A First-Order Perturbation Analysis of Crack Trapping by Arrays of Obstacles

A first-order perturbation analysis is presented for the configuration of an initially straight crack front which is trapped against forward advance by contact with an array of obstacles {i.e., regions of higher fracture toughness than their surroundings). The problem is important to the micromechanics of crack advance in brittle, locally heterogeneous solids. The formulation is based on a linear perturbation result for the stress intensity factor distribution along the front of a half-plane crack when the location of that front differs moderately from a straight line. The trapping solutions for a periodic array of blocking rectangular obstacles are given using an analogy to the plane stress Dugdale/BCS elastic-plastic crack model. For a periodic array of obstacles with a given spacing and size in the direction parallel to the crack front, the obstacle shape may affect the limit load at which the crack breaks through the array. When such effects are examined within the range of validity of the linear perturbation theory, it is found that obstacles whose cross-sections fully envelop a critical reference area give the maximum limit load while others are broken through at lower load levels. We also formulate a numerical procedure using the FFT technique and adopting a "viscoplastic" crack growth model which, in an appropriate limit, simulates crack growth at a critical stress intensity factor. This is applied to show how a crack front begins to surround and penetrate into various arrays of round obstacles (with a toughness ratio of 2) as the applied load is gradually increased. The limitations of the first-order analysis restrict its validity to obstacles only slightly tougher than the surrounding elastic medium. Recently, Fares (1988) analyzed the crack trapping problem by a Boundary Element Method (BEM) with results indicating that the first-order linear analysis is acceptable when the fracture of toughness of the obstacles differs by a moderate amount from that of their surroundings (e.g., the toughness ratio can be as large as 2 for circular obstacles spaced by 2 diameters). However, the first-order theory is not only quantitatively inaccurate, but can make qualitatively wrong predictions when applied to very tough obstacles.