Issue 18

S. Marfia et alii, Frattura ed Integrità Strutturale, 18 (2011) 23-33 ; DOI: 10.3221/IGF-ESIS.18.03 27 0 0 dT dT dT d d                              p    (18) together with the Khun-Tucker conditions:     0, 0, 0 d d         σ σ   (19) It can be remarked that the frictional problem can be activated only when the damage is greater than zero. In fact, only in this case the microcracks, in which the unilateral and friction effects can occur, are present at the interface. About the evolution of the damage parameter D  , a model which accounts for the coupling of mode I of mode II of fracture is considered. In fact, the two quantities N  and T  , defined as the ratios between the first cracking relative displacement 0 N s and 0 T s and the full damage relative displacement f N s and f T s , are introduced: 0 0 0 0 0 0 , 2 2 N N N T T T N T f f cN cT N T s s s s G G s s         (20) where 0 N  and 0 T  are the peak stresses corresponding to the first cracking relative displacement and cN G and cT G are the specific fracture energies in mode I and mode II, respectively. Then, the parameter  , which relates the two modes of fracture, is defined as follows: 2 2 2 2 N T N T s s       s s   (21) where   T N T s s   s  . The relative displacement ratios are introduced as: 0 0 N T N T N T s s Y Y s s    (22) and the equivalent relative displacement ratio is considered: 2 2 N T Y Y Y   (23) Finally, the damage parameter is assumed to be a function of the history of the relative displacement as follows:       1 max 0,min 1, 1 history Y D D D Y           (24) Interface damage models with coupling An interface coupled model, obtained considering different ways of coupling the body and the interface damage, is proposed. Denoting with   I D x the coupled interface damage evaluated in a point x of the interface ,the coupling between the body damage and the interface damage is performed, in the firs case, ensuring that the interface damage is not lower than the body damage computed on the bond surface [11]:         max , I t D D D    x x x (26) In the second case, a representative area A of the interface and, in particular, of the third layer made of cohesive support material, is assumed to be decomposed in two parts, as represented in Fig. 1. In fact, when the body damage occurs, it induces the presence of a microfracture in the representative area of the surface, characterized by a corresponding area