrothermal stress field of a generally anisotropic plate under uniform heat flux and moisture concentration transfer obstructed by a hygrothermally insulated crack. For uncoupled diffusion of temperature and moisture, the solutions of both temperature and moisture are obtained directly from the Hilbert problem approach, and are treated as the particular solutions to a pair of nonhomogeneous partial differential equations for an uncoupled hygrothermoelastic system. The associated homogeneous solutions are expressed in terms of three stress functions based on the complex variable approach of Lekhnitskii. With some identities concerning the eigenvalues and eigenvectors, the general expressions of the stress and displacement fields can then be found in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are expressed in terms of the heat flow, moisture concentration, material geometry, elastic and hygrothermal anisotropy. The simultaneous existence of mode I, II and III fracture is found to be due to material inherent anisotropy. Special cases for isotropic and orthotropic materials are also discussed.     

Effects of fluid layer at micropolar orthotropic boundary surface

Effects of a fluid layer at a micropolar orthotropic elastic solid interface to a moving point load have been studied. After using the Fourier transform an eigen value approach has been employed to solve the problem. The displacement, microrotation and stress components for a micropolar orthotropic elastic solid so obtained in the physical domain are computed numerically by applying the numerical inversion technique. Micropolarity and anisotropy effects along with that of the depth of the fluid layer on various expressions have been depicted graphically for a particular model. Some special cases of interest have been presented     

A generalized method for characterizing elastic anisotropy in solid living tissues

The microstructure of cortical bone may exhibit either transverse isotropic or orthotropic symmetry, thus requiring either five or nine independent elastic stiffness coefficients (or compliances), respectively, to describe its elastic anisotropy. Our previous analysis to describe this anisotropy in terms of two scalar quantities for the transverse isotropic case is extended here to include orthotropic symmetry. The new results for orthotropic symmetry are compared with previous calculations using the transverse isotropic analysis on the same sets of anisotropic elastic constants for bone, determined either by mechanical or by ultrasonic experiments. In addition, the orthotropic calculation has been applied to full sets of orthotropic elastic stiffness coefficients of a large variety of wood species. Although having some resemblance to plexiform bone in microstructural organization, there is a dramatic difference in both the shear and the compressive elastic anisotropy between the two materials: wood is at least one order of magnitude more anisotropic than bone.     

Singularity analysis of anisotropic multimaterial corners

Singular stress states induced at the tip of linear elastic multimaterial corners are characterized in terms of the order of stress singularities and angular variation of stresses and displacements. Linear elastic materials of an arbitrary nature are considered, namely anisotropic, orthotropic, transversely isotropic, isotropic, etc. Thus, in terms of Stroh formalism of anisotropic elasticity, the scope of the present work includes mathematically non-degenerate and degenerate materials. Multimaterial corners composed of materials of different nature are typically present at any metal-composite, or composite-composite adhesive joint. Several works are available in the literature dealing with a singularity analysis of multimaterial corners but involving (in the vast majority) only materials of the same nature (e.g., either isotropic or orthotropic). Although many different corner configurations have been studied in literature, with almost any kind of boundary conditions, there is an obvious lack of a general procedure for the singularity characterization of multimaterial corners without any limitation in the nature of the materials. With the procedure developed here, and implemented in a computer code, multimaterial corners, with no limitation in the nature of the materials and any homogeneous orthogonal boundary conditions, could be analyzed. As a particular case, stress singularity orders in corners involving extraordinary degenerate materials are, to the authors' knowledge, presented for the first time. The present work is based on an original idea by Ting (1997) in which an efficient procedure for a singularity analysis of anisotropic non-degenerate multimaterial corners is introduced by means of the use of a transfer matrix.