Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

Crack propagation under mode II dominance at stainless steel–epoxy interfaces with residual stresses and micro-scale roughness

This report describes investigations of grain boundary groove effects on mode II dominated interface fracture. The study focused on a specific interface between stainless steel and an epoxy adhesive. First, a finite element model was developed to simulate residual stresses and crack propagation. Second, the simulation results were compared with the experimental results from a previous study (Kanerva et al., 2013. Eng. Fract. Mech. 99, 147-158). Additional measurements were performed using atomic force microscopy. Based on the simulation, a 100-fold toughening effect due to the grain boundaries was determined. Implementation of flaws, in the form of interfacial voids, decreased the toughening effect by 35% and increased the mode II dominance significantly. The work underlines the practical importance of complete wetting by the adhesive and its necessary adherence to the grain boundary groove walls.     

Dynamic interaction between a sub-interface crack and the interface in a bi-material: time-domain BEM analysis

Summary Transient response of a sub-interface crack in a bi-material is studied with emphasis on the dynamic interaction between the crack and the interface, by combining the traditional time-domain displacement boundary element method (BEM) and the non-hypersingular traction BEM. Computations are performed for an unbounded bi-material with a crack subjected to impact tensile loading on its faces or incident impact waves and a bounded rectangular bi-material plate under remote impact tensile loading. Numerical results of the dynamic stress intensity factors (DSIFs) and dynamic interface tractions are presented for various material combinations and crack locations. It is shown that pronounced increases in DSIFs and the interface tractions may be caused in some cases because of the dynamic interaction between the crack and the interface.This work was initialized during the second author's stay at Institute of Mechanics, TU Darmstadt, Germany under the support of the Alexander von Humboldt Foundation. Discussion on the BEM formulation with Dr. Seelig is gratefully acknowledged. The first two authors are also grateful for the partial support by the China National Natural Science Foundation under Grant No. 10025211 and the NJTU Scientific Paper Fund (PD195).     

Interface crack between piezoelectric and elastic strips

Summary An interface crack between piezoelectric and elastic strips is analyzed using the theory of linear piezoelectricity. The combined out-of-plane mechanical and in-plane electrical loads are applied to the layered strip. Fourier transforms are used to reduce the problem to a pair of dual integral equations, which is then expressed in terms of a Fredholm integral equation of the second kind. The stress intensity factor is determined, and numerical analysis is performed and discussed. Received 22 September 1999; accepted for publication 3 May 2000     

Interface crack between a compressible elastomer and a rigid substrate with finite slippage

We study the deformation of a crack between a soft elastomer and a rigid substrate with finite interfacial slippage. It is assumed that slippage occurs when the interfacial shear traction exceeds a threshold. This leads to a slip zone ahead of the crack tip where the shear traction is assumed to be equal to the constant threshold. We perform asymptotic analysis and determine closed-form solutions describing the near-tip crack opening displacement and the corresponding stress distributions. These solutions are consistent with numerical results based on finite element analysis. Our results reveal that slippage can significantly affect the deformation and stress fields near the tip of the interface crack. Specifically, depending on the direction of slippage, the crack opening profile may appear more blunted or sharpened than the parabola arising from for the case of zero interfacial shear traction or free slippage. The detailed crack opening profile is determined by the constant shear traction in the slip zone. More importantly, we find that the normal stress perpendicular to the interface can increase or decrease when slippage occurs, depending on the direction of slippage and the shear traction in the slip zone.     

Initial kinking of an interface crack between two elastic media under tension and shear

The initial kinking of a thin fracture process zone near the tip of an interface crack between two elastic media under plane strain is studied using the Wiener–Hopf method. The zone is modeled by the discontinuity plane of the normal displacement. This plane is assumed to emerge from the crack tip at an angle to the interface. The angle between the fracture process zone and the interface is determined from the condition that the potential energy is maximum in the zone. The dependence of the length of the zone and its angle on the external load and other parameters is analyzed in the cases of biaxial tension and pure shear. The results obtained are compared with theoretical and experimental data reported by other researchers     

