Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials

The antiplane shear deformation problem of two edge-bonded dissimilar isotropic wedges is considered. In the case when the sum of the two apex angles is equal to 2π, the problem reduces to that of two edge-bonded dissimilar materials with an interfacial crack subjected to concentrated antiplane shear tractions on the crack faces. An explicit expression is extracted for the stress intensity factor at the crack tip. In the special cases of different combinations of the apex angles, the obtained expression for the stress intensity factor may be simplified and relations of a simpler form are given for the stress intensity factor. It is shown that the stress intensity factor is dependent on the material properties as well as the geometry and loading. However, in special cases of equal apex angles as well as the case of similar materials the dependency of the stress intensity factor on the material properties disappears.     

On dynamic stress analysis for cracks in elastic materials with voids

The present paper is concerned with the development of a semi-analytical approach to the dynamic problem of the concentration of stresses near the edges of a crack located in a porous elastic space (two-dimensional problem) and subjected to a normal oscillating load applied to the crack faces. Our analysis is made in the context of the Goodman–Cowin–Nunziato (G–C–N) theory for porous media. In previous work we studied static crack problems for such materials; now we introduce an analysis of the relevant dynamic aspects. By using the Fourier transform, the problem is reduced in explicit form to a hyper-singular integral equation with a convolution kernel valid over the crack length. Then, we apply a collocation technique developed in our previous work to solve this equation, and study the stress intensity factor. The principal goal is to compare the stress intensity factor for the static and dynamic cases. We also compare our results with the case of an ordinary linear elastic medium.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

Torsion problem for elastic multilayered finite cylinder with circular crack

An axisymmetric tangent stress is applied to a lateral surface of a multilayered elastic finite cylinder with a fixed bottom face. The problem is solved for an arbitrary number of layers. The layers are coaxial, and the conditions of an ideal mechanical contact are fulfilled between them. A circular crack is situated parallel to the cylinder's faces in the internal layer with branches free from stress. The upper face of the cylinder is also free from stress. Concretization of the problem is done on examples of two-and three-layered cylinders. An analysis of cylinders' stress state is conducted and the stress intensity factor is evaluated depending on the crack's geometry, its location and ratio of the shear modulus. Advantages of the proposed method include reduction of the solution constants' number regardless of the number of layers, and presentation of the mechanical characteristics in a form of uniformly convergent series.     

SCATTERING OF ANTI-PLANE SHEAR WAVES BY A SINGLE CRACK IN AN UNBOUNDED TRANSVERSELY ISOTROPIC ELECTRO-MAGNETO-ELASTIC MEDIUM

A theoretical treatment of the scattering of anti-plane shear (SH) waves is provided by a single crack in an unbounded transversely isotropic electro-magneto-elastic medium. Based on the differential equations of equilibrium, electric displacement and magnetic induction intensity differential equations, the governing equations for SH waves were obtained. By means of a linear transform, the governing equations were reduced to one Helmholtz and two Laplace equations. The Cauchy singular integral equations were gained by making use of Fourier transform and adopting electro-magneto imperme ableboundary conditions. The closed form expression for the resulting stress intensity factor at the crack was achieved by solving the appropriate singular integral equations using Chebyshev polynomial. Typical examples are provided to show the loading frequency upon the local stress fields around the crack tips. The study reveals the importance of the electro-magneto-mechanical coupling terms upon the resulting dynamic stress intensity factor.     

Stress intensity factors for anisotropic layered composites: Part II—isolated double periodic cracks

The formulation in Part I (Theoret. Appl. Fracture Mech. 17, 205–219 (1992)) of this work for collinear cracks in alternate layers of an anisotropic laminate is extended to a system where each of the cracked layer contains a periodic array of parallel cracks. These cracks are also collinear such that they are periodic in two mutually perpendicular directions. Finite Fourier transform is applied at discrete points for one of the space variables reducing the problem to a singular integral equation. The unknown is expressible in terms of the crack opening displacement as in Part I. Displayed graphically are the normalized stress intensity factor, effective stiffness of the laminate, and the interlayer stresses. The local stress intensity factor for the double array crack system is always less than that for a single isolated crack depending on the periodicity ratio. The interlayer stresses directly ahead of the crack are elevated, the intensity of which increases with decreasing distance between the crack tip and interface. Increase in the thickness of the adjoining layer tends to decrease the interlayer stresses nearest to the crack tip, a result that is to be expected.     

