Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.     

Elastic wave scattering by a three-dimensional inhomogeneity in an elastic half space

In this paper we will consider scattering of elastic waves in a half space. The half space is an isotropic, linear and homogeneous medium except for a finite inhomogeneity. The T-matrix method (also called the “extended boundary condition method” or “null field approach”) is extended to derive expressions for the elastic field inside the half space and the surface field on the interface. The assumptions on the source that excites the half space are fairly weak. In the numerical applications found in this paper we assume a Rayleigh surface wave to be the incoming field, and we only compute the surface displacements. We make illustrations on some simple types of scatterers (spheres and spheroids; the latter ones can be arbitrarily oriented).     

Magnetoelectroelastic composite with poling parallel to plane of line crack under out-of-plane deformation

The three-dimensional field equations can in general be regarded as the sum of in-plane and out-of-plane deformation. The method for the general solution is the same for both although the boundary conditions could make a difference. If a particular solution in exact form may be found for the out-of-plane case, the same may not hold for the in-plane case. Hence, there may be a good reason for discussing the out-of-plane crack problem in certain situations that should be emphasized. Otherwise, the reason may lie in the exploration of possible application to the in-plane problem, a direct solution of which would have required a considerable effort. The contribution of this work rests on the new findings for the case of poling parallel to the crack in a magnetoelectroelastic composite made of BaTiO₃–CoFe₂O₄. The inclusions are BaTiO₃ and the matrix is CoFe₂O₄. Several new features of the solution were not expected before hand.Unlike in-plane deformation with poling normal to the crack plane, maximum crack growth enhancement is found to occur in the BaTiO₃–CoFe₂O₄ composite for a volume fraction of about 50%. Crack retardation increases as the volume fraction of the inclusions either increase or decrease. The occurrence of this same phenomenon in Mode I and II remain to be investigated. Poling direction of magnetic and electric field for line defects can have a significant effect on crack growth for magnetoelectroelastic materials. The foregoing conclusions are based on predictions made from the strain energy density criterion.     

Scattering of a plane longitudinal wave by an elastic quarter space

We study the two-dimensional problem of the scattering of a plane longitudinal wave incident in a homogeneous, isotropic, linearly elastic quarter space. The complex-valued amplitudes of the Rayleigh waves propagating on the free surfaces are plotted versus Poisson's ratio. Also plotted are the farfield scattering patterns for Poisson's ratio $\text{μ=}\text{1}\text{4}$ and $\text{1}\text{3}$.     

Contact problem for a plane crack under a normally incident antiplane shear wave

The contact-interaction problem for a stationary plane crack with friction between its edges under the action of a normal (to the crack plane) harmonic shear wave is addressed. Antiplane deformation conditions are considered. The distribution of contact forces and displacement discontinuity of crack edges are studied Published in Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 138–142, May 2007.     

Penny-shaped crack at the interface between elastic half-spaces under the action of a shear wave

The three-dimensional problem of elasticity for a bimaterial body with a penny-shaped crack at the interface under the action of a normal harmonic shear wave is solved by the boundary-element method. The distribution of displacements of crack faces and tractions and displacements at the interface is analyzed     

Diffraction of an elastic wave in a disc

A numerical solution is constructed for the axisymmetric problem of the diffraction of a plane longitudinal wave in a rigid disc (cylinder) of finite thickness. The disc is enclosed in an unbounded elastic medium; at the contact surface, the tangential stresses are limited by some constant. The incident wave moves along the axis of the cylinder and has the form of a semiinfinite washed-out step. At the same time, a solution is obtained to the corresponding static problem. A study was made of the dependence of the rate of motion of the cylinder and the stress field on the parameters of the problem. In particular, it is shown that the contact conditions have a considerable effect on the stress field only near the lateral surface. The results obtained can be useful for evaluating the errors in measurement of the stresses and velocities in an elastic medium, and possibly also in certain other cases.Deceased.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 3, pp. 139–150, May–June, 1972.     

