Issue 29

R. Serpieri et alii, Frattura ed Integrità Strutturale, 29 (2014) 284-292; DOI: 10.3221/IGF-ESIS.29.24 287 area is   1  k D dA , and a damage part, whose area is k D dA , k D being a damage parameter ranging from zero, in the case of no damage, to unity in the case of complete damage (Fig. 2). Figure 2 : Decomposition of each infinitesimal area of microplane into damaged and undamaged parts. Assuming a linear elastic relationship for the undamaged part and a Coulomb-type frictional-contact relationship on the damaged part, the following expression of the free energy is obtained:                     1 1 1 1 1 , , , , , , 1 2 2                         s K s s K s s s s f f d f d f k k k k k k k k s s s s N k k k k D D s s D D s s (1) where  k s the ratio between the area of the k th microplane and the sum of all such areas while, denoting with x  the negative part of x i.e.   / 2 x x x    , it results:                   0 0 0 kI kII I II kI kII f k s s K K s s s s                                             s n s R s s t K s s k k k k k d f k k s s s s s s (2) in (local) modes I and II and f k s denotes the frictional slip on the k th microplane . Note that all microplanes are characterized by the same stiffness. The interface stress is obtained as the derivative of the free energy with respect to the relative displacement s :     1 1              σ R K s K s s T d k k k k N k k k k D D (3) The frictional slip on each microplane is obtained through a non-associate Coulomb-type relationship defined by:             ,  sign         s   f k I kI II kII k k kII s K s K s s s s s f f f k k k s s s (4)     0                  , 0                  , 0        s s   k k k k s s f f k k (5) Where  k represents the slip-onset function and  denotes the friction coefficient. The evolution of damage variable k D , for each microplane, is defined to recover the two classical bilinear cohesive-laws in pure modes I and II depicted in Fig. 3, for the relevant local stress-displacement relation: In such relationships 0 I s a nd 0 II s r epresent the values of the relative displacements corresponding to the onset of damage in modes I and II, respectively, while cI s a nd cII s r epresent the ‘critical values’ corresponding to the attainment of complete damage. Furthermore, 0  I a nd 0  II d enote the strengths in modes I and II, respectively, and it is easy to