Dislocation layers and crack problems—I

In the present paper plane strain and anti-plane strain problems for an elastic half-space have been formulated in terms of dislocation layers in a novel manner. It is then shown how these formulations can be exploited in solving crack problems in the classical theory of elasticity.     

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.     

Small edge crack in a semi-infinite solid subjected to thermal stratification

The mode I stress intensity factor for a small edge crack in an elastic half-space is found when the space is in contact with two stratified fluids of different temperatures, the boundary between the fluids oscillating sinusoidally over the solid surface. The variation in the stress intensity factor, which may lead to thermal fatigue crack growth, is examined as a function of time, crack depth, amplitude and temporal frequency of oscillation, surface heat transfer coefficient and material properties of the half-space. It is shown how this ‘boundary layer’ solution may be applied to problems involving finite geometries.     

周期界面裂纹的弹性波散射问题研究

本文研究了分布于两个关元限空间的周期界面对垂直入射Ｐ波及ＳＨ波的散射问题，文中利用有限Ｆｏｕｒｉｅｒ变换将一个周期带内散射场的边值问题转化为求解一个带周期核的奇异积分方程，并对ＳＨ波入射的情形进行了详细的分析，求解了相应的异积分方程，最后给出裂纹尖端的应力强度因子的计算公式及远离裂纹时散射位移场的渐进形式，并对散场的动态特性进行了数值分析。     

Study of half-space elastic contact problems by caustics

The optical method of caustics has been successfully applied to several two dimensional problems of elasticity. Up to now, no complicated three dimensional problems of elasticity have ever been treated by this method. In this paper, the experimental technique of caustics is developed, the caustics are obtained by annealing the stress-frozen epoxy slices. In applying this technique to Boussinesq's problem of a normal force and Cerruti's problem of a tangential force on the plane surface of a half-space, the experimentally obtained caustics for these problems are seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of the rather complicated three dimensional elasticity problems, including crack problems, by the author's method is also possible.     

Scattering of antiplane shear waves by a submerged crack

The scattering of time harmonic antiplane shear waves by a crack situated parallel to the surface of a homogenious elastic half-space is considered. The problem is investigated for both clamped and stress-free conditions at the surface of the half-space. Singular integral equations are derived for each case and treated numerically by the Gauss-Chebyshev technique. Harmonic stress intensity factors and crack opening displacements are computed as functions of dimensionless frequency, submerged depth-to-crack width ratio, and angle of incidence. Results are obtained for a moderately wide range of frequencies and are set out in graphical form.     

Scaling behavior of thermal shock crack patterns and tunneling cracks driven by cooling or drying

Cracks driven by shrinkage due to cooling or drying arrange themselves via mutual interaction. For parallel straight crack arrays driven by idealized transient shrinkage fields the scaling behavior in an infinite half-space is derived analytically by means of fracture mechanics bifurcation analysis with two plausible scaling assumptions. Crack spacing in thermal shock crack patterns has been found to be approximately proportional to the crack length and inversely proportional to the crack velocity. The spacing of tunneling cracks formed in a drying layer between plates scales as the 2/3rd power of layer thickness as a consequence of the specific interaction between the tunneling cracks. The difference in scaling behavior in the two cases is explained by the dimensionality of the geometrical setup determined by the boundary condition rather than by different physical processes. In either case, good agreement between theory and experiments is found.     

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.     

Elliptical crack and punch problems solved by integral equation method for piezoelectric media

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric media with the crack-plane parallel to the plane of isotropy of the media and subjected to remote normal mechanical as well as electric loading is considered first. The problem has been successfully reduced to a pair of coupled integral equations that are suitable for the application of an integral equation method developed earlier for three-dimensional problems of LEFM. Solution to the mechanical displacement and electric potentials are obtained for prescribed uniform loadings and expressions for corresponding intensity factors and crack opening displacement are deduced. The above method has further been applied to solve the problem of a rigid flat-ended elliptical punch indenting a transversely isotropic piezoelectric half-space surface with the plane of isotropy parallel to the surface. Solutions to mechanical stress and electric displacement are obtained for prescribed constant normal displacement and constant electric potential interior to the elliptical region and expression for the total force required to maintain a prescribed indentation is deduced.     

Dynamic fracture behavior of a cracked piezoelectric half space under anti-plane mechanical and in-plane electric impact

Summary  The dynamic response of a cracked piezoelectric half-space under anti-plane mechanical and in-plane electric impacting loads is investigated in the present paper. In the study, the crack is assumed parallel to the free surface of the half-space. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in the Laplace transform domain, which are solved numerically. Then, a numerical Laplace inversion is performed and the dynamic stress and electric displacement factors are obtained as functions of time and geometry parameters. The dynamic energy release rate is derived for piezoelectric materials in terms of the electroelastic intensities and is displayed graphically. Received 5 January 2000; accepted for publication 28 June 2000     

Periodic array of traction-free interface cracks subjected to far-field uniform heat flow

A periodic array of interface cracks is subjected to a uniform heat flow in the far field. The crack opening displacements and complex stress intensity factors are determined, using analytic function theory, for the case that the upper half-space is less distortive than the lower half-space.     