Diffraction of an evanscent plane wave by a half plane

A proper analytic continuation of Sommerfeld's solution is shown to provide the solution to the problem of diffraction of an evanescent plane wave. This is done by a correct extension of a parameter (detour parameter) from real to complex values. Some peculiarities of this solution are discussed. A few representative three-dimensional graphs show the field magnitude in the vicinity of the edge.     

Stress intensity factor in a contact problem for a plane crack under an antiplane shear wave

The frictional contact interaction of the finite edges of a plane crack under the action of a normally incident harmonic shear wave that produces antiplane deformation is studied. The influence of the forces of contact interaction on the stress intensity factor is analyzed Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 115–119, September 2007.     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.     

Torsion of an elastic space with a semi-infinite conical crack

The axisymmetric problem of stress concentration near a conical crack in an infinite elastic space with a rotation center is addressed. The problem is reduced to an integro-differential equation. Its exact solution is obtained. An expression for the stress intensity factor in crack neighborhood is derived and numerically analyzed for different positions of the rotation center and the crack opening angles __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 99–107, February 2007. For the centenary of the birth of G. N. Savin.     

Problem on an inhomogeneity in a linearly elastic space or half-space

For a weakly contrasting anisotropic inhomogeneity in a linearly elastic homogeneous space or half-space, using the perturbation method, we obtain an approximate solution and estimate its accuracy. In the case of inhomogeneity of arbitrary contrast, we reduce the problem to a system of integral equations. In the general case, it is easy to compose the procedure for solving this problem approximately. In the special case of a homogeneous anisotropic ellipsoidal inhomogeneity in space, the strain state inside the inhomogeneity turns out to be homogeneous, and we thus obtain the exact solution of the problem.     

The penny-shaped crack at a bonded plane with localized elastic non-homogeneity

This paper examines the axisymmetric problem pertaining to a penny-shaped crack which is located at the bonded plane of two similar elastic halfspace regions which exhibit localized axial variations in the linear elastic shear modulus, which has the form G(z)=G₁+G₂e^±ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The resulting mixed boundary value problem associated with the penny-shaped crack is reduced to a Fredholm integral equation of the second kind which is solved in a numerical fashion to generate the crack opening mode stress intensity factor at the tip.     

Diffraction of an antiplane shear wave by two coplanar griffith cracks in an infinite elastic medium

The scattering of a time-harmonic antiplane shear wave by two parallel and coplanar Griffith cracks embedded in an infinite elastic medium is considered. The input wave normally impinges on the cracks. Fourier transformations are utilized to reduce the problem to two simultaneous integral equations which can be solved by the series expansion method. The dynamic stress intensity factors are numerically computed.     

Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal

The present paper is devoted to determining the crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal. A pair of equal but opposite line loadings is assumed to be exerted on the upper and lower crack lips. By applying the Dugdale hypothesis together with the elastic results for a half-infinite crack, the extent of the plastic zone in the crack front is estimated. The normal stress outside the enlarged crack and crack surface displacements are explicitly presented, via the principle of superposition. The validity of the present solutions is discussed analytically by examining the overall equilibrium of the half-space.     

Transient contact shear stress in a layered elastic quarter space subjected to anti-plane shear loads

In this paper the transient behaviour of a contact shear stress in a layered elastic quarter space subjected to anti-plane shear loads is investigated. The loads are suddenly applied to upper and side edges of the layer. The effects of the reflected waves, the loaded position and the material properties to the contact shear stress are shown graphically.     

Scattering of plane, elastic waves by a plane crack of finite width

The diffraction of time-harmonic, vertically polarized, plane elastic waves by a crack of finite width is investigated with the aid of the integral-equation method. Using the integral representation for the particle displacement of the scattered field together with the constitutive equation, it is shown that the resulting integral equations uncouple for this kind of obstacle. In them, the amount by which the components of particle displacement jump across the crack occur as unknown quantities. The integral equations are solved numerically. Normalized power scattering characteristics and scattering cross-sections are computed.The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).