Moving interfacial crack between piezoelectric ceramic and elastic layers

An analysis is performed for the problem of a finite Griffith crack moving with constant velocity along the interface of a two-layered strip composed of a piezoelectric ceramic and an elastic layers. The combined out-of-plane mechanical and in-plane electrical loads are applied to the strip. Fourier transforms are used to reduce the problem to a pair of dual integral equations, which is then expressed in terms of a Fredholm integral equation of the second kind. The dynamic stress intensity factor(DSIF) is determined, and numerical results show that DSIF depends on the crack length, the ratio of stiffness and thickness, and the magnitude and direction of electrical loads as well as the crack speed. In case that the crack moves along the interface of piezoelectric and elastic half planes, DSIF is independent of the crack speed.     

Non-local theory solution for in-plane shear of through crack

A non-local theory of elasticity is applied to obtain the plane strain stress and displacement field for a through crack under in-plane shear by using Schmidt's method. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses finite at crack tip. Both the angular variations of the circumferential stress and strain energy density function are examined to associate their stationary value with locations of possible fracture initiation. The former criterion predicted a crack initiation angle of 54° from the plane of shear for the non-local solution as compared with about 75° for the classical elasticity solution. The latter criterion based on energy density yields a crack initiation angle of 80° for a Poisson's ratio of 0.28. This is much closer to the value that is predicted by the classical crack tips solution of elasticity.     

Perturbation approaches of a planar crack in linear elastic fracture mechanics: A review

One current challenge of linear elastic fracture mechanics (LEFM) is to take into account the non-linearities induced by the crack front deformations. For this, a suitable approach is the crack front perturbation method initiated by Rice (1985). It allows to update the stress intensity factors (SIFs) when the crack front of a planar crack is perturbed in its plane. This approach and its later extensions to more complex cases are recalled in this review. Applications concerning the deformation of the crack front when it propagates quasistatically in a homogeneous or heterogeneous media have been considered in brittle fracture, fatigue or subcritical propagation. The crack shapes corresponding to uniform SIF have been derived: cracks with straight or circular fronts, but also when bifurcations exist, with wavy front. For an initial straight crack, it has been shown that, in homogeneous media, in the quasistatic case, perturbations of all lengthscales progressively disappear unless disordered fracture properties yields Family and Vicsek (1985) roughness of the crack front. Extension of those perturbation approaches to more realistic geometries and to coalescence of cracks is also envisaged.     

Stability of an intersonic shear crack to a perturbation of its edge

A weight function matrix is developed for obtaining the stress singularity coefficients at the edge of a plane crack, moving uniformly at an intersonic speed while subjected to arbitrary shear loading. This is then utilised for deriving, to first order, the perturbations of these coefficients associated with a small spatially and temporally varying perturbation of its edge. The perturbation solution is employed, in conjunction with a simple fracture criterion, to investigate the stability of a uniformly moving intersonic crack, subjected to following loads.     

The sliding interface crack with friction between elastic and rigid bodies

The paper analyzes the frictional sliding crack at the interface between a semi-infinite elastic body and a rigid one. It gives solutions in complex form for non-homogeneous loading at infinity and explicit solutions for polynomial loading at the interface. It is found that the singularities at the crack tips are different and that they are related to distinct kinematics at the crack tips. Firstly, we postulate that the geometry of the equilibrium crack with crack-tip positions b and a is determined by the conditions of square integrable stresses and continuous displacement at both crack tips. The crack geometry solution is not unique and is defined by any compatible pair (b,a) belonging to a quasi-elliptical curve. Then we prove that, for an equilibrium crack under given applied load, the “energy release rate” G_tip, defined at each crack tip by the J_ε-integral along a semi-circular path, centered at the crack tip, with vanishing radius ε, vanishes. For arbitrarily shaped paths embracing the whole crack, with end points on the unbroken zone, the J-integral is path-independent and has the significance of the rate, with respect to the crack length, of energy dissipated by friction on the crack.     

