Issue 41

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 41 (2017) 175-182; DOI: 10.3221/IGF-ESIS.41.24 179 where ߙ k represents the angle of the active k asperity surface and  is the coefficient of friction of the Coulomb law. If roughness is not taken into account, only friction is present and, therefore, Eqs. (15) become simpler:       t n t F G (16) Figure 2 : Schematic representation of the model used to describe the roughness of the crack surfaces. (a) Local tangent-normal coordinate system at a contact point pair (shown separately for clarity). (b) Saw-tooth asperity model, with constant angle and asperity length. Relationships for stresses between the two coordinate systems are also shown. Compliance matrix To efficiently solve the problem, a compliance matrix C that relates the stresses, arising at the collocation points from the applied loads, to the values of the relative displacements of the crack at the integration points can be obtained. The details of this procedure are presented in [6]. Displacements along the crack, in an incremental form, are related to stresses by the following expression:          [ , ] [ , ] n n T t t T d d d d d d w w C (17) where we have collected the increments of stresses and displacements in ordered vectors, such that, for instance, we have d w t = [ dw 1t , ... dw Nt ] T , with subscripts i = 1,..., N referring to the integration points along the crack. Now the problem in (6)-(7) takes the following formulation:       ( )[ , ] [ , ] T t EP n T d d d d I CE T w w C (18) where  is the identity matrix and E EP is the matrix containing the terms described in Eq. (14). We have introduced a transformation matrix T to express crack displacements at the collocation points in terms of those at the integration points, so that the constitutive law in (13) can be enforced. This matrix is built using Krenk’s interpolation formula [11], in order to obtain the values of the unknown functions in points along the crack which are different from the integration points. Eq. (18) is a system of non-linear algebraic equations in the 2N unknown incremental relative displacements, to be solved by taking small increments in far-field loading. The stiffness matrix E EP needs to be updated at each step, using the displacement-stress configuration from the previous step. By using the compliance matrix, the dislocation densities at each step are not needed to be integrated in order to compute the relative displacements along the crack and, therefore, the efficiency of the algorithm is increased.