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Analytical displacement and stress fields with stress concentration factors (SCFs) are derived for linearly elastic annular regions subject to inhomogeneous boundary conditions: an infinite class of the mth order polynomial antiplane tractions or displacements. The solution of the Laplace equation governing the out-of-plane problem covers both rigid and void circular inclusions forming the core of the annulus. The results show first that the SCF and the loading order are inversely proportional. In particular, the SCF approaches value 2 when either the outer boundary of the annulus tends to infinity or the order of the polynomial loading increases. Second, the number of peculiar points on the inner contour having null stress increases with the increasing loading order. The analytical solution is confirmed and extended to noncircular enclosures via finite element analysis by exploiting the heat-stress analogy. The results show that the closed-form solution for a circular annulus can be used as an accurate approximation for noncircular enclosures. Altogether, the results shown can be exploited for analyzing complex loading conditions and/or multiple rigid or void inclusions for enhancing the design of hollow and reinforced composites materials.

In elasticity theory, a neutral inhomogeneity is defined as a foreign body which can be introduced into a host solid without disturbing the stress field in the solid. The existence of circular neutral elastic nano-inhomogeneities has been established for both antiplane shear and plane deformations when the interface effect is described by constant interface parameters, and the surrounding matrix is subjected to uniform external loading. It is of interest to determine whether noncircular neutral nano-inhomogeneities can be constructed under the same conditions. In fact, we prove that only the circular elastic nano-inhomogeneity can achieve neutrality under these conditions with the radius of the inhomogeneity determined by the corresponding (constant) interface parameters and bulk elastic constants. In particular, in the case of plane deformations, the (uniform) external loading imposed on the matrix must be hydrostatic in order for the corresponding circular nano-inhomogeneity to achieve neutrality. Moreover, we find that, even when we relax the interface condition to allow for a nonuniform interface effect (described by variable interface parameters), in the case of plane deformations, only the elliptical nano-inhomogeneity can achieve neutrality.

In this paper, we derive the null-field integral equation for an infinite medium containing circular holes and/or inclusions with arbitrary radii and positions under the remote antiplane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. By moving the null-field point to the boundary, singular and hypersingular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. Besides, the boundary-layer effect is eliminated owing to the introduction of degenerate kernels. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The method is basically a numerical method, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The exact solution for a single inclusion is derived using the present formulation and matches well with the Honein et al.’s solution by using the complex-variable formulation (Honein, E., Honein, T., and Hermann, G., 1992, Appl. Math., 50, pp. 479–499). Several problems of two holes, two inclusions, one cavity surrounded by two inclusions and three inclusions are revisited to demonstrate the validity of our method. The convergence test and boundary-layer effect are also addressed. The proposed formulation can be generalized to multiple circular inclusions and cavities in a straightforward way without any difficulty.

The transient response of a finite bimaterial plate with a crack perpendicular to and terminating at the interface is analyzed for two types of boundaries (free-free and clamped-clamped). The crack surface is loaded by arbitrary time-dependent antiplane shear impact. The mixed initial-boundary value problem is reduced to a singular integral equation of a generalized Cauchy kernel for the crack tearing displacement density or screw dislocation density. The Gauss-Jacobi quadrature technique is employed to numerically solve the singular integral equation, and then the dynamic stress intensity factors are determined by implementing a numerical inversion of the Laplace transform. As an example, numerical calculations are carried out for a cracked bimaterial plate composed of aluminum (material I) and epoxy or steel (material II). The effects of material properties, geometry, and boundary types on the variations of dynamic stress intensity factors are discussed in detail. Results indicate that an overshoot of the normalized stress intensity factor of the crack tip at the interface decreases for a cracked bimaterial plate, and the occurrence of which is delayed for a cracked aluminum/epoxy plate compared to a pure aluminum plate with the same crack.

