This study attempts to clarify the conditions under which the J-integral is path-independent in U-and V-shaped notch problems. The key is to determine the contribution to the J-integral evaluated in the global coordinate system from the second component of the Jk-vector evaluated in the local coordinate system along the traction-free surfaces that form part of the integration path. It is found that the global J-integral is path-independent only if the projected contribution from the J2-integral to J vanishes. The J-integral for a V-shaped notch is, strictly speaking, pathdependent even under remote symmetrical loading. This is due to the fact that, unlike in the case of a line-or planecrack, the value of the J-integral calculated along a closed contour surrounding the V-shaped notch is dependent on the selection of the starting and ending points on the notch surface. In other words, the traction-free surface of the V-shaped notch does contribute to the J-integral due to the non-zero projected values induced from the J2-integral. For a U-shaped notch, the path-independence of the J-integral is established if the integration path completely encloses the notch root. This is because both the upper and lower notch surfaces of the U-shaped notch are parallel to the geometric symmetrical line (the x1-axis) and hence the projected values from the J2-integral vanish. Furthermore, it is found that the small arc at the root of the notch (whether U-or V-shaped) also contributes to the J-integral even if the remote loading is symmetrical. These conclusions are derived by detailed analytical manipulations and by numerical examples; the analytical solution obtained by Lazzarin and Tovo [Int. J. Fracture 78 (1996) 3] and Lazzarin et al. [Int. J. Fracture 91 (1998) 269] for stress field in the vicinity of a notch root in an infinite elastic plane is used to calculate the contribution induced from the arc with different radii. Some useful results for studying the fracture and fatigue of notches are discussed.

This paper has two goals. First, we propose the 'pseudotraction electric displacement' method for solving the interaction problem of multiple parallel cracks in transversely isotropic piezoelectric ceramics. Second, we present a fundamental understanding for the role that the electric displacement loading plays in the interaction problem. Detailed comparisons between the results under the compound mechanical electric loading conditions and those derived under purely mechanical loading conditions are performed. It is shown that the mechanical fracture parameters such as the stress intensity factors are no longer independent of the electric loading as they wofiuld be in single crack problems. Quite contrary, the electric displacement loading has a signicant in uence on the stress intensity factors, the total potential energy release rate and the mechanical strain energy release rate. This important conclusion is mainly due to the interaction effect, i.e., one of the multiple cracks releases the stresses and disturbs the electric fields near the other crack. It is also found that there are some special relative locations for the multiple parallel cracks at which the electric displacement loading has no effect on the Mode I stress intensity factor. However, the mechanical strain energy release rate has no such a property.

The primary goal of this research is to show the fundamental features of an interface crack in metal/piezoelectric bimaterials via Stroh's theory [Phil. Mag. 7 (1958) 625]. Based on the previous works [Phil. Mag. 7 (1958) 625; J. Mech. Phys. Solids 40 (1992) 739; Int. J. Fract. 119 (2003) L41; Singularities and neartip field intensity factors of piezoelectric interface cracks, J. Mech. Phys. Solids (in press)] and by considering a metal as a special piezoelectric material withextremely small piezoelectricity and extremely large permittivity (conductor), we obtain the two dominant parameters e and j for description of interface cracktip singularity in such bimaterials. Numerical results show that almost all of such bimaterials have the feature that the first parameter e vanishes whereas the second parameter j remains non-zero. An interface crack in these bimaterials always possesses the stress singularity r-1/±k at the crack tip. From the physical point of view, this implies that an interface crack in such bimaterials shows a feature with non-oscillating ingularity, which is far apart from previous results [Prik. Mat. Mekh. 39 (1975) 145; V.Z. Parton, B.A. Kudryavtsev, Electromagnetoelasticity, Gordon and BreachSc ience Publishers, New York, 1988; J. Mech. Phys. Solids. 51 (2003) 921] and our classical understanding in dissimilar elastic or anisotropic materials [Bull. Seism. Soc. Am. 49 (1959) 119; ASME J. Appl. Mech. 32 (1965) 400]. On the other hand, there is one exceptional metal/piezoelectric bimaterial withnon-z ero e and vanishing j, the oscillating stress singularity r-1/2ie at the crack tip is reached in this bimaterial. Consequently, metal/piezoelectric bimaterials are categorized into two classes: one withnon-zero j and vanishing e could be called as j-class metal/piezoelectric bimaterials and the other one with non-zero e and vanishing j could be termed as e-class metal/piezoelectric. Analysis of the crack-tip generalized stress field is performed. Of great significance is that: if a purely electric-induced interface crack growthoc curs in metal/piezoelectric bimaterials, it is most likely due to the shear mode II fracture rather than the open mode I, and then a purely remote electrical loading enhances the interface crack extension in such bimaterials. Only when an external tensile loading is applied, could the mode I fracture play a dominant role.

