Infinite Isotropic Elastic Medium Research Articles

Plane strain problem of an elastic matrix containing multiple Gurtin–Murdoch material surfaces along straight segments

This paper presents the study of the plane strain problem of an infinite isotropic elastic medium subjected to far-field load and containing multiple Gurtin–Murdoch material surfaces located along straight segments. Each material segment represents a membrane of vanishing thickness characterized by its own elastic stiffness and residual surface tension. The governing equations, the jump conditions, and the surface tip conditions are reviewed. The displacements in the matrix are sought as the sum of complex variable single-layer elastic potentials whose densities are equal to the jumps in complex tractions across the segments. The densities are found by solving the system of coupled hypersingular boundary integral equations. The approximations by a series of Chebyshev’s polynomials of the second kind are used with the square root weight functions chosen to satisfy the tip conditions automatically. Numerical examples are presented to illustrate the influence of dimensionless parameters and to study the effects of interactions.

An accurate semi-analytical method for the treatment of an eccentric annular crack embedded in an infinite isotropic elastic medium under arbitrary internal pressure

This work provides a semi-analytical method for the calculation of the stress intensity factor (SIF) associated with an internally pressurized eccentric annular crack embedded in an infinite isotropic elastic solid. The inner and outer edges of the crack consist of the circumferences of two eccentric circles and the internal pressure exerted on the crack faces has an arbitrary distribution. Using Somigliana formulation a hypersingular integral equation governing the proposed crack problem is obtained. Next, by the employment of a conformal mapping the considered crack is mapped on to a concentric annular crack. In the mapped domain, with the aids of Fourier expansion of the displacement discontinuity function across the crack faces together with the notion of the Hadamard finite-part integral, the hypersingular integral equation is converted to a system of linear algebraic equations for the unknown constants of the expansion. Upon the determination of these constants, the angular dependent SIFs along both the inner and outer edges of the crack are obtained accurately. The accuracy of the current formulation is showcased through the re-examination of the simple concentric annular crack problems considered previously in the literature. In the presence of eccentricity, the effects of such geometrical parameters as the ratio of the radii of the inner to outer eccentric circular edges of the crack and the normalized eccentricity on the values of the SIFs along both the inner and outer edges of the crack are studied. As it will be seen the material properties of the elastic matrix do not affect the values of SIFs.

Unconventional cracks with dislocation-free zones

Using the method of continuously distributed dislocations, we study the dislocation-free zones for unconventional internal cracks in an infinite isotropic elastic medium under uniform remote anti-plane shear stress. The unconventional cracks are represented by a symmetric lip crack with two cusp tips and a symmetric airfoil crack with a single cusp tip. For each case, the equilibrium condition is formulated in terms of a single singular integral equation in the image plane. The resulting singular integral equation is solved numerically to arrive at the dislocation distribution function and the total number of dislocations in the plastic zone, the condition on remote loading and the local mode III stress intensity factor at the cusp tip. In the absence of the dislocation-free zones, the results are valid for the Dugdale model of these unconventional cracks.

Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

INFLUENCE OF THE HIGHER ORDER TERMS IN WILLIAMS’ SERIES EXPANSION OF THE STRESS FIELD ON THE STRESS-STRAIN STATE IN THE VICINITY OF THE CRACK TIP. PART II

The description of mechanical fields at the vicinity of a bi-dimensional crack-tip can be performed using the classic Williams asymptotic series expansion. While the general structure is well known, complete expressions are rarely available for specific crack problems. The paper is devoted to the multi-parameter description of the stress field in the vicinity of two collinear crack of different length in an infinite isotropic elastic medium subjected to 1) Mode I loading; 2) Mode II loading; 3) mixed (Mode I + Mode II) mode loading. The multi-parameter asymptotic expansions of the stress field are obtained. The procedure used in the paper relates Williams series coefficients and the complex potentials of the plane elasticity. The amplitude coefficients of the multi-parameter series expansion are found in the closed form. Having obtained the coefficients of the Williams series expansion one can keep any preassigned number of terms in the asymptotic series. Asymptotic analysis of number of the terms in the Williams asymptotic series which is necessary to keep in the asymptotic series at different distances from the crack tip. It is shown that the more distance from the crack tip the more terms in the Williams asymptotic expansion need to be kept. Complete closed-form expressions can be used to derive, test and improve numerical and experimental approaches involving higher order terms in crack-tip expansions.

