Elastic–plastic analysis of interaction between an interfacial crack and a subinterfacial microcrack in bi-materials

In the present work the finite element method is used to analyze the effect of interaction between an interfacial crack and a microcrack in ceramic/aluminum bi-materials. The behaviour is analyzed by the determination of the J integral, the plastic zone at the tips of the interfacial crack and the microcrack. The effects of longitudinal and transversal distance between the tips of the two cracks and the rotation of the microcrack are examined. The obtained results allow us to deduce a correlation between the position of the microcrack and the J integral and the plastic zone.The obtained results shows that the J integral at the interfacial crack tip reaches a maximum value when the microcrack is moved in the vicinity of the interfacial crack. With this distance the effect of interaction is marked more; the stress field at the microcrack tip and that of the interfacial crack generates only one plastic zone at the interfacial crack tip. The maximum size of the plastic zone is localised at the interfacial crack tip. Those of the two tips of the microcrack are very weak and even negligible in front of the zone plasticized at the interfacial crack tip.     

The interaction between screw dislocations and an interfacial blunt crack and a sharp crack

The interaction between screw dislocations and an interfacial blunt crack and a sharp crack under loads at infinity is dealt with. Utilizing the Muskhelishvili complex variable method, the closed form solutions are derived. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The results show that the shielding effect increases with the increase of the shear modulus and the distance between the two cracks, but decreases with the increase of dislocation azimuth. The critical loads at infinity for dislocation emission increase with the increment of the emission angle and the distance.     

The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading

The primary objective of this paper is to study the influence of the electroelastic interactions on the stress intensity factor in bonded layers of piezoelectric and orthotropic materials containing a crack along the interface under antiplane shear. Attention is given to a two-layer hybrid laminate formed by adding a layer of piezoelectric ceramic to a unidirectional graphite/epoxy composite or an aluminum layer. Electric displacement or electric field is prescribed on the surfaces of the piezoelectric layer. The problem is formulated in terms of a singular integral equation which is solved by using a relatively simple and efficient technique. A number of examples are given for various material combinations. The results show that the effect of the electroelastic interactions on the stress intensity factor and the energy release rate can be highly significant. This revised version was published online in July 2006 with corrections to the Cover Date.     

The interaction between a crack and an inclusion

A numerical method described recently[1] is used here to obtain the stress intensity factor for a crack near an inclusion. Results for the variation of the stress intensity factor with the distance of the crack tip from the inclusion, are shown graphically.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

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.     

The interaction between a penny-shaped crack and a spherical inhomogeneity in an infinite solid under uniaxial tension

Summary This paper presents an approximate three-dimensional analytical solution to the elastic stress field of a penny-shaped crack and a spherical inhomogeneity embedded in an infinite and isotropic matrix. The body is subjected to an uniaxial tension applied at infinity. The inhomogeneity is also isotropic but has different elastic moduli from the matrix. The interaction between the crack and the inhomogeneity is treated by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. The stress intensity factor at the boundary of the penny-shaped crack and the stress field inside the inhomogeneity are evaluated in the form of a series which involves the ratio of the radii of the spherical inhomogeneity and the crack to the distance between the centers of inhomogeneity and crack. Numerical calculations are carried out and show the variation of the stress intensity factor with the configuration and the elastic properties of the matrix and the inhomogeneity.     

The interaction between a dislocation and a crack

The solution is given in closed form to the problem. of the elastic field round a dislocatiop ,and a,; crack in any relative position when the field is independent of one of the three cartesian co-ordinates,, The, analysis holds for the most general anisotropy.

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Zusammenfassung Die Lösung wird dem Problem des elastischen Feldes rund um eine Verschiebung und einen Riss in irgendeiner relativen Stellung in geschlossener Form gegeben, wenn das Feld von den drei kartesischen Koordinaten unabhängig ist. Die Analyse gilt fiir die generalste Anisotropie.

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Résumé Le pròbleme du champ de tension élastique autour d' une dislocation et d' une fissure en positions relatives quelconques est résolu sous une forme générale dans le cas d' une champ élastique indépendant d' une des trois coordonnees cartésiennes.L' analyse effectuée couvre le cas de 1' anisotropie la plus générale.