The path-independence of the J-integral considering discontinuities (e.g. microcracks) in a near-tip stress field is studied in detail. The well-known integral defined by Rice and the related Jk-vector discussed by Bergez, Cherepanov, and Herrmann and Herrmann are evaluated, respectively, along three different closed contours. The first of them is surrounding the tip of a semi-infinite crack only, the second encloses the discontinuities completely, while the third encloses both the tip and the discontinuities. It is found that there is a simple, but universal relation among three values of the J-integral corresponding to the contributions induced from the semi-infinite crack tip, the discon-tinuities and the remote stress field, respectively. This means that the interaction effect between a macrocrack and discontinuities can be considered as the redistribution of the J-integral arising from the existence of the discontinuities.

A semi-infinite interface macrocrack interacting with multiple oriented subinterface microcracks in the process zone near the macrocrack tip in metal/piezoelectric bimaterials is studied. After deriving the elementary solutions for a semiin

The J-integral analysis is performed for the plane problems of multiple subinterface microcracks near the tip of an interface macrocrack in bimaterial solids. The analysis starts from a general solution based on the "pseudo-traction" method which has been addressed thoroughly in homogeneous cases. The contribution to the J-integral induced from the subinterface microcracks is shown in a consistent relation with those induced from an interface macrocrack tip and the remote stress field. A new technique is developed to evaluate the second component (being expressed by J*2 in this paper) of the well-known Jk-vector of a subinterface crack for considering a contour enclosing the whole crack, which is necessary to evaluate the contribution to the J-integral arising from the subinterface microcracks. The consistency of the J-integral for numerical examples is proved, where two kinds of material combinations presented by Hutchinson et at. [ASME J. Appl. Mech., 1987, 54, 828-832] are considered. Some discussion and conclusions are then given which seem very useful in the investigation of microcrack shielding problems in bimaterial cases.

This paper addresses and alternative description for brittle solids with strongly interacting microcracks. The basic idea starts from the M-integral analysis customatil used in single crack problems. As an initial attempt, the discussion is limited to the infinite two-dimensional cases and the microcracks are assumed to be stationary. It is proved from the global-local contributed from the two components of the Jk vector (Knowles and Sternberg, 1972, Budiansky and Rice, 1973) and the coordinates of each microcrack center. The later is concerned not only with the crack tip SIFs, but also with the contribution arising from the traction-free surfaces of each crak (Herrmann and Herrmann, 1981). A detailed proof for the vanishing nature of the Jk vector along a closed contour surrounding all the microcracks is presented, from which the confsion about the dependence of the M integral on the origin selection of global coor-dinates is clarified. Two numerical examples are shown in tables and figures tocomfirm the dervied conclusions. It is shown that the M integralis equivalent to the decrease of the total potential energy of the microcracking soilds although the strongly interacting situations are taken into account. Therefore, a simple relation between the M integral and the L integral is established under the assumption mentioned above. It is concluded that the M-integral analysis, from the physical point of view, does play important role and provide an effective measure in evaluatin the damage level of brittle solids with strongly interacting and randomly distributed microcracks. Although only the stationary microcracks are considered in the present investigation, the derived conclusions could actually be extended to treat much more useful problems, in which the multi-cracks may become critical and may grow during loading histories.

The J-integral analysis is presented for the interaction problem between a semi-infinite interface crack and subinterface matrix microcracks in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack subjected to different loads and the fundamental solutions for an edge dislocation beneath the interface, the interaction problem is deduced to a system of singular integral equations with the aid of a superimposing technique. The integral equations are then solved numerically and a conservation law among three values of the J-integral is presented, which are induced from the interface crack tip, the microcracks and the remote field, respectively. The conservation law not only provides a necessary condition to con

This paper focuses on characterizing damaged anisotropic piezoelectric materials by using the principles of fracture mechanics. The interpretation and evaluation of the two components of the Jk-vector along contours enclosing strongly interacting microcracks in two-dimensional piezoelectric materials are presented. The conservation laws of the Jk-vector established by Budiansky and Rice (ASME J. Appl. Mech. 40 (1973) 201) for traditional non-piezoelectric materials and extended by Chen and Hasebe (Int. J. Fract. 89 (1998) 333) and Chen (Int. J. Solids Struct. 38 (2000a) 3193 and 38 (2000b) 3213) for interacting multiple cracks are reexamined for anisotropic piezoelectric materials containing inter-acting multiple cracks. The interaction problem for arrays of arbitrarily orientated and distributed microcracks subjected to mechanical and electrical loading is studied in detail. The contribution of the second component of the Jk-vector, evaluated in the local coordinate system that is attached to each microcrack, to the Jk-vector evaluated in the global coordinate system is calculated. It is found that the conservation laws of the Jk-vector are still valid in damaged piezoelectric materials, although in the present problem the elastic and electric fields are coupled which add compli-cations to the original formulations by Budiansky and Rice. In other words, the total contributions from the micro-cracks to both components of the Jk-vector evaluated in the global system vanish, provided that the selected closedcontour encloses all microcracks and/or discontinuities.