Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Stress intensity factor for a crack normal to an interface between two orthotropic materials

In this paper, the problem of a crack normal to an interface in two joined orthotropic plates is studied as a plane problem. Body force method is used to investigate dependence of the stress intensity factor on the elastic constants: E _x1, E _y1, G _xy1, V _xy1 for material 1 and E _x2, E _y2, G _xy2, V _xy2 for material 2. A particular attention is paid to simplifying kernel functions, which is used in the body force method, so that all the elastic constants involved can be represented by three new parameters: H ₁, H _2I, H ₃ for the mode I deformation and H ₁, H _2II, H ₃ for the mode II deformation. From the kernel function so obtained it is found that the effects of the eight elastic constants on the stress intensity factors can be expressed by the three material parameters, H ₁, H _2I, H ₃ and H ₁, H _2II, H ₃, respectively for K _I and K _II. Furthermore, it is also found that the dependence of K _I on H ₁, H _2I, H ₃ is exactly the same as the dependence of K _II on H ₁, H _2II, H ₃. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Stress intensity factors of square crack inclined to interface of transversely isotropic bi-material

This paper analyzes a square crack in a transversely isotropic bi-material solid by using dual boundary element method. The square crack is inclined to the interface of the bi-material. The fundamental solution for the bi-material solid occupying an infinite region is incorporated into the dual boundary integral equations. The square crack can have an arbitrary angle with respect to the plane of isotropy of the bi-material occupying either finite or infinite regions. The stress intensity factor (SIF) values of the modes I, II, and III associated with the square crack are calculated from the crack opening displacements. Numerical results show that the properties of the anisotropic bi-material have evident influences on the values of the three SIFs. The values of the three SIFs are further examined by taking into account the effect of the external boundary of the internally cracked bi-material.     

Moving crack at the interface between two dissimilar magnetoelectroelastic materials

Summary Analytical solutions for an anti-plane Griffith crack moving at the interface between two dissimilar magnetoelectroelastic media under the conditions of permeable crack faces are formulated using the integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic materials. Expressions for stresses, electric displacements, and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities, and it is found that the dynamic stress intensity factor (DSIF), the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF) are independent of the remote electromagnetic loads. The moving speed of the crack has influence on the DEDIF and the DMIIF. When the crack is moving at lower speeds 0 ≤ M≤ M_c1 or higher speeds M_c2 < M < 1, the crack will propagate along its original plane, while in the range of M_c1 < M < M_c2 , the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Boundary element analysis for an interface crack between dissimilar elastoplastic materials

The boundary element method (BEM) is presented for elastoplastic analysis of cracks between two dissimilar materials. The boundary integral equations and integral representation of stress rates are written in such a form that all integrals can be evaluated by the regular Gaussian quadrature rule. An advanced multidomain BEM formulation is suggested for the solution of analysed problems where the substantial reduction of stiffness matrix is observed. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the stresses. The boundary and the yielding zone are discretized by elements with quadratic approximations. In numerical examples the path independence of the J- and L-integrals for a straight interface crack and a circular arc-shaped interface crack are investigated, respectively. The influence of the different values of Young's modulus on the J-integral, shape and size of plastic zones is treated too.     

Penny-shaped crack in a transversely isotropic solid

An analytical solution is given for the displacement and stress distribution produced in the interior of a transversely isotropie solid containing a penny-shaped crack situated in an elastic symmetry plane and axially-loaded. Curves of numerical results are presented for the stress intensity factor and the normal displacement. They show the influence of this type of anisotropy.     

Some crack problems in transversely isotropic solids

Summary. Crack problems in transversely isotropic solids are reexamined from a new point of view. It is shown that, when the crack is on the isotropic plane, the asymptotic forms of the elastic crack-tip fields are identical with those in orthotropic media. The equivalent inclusion method in conjunction with Eshelbys S tensor of a strongly oblate spheroid in transversely isotropic materials is used to solve penny-shaped crack problems. The stress intensity factors corresponding to uniform tension and shear are determined, respectively. Griffiths energy criterion for brittle cracking and Irwins energy release rate are discussed in the present context. Finally, the weight function for an axisymmetrically loaded penny-shaped crack is derived. It is found that the axisymmetric weight function is independent of the material constants and is identical with the isotropic case.AcknowledgementThis work was supported in part by the National Science Council of Taiwan.     

