The indentation of a precompressed penny-shaped crack

The paper examines the axisymmetric problem related to the indentation of the plane surface of a penny-shaped crack by a smooth rigid disc inclusion. The crack is also subjected to a far-field compressive stress field which induces closure over a part of the crack. The paper presents the Hankel integral transform development of the governing mixed boundary value problem and its reduction to a single Fredholm integral equation of the second kind and an appropriate consistency condition which considers the stress state at the boundary of the crack closure zone. A numerical solution of this integral equation is used to develop results for the axial stiffness of the inclusion and for the stress intensity factors at the tip of the penny-shaped crack.     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

Effect of elastic coating on fracture behaviour of piezoelectric fibre with a penny-shaped crack

In this paper, the problem of a penny-shaped crack in a piezoelectric fibre with an elastic coating is investigated. By using the potential function method and Hankel transform, this problem is formulated as the solution of a system of dual integral equations which are reduced to a Fredholm integral equation of the second kind. Numerical studies are conducted to investigate the effect of the thickness and the elastic material properties of the coating on the fracture behavior of piezoelectric fibre composites.     

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.     

Solution of nonstationary problems for composite bodies with cracks by the method of integral equations

We study the problem of nonstationary loading of a plane crack in a bimaterial body formed as a result of perfect bonding of two elastic half spaces made of different materials. In the spectral region of the Fourier transformation with respect to time, the problem is reduced to boundary integral equations for the functions of dynamic crack opening displacements. In deducing the equations, we satisfy the conditions of conjugation of the half spaces. As a result of the numerical solution of equations and finding the originals, we get the time dependences of the stress intensity factors in the vicinity of a penny-shaped crack perpendicular to the interface of materials for various profiles of normal dynamic loads and various ratios of the moduli of elasticity of the components of the analyzed composite.     

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.     

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.     

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.     

The contact problem for an open penny-shaped crack under normally incident tension-compression wave

This paper concerns fracture dynamic problems for elastic cracked solids with allowance for crack faces contact interaction. The contact problem for a penny-shaped crack with an initial opening under normally incident tension-compression wave is solved by the method of boundary integral equations. The contact forces and the displacement discontinuity of the crack faces are studied. The solution is compared with those obtained without allowance for crack faces contact interaction for various shapes of the initial opening.     

Thermally-induced interlaminar crack-tip singularities in laminated anisotropic composites

Thermally-induced stress singularities of an interlaminar crack in a fiber-reinforced composite laminate under a state of generalized plane deformation are examined within the framework of steady-state anisotropic thermoelasticity. The crack is assumed to be embedded within a matrix-rich interlaminar region of the composite. The Fourier integral transform technique and the flexibility/stiffness matrix method are introduced to formulate the current mixed boundary value problem. As a result, two sets of simultaneous Cauchy-type singular integral equations of the first kind are derived for the heat conduction and thermoelasticity. Within the context of linear elastic fracture mechanics, the mixed-mode thermal stress intensity factors are defined in terms of the solutions of the corresponding integral equations. Numerical results are presented, addressing the effects of laminate stacking sequence, crack location, and crack surface partial insulation on the values of thermal stress intensity factors.     

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.     

Crack transient torsional wave interaction in an elastic bi-material

An infinitely long cylindrical elastic solid of finite radius (fibre or inner medium) is imbedded in another infinite elastic cylindrical solid (matrix or outer medium) to form an elastic bi-material composite perfectly bonded at their bi-material interface. The early-time response of the composite to a sudden twist applied axisymmetrically over a stationary penny-shaped crack which appears over a cross-section of the fibre and terminates orthogonally at the bi-material interface, is investigated. The magnitude of the applied twist is assumed to be less than that for which debonding or crack propagation can take place. For a bi-material composite the square of whose matrix shear-wave speed is less than twice the square of whose fibre shear-wave speed, uniform asymptotic analytic results, whose accuracy can be improved by taking higher order terms in the solution, are obtained for the time-dependent surface displacement gradient normal to the plane of the crack and outside the crack region, the dynamic stress intensity factor, and also the stress singularity strength eigenvalue-all as functions of the fibre-matrix material disparities.     

Point temperature solution for a penny-shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium

The solution of an impermeable penny-shaped crack subjected to a concentrated thermal load (prescribed point temperature) applied arbitrarily at the crack surfaces is derived using the generalized potential theory method. The integral equation governing the temperature field is found to have the same structure as that for the elastic punch problem and the integro-differential equations related to the electroelastic field are similar to that reported for the elastic crack problem. Significant solutions to these integro-differential equations are obtained by generalizing the previous results available in literature. Exact three-dimensional expressions for the full-space thermo-electro-elastic field are finally obtained by simple differentiation, all in terms of elementary functions. The exact analysis for a permeable crack is also presented and discussed. The obtained point temperature solutions play an important role in the related BEM analysis.     

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.     

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.     

Torsional impact response of a penny-shaped interface crack in a layered composite

The torsional impact response of a penny-shaped interface crack in a layered composite is considered in this study. The geometry of the composite consists of two bonded dissimilar elastic layers which are sandwiched between two half-spaces made of a different material. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometry parameters is discussed.     

Penny-shaped interface crack between dissimilar nonhomogeneous elastic layers under axially symmetric torsion

Summary The problem of axially symmetric torsion for dissimilar nonhomogeneous bonded elastic layers containing a penny-shaped interface crack is considered. The mixed boundary value problem is reduced to solving a Fredholm integral equation of the second kind. The Fredholm integral equation is solved numerically by reducing it to a system of simultaneous algebraic equations. Numerical results for the stress intensity factor are presented in the form of graphs.     

Time-domain boundary element analysis of elastic wave interaction with a crack

The mathematical formulation of the problem of transient wave interaction with a crack in a homogeneous, isotropic, linearly elastic solid has been reduced to the solution of an integral equation over the insonified crack face. The integral equation relates the unknown crack-opening displacement, which depends on time and position, to the incident wave field. The integral equation has been solved numerically by a time-stepping method in conjunction with a boundary element discretization of the crack surface. For normal incidence of a longitudinal step-stress wave on a penny-shaped crack, results as functions of time have been obtained for the crack-opening displacement, the elastodynamic Mode-I stress intensity factor and the scattered far-field.     

Application of Boundary Integral Equations to Elastodynamics of an Interface Crack

We considers application of boundary integral equations to the problem of an interface crack between two elastic half-spaces with different mechanical properties under dynamic loading. The derived system of equations allows evaluation of the displacements at the crack faces, and the traction and the displacements at the interface.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.