An interface crack with an arti®cial contact zone at the right-hand side crack tip between two piezoelectric semi-in®nite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x 2 , the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the wellknown oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial.

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An interface crack of ®nite length is considered between two semi-in®nite planes with an arti®cial contact zone at one of the two crack tips. A transcendental equation and certain simple asymptotic formulas are established for the real contact zone (in the Comninou-Dundurs sense) in terms of the stress intensity factors (SIFs) of the considered model. In these terms analytical expressions are also provided for the energy release rate and for the SIF of the classical interface crack model with an oscillating singularity at the crack tip. The appropriate length of the arti®cal contact zone is shown to be attainable on the basis of the analysis of the stresses at the crack tip. The use of the proposed model is suggested for integrity assessment of inhomogeneous structural elements of composites containing interface cracks.Interfacial and intergranular fracture processes are common in composite materials and they strongly in¯uence the material overall strength properties. This is the main reason why special attention has been devoted to the interface crack problem in the literature, starting from the fundamental papers [1], [2], [3], [4], [5]. They initiated the classical interface crack model with oscillating singularities at the crack tips. This model was essentially developed in the recent works [6] and [7], and was applied to the problems of interface crack kinking and propagation [8], [9], [10], [11], [12]. Contact models for interface cracks were initially introduced and investigated numerically by Comninou [13±15] and Dundurs and Comninou [16]. Analytical treatments of this approach have been performed in [17], [18, 22], [19±21]. An overview of some results related to interface cracks was given in [23].In spite of an existing great number of essential results of the problem in question, some dif®culties in de®nition of the fracture mechanical parameters at interface crack tips have still not been overcome. The stress intensity factors (SIFs) of the classical interface crack model have improper dimensions and their numerical determination requires some interpolation [10] that is appropriate only for small values of the bimaterial constant. The use of the contact zone approach (in the Comninou-Dundurs sense) is connected with dif®culties concerning the determination of the contact zone length, and the solution of the corresponding nonlinear boundary value problem calls essential problems for ®nite sized composites.In Ref.[25], an arti®cial contact zone for an interface crack has been introduced, and the exact analytical solution of the corresponding mathematical problem has been found. The contact zone result [13,18] was obtained as a particular case of this solution. Contrary to the traditional contact zone approach, the solution expounded in [25] suggested an appropriate

Summary. The intent of this paper is to apply the technique of discrete Fourier transforms (DFT) to the computation of the stress and strain fields around holes in an externally loaded two-dimensional representative volume element (RVE). This is done to show that DFT is capable to handle geometries with rather sharp corners as Well as steep gradients in material properties which is of importance for modeling changes in micro-morphology. To this end DFT is first briefly reviewed. In a second step it is applied to the appropriate equations which characterize a linear-elastic as well as a time-independent elastic-plastic, heterogeneous material subjected to external loads. The equivalent inclusion technique is used to derive a functional equation which, in principle, allows to compute numerically the stresses and strains within an RVE that contains heterogeneities of arbitrary shape and arbitrary stiffness (in comparison to the surrounding matrix). This functional equation is finally specialized to the case of circular and elliptical holes of various slenderness which degenerate into Griffith cracks in the limit of a vanishing minor axis. The numerically predicted stresses and strains are compared to analytical solutions for problems of the Kirsch type (a hole in an large plate subjected to tension at infinity) as well as to finite element studies (for the case of time-independent elastic/plastic material behavior).