     

Impact behavior of an inclined edge crack in a layered medium with a graded nonhomogeneous interfacial zone: antiplane deformation

Summary The impact response of an inclined edge crack in a layered medium with a functionally graded interfacial zone is investigated under the state of antiplane deformation. The interfacial zone is modeled by a nonhomogeneous interlayer having the power-law variations of shear modulus and mass density between the coating and the substrate of dissimilar homogeneous properties. Based on the Laplace and Fourier integral transform technique and the coordinate transformations of basic field variables, the transient crack problem is reduced to the solution of a singular integral equation with a generalized Cauchy kernel in the Laplace transform domain. The crack-tip response in the physical domain is recovered through the inverse Laplace transform to evaluate the dynamic mode III stress intensity factors as functions of time. The peak values of the dynamic stress intensity factors are further obtained versus the crack orientation angle, addressing the effects of crack obliquity on the overshoot characteristics of the transient crack-tip behavior for various combinations of material and geometric parameters of the layered medium.     

A nonlinear bilayer beam model for an interfacial crack in dielectric bimaterials under mechanical/electrical loading

A bilayer beam model is extended to study the fracture behavior of dielectric interfacial cracks. In this model, a semi-infinite crack with an original opening value is oriented along the interface between two dielectric layers which are under mechanical/electrical loading. Taking into account the effect of the electrostatic traction on the interfacial crack, a nonlinear analytical solution is derived, along with also a developed finite element analysis method where a special constitutive equation for the capacitor element in ANSYS is utilized to simulate the electrostatic tractions. Both the analytical and numerical solutions predict the same results which further show that the elastic and dielectric mismatches can play a significant role in the interfacial cracking behavior under mechanical and electrical loading. Furthermore, the electrostatic tractions may cause hysteresis loops in the curve of crack opening versus applied mechanical displacement or versus applied electric voltage. An applied mechanical load is the driving force for the interfacial cracking, while an applied electric field retards it.     

Analytical solution for an interfacial crack subjected to dynamic anti-plane shear loading

Summary A closed form solution for an interfacial crack problem is obtained. The crack lies on the interface of two bonded dissimilar orthotropic strips. Its surfaces are subjected to a dynamic anti-plane shear traction. Separation of variables technique is employed, to reduce the problem to a singular system of triple series equations, then to a singular integral equation. That is solved exactly, such that the asymptotic stress field distribution and the stress intensity factor are obtained in closed form expressions. The validity of the solution is proved. Further, a parametric study is introduced to investigate the effects of elastic and geometric characteristics of the composition on the values of the dynamic stress intensity factor (DSIF).     

The interaction between a crack and a dislocation dipole

This paper presents the results of an analysis which considers the interaction between a semi-infinite crack and a dislocation dipole. Applying the operator derived by Denda [1] to the crack/dislocation interaction solution developed by Lo [2], explicit expressions are obtained for the stress intensity factors at the tip of the crack. Results are computed and discussed for a variety of geometrical configurations, with the intent of developing an understanding of the effects of position and orientation of the dislocation dipole on crack tip shielding and antishielding. The solution can be used as a Green's function in semi-empirical analyses such as the one proposed by Chudnovsky [3], where the interaction between a crack in a polymeric material and the damage (crazing) which surrounds it is solved by experimentally measuring the crack opening displacements of the crazes and calculating the amount of toughening caused by the damage.

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Résumé On présente les résultats d'une analyse prenant en compte l'interaction entre une fissure semi-infinie et une paire de dislocations. En appliquant un opérateur dû à Denda à la solution d'interaction fissure/dislocation proposée par Lo, on obtient des solutions explicites pour les facteurs d'intensité de contraintes à l'extrémité de la fissure. Les résultats sont calculés et discutés pour une gamme de configurations géométriques, dans le but de développer une intelligence des effets de la position et de l'orientation de la paire de dislocations sur l'effet de couverture ou de non-couverture à l'extrémité de la fissure. La solution peut être utilisée sous la forme d'une fonction de Green dans les analyses semi-empiriques telles que celles proposées par Chudnovsky, où l'interaction entre une fissure dans un matériau polymère et l'endommagement (criques) qui l'environne est résolue par une mesure expérimentale des COD des criques et par le calcul de l'intensité du durcissement associé à l'endommagement.

     

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

Summary. The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Möbius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.     

Dynamic fields near a collinear macrocrack-microcrack configuration under antiplane impact loading

Dynamic interaction between a macrocrack and a collinear, neighboring microcrack is investigated. The mathematical formulation presented here is two-dimensional, and for a state of antiplane strain. A time-domain boundary integral equation method is presented to calculate elastodynamic stress intensity factors generated by the incidence of stress (or displacement) pulses on single cracks and systems of two collinear cracks. As a check of the time-stepping technique presented here, numerical calculations have been carried out for pulse incidence on a single crack, and the results have been compared with those of other authors. For macrocrack-microcrack configurations the variation of the effective stress intensity factors at the tips of the macrocrack with time, with the microcrack location, and with the microcrack size will be displayed and discussed.