Micromechanics analyses of interface crack

A new mechanics model based on Peierls concept is presented in this paper, which can clearly characterize the intrinsic features near a tip of an interfacial crack. The stress and displacement fields are calculated under general combined tensile and shear loadings. The near tip stress fields show some oscillatory behaviors but without any singularity and the crack faces open completely without any overlapping when remote tensile loading is comparable with remote shear loading. A fracture criterion for predicting interface toughness has been also proposed, which takes into account for the shielding effects of emitted dislocations. The theoretical toughness curve gives excellent prediction, as compared with the existing experiment data. The project supported by the National Natural Science Foundation of China     

Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width

The problem of an orthotropic strip containing two collinear cracks normal to the strip boundaries is considered. The Fourier series method is used to reduce the associated boundary value problem to triple series equations, then to a singular integral equation, which can be solved analytically. Under remote uniform antiplane shear loading, the stress field and the crack sliding displacement are determined analytically and stress intensity factors are also given in a closed form.     

Electro-mechanical analysis of an interfacial crack between a piezoelectric and two orthotropic layers

Summary  The problem of an interfacially cracked three-layered structure constructed of a piezoelectric and two orthotropic materials is analyzed using the theory of linear piezoelectricity and fracture mechanics. Anti-plane shear loading is considered, and the integral transform technique is used to determine the stress intensity factor. Numerical examples show the electro-mechanical effects of various material combinations and layer thicknesses on the stress intensity factor. Interesting results are obtained in comparison with earlier solutions for interfacially cracked piezoelectric structures. Received 29 December 2000; accepted for publication 3 May 2001     

Effects of pre-stress on crack-tip fields in elastic,incompressible solids

A closed-form asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously pre-stressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial pre-stress is assumed with stress axes parallel and orthogonal to the crack faces. Two boundary conditions are considered on the crack faces, namely a constant pressure or a constant dead loading, both preserving an homogeneous ground state. Starting from this configuration, small superimposed Mode I or Mode II deformations are solved, in the framework of Biot's incremental theory of elasticity. In this way a definition of an incremental stress intensity factor is introduced, slightly different for pressure or dead loading conditions on crack faces. Specific examples are finally developed for various hyperelastic materials, including the J₂-deformation theory of plasticity. The presence of pre-stress is shown to strongly influence the angular variation of the asymptotic crack-tip fields, even if the nominal stress rate displays a square root singularity as in the infinitesimal theory. Relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence.     

Fracture mechanics of plates and shells applied to fail-safe analysis of fuselage Part I: Theory

In this paper, a general theory on the asymptotic field near the crack tip for plates and shells with and without shear deformation effect is established. It is found that four stress intensity factors, two for symmetrical and antisymmetrical stretching and two for symmetrical and antisymmetrical bending, are required to describe arbitrary asymptotic fields near the crack tip for plates without shear deformation. An additional stress intensity factor is required for the transverse shearing force induced by antisymmetrical bending when the shear deformation is included in the analysis. It is also proven by means of the complex variable technique that for problems of plates with shear deformation, there exist similarities in the asymptotic expressions of moments and membrane forces and also in the asymptotic expressions of in-plane displacements and rotations of the mid-surface. The energy release rate associated with crack growth in the direction of the crack line can be expressed in terms of stress intensity factors by means of Irwin's method of work and energy associated with a virtual crack extension. A combined stress intensity factor can be defined through the total energy release rate. The theory of the fracture of plates is generalized and applied to the study of problems in the fracture of shells. An example of an infinitely long cylindrical shell with a circumferential crack subjected to remote axial tension is given to demonstrate the application of the theory and to test the accuracy of the numerical analysis used for the problem.     

The dynamic stress intensity factor analysis of adhesively bonded material interface crack with damage under shear loading

This paper studies the dynamic stress intensity factor (DSIF) at the interface in an adhesive joint under shear loading. Material damage is considered. By introducing the dislocation density function and using the integral transform, the problem is reduced to algebraic equations and can be solved with the collocation dots method in the Laplace domain. Time response of DSIF is calculated with the inverse Laplace integral transform. The results show that the mode Ⅱ DSIF increases with the shear relaxation parameter, shear module and Poisson ratio, while decreases with the swell relaxation parameter. Damage shielding only occurs at the initial stage of crack propagation. The singular index of crack tip is -0.5 and independent on the material parameters, damage conditions of materials, and time. The oscillatory index is controlled by viscoelastic material parameters.     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

Scattering of anti-plane shear waves by a single crack in an unbounded trasversely isotropic electro-magneto-elastic medium

