Infinite Isotropic Elastic Medium Research Articles

Fast-sum method for the elastic field of three-dimensional dislocation ensembles

The elastic field of complex shape ensembles of dislocation loops is developed as an essential ingredient in the dislocation dynamics method for computer simulation of mesoscopic plastic deformation. Dislocation ensembles are sorted into individual loops, which are then divided into segments represented as parametrized space curves. Numerical solutions are presented as fast numerical sums for relevant elastic field variables (i.e., displacement, strain, stress, force, self-energy, and interaction energy). Gaussian numerical quadratures are utilized to solve for field equations of linear elasticity in an infinite isotropic elastic medium. The accuracy of the method is verified by comparison of numerical results to analytical solutions for typical prismatic and slip dislocation loops. The method is shown to be highly accurate, computationally efficient, and numerically convergent as the number of segments and quadrature points are increased on each loop. Several examples of method applications to calculations of the elastic field of simple and complex loop geometries are given in infinite crystals. The effect of crystal surfaces on the redistribution of the elastic field is demonstrated by superposition of a finite-element image force field on the computed results.

Interaction between coplanar elliptic cracks—II shear loading

Consider two equal, coplanar elliptical cracks embedded in an infinite isotropic homogeneous elastic medium with major axes collinear and occupying the regions (see Fig. 1 )

Computer-aided quantifier elimination in crack problems under constraints for the stress intensity factors

It is suggested that crack problems under constraints, concerning the values of the parameters involved, be studied by the method of quantifier elimination in such a way that quantifier-free formulae can be derived assuring the validity of a restriction about the stress intensity factors (usually that such a factor does not exceed a critical value) for all possible values of the parameters under the applicable constraints. This permits safe conclusions about the possible fracture of a cracked specimen (under constraints on the parameters involved) for every set of values of the geometry/loading/fracture parameters. The elementary crack problem of a single straight crack in an infinite plane isotropic elastic medium is used as the vehicle for the illustration of the present approach and the related quantifier-free formula is easily derived. A simple case of a periodic array of cracks is also considered in brief. The classical algebraic tools of resultants and Sturm's sequences (together with Sturm's theorem) as well as the computer algebra system Maple V have been used in the present computations. More general results, with the help of Collins' famous cylindrical algebraic decomposition method, in more complicated crack problems, can also be obtained.

Interaction of a penny-shaped crack with an elliptic crack

Three-dimensional problem of crack-microcrack interaction is solved. Both the crack and microcrack are embedded in an infinite isotropic elastic medium which is subjected to constant normal tension at infinity. One of the cracks is circular while the other is elliptic and they are coplanar and are positioned in such a way that the axis of the elliptic crack passes through the centre of the circular crack. A recently developed integral equation method has been used to solve the corresponding two dimensional simultaneous dual integral equation involving the displacement discontinuity across the crack faces that arises in such an interaction problem. A series of transformations first reduce them to a quadruplet infinite system of equations. A series solution is finally obtained in terms of crack separation parameter β which depends on the separation of the crack microcrack centre. Analytical expression for the stress intensity factors have been obtained up to the order β6. Numerical values of the interaction effect have been computed for and results show that interaction effects fluctuate from shielding to amplification depending on the location of each crack with respect to the other and crack tip spacing as well as the aspect ratio of the elliptic crack. The short range interaction can play a dominant role in the prediction of crack microcrack propagation.

On non-uniqueness of the inverse problem for a seismic source-II. Treatment in terms of polynomial moments

SUMMARY In a companion paper (Pavlov 1994) we demonstrated and examined non-uniqueness of the inverse problem for a 3-D seismic source of indigenous type in terms of equivalent force fi(x, t). In this paper we study the non-uniqueness of the inverse problem in terms of polynomial moments (PM) with respect to spatial and temporal variables of both the rate moment tensor mij(x, t) and the rate equivalent force fi(x, t). The study includes two steps. In the first step we obtain the representation for general solution of the inverse problem for the PM of fi(x, t) of total degree n. To do this we use the linear equations for the PM of fi(x, t) derived here from the representation theorem for displacements in an infinite homogeneous isotropic elastic medium. Because of linearity of the problem its general solution is represented as the sum of two members: (1) a particular solution of the inhomogeneous problem and (2) the general solution of the homogeneous one. We obtain the explicit expression for the second member of the sum. It is equal to zero for n= 1, 2 and contains n(n2 - 1)/2 - 6 free parameters for n×3. In the second step we obtain the general representation for the solution of the inverse problem in terms of the PM of mij(x, t) of total degree n. The solution is unique for n= 0, 1 and contains (n+ 2)(n2+n - 3) free parameters for n× 2. We also consider the inverse problem for a static source. Non-uniqueness of the problem is characterized by the set of null sources (i.e. a static source that does not produce displacements outside the source region). Representations are obtained for the PM of a null source. The PM of the final equivalent force fsti(x) of a null source of degree n are equal to zero for n= 0, 1 and are expressed through 3n(n+ 1)/2 arbitrary parameters for n× 2. The PM of degree n of the final-moment density tensor mstij(x) of a null source is equal to zero for n= 0 and is expressed through n2+n - 1 arbitrary parameters for n× 1.

