Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

Three-dimensional stress intensity factors of a central square crack in a transversely isotropic cuboid with arbitrary material orientations

In this paper, we present the dual boundary element method (dual-BEM) or single-domain BEM to analyze the mixed three-dimensional (3D) stress intensity factors (SIFs) in a finite and transversely isotropic solid containing an internal square crack. The planes of both the transverse isotropy and square crack can be oriented arbitrarily with respect to a fixed global coordinate system. A set of four special nine-node quadrilateral elements are utilized to approximate the crack front as well as the outer boundary, and the mixed 3D SIFs are evaluated using the asymptotic relation between the SIFs and the relative crack opening displacements (COD) via the Barnett–Lothe tensor.Numerical examples are presented for a cracked cuboid which is transversely isotropic with any given orientation and is under a uniform vertical traction on its top and bottom surfaces. The square crack is located in the center of the cuboid but is oriented arbitrarily. Our results show that among the selected material and crack orientations, the mode-I SIF reaches the largest possible value when the material inclined angle ψ₁=45° and dig angle β₁=45°, and the crack inclined angle ψ₂=0° and dig angle β₂=0°. It is further observed that when the crack is oriented vertically or nearly vertically, the mode-I SIF becomes negative, indicating that the crack closes due to an overall compressive loading normal to the crack surface. Variation of the SIFs for modes II and III along the crack fronts also shows some interesting features for different combinations of the material and crack orientations.     

Stress intensity factors for an inclined edge crack in a semiplane

Mode I and II Stress Intensity Factors under uniform general biaxial loadings were derived for an inclined edge crack in a semiplane. By interpolating Finite Element results in the angular range [0°÷80°], analytical expressions were obtained for both K_I and K_II with an accuracy better than 1%. Influence coefficients were defined in the crack reference frame thus highlighting the coupling effects between Modes I and II due to the loss of symmetry when the crack is not normal to the surface.     

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.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

Stress intensity factors of a surface crack near an interface end

Due to the singular behavior of the stress field near the interface edge of bonded dissimilar materials, fracture generally initiates near the interface edge, or just from the interface edge point. In this paper, an edge crack near the interface, which can be considered as being induced by the edge singularity and satisfying two conditions, is analyzed theoretically, based on the singular stress field near the interface edge and the superposition principle. It is found that the stress intensity factor can be expressed by the stress intensity coefficient of the edge singular stress field, the crack length, the distance between the interface and the crack, as well as the material combination. Boundary element method analysis is also carried out. It is found that the theoretical result coincides well with the numerical result when the crack length is small. Therefore, the theoretical representation obtained by this study can be used to simply evaluate the stress intensity factor of an edge singularity induced crack for this case. However, when the crack length becomes larger than a certain value, a significant difference appears, especially for the case with large edge singularity.     

Coupled horizontal and rocking vibrations of a rigid circular plate on a transversely isotropic bi-material interface

This paper investigates the coupled rocking and horizontal vibratory response of a rigid circular plate embedded in viscoelastic, transversely isotropic, three-dimensional unbounded media. The boundary-value problem corresponding to the case of distributed horizontal and rocking ring loads at a bi-material interface is solved to obtain the required influence functions for the solution of the present problem. The case of an embedded rigid plate is formulated in terms of a discretized integral equation, which couples the rigid body displacements of the plate with the tractions acting over its contact surface through a set of displacement influence functions. The system of resulting discretized integral equations is solved numerically. The solution results in the tractions over each disc element. This paper carefully takes into account the coupling of the rocking and horizontal responses of the plates that is typical of non-homogeneous interfaces, i.e., their horizontal displacements due to rocking moments and their rotations due to horizontal loads. The dynamic direct and cross compliances of the embedded plate are shown for different governing parameters such as frequency of excitation and bi-material configuration. The present results are useful to the study of dynamic response of deeply buried foundations and anchors in non-homogeneous soils.     

Singularities of an inclined crack terminating at the bi-material interface in a Reissner plate

On the basis of Reissner’s plate theory, the stress singularities at the tip of an arbitrarily inclined semiinfinite crack terminating at the interface of two dissimilar materials are investigated in the present paper. Using the eigenfunction expansion method, the eigenequation of the corresponding problem is derived explicitly by directly solving the governing equations of Reissner’s plate theory in terms of three generalized displacement components. In this paper, the focus is on the calculation of the singularity order as a fundamental quantity in fracture mechanics. The singularity orders of the moments and shear force at the crack tip are determined by the dominant eigenvalues whose real parts lie between 0 and 1. The influences of the bi-material parameters and the crack inclination angle on the moment and shear force singularity orders are discussed in detail. Specifically, the variations of the shear force singularity order with the bi-material parameter and the crack inclination angle are examined in detail. It is proved that the shear force singularity order is a completely monotonic function of the bi-material parameter and the inclination angle. Some numerical results are given in order to prove the validity of the present study.     

Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

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.     

Stress intensity factor for transient thermal stresses in a transversely isotropic infinite body with an external circular crack

Summary The present work deals with the transient thermal stress in a transversely isotropic infinite body with an external circular crack. The surface cooling of the crack depends on position and time. Since it it usually very difficult to obtain an analytical solution for the temperature field, a finite difference formulation with respect to a tive variable is introduced. In the first step, applying this method to the general heat conduction equation in an orthotropic body, a very compact difference equation with respect to the spatial variables is obtained. In the second step, this method is applied to the transient thermoelastic problem in a transversely isotropic infinite body with an external circular crack subjected to heat exchange on the crack surface. Thermal stresses are analyzed by means of the transversely isotropic potential functions method.With 7 Figures     

The stress intensity factors for an infinitely long transversely isotropic, thick-walled cylinder which contains a ring-shaped crack

In this study the elastostatic axisymmetric problem for a long thick-walled transversely anisotropic cylinder containing a ring-shaped internal crack is analyzed. The problem is reduced to a singular integral equation which has a simple Cauchy kernel as the dominant part by using Hankel and Fourier transform techniques. These equations are then solved numerically and the stress intensity factors are calculated.The results are given for different transversely anisotropic materials and crack geometries.     

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.     

Stress intensity factors of microstructurally small crack

The stress intensity factor (SIF) is widely used for evaluating integrity of cracked components. Averaging the anisotropy of each crystal, the macroscopic behavior of polycrystalline materials is isotropic and homogenous in terms of elastic deformation. However, the anisotropic and/or inhomogeneous property influences on the stress field around a crack if the crack size is small in comparison with the grain. Thus, the SIF of the microstructurally small crack may differ from that in the isotropic body. In present study, the effect of anisotropic/inhomogeneous elasticity on the SIF is investigated by using the finite element analysis (FEA). At first, the SIFs of semi-circular crack in a single crystal and a polycrystalline material are calculated. These reveal that the magnitude of SIF is dependent not only on the crystal orientation but also on the deformation constraint by the neighboring crystals. Then, the statistical scatter of SIF due to the random orientation of crystal orientation in a polycrystal is examined by a Monte Carlo simulation.     

Finite element analysis of stress intensity factors for cracks at a bi-material interface

The stress singularity at the tip of a crack, either lying along or perpendicular to the interface of the two materials, is first investigated by the complex variable method. The order of the singularity is shown to be dependent on both the crack geometry and two parameters , which are related to the four elastic constants of the two materials. A hybrid crack element is constructed to properly account for the crack tip singularity. The stress intensity factors and energy release rate for cracks in different bi-material continua are then calculated using the finite element method. The results show that the present finite element analysis makes possible a highly accurate and efficient numerical solution of fracture mechanics problems.

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Résumé On étudie la singularité de la contrainte à l'extrémité d'une fissure située le long de ou perpendiculairement à l'interface de deux matériaux, en recourant d'abord à la méthode des variables complexes. On montre que l'ordre de la singularité dépend à la fois de la géométrie de la fissure et de deux paramètres et , en relation avec les quatre constantes élastiques des deux matériaux. On construit un élément de fissure hybride propre à tenir compte de la singularité d'extrémité de fissure, et on calcule par éléments finis les facteurs d'intensité des contraintes et le taux de relaxation de l'énergie, pour des fissures dans différents continuum à deux matériaux. Les résultats montrent que les techniques actuelles d'analyse aux éléments finis permettent de trouver une solution numérique efficace et de haute précision aux problèmes de mécanique de la rupture.

     

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.     

Dynamic mode-III interface crack in a bi-material strip

An interfacial crack is placed within a two-phase elastic strip subjected to an out-of-plane loading. In the unperturbed state, the crack propagates with a constant speed V along the interface. The Dirichlet boundary conditions are applied to the upper and lower sides of the strip. The exterior boundary is subjected to a regular small perturbation; in addition, it is assumed that the crack speed changes by a small amount ${\varepsilon \phi^{\prime}(t)}$ , where ${\phi}$ is a smooth function of time t. The asymptotic model presented in this paper delivers an approximation for the stress-intensity factor and an integro-differential equation for the perturbation function ${\phi}$ . A particular feature of the model is in the use of skew-symmetric dynamic weight functions, attributed to the interfacial crack problem in a strip.