Application of a conservation integral to an interface crack interacting with singularities

A mutual integral, which has the conservation property is applied to the problem of an interfacial crack. The stress intensity factors K ₁, K ₂, K ₃ and T-stress for the problem in an infinite medium are easily obtained by using the mutual integral without solving the boundary value problem. In doing so the auxiliary solutions are required and they are taken from the known asymptotic solutions. This method is amenable to numerical evaluation of the stress intensity factors and T-stress if the interfacial crack in a finite medium is considered.     

Elastic-plastic analysis of an arbitrarily-oriented mode III crack touching an interface

In this paper we investigate a semi-infinite crack terminating at an arbitrarily oriented interface between two elastic-plastic materials under an anti-plane shear loading. An analytical solution is first developed for general power-law hardening materials under a mode III loading. If both materials have the same hardening exponent, the formulation results in a nonlinear eigenequation which can be solved numerically. The stress singularities are determined as a function of two material constants: the hardening exponent n and parameter G which represents the relative resistance of the two materials. In addition to the power of the singularity, the stress, strain and displacement asymptotic fields are also determined. If the hardening exponents are not the same, the leading order terms of an expansion model ensure the stress continuity across the interface. The results show that the stress singularity mainly depends upon the material having the larger hardening exponent, with the highest stresses in the material having the smaller hardening exponent. By taking the hardening exponent n , the perfectly plastic bimaterial problem is studied. It has been found that if the crack lies in the less stiff material, the entirely plastic asymptotic fields around the crack tip can be determined. On the other hand, if the crack lies in the stiffer material, the crack-tip fields are partially elastic and partially plastic. For both cases, unique asymptotic fields can be determined explicitly. For those cases when the materials present a strain hardening property, different mathematical models are established.     

On mode I and mode II energy release rates of an interface crack

The opening (mode I) and sliding (mode II) components of the energy that is released during an incremental extension of an interface crack between two different elastic materials are evaluated by the Irwin's crack closure method. Each component of the energies (G _I and G _II ) is expressed in terms of the functions of the length of the incremental crack extension (a) and the real and imaginary part of the complex stress intensity factor defined by Malyshev and Salganik. It is found that values of G _I /a and G _II /a oscillate violently when a approaches zero and that, hence, in contrast with the case for homogeneous materials, each energy release rate should be defined as G _I /a and G _II /a for an actual crack growth step size.     

A new conservation integral for an arbitrarily kinked interfacial crack

A new conservation integral consisting of the path and area integral, for an arbitrarily kinked interfacial crack is proposed. The new conservation integral is shown to have the physical meaning of the energy release rate. We present two numerical examples to verify its usefulness; one is the problem of an interfacial crack with a parabolic-curved kink, and the other is the problem of a circular arc-shaped interfacial crack with a straight kink.     

Vibration in mode III of an elastic layer containing a crack perpendicular to the boundary edge

A mathematical formulation of the problem of stresses and displacements in an elastic layer which contains a crack perpendicular to the boundary and subjected to a vibrating stress, in mode III, is developed. The boundary conditions for the case of free loading at the edges of the strip are used to obtain a solution to this problem. The problem is reduced to the solution of a Fredholm integral equation. In Part II the problem describing the case of rigid constraint at the edges of an elastic strip containing a vibrating external crack in mode II is reduced to the Fredholm integral equation of the second kind. In Part III the solution of the problem describing the case of a strip containing a vibrational crack in mode III and laying on a rigid boundary is presented.

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Résumé On développe une formulation mathématique du problème des containtes et déplacements dans une couche élastique comportant une fissure perpendiculaire au bord d'une bande et soumise à une sollicitation vibratoire de mode III. Les conditions aux limites de bord suivantes sont envisagées: (1) sollicitation libre en bordure de bande, (2) bridage sévère des bords de la bande et (3) maintien de la bande dans une fixation rigide. Dans les trois cas, le problème est ramené à la solution d'une intégrale de Fredholm.