Effective viscoelastic response of a unidirectional fiber composite with interfaces that may separate or slip according to uniform Needleman-type cohesive zones is analyzed. Previous work on the solitary elastic composite cylinder problem leads to a formulation for the mean response consisting of a stress-strain relation depending on the interface separation∕slip discontinuity together with an algebraic equation governing its evolution. Results for the fiber composite follow from the composite cylinders representation of a representative volume element (RVE) together with variational bounding. Here, the theory is extended to account for viscoelastic matrix response. For a solitary elastic fiber embedded in a cylindrical matrix which is an nth-order generalized Maxwell model in shear relaxation, a pair of nonlinear nth-order differential equations is obtained which governs the relaxation response through the time dependent stress and interface separation∕slip magnitude. When the matrix is an nth-order generalized Kelvin model in shear creep, a pair of nonlinear nth-order differential equations is obtained governing the creep response through the time dependent strain and interface separation∕slip magnitude. We appeal to the uniqueness of the Laplace transform and its inverse to show that these equations also apply to an RVE with the composite cylinders microstructure. For a matrix, which is a standard linear solid (n=2), the governing equations are analyzed in detail paying particular attention to issues of bifurcation of response. Results are obtained for transverse bulk response and antiplane shear response, while axial tension with related lateral Poisson contraction and transverse shear are discussed briefly. The paper concludes with an application of the theory to the analysis of stress relaxation in the pure torsion of a circular cylinder containing unidirectional fibers aligned parallel to the cylinder axis. For this problem, the redistribution of shear stress and interface slip throughout the cross section, and the movement of singular surfaces, are investigated for an interface model that allows for interface failure in shear mode.

Based on the anisotropic elasticity theory and Lekhnitskii’s complex potential functions, the analytical eigenequations of anisotropic bimaterial half-plane wedges under antiplane shear are derived in brief forms. The boundary surfaces of half-plane wedges can be combinations of free and/or clamped edges. The disappearance conditions of stress singularities can be obtained directly from the derived eigenequations, which can be applied to improve the safety of the structures. The interesting phenomenon on the periodic appearance of the singularity orders is proposed and discussed as well.

This study aims at determining the elastic stress and displacement fields around a crack in a microstructured body under a remotely applied loading of the antiplane shear (mode III) type. The material microstructure is modeled through the Mindlin-Green-Rivlin dipolar gradient theory (or strain-gradient theory of grade two). A simple but yet rigorous version of this generalized continuum theory is taken here by considering an isotropic linear expression of the elastic strain-energy density in antiplane shearing that involves only two material constants (the shear modulus and the so-called gradient coefficient). In particular, the strain-energy density function, besides its dependence upon the standard strain terms, depends also on strain gradients. This expression derives from form II of Mindlin’s theory, a form that is appropriate for a gradient formulation with no couple-stress effects (in this case the strain-energy density function does not contain any rotation gradients). Here, both the formulation of the problem and the solution method are exact and lead to results for the near-tip field showing significant departure from the predictions of the classical fracture mechanics. In view of these results, it seems that the conventional fracture mechanics is inadequate to analyze crack problems in microstructured materials. Indeed, the present results suggest that the stress distribution ahead of the tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the classical results. The latter can be explained physically since materials with microstructure behave in a more rigid way (having increased stiffness) as compared to materials without microstructure (i.e., materials governed by classical continuum mechanics). The new formulation of the crack problem required also new extended definitions for the J-integral and the energy release rate. It is shown that these quantities can be determined through the use of distribution (generalized function) theory. The boundary value problem was attacked by both the asymptotic Williams technique and the exact Wiener-Hopf technique. Both static and time-harmonic dynamic analyses are provided.

We develop a rigorous solution to the antiplane problem of a circular inhomogeneity embedded within an infinite isotropic elastic medium (matrix) under the assumption of nonuniform remote loading. The bonding at the inhomogeneity/matrix interface is assumed to be homogeneously imperfect. We examine both the case of a single circular inhomogeneity and the more general case of a three-phase circular inhomogeneity. General expressions for the corresponding complex potentials are derived explicitly in both the inhomogeneity and in the surrounding matrix. The analysis is based on complex variable methods. The solutions obtained demonstrate the effect of the prescribed nonuniform remote loading on the stress field within the inhomogeneity. Specific solutions are derived in closed form which are verified by comparison with existing solutions.