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Résumé On étudie l'interaction dynamique entre une macrofissure et une microfissure colinéaire voisine. La formulation mathématique présentée est à deux dimensions et relative à un état de déformations antiplanaires. On présente une méthode à intégrales permettant de calculer les facteurs d'intensité de contraintes élastodynamiques dus à l'effet de pics de contraintes ou de déplacements sur des fissures simples ou sur des systèmes de deux fissures colinéaires. Pour contrôler la technique présentée, des calculs numériques sont effectués pour l'effet d'une impulsion sur une fissure simple, et les résultats sont comparés avec ceux d'autres auteurs. Pour des configurations macro-microfissure, on établit et on discute la variation des facteurs d'intensité de contraintes aux extrémités de la macrofissure en fonction du temps, de la position de la microfissure et de sa dimension.

     

Electromagnetoelastic behavior induced by a crack under antiplane mechanical and inplane electric impacts

Based on the piezoelectromagnetism theory, the dynamic problem of a crack of finite length embedded in a polarized ceramic under the action of antiplane mechanical impact and inplane electric impact is considered. The basic equations are simplified to two decoupled wave equations. Integral transform techniques are employed to reduce the associated initial-boundary value problem to integral equations. By using numerical methods, the resulting integral equations are solved and dynamic field intensity factors are presented graphically. A comparison of the dynamic results and the quasi-static results for dynamic field intensity factors near the crack tips is made, and the effect of mechanical impact on the dynamic magnetic field intensity factor is examined.     

Anisotropic effective moduli of microcracked materials under antiplane loading

This study focuses on the prediction of the anisotropic effective elastic moduli of a solid containing microcracks with an arbitrary degree of alignment by using the generalized self-consistent method (GSCM). The effective elastic moduli pertaining to anti-plane shear deformation are discussed in detail. The undamaged solid can be isotropic as well as anisotropic. When the undamaged solid is isotropic, the GSCM can be realized exactly. When the undamaged solid is anisotropic it is difficult to provide an analytical solution for the crack opening displacement to be used in the GSCM, thus an approximation of the GSCM is pursued in this case. The explicit expressions of coupled nonlinear equations for the unknown effective moduli are obtained. The coupled nonlinear equations are easily solved through iteration.     

Comparative study of time and frequency domain BEM approaches in frictional contact problem for antiplane crack under harmonic loading

Two different boundary element methods (BEM) for crack analysis in two dimensional (2-D) antiplane, homogeneous, isotropic and linear elastic solids by considering frictional contact of the crack edges are presented. Hypersingular boundary integral equations (BIE) in time-domain (TD) and frequency domain (FD), with corresponding elastodynamic fundamental solutions are applied for this purpose. For evaluation of the hypersingular integrals involved in BIEs a special regularization process that converts the hypersingular integrals to regular integrals is applied. Simple regular formulas for their calculation are presented. For the problems solution while considering frictional contact of the crack edges a special iterative algorithm of Udzava's type is elaborated and used. Numerical results for crack opening, frictional contact forces and dynamic stress intensity factors (SIFs) are presented and discussed for a finite III-mode crack in an infinite domain subjected to a harmonic crack-face loading and considering crack edges frictional contact interaction using the TD and FD approaches.     

The characteristics of an inclined crack in anisotropic paperboard under biaxial loading

In this paper, a fracture mechanics method was applied for the evaluation of crack behaviour in anisotropic paperboard subjected to biaxial uniform loading. The experiment was performed to determine the crack propagation angle and the fracture strength of paperboard under biaxial loading with the cruciform specimen optimized by FEM simulation. The effects of biaxial loads on the critical stress ratio and crack propagation angle for various inclination angles were investigated. The experimental results were compared with theoretical results, which were calculated by using the Normal Stress Ratio Criteria. The experimental results for crack propagation angle and critical stress show good agreement with theoretical results.     

On a crack problem in an infinite circular cylinder with a penny-shaped crack under a transient thermal loading

In this paper we deal with finding the stress intensity factors under the transient thermal loading in a circular cylinder with infinite length containing a penny-shaped crack. Variations of the stress intensity factors with time, which are closely related with the crack propagations, are obtained and illustrated in figures. From these figures we can obtain useful suggestions respecting crack propagation.

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Résumé On traite, dans cette étude, de la recherche des facteurs d'intensité de contraintes dans un cylindre circulaire de longueur infinie comportant une fissure en forme d'angle et soumise à sollicitation thermique en régime transitoire.On obtient, et on donne des valeurs à titre d'exemples pour les variations de facteur d'intensité des contraintes en fonction du temps, qui sont en relation étroite avec la propagation de la fissure. Ces valeurs conduisent à des suggestions utiles en ce qui regarde la propagation d'une fissure.