Anti-plane shear crack in a magnetoelectroelastic layer sandwiched between dissimilar half spaces

The crack problem of a magnetoelectroelastic layer bonded to dissimilar half spaces under anti-plane shear and in-plane electric and magnetic loads is considered. Fourier transforms are used to reduce the mixed boundary value problems of the crack, which is assumed to be permeable, to simultaneous dual integral equations, and then expressed in terms of Fredholm integral equations of the second kind. Numerical results show that the stress intensity factors are influenced by the magnetoelectric interactions and the geometry size ratio.     

Mode III fracture of a magnetoelectroelastic layer: exact solution and discussion of the crack face electromagnetic boundary conditions

This paper develops a closed-form solution for anti-plane mechanical and in plane electric and magnetic fields in a magnetoelectroelastic layer of finite thickness. Explicit expressions for the stresses, electric fields, and magnetic fields, together with their intensity factors are obtained for the extreme cases for impermeable and permeable cracks. Solutions for some special cases, such as a magnetoelectroelastic layer with infinite thickness, are also obtained. Applicability of the crack face electromagnetic boundary conditions is discussed. It is found that the crack profile is important in obtaining the correct electromagnetic fields and their intensity factors. The stress intensity factor, however, does not depend on the crack face electromagnetic boundary condition assumptions.     

A moving interface crack between two dissimilar functionally graded strips under plane deformation with integral equation methods

In this paper a finite interface crack with constant length (Yoffe-type crack) propagating along the interface between two dissimilar functionally graded strips with spatially varying elastic properties under in-plane loading is studied. By utilizing the Fourier transformation technique, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters, the graded parameter, the strip thickness and crack speed on the stress intensity factors and the probable kink angle are investigated. The numerical results show that the graded parameters, the thicknesses of the functionally graded strips and the crack speed have significant effects on the dynamic fracture behavior of functionally graded material structures.     

Fracture analysis of a penny-shaped magnetically dielectric crack in a magnetoelectroelastic material

In this paper we investigate the magnetoelectroelastic behavior induced by a penny-shaped crack in a magnetoelectroelastic material. The crack is assumed to be magnetically dielectric. A closed-form solution is derived by virtue of Hankel transform technique with the introduction of certain auxiliary functions. Field intensity factors are obtained and analyzed. The results indicate that the stress intensity factor depends only on the mechanical loads. However, all the other field intensity factors depend directly on both the magnetic and dielectric permeabilities inside the crack as well as on the applied magnetoelectromechanical loads and the material properties of the magnetoelectroelastic material. Several special cases are further discussed, with the reduced results being in agreement with those from literature. Finally, according to the maximum crack opening displacement (COD) criterion, the effects of the magnetoelectromechanical loads and the crack surface conditions on the crack propagation and growth are evaluated.     

裂尖具线性分布约束应力的运动裂纹模型及其解析解

以恒定速度运动的Griffith裂纹解析解为著名的Yoffe解。静止裂纹的条状屈服模型即Dugdale模型,将其推广到运动裂纹模型时发现,当裂纹运动速度跨越Rayliegh波速时,裂纹张开位移COD趋于(∞,且表现为间断。通过在裂尖引入一个约束应力区及两个速度效应函数,假设约束应力为线性分布,采用复变函数方法,求得动态应力强度因子SIF与裂纹张开位移COD的解析解。新的结果,在Rayleigh波速下裂纹张开位移连续且为有限值。给出裂纹张开位移的一些数值结果,获得了一些有意义的结论。     

Interaction of a penny-shaped crack with an elliptic crack under shear loading

The problem of interaction between equal coplanar elliptic cracks embedded in a homogeneous isotropic elastic medium and subjected to shear loading was solved analytically by Saha et al. (1999) International Journal of Solids and Structures 36, 619–637, using an integral equation method. In the present study the same integral equation method has been used to solve the title problem. Analytical expression for the two tangential crack opening displacement potentials have been obtained as series in terms of the crack separation parameter _i up to the order _i⁵,(i=1,2) for both the elliptic as well as penny-shaped crack. Expressions for modes II and III stress intensity factors have been given for both the cracks. The present solution may be treated as benchmark to solutions of similar problems obtained by various numerical methods developed recently. The analytical results may be used to obtain solutions for interaction between macro elliptic crack and micro penny-shaped crack or vice-versa when the cracks are subjected to shear loading and are not too close. Numerical results of the stress-intensity magnification factor has been illustrated graphically for different aspect ratios, crack sizes, crack separations, Poisson ratios and loading angles. Also the present results have been compared with the existing results of Kachanov and Laures (1989) International Journal of Fracture 41, 289–313, for equal penny-shaped cracks and illustrations have been given also for the special case of interaction between unequal penny-shaped cracks subjected to uniform shear loading.     

