13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- the crack terminating at the interface of two different materials is, for infinitesimally small crack increment, zero or infinite (depending on the singularity type). Thus the classical Griffith approach cannot be used. To bypass this problem, a theory of Finite Fracture Mechanics (FFM) can be employed – see e.g. references [20,21]. Infinitesimal crack increment is replaced by a finite crack increment for which the change of the potential energy can be calculated. The essence of the FFM consists in the key assumption that crack propagation is a discontinuous process occurring in finite steps, rather than continuously and smoothly as in the traditional LEFM theory [20,22-24]. Mathematically, instead of using the differential form of the Griffith energy balance, the integral formulation of the Griffith criterion is applied. Such approach is of particular importance, for instance, in the case of a crack crossing thermo-elastically mismatched interfaces, where the energy release rate is either zero or infinite and, as a consequence, the differential approach fails. For example, if the crack penetrates from material layer 1 to layer 2, the concept of FFM states that the crack will follow the path which maximizes the additional energy ΔWp released in the fracture process [24], as given by: ( )2 p p c p W G a Δ =δΠ − . (1) Here, Gc (2) is the toughness of the next layer to which crack penetrates, δΠp is a change of the potential energy between the original and new crack position, and ap stands for an increment of the new crack. Hereafter the concept of the Finite fracture mechanics is applied. Matched asymptotic expansions procedure, see e.g. [17-21], is used to derive the change of potential energy due to the perturbation caused by a single or branched crack extension of total length ap = ab or a straight penetrating crack extension of length ap (in several possible crack propagation directions φp – see Figure 3). a) b) c) Figure 3. Scheme of the a) crack terminating at the interface of M1 and M2, b) single crack deflection and c) crack bifurcation (branching) and local coordinate systems in the inner domain, where the crack extension length ap=ab/2+ ab/2 = 1 M2 M1 y1 y‘2 y2 y‘1 90- φp y“1 -(90- φp) ab/2 ab/2 y“2 Main crack y1 y‘2 y2 y‘ 1 90- φp ap Main crack M2 M1 Ωin Ωin x1 x2 M2 M1 Magnification 1/ε ×
RkJQdWJsaXNoZXIy MjM0NDE=