The elastostatic axisymmetric problem of a sphere containing a penny-shaped crack in a nonequatorial plane

The axisymmetric problem of a sphere containing a penny-shaped crack in a nonequitorial plane is solved with the use of Bousinesq stress functions. Two coordinate systems—oblate spheroidal for representing the crack surface and spherical polars for the spherical surface, translated along the z-axis with respect to each other—are used to satisfy boundary conditions. Integral representations and transformations of harmonic functions are used to relate stress functions in the two coordinate systems. This procedure-leads to a system of algebraic equations which is solved, for axisymmetric tractions on both the surfaces. Graphical results are presented for a specific loading case.     

Stress concentration around a strongly oblate cavity in a cubic crystal and its associated crack problems

The stress distribution around a strongly oblate spheroidal cavity in a cubic crystal is determined by the equivalent inclusion method. The stress concentration factor is shown to be a product of two factors: one factor is purely geometric; the other factor depends on the material properties. By allowing the aspect ratio of the cavity to approach zero, the stress intensity factor of the associated penny-shaped crack is deduced. The energy release rates of cracks on {1 0 0} planes are computed for different growth directions. Theses results are found to be correlated well with Zener’s anisotropy factor.     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

Macroscopic yield criteria for ductile materials containing spheroidal voids: An Eshelby-like velocity fields approach

Making use of limit analysis theory, we derive a new expression of the macroscopic yield function for a rigid ideal-plastic von Mises matrix containing spheroidal cavities (oblate or prolate). Key in the development of the new criterion is the consideration of Eshelby-like velocity fields which are built by taking advantage of the solution of the equivalent inclusion problem in which the eigenstrains rate are unknown for the plasticity problem. These heterogeneous trial velocity fields contain non-axisymmetric components which prove to be original in the context of limit analysis of hollow spheroid. After carefully computing the macroscopic plastic dissipation and implementing a minimization procedure required by the use of the Eshelby-like velocity fields, we derive, for the porous medium, a two-field estimate of the anisotropic yield criterion whose closed-form expression is provided. This estimate is compared to existing criteria based on limit analysis theory. Interestingly, in contrast to these criteria, the new results predict a significant effect of shear loadings in the particular case of ductile materials weakened by penny-shaped cracks.     

Fracture analysis of a piezoelectric layer with a penny-shaped and energetically consistent crack

The problem of an eccentric penny-shaped crack embedded in a piezoelectric layer is addressed by using the energetically consistent boundary conditions. The Hankel transform technique is applied to solve the boundary-value problem. Then two coupling Fredholm integral equations are derived and solved by using the composite Simpson’s rule. The intensity factors of stress, electric displacement, crack opening displacement and electric potential together with the energy release rate are further given. The effects of the thickness of a piezoelectric layer and the discharge field inside the penny-shaped crack on the fracture parameters of concern are discussed through numerical computations. The observations reveal that an increase of the discharge field decreases the stress intensity factor and the energy release rate. An eccentric penny-shaped crack is easier to propagate than a mid-plane one in a piezoelectric layer, and the geometry of the crack along with the layer thickness have significant influences on the electrostatic traction acting on the crack faces. The solutions for a penny-shaped dielectric crack in an infinite or a semi-infinite piezoelectric material can be obtained easily.     

Asymptotic stresses around a crack tip at the interface between planar or cylindrical bodies

Closed-form equations are derived for the asymptotic stresses in the neighborhood of a crack tip impinging on an interface between two isotropic materials. The symmetric problem is considered and follows from an exact elasticity solution formulated by Gupta [1]. The equations are valid for the planar problem, where the interface is straight and also for an axisymmetric problem in the presence of an annular or penny-shaped crack. The equations may serve to establish a tentative criterion that defines the subsequent direction of a crack impinging on a bimaterial interface. The ambiguity of the asymptotic stress state is highlighted and plausible application of the results is discussed.The U.S. Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Dugdale plastic zone of a penny-shaped crack in a piezoelectric material under axisymmetric loading

The Dugdale plastic zone ahead of a penny-shaped crack in a piezoelectric material, subjected to electric and axisymmetric mechanical loadings, is evaluated analytically. Hankel transform is employed to reduce the mixed boundary-value problem of the penny-shaped crack to dual integral equations, which are solved exactly under the assumption of electrically permeable crack face conditions. A closed-form solution to the mixed boundary-value problem is obtained to predict the relationship between the length of the plastic zone and the applied loading. The stress distribution in and outside of the yield zone has been derived analytically, and the crack opening displacement has been investigated. The electric displacement has a constant value in the strip yield zone. The current Dugdale crack model leads to non-singular stress and electric fields near the crack front, and it is observed that the material properties affect the crack opening displacement.     

Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system

A penny-shaped crack at the interface of a piezoelectric bi-material system is considered. Analytical general solutions based on Hankel integral transforms are used to formulate the mixed-boundary value problem corresponding to an interfacial crack and the problem is reduced to a system of singular integral equations. The integral equations are further reduced to two systems of algebraic equations with the aid of Jacobi polynomials and Chebyshev polynomials. Thereafter, the exact expressions for the stress intensity factors and the electric displacement intensity factor at the tip of a crack are obtained. Selected numerical results are presented for various bi-material systems to portray the significant features of crack tip fracture parameters and their dependence on material properties, poling orientation and electric loading.     

Boundary integral equations in elastodynamics of interface cracks

The paper concerns the validation of a method for solving elastodynamics problems for cracked solids. The proposed method is based on the application of boundary integral equations. The problem of an interface penny-shaped crack between two dissimilar elastic half-spaces under harmonic loading is considered as an example.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

On Boussinesq’s problem for a cracked halfspace

This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.     

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half-spaces

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half spaces is studied. The problem is first reduced to a system of dual integral equations which are further reduced to the solution of Riemann-Hilbert problem. Further stresses at the rims of cracks and in the vicinity have been evaluated.     

Thermal fracture of bonded dissimilar media containing a penny-shaped crack

The problem of uniform heat flow disturbed by an insulated penny-shaped crack along the common plane between two semi-infinite elastic media with different thermo-mechanical properties is formulated in terms of two potential functions in a half-space which in turn is reduced to a plane problem solvable by Muskhelishvili's method in complex function theory. Explicit expressions for the stress-intensity factors and the local stress field are derived and used in conjunction with Griffith's energy criterion to obtain the critical temperature gradient which motivates and produces initial crack extension along the bonding surface. The stress analysis involved is also applicable to a penny-shaped crack between two dissimilar solids under shear loadings.

     

Dynamic stress intensity factors studied by boundary integro-differential equations

The boundary integro-differential equation method is illustrated by two numerical examples concerning the study of the dynamic stress intensity factor around a penny-shaped crack in an infinite elastic body. Harmonic and impact load on the crack surface has been considered. Applying the Laplace transform with respect to time to the governing equations of motion the problem is solved in the transformed domain by the boundary integro-differential equations. The Laplace transformed general transient problem can be used to solve the steady-state problem as a special case where no numerical inversion is involved.     

多层介质硬币形(Penny-Shaped)交界裂纹的弹性波散射

本文研究多层介质硬币形交界裂纹的弹性波散射.文中采用Hankel积分变换,得到了含有硬币形交界裂纹多层介质模型的散射波传递矩阵,并将散射问题为转化求解矩阵形式的对偶积分方程.作为特例,文中给出了单一弹性层与半空间的硬币形交界裂纹的弹性波散射远场模式,并计算了几组不同弹性常数组合情形下的远场模式的幅频特性曲线,其结果表明有共振峰存在.     

Slow torsional oscillations: The reissner-sagoci and crack problems at low frequencies

The axisymmetric elastic field produced by slowly forced torsional oscillations of a finite circular fiaw in a certain inhomogeneous medium is sought. The problem is reduced to a system of integral equations, which system is shown to cover intrinsically, the two separate cases of the flaw in the form of a penny-shaped crack and the flaw in the form of a rigid disc. The solutions are given in series of the frequency factor. Estimates of the radius of convergence are given as functions of the inhomogeneity parameter. For the flaw in the form of a rigid disc, the solution of the integral equations gives the normal displacement gradient just above the disc, from which simple integration gives the moment of the applied forces necessary to oscillate the disc. In the case of the flaw in the form of a penny-shaped crack, the solution gives the normal displacement over the crack region. This is then used to obtain the surface shear stress just outside the crack rim, from which is obtained the stress intensity factor. These physical results are all given as functions of the frequency factor and inhomogeneity parameter.     

Elastodynamics of a crack on the bimaterial interface

The paper is an application of boundary integral equations to the problem of a crack located on the bimaterial interface under time-harmonic loading. A system of linear algebraic equations is derived for solving the problem numerically. The distributions of the displacements and tractions at the bimaterial interface are obtained and analysed for the case of a penny-shaped crack under normal tension-compression wave. The dynamic stress intensity factors (normal and shear modes) are also computed. The results are compared with those obtained for the static case.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

Scattering of a spheroidal particle illuminated by a gaussian beam

An approach to expanding a Gaussian beam in terms of the spheroidal wave functions in spheroidal coordinates is presented. The beam-shape coefficients of the Gaussian beam in spheroidal coordinates can be computed conveniently by use of the known expression for beam-shape coefficients, g(n), in spherical coordinates. The unknown expansion coefficients of scattered and internal electromagnetic fields are determined by a system of equations derived from the boundary conditions for continuity of the tangential components of the electric and magnetic vectors across the surface of the spheroid. A solution to the problem of scattering of a Gaussian beam by a homogeneous prolate (or oblate) spheroidal particle is obtained. The numerical values of the expansion coefficients and the scattered intensity distribution for incidence of an on-axis Gaussian beam are given.