Numerical solution for the T-stress in branch crack problem with infinitesimal branch length

This paper investigates the T-stress in a branch crack problem with infinitesimal branch length. The branch crack is composed of a main crack and many branches. The ratio of the lengths for branch to main crack is very small. A singular integral equation method is suggested to solve the problem numerically and the stress intensity factor and T-stress can be evaluated immediately. Many computed results for T-stress under different conditions for branches are presented. It is found from the computed results that the interaction for T-stress among branches is very complicated.     

Transient response of a Griffith crack between dissimilar piezoelectric layers under anti-plane mechanical and in-plane electrical impacts

The problem of an interface crack between dissimilar piezoelectric layers under mechanical and electrical impacts is formulated by using integral transform and Cauchy singular integral equation methods. The dynamic stress intensity factor and dynamic energy release rate (DERR) are determined through use of the obtained solutions and the effects of the loading ratio, the geometry of crack configuration and the combination of material parameters on the above two quantities are discussed. The numerical calculations indicate that the electrical load can promote or retard the crack growth depending on its magnitude, direction and the existence of the mechanical load and that with the increase of the value of ratio of two material parameters, some material parameters will inhibit the crack growth. On the other hand, some material parameters play the contrary roles. In addition, the geometry of the crack configuration has the significant effects on the DERR. Finally the results are compared with those obtained in a previous investigation.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. 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. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

An analytical and numerical study of a dynamically accelerating semi-infinite crack in a linear viscoelastic material

In a previous paper, a general method was presented for constructing the solution to the problem of a semi-infinite, mode III crack propagating dynamically through an infinite, general linear viscoelastic body. The only restrictions placed upon the crack tip speed were that it have constant sign and in magnitude not exceed the glassy shear wave speed. In the present contribution, those previous analytical results are applied to a study of dynamic unsteady crack growth in a linear viscoelastic body. In particular, a numerical algorithm for computing the stress intensity factor is given along with example simulations of running cracks using the Achenbach-Chao viscoelastic model and a stress intensity factor (SIF) fracture criterion. We also compare the transient SIF with the dynamic steady state SIF, and examine the transition to constant crack speed for a dynamically accelerating crack in a viscoelastic material.     

A mode I crack parallel to the interfaces in a periodically layered medium

The plane problem of a single crack in a periodically layered bimaterial composite is considered. For the case of a long crack loaded by opening normal tractions, the universal relation obtained between the Mode I and Mode II stress intensity factors show that the most dangerous crack location lies in the midplane of the layer. This crack location of the Mode I finite length crack is examined in detail. A closed form expression of the Green's function for a single dislocation is derived and the problem is reduced to a singular integral equation of the first kind. The study of the dependence of the normalized stress intensity factor upon the crack length reveals a wavy nonmonotonic behavior. A simple analytic formula for the limiting case of a semi-infinite crack is derived. It is found to be valid for a broad range of parameters.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

A complex variable boundary element method for antiplane stress analysis around a crack in some nonhomogeneous bodies

Abstract

A boundary element method based on the Cauchy integral formulae, i.e. a complex variable boundary element method (CVBEM), is proposed for the numerical solution of an antiplane crack problem involving an elastic body with shear modulus that varies continuously in space. The shear modulus assumes a certain form which is quite general to allow for multiparameter fitting of its variation. The method reduces the problem to a system of linear algebraic equations and can be readily implemented on the computer. For clarity, the CVBEM formulation is first carried out for a straight crack and then its extension to include an arbitrary curved crack is indicated.     

