Issue 12

S. Marfia et alii, Frattura ed Integrità Strutturale, 12 (2010) 13-20; DOI: 10.3221/IGF-ESIS.12.02 18 where the starting value of D  is zero, 0  is the initial threshold damage strain and k is a material parameter associated to the damage energy. About the evolution of the interface damage parameter D  , a model which accounts for the coupling of mode I of mode II of fracture is considere d [13, 14]. 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: (12) where 0 N  and 0 T  are the peak stresses on 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: (13) Then, the relative displacement ratios are introduced: (14) and the equivalent relative displacement ratio is considered: (15) Finally, the damage parameter is assumed to be a function of the history of relative displacement as follows: (16) where the parameter D   can be expressed by the relationship : (17) Note that the Eq. (17) allows to obtain a linear stress - relative displacement relationship when pure mode I or pure mode II is activated. In fact, setting for instance 0 T s  , formul a (17) becomes: (18) Thus, in the softening phase, the normal stress at the interface results: (19) which is a linear relation between the stress N  and the relative displacement N s . Analogously, setting 0 N s  , the tangential stress is related to T s by the linear relationship: (20) Indeed, as previously emphasized, the interface damage depends also on the state of deterioration of the support material, i.e. on the damage occurring in the body 1  . Thus, the body damage D  has to be evaluated on the surface corresponding to the interface  . Once the interface damage   D   x and the body damage   D   x , are evaluated, the damage state of the interface   c D   x is set as the maximum between the two obtained values, i.e.: 0 0 0 0 0 0 , 2 2 N N N T T T N T f f N cN T cT s s s s s G s G         2 2 2 2 N T N T s s       s s 0 0 N T N T N T s s Y Y s s   2 2 N T Y Y Y       max min 1, history D D       1 1 Y D Y           0 1 1 1 N N N N N N N Y s s D Y s               0 0 1 f N N N N N N N f N N K D K s s s s s s             0 0 1 f T T T T T T T f T T K D K s s s s s s        