The impact response of bonded media with a functionally graded interlayer weakened by a pair of two offset interfacial cracks is investigated under the condition of antiplane deformation. The material nonhomogeneity in the graded interlayer is represented in terms of power-law variations of shear modulus and mass density between the dissimilar, homogeneous half-planes. Laplace and Fourier integral transforms are employed to reduce the crack problem to solving a system of Cauchy-type singular integral equations in the Laplace domain. The crack-tip behavior in the physical domain is recovered through the inverse Laplace transform to evaluate the dynamic mode III stress intensity factors as a function of time. As a result, the transient interaction of the offset interfacial cracks spaced apart by the graded interlayer is illustrated. The peak values of the dynamic stress intensity factors are also presented versus offset crack distance, elaborating the effects of various material and geometric parameters of the bonded system on the overshoot characteristics of the transient behavior in the near-tip regions, owing to the impact-induced interaction of singular stress fields between the two cracks.

In this paper, a crack in a strip of a viscoelastic functionally graded material is studied under antiplane shear conditions, The shear relaxation function of the material is assumed as μ = μ 0 exp (βy/h)f(t), where h is a length scale and f(t) is a nondimensional function of time t having either the form f(t) = μ∞/μ 0 + (1 - μ α /μ 0 )exp(-t/t 0 ) for a linear standard solid, or f(t)=(t 0 /t) q for a power-law material model. We also consider the shear relaxation function μ = μ 0 exp (βy/h)[t 0 exp (δy/h)/t] q in which the relaxation time depends on the Cartesian coordinate y exponentially, Thus this latter model represents a power-law material with position-dependent relaxation time. In the above expressions, the parameters β, μ 0 , μ∞, t 0 ; δ, q are material constants. An elastic crack problem is first solved and the correspondence principle (revisited) is used to obtain stress intensity factors for the viscoelastic functionally graded material. Formulas for stress intensity factors and crack displacement profiles are derived. Results for these quantities are discussed considering various material models and loading conditions.

This paper treats the effective antiplane shear response of a composite consisting of fibers that interact with the matrix through nonlinear Needleman-type cohesive zones. The first paper (Part I) examines effective axial tension response. The composite cylinders representation of a representative volume element (RVE) is employed throughout. For antiplane shear loading the elastic field solution for a single composite cylinder is found in the form of a series expansion whose coefficients are governed by an infinite set of nonlinear equations. Bounds on the total potential energy and the total complementary energy of an RVE do not coincide although they are shown to differ by a term of order Oc4 where c is the fiber volume concentration. Interaction effects due to finite volume concentration, coupled with nonlinear interface characterization, are shown to precipitate instability in composite response. [S0021-8936(00)02104-8]

This paper treats the effective axial tension response of a composite consisting of fibers that debond from the matrix according to nonlinear Needleman-type cohesive zones. A second, related paper (Part II) treats effective antiplane shear response. The composite cylinders representation of a representative volume element (RVE) is employed throughout. For axial tension loading a simple rotationally symmetric boundary value problem for a single composite cylinder is solved. Bounds on the total potential energy and the total complementary energy are shown to coincide and an exact solution for axial extension and Poisson contraction of an RVE of the composite is obtained. Nonlinear interfacial debonding, however, is shown to have a negligible effect on extensional response and only a small, though potentially destabilizing, effect on Poisson contraction response. [S0021-8936(00)02004-3]

