Issue 47

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 401-407; DOI: 10.3221/IGF-ESIS.47.30 403 F ORMULATION Description of the model n order to account for the effects of friction and roughness, we make use of an interface model formulated as a constitutive relationship between opposing points along the crack, written in the form of a classic elastic-plastic law of small strain plasticity. Globally, the crack is smooth and frictionless and the surface interference is modelled by means of bridging stresses between the crack surfaces. In this fashion, we can obtain a straightforward implementation within the distributed dislocation technique, as the bridging stresses, computed at a discrete number of points, are simply added to the stress distribution resulting from the remote loads. Let us define the relative displacement (increments) between two opposing points of the crack as composed of a recoverable elastic part dw i E , related to the remote stress field, and a non-recoverable plastic part dw i P , which accounts for frictional sliding and dilatancy. The stresses on the crack interface are related to the displacement increments by means of interface stiffnesses E ij EP : , , EP i ij j n d d i j t E w    (1) where t,n denote, respectively, the tangential and normal directions with respect to the nominally flat crack surface (Fig. 1a). In order to compute the interface stiffness, we need to introduce a slip function F and a slip potential G . It can be noticed that, since Coulomb’s friction law is non-associated, F and G do not coincide, and the direction of slip is given by the gradient of G . Therefore, we can express the non-recoverable displacement increment as: 0 if F 0 or dF=0 if F= dF=0 p p i i i dw dw G        (2) while the interface stiffness E ij EP is computed according to the following equation: if F 0 or dF=0 if F=dF=0 iq pj p q EP EP ij ij ij pq p q F G E E E E E E F G E                 (3) E ij is the elastic interface stiffness, which is taken two to four orders of magnitude greater than the elastic modulus of the medium itself, in order to assure the impenetrability between the crack surfaces and ensure good numerical compatibility. The surface roughness is described through a saw-tooth model, characterised by a constant angle  , a mean length of the asperities equal to 2 L and a height h . The coefficient of Coulomb’s friction f is kept constant. With the previous assumptions, we obtain the following formulation of F and G : 1 2 1 sin cos ( cos sin ) sin cos n t n t n t F f f G                         (4) We can define a crack size parameter by considering the ratio c/L , where ideally c/ 2 L approaching the unity identifies the case of a short crack. The parameters used to describe the surface roughness are somehow related to the specific material, and the scale length might differ of several orders of magnitude. For instance, the fracture process in concrete is influenced by particles and voids whose sizes are in the order of few nanometres, but if the aggregate is particularly coarse the size of the irregularities can go up to some millimetres [18]. It is also common to evaluate roughness through a digitalisation of the crack profiles, and consider it as a high-frequency shortwave length component of the measured surface, characterised through average measures or peak-to-valley heights [19]. The numerical algorithm In this Sub-Section, we briefly summarise the numerical algorithm used in the present work, which is based on the application of the distributed dislocation technique, combined with the interface model introduced above. We begin by I