Obtaining mode mixity for a bimaterial interface crack using the virtual crack closure technique

We review, unify and extend work pertaining to evaluating mode mixity of interfacial fracture utilizing the virtual crack closure technique (VCCT). From the VCCT, components of the strain energy release rate (SERR) are obtained using the forces and displacements near the crack tip corresponding to the opening and sliding contributions. Unfortunately, these components depend on the crack extension size, Δ, used in the VCCT. It follows that a mode mixity based upon these components also will depend on the crack extension size. However, the components of the strain energy release rate can be used for determining the complex stress intensity factors (SIFs) and the associated mode mixity. In this study, we show that several—seemingly different—suggested methods presented in the literature used to obtain mode mixity based on the stress intensity factors are indeed identical. We also present an alternative, simpler quadratic equation to this end. Moreover, a Δ-independent strain energy release based mode mixity can be defined by introducing a “normalizing length parameter.” We show that when the reference length (used for the SIF-based mode mixity) and the normalizing length (used for Δ-independent SERR-based mode mixity) are equal, the two mode mixities are only shifted by a phase angle, depending on the bimaterial parameter ε.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

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.     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

Dynamic penny-shaped crack in a transversely isotropic material

The scattering of a harmonic longitudinal wave by a penny-shaped crack in a transversely isotropic material is investigated using the techniques of Hankel transform. The wave impinges normally on the crack surfaces. A complete contour integration is employed to simplify the expressions of the results. An exact expression of the dynamic stress-intensity factor is obtained as a function of the frequency factor and the anisotropic material constants. The normalized dynamic stress-intensity factor is shown to have different maximum values at different wave frequencies for the sample composite and metallic materials. The distortion of the dynamic crack shape and the displacement at the crack center are also shown to be dependent of the wave frequency and the anisotropy of the material.     

A transversely isotropic composite with a penny-shaped crack

Summary We consider the problem of determining the stress intensity factor and the crack energy in a transversely isotropic composite medium, containing a penny-shaped crack. We assume that the crack surface is perpendicular to the bond face and the crack is opened by constant internal pressure. By use of integral transform, we reduce the problem to solving a Fredholm integral equation of the second kind. Numerical results are given for the combination of some practical materials such as magnesium and cadmium. The effect of transverse isotropy upon the stress intensity factor, the crack energy and the deformation on the crack surface is discussed.

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Ein transversal, isotropes, komposites Medium mit einem münzenförmigen Riß

Zusammenfassung Das Problem der Bestimmung des Spannungsintensitätsfaktors und der Rißenergie, in einem transversalen, isotropen, kompositen Medium mit einem münzenförmigen Riß, wird betrachtet. Es wird vorausgesetzt, daß die Rißoberfläche normal zur Verbundfläche liegt, und der Riß sich durch konstanten inneren Druck öffnet. Durch Anwendung einer Integraltransformation, wird das Problem auf die Lösung einer Fredholmschen Integralgleichung zweiter Art reduziert. Numerische Ergebnisse werden für die Kombination einiger Materialien, wie Magnesium und Cadmium angegeben. Der Einfluß der transversalen Isotropie auf den Spannungskonzentrationsfaktor, die Rißenergie und die Deformation an der Rißoberfläche werden diskutiert.

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With 3 Figures

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This work is supported by the Board of Scientific and Industrial Research, Orissa (India).     

Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.     

Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials

Explicit analytical expressions are obtained for the longitudinal and transverse effective spring stiffnesses of a planar periodic array of collinear cracks at the interface between two dissimilar isotropic materials; they are shown to be identical in a general case of elastic dissimilarity (the well-known open interface crack model is employed for the solution). Since the interfacial spring stiffness can be experimentally determined from ultrasound reflection and transmission analysis, the proposed expressions can be useful in estimating the percentage of disbond area between two dissimilar materials, which is directly related to the residual strength of the interface. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness are clearly represented in the solution. It is shown that in general the crack interaction weakly depends on material dissimilarity and, for most practical cases, the crack interaction is nearly the same as that for crack arrays between identical solids. This allows approximate factorization of the effective spring stiffness for an array of cracks between dissimilar materials in terms of an elastic dissimilarity factor and two factors obtained for cracks in a homogeneous material: the effective spring stiffness for non-interacting (independent) cracks and the crack interaction factor. In order to avoid the effect of the crack surface interpenetration zones on the effective spring stiffness, the range of the tensile to transverse load ratios is obtained under the assumption of small-scale contact conditions. Since real cracks are often slightly open (due to prior loading history and plastic deformation), it is demonstrated that for ultrasound applications the results obtained are valid for most practical cases of small interfacial cracks as long as the mid-crack opening normalized by the crack length is at least in the order of 10⁻⁵.