A theoretical treatment of the scattering of anti-plane shear (SH) waves is provided by a single crack in an unbounded transversely isotropic electro-magneto-elastic medium. Based on the differential equations of equilibrium, electric displacement and magnetic induction intensity differential equations, the governing equations for SH waves were obtained. By means of a linear transform, the governing equations were reduced to one Helmholtz and two Laplace equations. The Cauchy singular integral equations were gained by making use of Fourier transform and adopting electro-magneto impermeable boundary conditions. The closed form expression for the resulting stress intensity factor at the crack was achieved by solving the appropriate singular integral equations using Chebyshev polynomial. Typical examples are provided to show the loading frequency upon the local stress fields around the crack tips. The study reveals the importance of the electro-magneto-mechanical coupling terms upon the resulting dynamic stress intensity factor. Contributed by SHEN Ya-peng Foundation item: the National Natural Science Foundation of China (10132010, 50135030) Biographies: DU Jian-ke (1970∼)     

Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer

In this paper we develop closed form solutions for anti-plane mechanical and in-plane electric and magnetic fields for two collinear cracks in magneto-electro-elastic layer of finite thickness under the conditions of permeable crack faces using integral transform method. The anti-plane mechanical shear or displacement and in-plane electrical and magnetic loading are applied to the top and bottom surfaces of the layer for the two cases considered. Expressions for shear stresses, electric displacements and magnetic inductions in the vicinity of the cracks are derived as well as intensity factors for two cracks in magneto-electro-elastic layer. Numerical results for stress intensity factors and energy release rate are shown graphically.     

Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading

A plane problem for a tunnel electrically permeable interface crack between two semi-infinite piezoelectric spaces is studied. A remote mechanical and electrical loading is applied. Elastic displacements and potential jumps as well as stresses and electrical displacement along the interface are presented using a sectionally holomorphic vector function. It is assumed that the interface crack includes zones of crack opening and frictionless contact. The problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved analytically. From the obtained solution, simple analytical expressions are derived for all mechanical and electrical characteristics at the interface. A quite simple transcendental equation, which determines the point of separation of open and close sections of the crack, is found. For the analysis of the obtained results, the main attention is devoted to the case of compressive-shear loading. The analytical analysis and numerical results show that, even if the applied normal stress is compressive, a certain crack opening zone exists for all considered loading values provided the shear field is present. It is found that the shear stress intensity factor at the closed crack tip and the energy release rates at the both crack tips depend very slightly on the magnitude of compressive loading.     

The stress intensity factor history due to three-dimensional transient loading on the faces of a crack

Aprocedure is described for determining dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body, with the crack faces subjected to tractions that result in variation of the stress intensity factor along the crack edge. The procedure is based on integral transform methods and the properties of analytic functions of a complex variable. The procedure is illustrated for the case of a pair of opposed line loads suddenly applied on the crack faces along a line perpendicular to the crack edge. An exact expression is obtained for the resulting mode I stress intensity factor as a function of time for any point along the crack edge. Some features of the solution, as well as possible extensions of the procedure, are discussed.     

The dynamic behavior of two collinear interface cracks in magneto-electro-elastic materials

In this paper, the dynamic behavior of two collinear symmetric interface cracks between two dissimilar magneto-electro-elastic material half planes under the harmonic anti-plane shear waves loading is investigated by Schmidt method. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. To solve the triple integral equations, the jump of the displacements across the crack surface is expanded in a series of Jacobi polynomials. Numerical solutions of the stress intensity factor, the electric displacement intensity factor and the magnetic flux intensity factor are given. The relations among the electric filed, the magnetic flux field and the stress field are obtained.     

Stress intensity factors for circular hole and inclusion using finite element alternating method

A centre cracked plate subjected to remote tensile and shear loading is considered for the analysis. Effect of circular hole and influence of shrunk fit inclusion on stress intensity factors are studied. Multiply connected domain boundary value problem is solved using finite element alternating method (FEAM). Parametric studies involving drilled hole/inclusion sizes and locations are investigated. Energy release rates evaluated using the stress field obtained by FEAM are in good agreement with other methods. The optimum location in reducing the stress intensity factor with hole/inclusion is obtained and located at a distance 20% of semi-crack length from crack tip on the side opposite the ligament for Mode-I loading and it is also observed that the location is almost invariant of hole sizes. For Mode-II loading, the optimum location for the hole is located at a distance about 23% of semi-crack length from the middle of the crack along the transverse direction.     

Electroelastic analysis of a penny-shaped crack in a piezoelectric ceramic under mode I loading

Following the theory of linear piezoelectricity, we consider the electroelastic problem for a piezoelectric ceramic with a penny-shaped crack under mode I loading. The problem is formulated by means of Hankel transform and the solution is solved exactly. The stress intensity factor, energy release rate and energy density factor for the exact and impermeable crack models are expressed in closed form and compared for a P-7 piezoelectric ceramic. Based on current findings, we suggest that the energy release rate and energy density factor criteria for the exact crack model are superior to fracture criteria for the impermeable crack model.