     

Energy release rate approximation for edge cracks using higher-order topological derivatives

Sub-micron films deposited on a flexible substrate are now commonly used in electronic industry. The main damaging mode of these systems is a multi-cracking of the film under the action of thermal and mechanical stresses. This multi-cracking phenomenon is described using the coupled criterion based on the simultaneous fulfilment of an energy and a stress criteria. The coupled criterion is implemented in a representative volume element and it allows to decide whether the stress or the energy condition governs the cracking mechanism. It is found that the energy conditions predominates for very thin films whereas the stress condition can take place for thicker films. The initial density of cracks is determined and is in good agreement with the experimental measures. Further subdivisions, when increasing the load, are also predicted. Moreover, under some conditions, a master curve can rule the density of cracks function of the applied strain, showing a good agreement between predictions and experiments for a wide range of film thicknesses.     

Singular fields of a mode III interface crack in a power law hardening bimaterial

A high order of asymptotic solution of the singular fields near the tip of a mode III interface crack for pure power law hardening bimaterials is obtained by using the hodograph transformation. It is found that the zero order of the asymptotic solution corresponds to the assumption of a rigid substrate at the interface, and the first order of it is deduced in order to satisfy completely two continuity conditions of the stress and displacement across the interface in the asymptotic sense. The singularities of stress and strain of the zero order asymptotic solutions are –1/(n ₁+1) and –n/(n ₁+1) respectively (n=n ₁, n ₂ is the hardening exponent of the bimaterials). The applicability conditions of the asymptotic solutions are determined for both zero and first orders. It is proved that the Guo-Keer solution [23] is limited in some conditions. The angular functions of the singular fields for this interface crack problem are first expressed by closed form.     

Steady state propagation of a mode III interface crack between dissimilar viscoelastic media

The steady state propagation of a semi-infinite crack between two dissimilar viscoelastic solids is considered. By means of the Wiener-Hopf technique, the stress intensity factor is found as a function of the crack tip velocity and the material parameters. Results for an interface crack between an elastic and a viscoelastic medium are obtained as a special case. Various limiting cases are examined as a check on the accuracy of the results. Finally, graphs are presented which examine the salient features of the stress intensity factor.     

Energy release rate calculations by the finite element method

The present paper outlines a finite element method for calculating the energy release rate. The method is based on a continuum mechanics formulation of the virtual crack extension principle and can be used with linear elastic materials as well as materials following the deformation theory of plasticity. The formulation is easily incorporated into a finite element program, but can also conveniently be used as part of a post-processing program, which uses stress and displacement data from a finite element analysis to calculate the energy release rate. The present formulation can be used as a unified approach for 2-D and 3-D fracture problems and includes the effect of body forces and traction loading on the crack faces.     

Fracture dynamics analysis of mode III interface moving crack on functionally graded coating

The interface moving crack between the functionally graded coating and infinite substrate structure with free boundary is investigated in this paper. By application of the interface bonding conditions of the two media, all the quantities have been represented by means of a single unknown function. With the help of the exponent model of the shear modulus and density, the dual integral equation of moving crack problem is obtained by Fourier transform. The displacement is expanded into series form using Jacob Polynomial, and then the semi‐analytic solution of dynamic stress intensity factor is derived by Schmidt method. Dynamic stress intensity factor is influenced by those parameters such as crack velocity, graded parameter and coating height.     

The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body

The Energy Release Rate (ERR) for the quasi-static problem of a semi-infinite mode I crack propagating through an inhomogeneous isotropic linearly viscoelastic body is examined. The shear modulus is assumed to have a power-law dependence on depth from the plane of the crack and a very general behavior in time. A Barenblatt type failure zone is introduced in order to cancel the singular stress and a formula for the ERR is derived which explicitly displays the combined influences of material viscoelasticity and inhomogeneity. The ERR is calculated for both power-law material and the standard linear solid and the qualitative features of the ERR are presented along with numerical illustrations.     

