Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 279 with the findings from the previous analysis. Moreover, the reformulation is convenient for the numerical implementation, as it allows for the introduction of a unique model for loading and unloading, as well as for debonding and contact conditions, which is not possible with the original formulation. In this reformulation, a predefined Helmoltz energy is introduced, and the inelastic nature of the decohesion process is accounted for by means of damage variables. We first assume an additive decomposition of the elastic energy  e at the interface into a normal and tangential contributions, i.e. ( ) ( ) e N N T T g g      (25) where each contribution is associated with one characteristic deformation mode (i.e. with the displacement jump or, equivalently, the gap g in the normal or tangential directions). The material damage is also suitably approximated through a set of scalar-valued damage parameters, where different damage mechanisms are coupled multiplicatively. Accordingly, the total Helmoltz energy of the interface reads       1 1 ( ) 1 1 ( ) N T N T N N N N T T T T d d g d d g          (26) where the damage variables [0;1] j i d  , with i=N,T and j=N,T . By applying the Coleman and Noll procedure the traction vector T becomes       ( ) ( ) 1 1 1 1 N T N T N N T T N N T T g g d d d d             T g g (27) Based on a quadratic assumption for the elastic energies,  N and  T in Eq. (25) can be expressed as 2 1 ( ) 2 N N N N g K g   (28a) 2 1 ( ) 2 T T T T g K g   (28b) which yields the following equations for the derivatives of the normal and tangential elastic energies ( ) N N N N g K g     n g (29a) ( ) T T T T g K g     t g (29b) The combination of Eq. (27) and (29a, b) gives       1 1 1 1 N T N T N N N N T T T T d d K g d d K g       T n t (30) or equivalently N T p p   T n t (31) From a comparison between Eq. (30) and (31) the normal and tangential tractions are expressed as    1 1 N T N N N N N p d d K g    (32a)    1 1 N T T T T T T p d d K g    (32b) Based on this general formulation, a new thermodynamically consistent version for the improved exponential model [7] can be obtained straightforwardly as follows. From a comparison between Eq. (1), (2) and (32a,b), the evolution of the damage variables in the normal and tangential directions is governed by the following equations 2 2 max max 2 , N T N T N T K K g g     (33)