Dynamic stability of a propagating crack

In this work we investigate the stability of a straight two-dimensional dynamically propagating crack to small perturbation of its path. Willis and Movchan (J. Mech. Phys. Solids 43 (1995) 319; J. Mech. Phys. Solids 45 (1997) 591) constructed formulae for the perturbations of the stress intensity factors induced by a small three-dimensional dynamic perturbation of a nominally plane crack. Their solution is exploited here to derive equations for the in-plane and out-of-plane perturbations of the crack path making use of the Griffith fracture criterion and the principle of “local symmetry” (i.e the crack propagates so that local K_II=0). We consider a crack propagating in a body loaded by a pair of point body forces and subjected to a remote uniaxial stress, aligned with the direction of the unperturbed crack. We assume that the loading follows the crack as the crack advances and is such that the unperturbed crack is subjected to Mode I loading. We perform an analysis of the stability of the dynamic crack in a similar way as in earlier work (Obrezanova et al., J. Mech. Phys. Solids 50 (2002) 57) on the quasistatically advancing crack. We present numerical results illustrating the influence of the crack velocity on the crack stability. Numerical computations of the possible crack paths have been performed which show that at velocities of crack propagation exceeding about one-third of the speed of Rayleigh waves the crack may admit one or more oscillatory modes of instability.     

Theoretical and experimental studies on the extension of cracks subjected to concentrated loading near their faces to compare the criteria for mixed mode brittle fracture

The criteria of maximum tangential stress (MTS), maximum tangential principal stress (MTPS), maximum tangential strain (MTSN) and strain energy density (SED) are applied to problems of infinite sheets with straight cracks loaded at or very near their faces by in-plane point loads. Prediction of the direction of initial crack extension, critical load and unstable crack path are made. Unlike the results reported earlier by various investigators predictions by the four criteria differ substantially. The accuracy in respect of prediction of crack path is checked experimentally for a particular loading. The results indicate that a modification of the definition of the criteria of MTSN and SED is beneficial.     

The stress intensity factor history due to three-dimensional transient loading on the faces of a crack

Aprocedure is described for determining dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body, with the crack faces subjected to tractions that result in variation of the stress intensity factor along the crack edge. The procedure is based on integral transform methods and the properties of analytic functions of a complex variable. The procedure is illustrated for the case of a pair of opposed line loads suddenly applied on the crack faces along a line perpendicular to the crack edge. An exact expression is obtained for the resulting mode I stress intensity factor as a function of time for any point along the crack edge. Some features of the solution, as well as possible extensions of the procedure, are discussed.     

Oblique edge crack in an anisotropic material under antiplane shear

An oblique edge crack in an anisotropic material under antiplane shear loadings is investigated. The antiplane problems are formulated based on a linear transformation method. An anisotropic solid containing an edge crack subjected to concentrated forces is first considered. The stress intensity factor for the edge crack with concentrated forces is obtained from the solution of the transformed edge crack in an isotropic material which is solved by using conformal mapping technique and complex function theory. The solution of the edge crack under concentrated loads is used to construct the stress intensity factor for the oblique edge crack in the anisotropic material subjected to antiplane distributed loads. Some numerical computations are carried out to calculate the stress intensity factors for the edge crack in inclined orthotropic materials subjected to point forces as well as distributed tractions.     

Some new aspects of boundary conditions at cracks in piezoelectrics

Fracture mechanical investigations of piezoelectric materials as components of smart structures have become popular in the last 30?years. In the early years of research, boundary conditions at crack faces have been adopted from pure mechanical systems under the assumption that boundaries were traction free. From the electrostatic point of view, cracks have been assumed to be either free of charge or fully permeable. Later, limitedly permeable crack boundary conditions have become popular among the community, nevertheless still assuming traction-free crack faces. Recently, the theoretical framework has been extended to include electrostatically induced mechanical tractions in crack models yielding a significant crack closure effect. However, these models are still simple, neglecting, e.g., the piezoelectric field coupling. In this work, we present an extended model for crack surface tractions yielding some interesting effects. In particular, the orientation of the electrical field with respect to the poling axis becomes important. Furthermore, applying a collinear stress parallel to the crack faces influences the Mode-I stress intensity factor and a Mode-II shear loading couples to the Mode-I SIF.     