Energy release rate along a three-dimensional crack front in a thermally stressed body

Based on a line-integral expression for the energy release rate in terms of crack tip fields, which is valid for general material response, a (area/volume) domain integral expression for the energetic force in a thermally stressed body is derived. The general three-dimensional finite domain integral expression and the two-dimensional and axisymmetric specializations for the energy release rate are given. The domain expression is naturally compatible with the finite element formulation of the field equations. As such it is ideally suited for efficient and accurate calculation of the pointwise values of the energy release rate along a three-dimensional crack front. The finite element implementation of the domain integral corresponds to the virtual crack extension technique. Procedures for calculating the energy release rate using the numerically determined field solutions are discussed. For illustrative purposes several numerical examples are presented.

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Résumé En sa basant sur une intégrale simple exprimant le taux de relaxation d'énergie afférant aux champs de contrainte à l'extrémité d'une fissure, expression applicable à la réponse d'un matériau quelconque, on a déduit une intégrale de domaine (superficielle ou volumique) décrivant l'énergie dans un corps soumis à contraintes thermiques.On fournit l'intégrale générale relative à un domaine fini tridimensionnel et á des cas particuliers bidimensionnels et axisymétriques, exprimant le taux de relaxation d'énergie. L'intégrale de domaine est naturellement compatible avec une formulation par éléments finis des équations de champ. Comme telle, elle convient idéalement pour un calcul facile et précis des valeurs ponctuelles du taux de relaxation de l'énergie le long du front d'une fissure tridimensionnelle. L'implantation d'éléments finis dans l'intégrale de domaine correspond à une technique d'extension virtuelle de la fissure. On discute des procédures de calculs du taux de relaxation de l'énergie, qui utilisent les solutions relatives au champ déterminées par voie numérique. A titre d'illustration, on présente plusieurs exemples numériques.

     

Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.     

The shielding effect of the imperfect interface on a mode III permeable crack in a layered piezoelectric sensor

The mechanical model is established for a piezoelectric sensor with a mode III permeable crack parallel to the imperfect interface. Fracture analysis is performed by the standard methods of Fourier transform and singular integral equation. Three conclusions are drawn: (a) the imperfect interface has a shielding effect on the crack parallel and very near to it; (b) the shielding effect depends on the structural stiffness and the distance between the crack and interface; (c) for the electrically permeable crack, mechanical imperfection has more remarkable shielding effect than dielectric imperfection does.     

Dynamic asymptotic mode III crack tip field in rate dependent materials

The asymptotic field at a dynamically growing crack tip in strain-rate sensitive elastic-plastic materials is investigated under anti-plane shear loading conditions. In the conventional viscoplasticity theory, the rate sensitivity is included only in the flow stress. However, it is often found that the yield strength is also affected by previous strain rates. The strain rate history effects in metallic solids are observed in strain rate change tests in which the flow stress decreases gradually after a rapid drop in strain rate. This material behavior may be explained by introducing the rate sensitivity in the hardening rule in addition to the flow rule. The strain-rate history effect is pronounced near the propagating crack where the change of strain rates take place. Effects of the rate dependency in the flow rule and the hardening rule on the crack propagation are analyzed. The order of the stress singularity in the asymptotic field is determined in terms of material parameters which characterize the rate sensitivity of the material. The results show that an elastic sector is present in the wake zone when the rate-dependency is considered only in the hardening rule. Terminal crack propagation speed is determined by applying the critical stress fracture criterion and the critical strain criterion to the asymptotic fields under the small scale yielding condition.     

Creep crack initiation at a free edge of an interface between submicron thick elements

To clarify the mechanics of time-dependent crack initiation at an interface edge in submicron thick elements due to creep, delamination experiments are conducted using a micro-cantilever bend specimen with a tin/silicon interface edge. After the specimen time-dependently deforms under a constant load, a delamination crack is initiated at the Sn/Si interface edge. In addition, the steady state creep property of Sn is estimated by performing an inverse analysis using a finite element method based on creep deformation experiments conducted for different specimens. Stress analysis using the obtained creep property reveals that stress and strain rate singularities exist at the Sn/Si interface edge under creep deformation. The intensity of the singular field time-dependently increases as the creep region expands, and eventually it becomes a steady state. The stress and strain rate intensities at the steady state correlate well with the crack initiation life, which indicates that the singular stress field near the interface edge governs the creep crack initiation.