Contact problem for the flat crack under two normally incident shear H-waves with wave mode-shifting

The frictional contact interaction of the finite crack edges in an infinite plane is studied for the case of normal incidence of two harmonic shear H-waves with multiple mode-shifted frequencies. Boundary integral equation method and constrained optimization algorithm are used for the problem solution. The forces of contact interaction and displacement discontinuity are analyzed. Influence of the waves frequencies on the stress intensity factor for different normalized wave numbers and coefficients of friction is considered here.     

Elastic–plastic and elastic anti-plane shear of two parallel cracks

Two infinite interacting parallel cracks in an elastic–plastic and in an elastic body under anti-plane strain (mode III) loading conditions are considered. The body is subjected to vanishing remote loading and the cracks are traction free. Closed-form solution is found for the elastic–plastic problem in terms of elementary functions, where the shape of the plastic boundary is obtained. The complete stress distribution is obtained in an inverse form i.e. physical coordinates are functions of stresses.     

Crack growth of an interface crack between two dissimilar magneto-electro-elastic materials under anti-plane mechanical and in-plane electric magnetic impact

This paper examines the dynamic response of an interface crack between two dissimilar magneto-electro-elastic materials subjected to the mechanical and electric magnetic impacts. The magneto-electric impermeable boundary conditions are adopted. Laplace and Fourier transforms and dislocation density functions are employed to reduce the mixed boundary value problem to Cauchy singular integral equations in Laplace transform domain, which are solved numerically. Lots of numerical results are given graphically in time domain. The effects of electric impact loading and magnetic impact loading on dynamic energy density factors are discussed. Crack growth and propagation is predicted. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

A semi-infinite interfacial crack between two bonded dissimilar elastic strips

Summary  This paper is concerned with a semi-infinite interfacial crack between two bonded dissimilar elastic strips with equal thickness. Solutions for the complex stress intensity factor (SIF) and energy release rate (ERR) are obtained in closed form under in-plane deformations. During the procedure, the mixed boundary-value problem is reduced by means of the conformal mapping technique to the standard Riemann–Hilbert problem. In some limiting cases, the present solutions can cover the results found in literature. Received 21 February 2002; accepted for publication 2 July 2002 X.-F Wu's work was supported in part by the Milton E. Mohr Research Fellowship (2001, 2002) of the Engineering College at University of Nebraska-Lincoln.     

Initial kinking of an interface crack between two elastic media

The initial kinking of a thin fracture process zone near the crack tip under plane strain is studied using the Wiener-Hopf method. The crack is located at the interface between dissimilar elastic media. The fracture process zone is modeled by a straight line of normal displacement discontinuity emerging from the crack tip at an angle to the interface. The angle between the process zone and the interface is determined from the condition of strain energy maximum in the process zone. The dependences of the length and angle of the process zone on the external load and other parameters of the problem are studied. The results are compared with theoretical and experimental data obtained by other researchers __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 28–41, October 2007.     

Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence

A Finite Element (FE) model is proposed to study the interaction between in-plane elastic waves and a crack of different orientations. The crack is modeled by an interface of unilateral contact with Coulombs friction. These contact laws are modified to take into account a pre-stress _σ0${\sigma }_0$ that closes the crack. Using the FE model, it is possible to obtain the contact stresses during wave propagation. These contact stresses provide a better understanding of the coupling between the normal and tangential behavior under oblique incidence, and explain the generation of higher harmonics. This new approach is used to analyze the evolution of the higher harmonics obtained as a function of the angle of incidence, and also as a function of the excitation level. The pre-stress condition is a governing parameter that directly changes the nonlinear phenomenon at work at the interface and therefore the harmonic generation. The diffracted fields obtained by the nonlinear and linear models are also compared.     

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     

Effect of plastic zone on the fatigue crack propagation behavior between two fatigue cracks

In this study, the fatigue crack propagation behavior in the stress interaction field between two different fatigue cracks is studied by experiment and finite element analysis. In the experiment, the offset distance between two cracks and the applied stress are varied to create different stress interaction fields. The size of the plastic zone area is used to examine the crack propagation path and rate. Three types of crack propagation in the interaction field were found by experiment, and the crack propagation behavior of two cracks was significantly changed as different stresses were applied. The size of the plastic zone obtained by finite element analysis can be used to explain crack propagation behavior qualitatively.