Perturbation analysis of Mode III interfacial cracks advancing in a dilute heterogeneous material

The paper addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as microcracks, rigid line inclusions and voids. It is assumed that the bonding between dissimilar elastic materials is weak so that the interface is a preferential path for the crack. The perturbation analysis is made possible by means of the fundamental solutions (symmetric and skew-symmetric weight functions) derived in Piccolroaz et al. (2009). We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate “effective” tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects. For a stable propagation of the crack, the perturbation of the stress intensity factor induced by the defects is then balanced by the elongation of the crack along the interface, thus giving an explicit asymptotic formula for the calculation of the crack advance. The method is general and applicable to interfacial cracks with general distributed loading on the crack faces, taking into account possible asymmetry in the boundary conditions.The analytical results are used to analyse the shielding and amplification effects of various types of defects in different configurations. Numerical computations based on the explicit analytical formulae allows for the analysis of crack propagation and arrest.     

A dynamic model of bridging fiber pull-out of composite materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane is carried out. In this paper a dynamic model of bridging fiber pull-out is presented for analyzing the distributions stress and displacement of composite materials with the internal central crack under the loading conditions of an applied non-uniform stress and the traction forces on crack faces yielded by the fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, the fiber breaks and hence the crack propagation should occur in self-similar fashion. By reducing the dynamic model to the Keldysh–Sedov mixed boundary value problem, a straightforward and easy analytical solution can be attained. When the crack extends, its fibers continue to break. Analytical study on the crack extension under the action of an inhomogeneous point force Px/t, Pt is obtained for orthotropic anisotropic body, respectively; and it can be utilized to attain the concrete solutions of the model by the ways of superposition.     

The fracture mechanics of slit-like cracks in anisotropic elastic media

U method of continuously distributed dislocations, the problem of a slit-like crack in an arbitrarily-anisotropic linear elastic medium stressed uniformly at infinity is formulated and solved. The crack faces may be either freely-slipping or loaded by arbitrary equal and opposite tractions. If there is no net dislocation content in the crack, then the tractions and stress concentrations on the plane of the crack are independent of the elastic constants and the anisotropy; the same is true of the elastic stress intensity factors. The crack extension force depends on anisotropy only through the inverse matrix elements K _mg⁻¹, where [K] is the pre-logarithmic energy factor matrix for a single dislocation parallel to the crack front. Numerical results for crack extension forces are presented for three media of cubic symmetry.     

Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials

The antiplane shear deformation problem of two edge-bonded dissimilar isotropic wedges is considered. In the case when the sum of the two apex angles is equal to 2π, the problem reduces to that of two edge-bonded dissimilar materials with an interfacial crack subjected to concentrated antiplane shear tractions on the crack faces. An explicit expression is extracted for the stress intensity factor at the crack tip. In the special cases of different combinations of the apex angles, the obtained expression for the stress intensity factor may be simplified and relations of a simpler form are given for the stress intensity factor. It is shown that the stress intensity factor is dependent on the material properties as well as the geometry and loading. However, in special cases of equal apex angles as well as the case of similar materials the dependency of the stress intensity factor on the material properties disappears.     

The singular elastostatic field due to a crack in rubberlike materials

Within the framework of finite-strain elastostatics an asymptotic analysis is carried out in order to calculate the singular field near the crack tip in a slab under conditions of plane deformation. A class of Ogden-Ball hyperelastic rubberlike materials and general loading conditions ensuring vanishing tractions on the crack faces near the crack tip are considered. It is shown that the singular deformation field near the crack tip can be specified by applying a rigid body rotation with a subsequent parallel translation to a so-called canonical field. The adjective canonical is adopted here to denote the field with symmetrically opening crack faces, just resembling the displacement field of the symmetric mode in linear elastic fracture mechanics. No analogy with the antisymmetric mode is possible, and the crack equilibrium criterion requires only one stress intensity factor to be determined.     

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).     

Stability analysis of crack path using the strain energy density theory

Prediction of crack growth path is a pre-requisite for estimating the final shape of broken solids and structures. Crack path in broken specimens provides information for the loading conditions just before fracture. Experiments on brittle materials, pre-cracked specimens of the same geometry under similar loading conditions, however, may yield different crack trajectories at times. The existing theories for the prediction of the crack path are based on the perturbation method combining the analytical and finite elements methods. They require a knowledge of the toughness equations. Moreover, they can only be applied to specimens with simple geometry and loadings.A different approach is used in the present work. The finite element technique is used to calculate the strain energy density (SED) contours. The predicted trajectory of the crack during unstable propagation is assumed to coincide with the minimum of the strain energy density function according to the SED criterion.The degree of crack path stability depends on the sharpness of the SED oscillations. This simple method offers a reliable prediction of the crack path stability for two as well as three-dimensional problems with complex geometry structures and arbitrary loadings. To be specific, both the TPB and DCB specimens are analysed. The findings are in good agreement with the theoretical and experimental results in the literature.     