Green’s functions for anisotropic elastic multilayered media subjected to antiplane shear deformation are presented in this study. The antiplane shear deformation due to a concentrated shear force and screw dislocation in an arbitrary layer was investigated in detail. A linear coordinate transformation is introduced in this study to simplify the problem. The linear coordinate transformation reduces the anisotropic multilayered problem to an equivalent isotropic problem without complicating the geometry of the problem. Explicit analytical solutions were derived using the Fourier transform and the series expansion technique. The complete solutions for the multilayered problem consist only of the simplest solutions obtained from an infinite homogeneous medium with concentrated loadings. Numerical results for the full-field stress distribution in multilayered media subjected to a point body force are presented. These numerical results were compared with the solutions obtained by considering the multilayered medium as one layer with effective elastic constants determined from the averaged material constants of the multilayered medium. It is found that the shear stress τyz of the homogeneous one layer solution is a very good approximation of the result for the multilayered medium; however, the shear stress τxz in these two solutions has a large discrepancy due to the fact that τxz is discontinuous at the interfaces of the multilayered medium. [S0021-8936(00)01703-7]

The analytical solution for an infinitely long borehole in an isotropic, poroelastic medium, inclined to the far-field principal stresses, is presented. The solution utilizes a loading decomposition scheme which leads to three fundamental problems: a poroelastic plane-strain, an elastic uni-axial, and an elastic antiplane shear problem.

In this study, the transient analysis of dynamic antiplane crack propagation with a constant velocity in a layered medium is investigated. The individual layers are isotropic and homogeneous. Infinite numbers of reflected cylindrical waves, which are generated from the interface of the layered medium, will interact with the propagating crack and make the problem extremely difficult to analyze. A useful fundamental solution is proposed in this study, and the solution can be determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The Cagniard’s method for Laplace inversion is used to obtain the transient solution in time domain. The exact closed-form transient solutions of dynamic stress intensity factors are expressed in compact formulations. These solutions are valid for an infinite length of time and have accounted for contributions from all the incident and reflected waves interaction with the moving crack tip. Numerical results of dynamic stress intensity factors for the propagation crack in layered medium are evaluated and discussed in detail.

The asymptotic stress field near the tip of a crack subjected to antiplane shear loading is analysed. The crack is growing quasi-statically along an elastic/elastic power-law creeping bimaterial interface. We find there is a separable solution with the following characteristics: for n < 3, where n is the power-law creeping exponent, the asymptotic stress field is dominated by the elastic strain rates and has an inverse square root singularity, r−1/2, where r is the distance from the current crack tip. For n ≥ 3, the near-tip stress and strain fields has a singularity of the form r−1/(n−1). The strength of this field is completely specified by the current crack growth rate, besides material properties, and is otherwise independent of the applied load and of the prior crack growth history.

Antiplane shear deformations of a cylindrical body, with a single displacement field parallel to the generators of the cylinder and independent of the axial coordinate, are one of the simplest classes of deformations that solids can undergo. They may be viewed as complementary to the more familiar plane deformations. Antiplane (or longitudinal) shear deformations have been the subject of the considerable recent interest in nonlinear elasticity theory for homogeneous isotropic solids. In contrast, for the linear theory of isotropic elasticity, such deformations are usually not extensively discussed. The purpose of the present paper is to demonstrate that for inhomogeneous anisotropic linearly elastic solids the antiplane shear problem does provide a particularly tractable and illuminating setting within which effects of anisotropy and inhomogeneity may be examined. We consider infinitesimal antiplane shear deformations of an inhomogeneous anisotropic linearly elastic cylinder subject to prescribed surface tractions on its lateral boundary whose only nonzero component is axial and which does not vary in the axial direction. In the absence of body forces, not all arbitrary anisotropic cylinders will sustain an antiplane shear deformation under such tractions. Necessary and sufficient conditions on the elastic moduli are obtained which do allow an antiplane shear. The resulting boundary value problems governing the axial displacement are formulated. The most general elastic symmetry consistent with an antiplane shear is described. There are at most 15 independent elastic coefficients associated with such a material. In general, there is a normal axial stress present, which can be written as a linear combination of the two dominant shear stresses. For a material with the cylindrical cross-section a plane of elastic symmetry (monoclinic, with 13 moduli), the normal stress is no longer present. For homogeneous materials, it is shown how the governing boundary value problem can be transformed to an equivalent isotropic problem for a transformed cross-sectional domain. Applications to the issue of assessing the influence of anisotropy and inhomogeneity on the decay of Saint-Venant end effects are described.