INFLUENCE OF THE HIGHER ORDER TERMS IN WILLIAMS’ SERIES EXPANSION OF THE STRESS FIELD ON THE STRESS-STRAIN STATE IN THE VICINITY OF THE CRACK TIP. PART I

The paper is devoted to the multi-parameter description of the stress fields in the vicinity of two collinear crack of different length in an infinite isotropic elastic medium subjected to 1) Mode I loading; 2) Mode II loading; 3) mixed (Mode I + Mode II) mode loading. The multi-parameter asymptotic expansions of the stress field in the vicinity of the crack tip in isotropic linear elastic media under mixed mode loading are obtained. The amplitude coefficients of the multi-parameter series expansion are found in the closed form. Having obtained the coefficients of the Williams series expansion one can keep any preassigned number of terms in the asymptotic series. Asymptotic analysis of number of the terms in the Williams asymptotic series which is necessary to keep in the asymptotic series at different distances from the crack tip. It is shown that the more distance from the crack tip the more terms in the Williams asymptotic expansion need to be kept.

P-wave Diffraction by a Crack Under Impact Load

The focus of this paper is centered on diffraction of P-wave by a Griffith crack in an infinite isotropic elastic medium on which impact load is applied. Suddenly applied equal but opposite normal stresses on crack faces causes the separation of the crack faces. Laplace and Fourier transforms have been employed to convert the problem to a Fredholm integral equation of second kind in the Laplace transform plane which has been solved by perturbation method. Stress intensity factor has been computed numerically by Laplace inversion using Zakian’s algorithm and plotted graphically against time for different isotropic materials.

Efficient numerical solution of the hydraulic fracture problem for planar cracks

An infinite isotropic elastic medium with a planar crack is considered. The crack is subjected to pressure of fluid injected inside the crack at a point of its surface. The fluid is viscous newtonian, incompressible, the medium is impermeable. Description of the crack growth is based on the lubrication equation (local balance of the injected fluid and the crack volume), the elasticity equation for crack opening caused by fluid pressure, Poiseulle equation related the fluid flux with crack opening and the pressure gradient, and the criterion of crack propagation of linear fracture mechanics. The crack growth is simulated by a series of discrete steps. Each step consists of three stages: increasing the crack volume by a constant crack size, crack jump to a new size defined by the fracture criterion, and filling the new crack configuration by the fluid presented in the crack. The problem is ill-posed and requires specific methods for numerical solution. The proposed method is based on an appropriate class of approximating functions for fluid pressure distributions on the crack surface and the theory of solution of ill-posed problems. Example of crack growth in a homogeneous and isotropic elastic medium is considered, influence of the fluid viscosity on the process of crack growth is studied.

Propagation of mechanical waves through a stochastic medium with spherical symmetry

We theoretically analyze the propagation of outgoing mechanical waves through an infinite isotropic elastic medium possessing spherical symmetry whose Lamé coefficients and density are spatial random functions characterized by well-defined statistical parameters. We derive the differential equation that governs the average displacement for a system whose properties depend on the radial coordinate. We show that such an equation is an extended version of the well-known Bessel differential equation whose perturbative additional terms contain coefficients that depend directly on the squared noise intensities and the autocorrelation lengths in an exponential decay fashion. We numerically solve the second order differential equation for several values of noise intensities and autocorrelation lengths and compare the corresponding displacement profiles with that of the exact analytic solution for the case of absent inhomogeneities.

Crack opening displacements under remote stress gradient: Derivation with a canonical basis of sixth order tensors

In this paper, we derive the crack opening displacement of a penny-shaped crack embedded in an infinite isotropic elastic medium and subjected to a remote constant stress gradient. The solution is derived by taking advantage of the solution of the equivalent ellipsoidal inclusion problem subjected to a linear polarization. The case of the penny-shaped crack is thereafter investigated by considering the case of a spheroidal cavity which has one principal axis infinitesimally small compared to both others. The derivation of the explicit solution for the inhomogeneity subjected to a remote stress gradient raises the problem of the inversion of a sixth order tensor. For the problem having a symmetry axis (this including the case of penny shaped crack), this problem can be tackled by using a decomposition on the canonical basis for transversely isotropic sixth order tensors.

Second-order in-plane dynamic perturbation of a crack propagating under shear loading

The title problem is solved, for a crack propagating in a slab which is composed of material which can be viscoelastic, “vertically stratified” and transversely isotropic with the axis in the “vertical” direction. The result is illustrated by specialising to the case of a stationary crack in the plane x3 = 0 of an infinite isotropic elastic medium, whose leading edge is at x1 = a cos(kx2), with |k|a ≪ 1.

The Influence Area of a Supersonically Expanding Spherical Inclusion

The expression for the elastic field of an expanding spherical inclusion in an infinite isotropic elastic medium is presented. The influence area of a supersonically expanding spherical inclusion is given in the form of expressions and graphs. And the difference of the influence area between supersonic expansion and subsonic expansion of a spherical inclusion is discussed.

