Issue 41

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 41 (2017) 175-182; DOI: 10.3221/IGF-ESIS.41.24 178 globally smooth, with the effect of friction and roughness built in a rigid-plastic constitutive relationship. Let us consider the relative tangential and normal displacements between two points situated on opposite positions on the crack surfaces, Fig. 2(a):         w w t x n y y x u u u u (9) where superscripts + and - indicate the superior and inferior surface of the crack, respectively. Tangential and normal relative displacement increments are additively composed of a recoverable elastic part dw i e and a non-recoverable plastic part dw i p :    , , e p i i i dw dw dw i t n (10) where subscript i is used to denote the vector components t and n . The stress that the interface supports is assumed to be related to the elastic part by the following expression:   e i i j j d E dw (11) where E ij is the interface stiffness (summation convention is applied to repeated indices). Here, we assume E ij = 0 for i ≠ j and assure that they are relatively large compared to the stiffness of the adjacent medium, by applying a penalty two to four orders of magnitude greater to the shear modulus G of the medium itself. The permanent part of the deformation is given by the following slip rule: 0 0 0 0 p i i if F or dF dw G if F dF              (12) where F is the slip function and G is the slip potential. It can be noted that, since the friction law is non associated, F and G do not coincide, and the direction of slip is given by the gradient of G . Combining (10)-(12), we obtain the fundamental relationship which connects the stresses along the crack to the relative displacements:   EP ij i j d E dw (13) where:                    if 0 or 0 if 0 EP ij ij iq pj p EP ij ij pq q p q E E F dF F G E E E E F dF F G E (14) We chose to describe the roughness by means of a saw-tooth model, with identically shaped asperity surfaces whose size is characteristic of the type of discontinuity being modelled, Fig. 2(b). Moreover, we assume that the amount of tangential sliding is small enough so that the asperity peak of one surface does not override that of the other surface. The slip function and slip potential are the following:                    sin cos ( cos sin ) sin cos n k k n k k n k t k t t F G (15)