We examined the effects of surface elasticity in a classical mode-III crack problem arising in the antiplane shear deformations of a linearly elastic solid. The surface mechanics are incorporated using the continuum based surface/interface model of Gurtin and Murdoch. Complex variable methods are used to obtain an exact solution valid everywhere in the domain of interest (including at the crack tip) by reducing the problem to a Cauchy singular integro-differential equation of the first order. Finally, we adapt classical collocation methods to obtain numerical solutions, which demonstrate several interesting phenomena in the case when the solid incorporates a traction-free crack face and is subjected to uniform remote loading. In particular, we note that, in contrast to the classical result from linear elastic fracture mechanics, the stresses at the (sharp) crack tip remain finite.

An analysis of antiplane shear interface cracks in a finite anisotropic composite body is presented. The analysis is done by a new complex-variable integral equation formulation based on the solutions of a dislocation and body force in an infinite composite body. Numerical results of the stress intensity factors are presented for the composite bodies with finite rectangular cross-sections under uniform shear. The composite bodies are formed by bonding an orthotropic material to an isotropic material. The numerical results show that there exists a lower bound for the stress intensity factor for a fixed crack-length-to-height ratio and that the lower bound is attained in the case of isotropic bimaterial.

A correspondence is established between the problem of an interface crack in mon-oclinic composites and that of an interface crack in isotropic composites. The interface crack considered is subjected to a combined tension-compression, in-plane shear and antiplane shear loading at the crack faces. Under the applied loading, the interface crack is assumed to be partially opened. Through the correspondence, quantities of interest such as stress intensity factors, sizes of the contact zones, for monoclinic composites can be obtained from the results of the isotropic interface crack problem.

The problem of an elastic plate containing a through crack and subjected to twisting moments or transverse shear loads is considered. By using a bending theory which allows the satisfaction of the boundary conditions on the crack surface regarding the normal and the twisting moments and the transverse shear load separately, it is found that the resulting asymptotic stress field around the crack tip becomes identical to that given by the elasticity solutions of the plane strain and antiplane shear problems. The problem is solved for uniformly distributed or concentrated twisting moment or transverse shear load and the normalized Mode II and Mode III stress-intensity factors are tabulated. The results also include the effect of the Poisson’s ratio and material orthotropy for specially orthotropic materials on the stress-intensity factors.

The diffraction of time harmonic antiplane shear waves by an edge crack normal to the free surface of a homogeneous half space is considered. The problem is formulated in terms of a singular integral equation with the unknown crack opening displacement as the density function. A numerical scheme is utilized to solve the integral equation at any given finite frequency. Asymptotic solutions valid at low and high frequencies are obtained. The accuracy of the numerical solution at high frequencies and of the high frequency asymptotic solution at intermediate frequencies are examined. Graphical results are presented for the crack opening displacement and the stress intensity factor as functions of frequency and the incident angle, Expressions for the far-field displacements at high and low frequencies are also presented.

The main objective of this paper is the investigation of the singular nature of the crack-tip stress field in a nonhomogeneous medium having a shear modulus with a discontinuous derivative. The problem is considered for the simplest possible loading and geometry, namely the antiplane shear loading of two bonded half spaces in which the crack is perpendicular to the interface. It is shown that the square-root singularity of the crack-tip stress field is unaffected by the discontinuity in the derivative of the shear modulus. The problem is solved for a finite crack and extensive results are given for the stress intensity factors.

Results from free streamline hydrodynamics are exploited in order to solve optimization problems for antiplane shear deformation, in which the stress concentration is to be minimized. These problems pertain to the optimum shapes for grooves cut into a half-space. We obtain results, which from the standpoint of the hydrodynamics problem, complement those presently in the literature. The solution is given in an integral form which in general must be evaluated by numerical methods, but that reduces to elliptic integrals for the special case of a notch whose faces meet the half-space boundary at right angles.