On non-uniqueness of the inverse problem for a seismic source-I. Treatment in terms of equivalent force

SUMMARY The inverse problem of the earthquake-source theory is considered. It is shown that for a seismic source of general type distributed over a 3-D spatial region in infinite homogeneous isotropic elastic medium the problem has no unique solution not only in terms of moment density tensor (Backus & Mulcahy 1976a,b), but also in terms of equivalent force. This fact is related to the existence of non-radiating seismic sources. (A non-radiating seismic source is the one that has non-zero equivalent force but produces no motion outside some ball containing the source region: the acoustic case was considered by Friedlander 1973). We show that in the case when the input data are specified to be the far-field displacements, the only solutions of the homogeneous inverse problem are non-radiating sources. This means that far-field displacements uniquely determine the near-field ones. Therefore, the adding of near-field data cannot improve the statement of the problem to avoid non-uniqueness. To constrain the class of admissible sources we consider 2-D indigenous seismic sources that have a moment density tensor distributed over a known smooth surface. We show that for an unclosed surface the inverse problem has a unique solution in terms of equivalent force. A counter example for the unit sphere shows that if a surface is closed the inverse problem may have no uniqueness.

Disclinations and Their Applications to Stress Analysis : Application of Wedge Disclination Dipole Field to Two-Dimensional Elastic Problem

This paper deals with application of the stress and displacement field of a wedge disclination dipole in an infinite isotropic elastic medium to analysis of a two-dimensional elastic problem by means of the indirect boundary element method. The disclination dipole has a singularity at its center, and hence we introduce the fictitious boundary method in order to avoid the reduction in the accuracy of the analysis which is caused by the singularity. Consequently, it is found that we can obtain sufficient accuracy if we choose a value 3 or greater as the coefficient of the distance of the fictitious boundary, and it is ascertained that the fields of the wedge disclination dipole can be applied to the analysis of the two-dimensional elastic problem.

Properties of type 6 transonic states with respect to grazing incidence reflection of bulk waves at planar free or clamped surfaces of half infinite elastically anisotropic media

Recognizing different characteristics of reflection of P, SV and SH waves at the planar surface of a half infinite isotropic elastic medium, the properties of Type 6 transonic states with respect to grazing incidence reflection of bulk waves are investigated by generalization of the case of grazing incidence reflection at Type 1 transonic states. It is shown that for grazing incidence reflection related to Type 6 transonic states, which are always composite exceptional, the reflection can have two different features, skimming reflection or repellent reflection. For a free surface, the reflection is skimming or repellant according to whether the incident wave is associated with an exceptional or composite exceptional limiting wave or a non-exceptional wave, respectively. It is also proven that the self-orthogonality of limiting waves and the completeness of the Stroh eigenvector system for Type 6 transonic states reveal a series of linear dependence/independence relations between the Stroh eigenvector components. These relations are significant for establishing a theoretical framework for the description of two component surface waves and spaces of simple reflection.

Numerical calculation of equilibrium critical thickness in strained-layer epitaxy

We calculate by numerical integration the energy of a strained layer in an infinite isotropic continuous elastic medium with and without a misfit dislocation dipole of pure edge character. The equilibrium critical thickness is the thickness at which these energies are the same. Solving for a range of values of misfit strain, we compare the results with some versions of the existing approximate models, and we are thereby able to indicate the best analytic expressions to use to predict equilibrium critical thickness.

Non-symmetric extension of a plane crack due to plane SH-waves in a prestressed infinite elastic medium

In an infinite isotropic elastic medium initially in a state of uniform anti-plane shear, the problem of non-symmetric extension of an infinitesimal flaw into a plane shear crack due to two identical linearly varying plane SH-waves with non-parallel wave fronts has been analyzed. Fracture is assumed to initiate at a point a finite time after the waves intersect there and the crack is assumed to extend non-symmetrically along the trace of the wave intersection. Following Cherepanov [10], Cherepanov and Afanas'ev [11] the general solution of the problem has been derived in terms of analytic function of complex variable. Numerical results have been presented to illustrate the nature of variation of the stress intensity factors and the rate of energy flux into the crack edges with the speed of the crack tips and also with the time after fracture initiation.

Three co-planar moving Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to three co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement around three co-planar Griffith cracks in an infinite isotropic elastic medium has also been considered. In both the cases, employing Fourier integral transform, the problems have been reduced to solving a set of four integral equations. These integral equations have been solved using finite Hilbert transform technique and Cook's result [16] to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.