Full field analysis of an anti-plane interface crack subjected to dynamic body forces

In this study, the transient full field response of an interface crack between two different media subjected to dynamic body force at one material is investigated. For time t < 0, the bimaterial medium is stress free and at rest. At t = 0, a concentrated anti-plane dynamic point loading is applied at the medium as shown in Fig. 1. The total wave field is due to the effect of this point loading and the scattering of the incident waves by the interface crack. An alternative methodology that is different from the conventional superposition method is used to construct the reflected, refracted and diffracted wave fields. A useful fundamental solution is proposed in this study and the full field solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying an exponentially distributed traction (in the Laplace transform domain) on the interfacial crack faces. The Cagniard–de Hoop method of Laplace inversion is used to obtain the transient solution in time domain. Exact transient closed form solutions for stresses and stress intensity factors are obtained. Numerical results for the time history of stresses and stress intensity factors during the transient process are discussed in detail.     

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.     

Crack initiation and growth in contact problems

The problem of crack initiation and growth in contact problems is studied by the strain energy density theory. This is accomplished by considering each point of the body as a possible source of crack initiation. The direction of crack initiation is determined by calculating the minimum value of the strain energy density function along a circular core area surrounding the point. The crack initiates from the point with the maximum of the local minimum values of the strain energy density function. Considered is a circular disc subjected to two equal and opposite forces and a circular cylindrical body pressed by a cylindrical punch. The stress field for the first case is obtained from the classical elasticity solution, while a finite element code is used for the second case. The locations of fracture initiation and the subsequent fracture trajectories for fast unstable fracture are determined.     

Experimental analysis of crack propagation stability in single edge notch specimens

An experimental technique for analyzing the crack propagation stability of single edge notch specimen is discussed. The testing rig and its calibration procedure are described. Analyzed are the crack path under biaxial loading and its correlation with theoretical predictions. The mode II influence on the crack path is also presented. Use is made of the computer program ANGCRK. Fracture criteria are applied to determine the stability of crack growth. Stability is assumed to prevail when the second term of the eigenfunction series expansion is negative. Otherwise, the crack growth could be unstable.     

In-plane perturbation of the tunnel-crack under shear loading II: Determination of the fundamental kernel

For any plane crack in an infinite isotropic elastic body subjected to some constant loading, Bueckner–Rice's weight function theory gives the variation of the stress intensity factors due to a small coplanar perturbation of the crack front. This variation involves the initial SIF, some geometry independent quantities and an integral extended over the front, the “fundamental kernel” of which is linked to the weight functions and thus depends on the geometry considered. The aim of this paper is to determine this fundamental kernel for the tunnel-crack. The component of this kernel linked to purely tensile loadings has been obtained by Leblond et al. [Int. J. Solids Struct. 33 (1996) 1995]; hence only shear loadings are considered here. The method consists in applying Bueckner–Rice's formula to some point-force loadings and special perturbations of the crack front which preserve the crack shape while modifying its size and orientation. This procedure yields integrodifferential equations on the components of the fundamental kernel. A Fourier transform in the direction of the crack front then yields ordinary differential equations, that are solved numerically prior to final Fourier inversion.     

Growth of an arc-crack in bonded dissimilar materials

The problem of growth of a crack lying along the interface of a circular inclusion embedded in an infinite plate is studied within the framework of linear elasticity. The plate is subjected to a uniform uniaxial stress at infinity at any angle of inclination relatively to the crack. The critical load for unstable crack growth, the angle of initial crack extension and the subsequent crack path are investigated using the strain energy density fracture criterion. The combined effect of crack length and orientation on the fracture stress is considered for the case of an aluminum-epoxy composite.     

The effect of load biaxiality on crack growth in non-linear materials

The influence of load biaxiality on the stress field and fracture behavior of a cracked plate is investigated. Considered is a square plate containing a central through the thickness crack and subjected to a biaxial loading perpendicular and parallel to the crack plane. The stress field of the plate is analyzed by a finite element code based on incremental plasticity and the von Mises yield condition. A method based on the strain energy density theory is used to determine the critical stress for crack initiation. It was found that the equi-biaxial loading mode induces the smallest plastic zones, while the critical applied stress for crack initiation becomes maximum. Quite the contrary happens for the shear loading system which causes the largest plastic zones and the minimum applied stress values fro crack growth. Results showing the dependence of the above quantities on the biaxiality of the applied stress are presented in graphical form.     

Path Independent Integral for an Elliptical Hole in a Plate under Tension for Plane Stress Deformation Theory

An exact expression is derived for a path independent integral surrounding an elliptical hole in an infinite plate with tensile tractions at infinity for plane stress loading conditions. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under proportional loading and small strain assumptions. This problem may serve as a simple classroom example for the derivation of a relationship between crack opening displacement and path independent integral for a nonlinear elastic material satisfying the Tresca yield condition.