Strain distribution in quantum dot of arbitrary polyhedral shape: Analytical solution

An analytical expression of the strain distribution due to lattice mismatch is obtained in an infinite isotropic elastic medium (a matrix) with a three-dimensional polyhedron-shaped inclusion (a quantum dot). The expression was obtained utilizing the analogy between electrostatic and elastic theory problems. The main idea lies in similarity of behavior of point charge electric field and the strain field induced by point inclusion in the matrix. This opens a way to simplify the structure of the expression for the strain tensor. In the solution, the strain distribution consists of contributions related to faces and edges of the inclusion. A contribution of each face is proportional to the solid angle at which the face is seen from the point where the strain is calculated. A contribution of an edge is proportional to the electrostatic potential which would be induced by this edge if it is charged with a constant linear charge density. The solution is valid for the case of inclusion having the same elastic constants as the matrix. Our method can be applied also to the case of semi-infinite matrix with a free surface. Three particular cases of the general solution are considered—for inclusions of pyramidal, truncated pyramidal, and “hut-cluster” shape. In these cases considerable simplification was achieved in comparison with previously published solutions.

A closed-form solution for composite tunnel linings in a homogeneous infinite isotropic elastic medium

This paper presents a closed-form solution for composite tunnel linings in a homogeneous infinite isotropic elastic medium. The tunnel lining is treated as an inner thin-walled shell and an outer thick-walled cylinder embedded in linear elastic soil or rock. Solutions for moment and thrust have been derived for cases involving slip and no slip at the lining–ground interface and lining–lining interface. A case involving a composite tunnel lining is studied to illustrate the usefulness of the solution.

On the transient elastic field for an expanding spherical inclusion in 3-D solid

The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.

On the uniformity of stresses inside an inhomogeneity of arbitrary shape

We consider an inhomogeneity of 'arbitrary' shape embedded within an infinite isotropic elastic medium (matrix) subjected to antiplane shear deformations under the assumption of uniform remote loading. The inhomogeneity-matrix interface is assumed to be imperfect, characterized by a single interface function. Under these assumptions, we present a novel method leading to the solution of the problem concerned with identifying the shape of the inhomogeneity and the form of the corresponding interface function which leads to a uniform interior stress field. The analysis is based on complex variable methods. Specific solutions are derived in closed form and verified by comparison with existing solutions. As a consequence of our analysis, we present an interesting result on the uniqueness (within a certain class of smooth curves) of the circular inhomogeneity as the only inhomogeneity which, under the given conditions, leads to a uniform interior stress field when the interface function is constant (homogeneously imperfect interface).

Interaction between coated inclusions and cracks in an infinite isotropic elastic medium

A domain integral equation method is presented for the analysis of the interaction between coated inclusions of arbitrarily shapes and cracks. Quadratic triangular and quadrilateral elements are employed to discretize the coated inclusion domains. Boundary integral equation method is used to investigate the interaction of coated inclusions and internal cracks. The interface between the coating and the matrix and that of the inclusion and the coating are meshed into quadratic boundary elements. For both methods, the cracks are discretized into quadratic discontinuous boundary elements. Special crack tip elements are employed to model the r variation of displacements near the crack tips. Stress intensity factors are evaluated using the well-known one point formulation.

Scattering From an Elliptic Crack by an Integral Equation Method: Normal Loading

The scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack-opening displacement. This integral equation has been further reduced to an infinite system of Fredholm integral equation of the second kind and the Fourier displacement potentials are expanded in terms of Jacobi’s orthogonal polynomials. Finally, proper use of orthogonality property of Jacobi’s polynomials produces an infinite system of algebraic equations connecting the expansion coefficients to the prescribed dynamic loading. The matrix elements contains singular integrals which are reduced to regular integrals through contour integration. The first term of the first equation of the system yields the low-frequency asymptotic expression for scattering cross section analytically which agrees completely with previous results. In the intermediate and high-frequency scattering regime the system has been truncated properly and solved numerically. Results of quantities of physical interest, such as the dynamic stress intensity factor, crack-opening displacement scattering cross section, and back-scattered displacement amplitude have been given and compared with earlier results.

On an Elastic Circular Inhomogeneity With Imperfect Interface in Antiplane Shear

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.

Transient Response of an Infinite Elastic Medium Containing a Spherical Cavity Subjected to Torsion

An exact closed-form solution is obtained for the transient response of an infinite isotropic elastic medium containing a spherical cavity subjected to torsional surface loading using the residual variable method. The main advantage of the present approach is that it eliminates the computational problems arising in the existing methods which are primarily based on Fourier or Laplace transformation techniques. Extensive numerical results for the circumferential displacements and shear stresses at various locations are presented graphically for Heaviside loadings. [S0021-8936(00)01102-8]