Three-dimensional crack in the interior of a half-space

In this note, integral equations for the problem of an internal plane crack of arbitrary shape in a three-dimensional elastic half-space are derived. The crack plane is assumed to beparallel to the free surface. Use is made of Mindlin's point force solution in the interior of a semi-infinite solid in deriving the integral equations for the problem.     

Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in a unified system from which the field equations for coupled thermoelasticity as well as for generalized thermoelasticity can be easily obtained as particular cases. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating with different theories of thermoelasticity.     

The propagation and arrest of an edge crack in anti-plane shear

The motion of an edge crack extending non-uniformly in an elastic half-space under conditions of anti-plane shear is investigated, taking the first reflected stress wave into account. An expression for the stress intensity factor at the crack tip is obtained, and an energy-balance crack propagation criterion is used to find the equation of motion of the tip. On solving this equation numerically, it is found that crack arrest occurs before the second reflected wave reaches the tip.     

Green functions for a compressible linearly non homogeneous half-space

Summary The paper presents a study of time-harmonic vibration of a half-space possessing a shear modulus linearly increasing with depth. Completing the previous paper [1], where the time-harmonic vibration of an incompressible half-space has been considered, the problem is now solved for a compressible as well as an incompressible material. The half-space is subjected to a vertical or horizontal surface load. The solution is represented in terms of Fourier-Bessel integrals containing functions of depth coordinate that are expressed through confluent hypergeometric functions. Numerical results concerning surface displacements due to a point force are given for a wide range of frequency variations and degree of non-homogeneity. The results show that, as compared to the homogeneous case, non-homogeneity can considerably increase vibration amplitudes at large distances from the applied force. Received 19 August 1996; accepted for publication 16, December 1996     

Harmonic vibrations of a half-space with a surface-breaking crack under conditions of out-of-plane deformation

The paper presents a solution of the problem of determining the stress state in an elastic isotropic half-space with a crack intersecting its boundary under harmonic longitudinal shear vibrations. The vibrations are excited by a regular action of a harmonic shear load on the crack shores. The solution method is based on the use of the discontinuous solution of the Helmholtz equation, which allows one to reduce the original problem to a singular integro-differential equation for the unknown jump of the displacement on the crack surface. The solution of this equation is complicated by the existence of a fixed singularity of its kernel. Therefore, one of the main results is the development of an efficient approximate method for solving such equations, which takes into account the true asymptotics of the unknown function. The latter allows one to obtain a high-precision approximate formula for calculating the stress intensity factor.     

Love wave dispersion in pre-stressed homogeneous medium over a porous half-space with irregular boundary surfaces

The paper investigates the existence of Love wave propagation in an initially stressed homogeneous layer over a porous half-space with irregular boundary surfaces. The method of separation of variables has been adopted to get an analytical solution for the dispersion equation and thus dispersion equations have been obtained in several particular cases. Propagation of Love wave is influenced by initial stress parameters, corrugation parameter and porosity of half-space. Velocity of Love waves have been plotted in several figures to study the effect of various parameters and found that the velocity of wave decreases with increases of non-dimensional wave number. It has been observed that the phase velocity decreases with increase of initial stress parameters and porosity of half-space.     

Effect of loosely bonded undulated boundary surfaces of doubly layered half-space on the propagation of torsional wave

This paper investigates the propagation of torsional wave in an initially stressed poroelastic layer with corrugated as well as loosely bonded boundary surfaces, sandwiched between a corrugated fiber-reinforced layer and a viscoelastic half-space under initial stress. The velocity equation has been obtained in closed form analytically and the substantial effect of affecting parameters on the phase velocity of torsional surface wave has been demonstrated numerically and graphically. Comparative study has been made to observe the effect of flatness parameter, reinforcement, viscoelasticity and porosity on the phase velocity, meticulously. Some particular cases have also been discussed and it is found that velocity equation is in well-agreement to the classical Love wave equation. Moreover, some remarkable observation has been made through numerical computation and graphical demonstration for fiber-reinforced layer of carbon fiber-epoxy resin, poroelastic layer of sandstone and a viscoelastic half-space.     

INFLUENCE OF RIGID BOUNDARY ON THE LOVE WAVE PROPAGATION IN ELASTIC LAYER WITH VOID PORES

In this paper, a mathematical model for Love wave propagation in a porous elastic layer under a rigid boundary resting over a poro-elastic half-space has been developed. The study shows that such a medium transmits two types of Love waves. The first front depends on the change in volume fraction of the pores whereas the second front depends upon the modulus of rigidity of the elastic matrix of the medium and is the same as the Love wave in an elastic layer over an elastic half-space. It is observed that the first front is many times faster than the shear wave in the medium with void pores due to the change in the volume fraction of the pores and is significant.     

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.