Analysis of a propagating crack in functionally graded materials with property variation angled to crack direction

The stress and displacement fields for a crack propagating in functionally graded materials (FGMs) with property variation angled to crack direction are obtained. The FGMs have a linear variation of shear modulus with a constant density and Poisson’s ratio. The solutions for higher order terms in the dynamic equilibriums are obtained by transforming the general differential equations to Laplace’s equations. Using the stress fields, the effects of the nonhomogeneity and the angled properties on stress components are investigated. In addition to, the contours of the constant maximum shear stress around the static and propagating crack tip are generated. The contours of the constant maximum shear stress around the static and propagating crack tip tilt toward the property gradation direction.     

An energetically consistent annular crack in a piezoelectric medium

The analytical treatment of an energetically consistent annular crack in a piezoelectric solid subjected to remote opening electromechanical loading is addressed. Potential functions and Hankel transform in combination with a robust technique are employed to reduce the solution of the mixed boundary value problem into a Fredholm integral equation of the second kind. The limiting case of a penny-shaped crack in a piezoelectric medium with energetically consistent boundary conditions over the crack faces is extracted for the first time. The electrical discharge phenomenon within the crack gap is modeled utilizing a non-linear constitutive law and the effects of the breakdown field on the energy release rate are delineated. The energy release rate, the electric displacement inside the crack gap, and the closing traction on crack faces are plotted for all possible geometries of a non-discharging annular crack.     

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r ^–3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary.     

Kinked crack in a bending trapezoidal plate

The interaction problem of a kinked crack and the edges of a bending trapezoidal plate which takes the effects of transverse shear deformation into account is presented. The research method is based upon the complex potential technique of Muskhelishvili using conformal mapping. Furthermore, for the analysis of the moment intensities at the tips of the kinked crack, the concept of dislocation distribution is applied. The integral equations for the stress disturbance problem along the line that is the presumed location of the kinked crack are then obtained as a system of singular integral equations with simple Cauchy kernels. As a consequence, the variation of moment intensity factors at the crack-tips is also illustrated.     

A crack in a linear-elastic material under mode II loading, revisited

A sharp crack in a two-dimensional infinite linear-elastic material, under pure shear (mode II) loading is re-examined. Several criteria have been proposed for the prediction of the onset and direction of crack extension along a path emanating from the tip of the initial crack. These criteria date back some three decades and are well documented in the literature. All the predictions from the different criteria are close and indicate that the crack extension takes a direction at an angle of ≈ −70° measured counterclockwise from the positive x -axis, in the case of a remotely applied positive shear stress. However, the possibility seems to have been overlooked that the crack extension may initiate not from the crack tip itself, but instead may initiate on the free surface at an infinitesimal distance behind the crack tip. The effect of crack tip plasticity on the relevant stresses in the region of the crack tip is investigated by the application of an elastic–plastic finite element program.     

Branch cracking and debonding at an end of a rigid stiffener under anti-plane shear stress

A semi-infinite body with a rigid stiffener on a part of the surface under uniform anti-plane shear stress is considered. This is a mixed boundary value problem, and a closed and exact solution is obtained. Stress concentration occurs at the ends of the stiffener; therefore a crack or a debonding may occur at the end of the stiffener. This paper investigates the competition between a crack or a debonding occurrence. It also investigates how far the debonding will extend. The maximum strain energy release rate is used as criterion for detecting a crack and a debonding initiation. Also the strain energy release rate just after crack initiation is investigated and the crack initiation angle is 140.8°. As the applied load, the following three kind of loading conditions are considered; constant loading, increasing loading and small cyclic loading.     

A model for fatigue crack growth in a threshold region

The prediction of fatigue crack growth at very low ΔK values, and in particular for the threshold region, is important in design and in many engineering applications. A simple model for cyclic crack propagation in ductile materials is discussed and the expression

$\text{da}\text{dN}\text{=}\text{2}{}^{\mathrm{1+n}}\text{(1?2v)(ΔK}{}^2{}_{\mathrm{eff}}\text{?ΔK}{}^2{}_{\mathrm{c,eff}}\text{)}\text{4(1+n)π σ}{}^{\mathrm{1?n}}{}_{\mathrm{yc}}\text{E}{}^{\mathrm{1+n}}\text{?}{}^{\mathrm{1+n}}{}_{\mathrm{f}}$

developed. Here, n is the cyclic strain hardening exponent, σ_yc is cyclic yield, and ε_f is the true fracture strain. The model is successfully used in the analysis of fatigue data BS 4360-50D steel.