Relating P‐wave attenuation to permeability

To relate P‐wave attenuation to permeability, we examine a three‐dimensional (3-D) theoretical model of a cylindrical pore filled with viscous fluid and embedded in an infinite isotropic elastic medium. We calculate both attenuation and permeability as functions of the direction of wave propagation. Attenuation estimates are based on the squirt flow mechanism; permeability is calculated using the Kozeny‐Carman relation. We find that in the case when a plane P‐wave propagates perpendicular to the pore orientation [Formula: see text], attenuation is always higher than when a wave propagates parallel to this orientation [Formula: see text]. The ratio of these two attenuation values [Formula: see text] increases with an increasing pore radius and decreasing frequency and saturation. By changing permeability, varying the radius of the pore, we find that the permeability‐attenuation relation is characterized by a peak that shifts toward lower permeabilities as frequency decreases. Therefore, the attenuation of a low‐frequency wave decreases with increasing permeability. We observe a similar trend on relations between attenuation and permeability experimentally obtained on sandstone samples.

Four co-planar Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to four co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement in an infinite isotropic elastic medium has also been considered. In both cases, employing the Fourier integral transform, the problems have been reduced to solving a set of five integral equations. These integral equations have been solved using the finite Hilbert transform technique to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.

An exact solution in gauge theory of strait disclinations in an isotropic elastic medium

In the framework of the gauge model of defects an exact vortex-like solution for an individual static strait wedge disclination with an arbitrary integer Frank index in an infinite isotropic elastic medium is obtained. The solution is topologically stable, continuous and yields elastic fields subject to standard boundary conditions.

Diffraction of elastic P waves by four equal coaxial circular rigid strips

The two-dimensional boundary value problem of diffraction of obliquely incident low-frequency elastic P waves by four equal coaxial circular rigid strips embedded in an infinite isotropic and homogeneous elastic medium is solved. Approximate expressions are derived for the far-field amplitudes and scattering cross section when the wavelengths are large compared to the radius of the circular strips.

Determination of dynamic stress intensity factors of multiple cracks

The dynamic stress field around a crack or cracks embedded in an infinite isotropic elastic medium subjected to a SH wave are determined. Based on the qualitatively similar features of crack and dislocation, the stress wave emitted from a vibrating screw dislocation can be regarded as a Green's function. By superimposing an array of dislocation and adjusting the distribution density to fulfill the boundary condition, a singular integral equation with kernels containing Bessel functions is derived, which can be solved by the Galerkin method. Dynamic stress intensity factors, which can be as much as 28 percent higher than the static value, are found to be the same as those results obtained by other investigators. The stress intensity factors of a set of infinite cracks of equal length are also calculated as an application of this method.

Determination of dynamic stress intensity factors of two finite cracks at arbitrary positions by dislocation model

The effects of antiplane shear waves on the stress intensity factors of two cracks at arbitrary positions in an infinite isotropic elastic medium are analysed. Based on the concept that the cracks can be simulated by dislocation models, the distribution densities of the dislocations as well as the stress field due to such dislocations are expressed by a system of singular integral equations with kernels containing Bessel functions which can be solved by Galerkin method. The results show that the dynamic stress intensity factors fluctuate with the distance between the two cracks and finally converge to the values of a single crack as the distance between the two cracks becomes very large.

Diffraction of P waves by two cracks at arbitrary position in an elastic medium

By means of dislocation model, the effects of P waves and SV waves on the stress intensity factors of two cracks at arbitrary positions in an infinite isotropic elastic medium are analysed. Each crack can be simulated by two arrays of edge dislocations: one vibrates on its glide plane and the other along its climbing direction. The dislocation distribution densities as well as the phase lags of the dislocations can be obtained by using the Galerkin method, by which a system of singular equations with kernels containing Bessel functions can be solved. The results show that both the dynamic stress intensity factors in mode one and mode two fluctuate with the distance between the two cracks and finally converge to the values of a single crack when the distance between the two cracks becomes very large.

The low-frequency scattering theory for a penetrable scatterer with an impenetrable core in an elastic medium

We consider an incident time harmonic longitudinal or transverse plane wave which propagates in an infinite isotropic and homogeneous elastic medium and it is scattered by a penetrable body containing an impenetrable core. Integral representations for the total displacement field as well as the normalized scattering amplitudes are provided. A systematic procedure for finding the low-frequency expansions for the displacement field, the scattering amplitudes and the scattering cross section is developed. As a result the scattering problem is reduced to a sequence of elastostatic problems which can be solved using the Papkovich-Grodski-Neuber potentials. The leading low-frequency terms of the scattering amplitudes and the cross section are evaluated. The proposed procedure is applied to the scattering of a plane elastic wave by a penetrable spherical scatterer with a concentric spherical rigid inclusion. Particular values of the physical parameters correspond to special scattering problems and they are included as such.

Locating a straight crack in an infinite elastic medium by using complex path-independent integrals

The Cauchy integral theorem and the relevant formula (or, equivalently, complex path-independent integrals) have been used in a long series of papers for the determination of zeros and poles of analytic and meromorphic functions. Here this approach is generalized to become applicable to the problem of location of a straight crack inside an infinite plane isotropic elastic medium. The complex path-independent integrals used here contain the first complex potential Φ(z) of Kolosov-Muskhelishvili, which can be obtained experimentally. The present method can be modified to apply to a variety of problems where discontinuity intervals of analytic (or, rather, sectionally analytic) functions are sought.