In a previous paper, the dynamic, steady-state propagation of a semi-infinite antiplane shear crack was considered for an infinite, general linearly viscoelastic body. Under the assumptions that the shear modulus is a positive, nonincreasing continuous and convex function of time, convenient, closed-form expressions were derived for the stress intensity factor and for the entire stress distribution ahead of and in the plane of the advancing crack. The solution was shown to have a simple, universal dependence on the shear modulus and crack speed from which qualitative and quantitative information can readily be gleaned. Here, the corresponding problem for a general, linearly viscoelastic layer is solved. An infinite series representation for the stress intensity factor is derived, each term of which can be calculated recursively in closed form. As before, a simple universal dependence on crack speed and material properties is exhibited.

A detailed finite element analysis is performed to model quasi-static crack growth under plane stress, small-scale yielding conditions in elastic-plastic materials characterized by isotropic power law hardening and the Huber-Von Mises yield surface. A nodal release procedure is used to simulate crack extension. Results pertaining to the influence of hardening on the extent of active yielding and the near-tip stress and deformation fields are presented. Clear evidence of an elastic unloading wake following the active plastic zone is found, but no secondary (plastic) reloading along the crack flank is numerically observed for any level of hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small microstructural distance behind the tip, is employed to investigate the nature of the J resistance curves under plane stress. In addition, the same criterion is employed to investigate the influence of hardening on the potential for stable crack growth under plane stress. It is found that predictions based on a perfectly plastic model may be unconservative in this respect, which is qualitatively similar to the conclusions reached in antiplane shear and Mode I plane strain.

The steady-state propagation of a semi-infinte, anti-plane shear crack is reconsidered for a general, infinite, homogeneous and isotropic linearly viscoelastic body. As with an earlier study, the inertial term in the equation of motion is retained and the shear modulus is only assumed to be positive, continuous, decreasing and convex. A Barenblatt type failure zone is introduced in order to cancel the singular stress, and a numerically convenient expansion for the dynamic Energy Release Rate (ERR) is derived for a rather general class of crack face loadings. The ERR is shown to have a complicated dependence on crack speed material properties with significant qualitative differences between viscoelastic and elastic material. The results are illustrated with numerical calculations for both power-law material and a standard linear solid. Keywords: Crack Growth; Fracture Mechanics; Dynamic Fracture.

In this paper we consider a problem of the response of an elastic half space subjected to an antiplane shear load. The load is suddenly applied and thereafter moves in an interval reciprocally as a trigonometric function of time. An analytical solution for the displacement is obtained in terms of single integration. It is shown that the discontinuity in the displacement occurs only for the case that the initial (maximum) speed of the load is greater than the speed of SH-wave. In this case the displacement has a finite jump on the leading wave front and a logarithmic discontinuity immediately behind the wave front which emanates from a point where the load speed comes up with SH-wave speed. Numerical calculations are carried out for several cases of the initial (maximum) speed of the load and are shown graphically.

A rigid rectangular foundation, embedded at an arbitrary depth below the surface of an elastic half space is subjected to a plane, transient SH-wave. The Laplace and Kantorovich-Lebedev transforms are applied to derive the equation of motion for the foundation during the initial time period required for an SH-wave to traverse the base width. The peak impulse response is found to occur during this time and the response there-after appears to be valid based on a comparison with the known, long-time limit. Consequently, the results presented here can be convolved with an earthquake accelerogram to yield an accurate foundation earthquake response.

The problem of determining the order to the stress singularity for an anti-plane shear crack at the interface of two bonded inhomogeneous elastic materials is considered. It had been conjectured previously that the usual square root singularity occurs if the spatially varying material properties are merely continuous across the bond line but not necessarily continuously differentiable. This paper presents a proof of that conjecture. Keywords: Anti- plane shear interface crack; Nonhomogeneous elastic material; Crack tip stress singularity.