Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 190 where E and  are the material Young's modulus and Poisson ratio, and I is the 6 6  identity matrix. The damage variable is bounded in the range   0 ,1 , where D = 0 correponds to the initial undamaged state of the material and D = 1 to the complete degraded state. Figure 8 : Axial stress x  ( x , y , z ) over the cross - section in the channel-shaped cantilever beam with warping restraints at the fixed end. For a given strain state of the material, the value assumed by D is computed by defining a damage associated variable, which governs its evolution. In particular, an equivalent strain eq,t  is defined on the basis of the positive part of the principal strains k  as: 2 3 =1 2 k k eq,t k             (39) To model the unilateral effect, two different damage variables are considered: D t describing the damage related due to tensile deformation states, and D c reproducing the damage in presence of compressive deformation states. The following evolution laws are stated for both:     eq, 0 B 0 1 A  D 1 A    ,    D 0          =         i t i i i i eq,t e i t , c      (40) where 0  is the initial damage threshold, and A i and B i are material parameters. The resulting damage variable D is a combination of the two variables D t and D c defined as:  D D D t t c c       (41) The weighting coefficients i  are defined as follows: 3 3 2 2 =1 =1 ( ) ( )           and           t,k t,k c,k c,k t,k c,k t k c k eq,t eq,t k k H H                 (42) with: 1     if      0 0      if      0         t,k c,k k t,k c,k H     where t,k  and c,k  are the principal strains evaluated on the basis of the two tensors t E and c E , defined as: -1 -1            and           t t c c    E Λ Σ E Λ Σ (43)