Issue 18

S. Marfia et alii, Frattura ed Integrità Strutturale, 18 (2011) 23-33 ; DOI: 10.3221/IGF-ESIS.18.03 25 where  C is the elastic tensor,  ε , p  ε and  e are the total strain, the plastic strain and the elastic strain tensors, respectively. The following plastic yield function is introduced:          2 2 1 2 1 2 y y y y f A B                         Ω σ (3) with 1  and 2  the principal stresses of the effective stress tensor Ω σ , y  the yield stress and A and B material parameters governing the shape of the yield function (a brunch of hyperbola). In particular, it is set A = 0.1 MPa and 100 B  . The evolution law of the plastic strain is: p f     Ω Ω ε σ   (4) which is completed with the classical loading-unloading Kuhn-Tucker conditions:     0, 0, 0 f f        σ σ   (5) The accumulated plastic strain   is introduced as 0 t p dt      ε  . As a damage softening constitutive law is introduced, localization of the strain and damage parameter could occur. In order to overcome this pathological problem, to account for the correct size of the localization zone and, also, to avoid strong mesh sensitivity in finite element analyses, a nonlocal constitutive law is considered. In particular, an integral nonlocal model is adopted for the damage in compression and in tension. The evolution of the compressive damage variable is governed by the following law:         3 2 3 2 2 3 max min 1, c c c history u u D D D                (6) with u  the final damage threshold in compression and   the nonlocal accumulated plastic strain, evaluated at the point x , as:         1 d d                x x y y x y (7) where y is a typical point of the body 1  and the weight function    x is set as:   2 2 1 R      x y x (8) with R the radius of the nonlocal integration domain and the symbol   denoting the positive part of the number  . The evolution of the tensile damage parameter is governed by an exponential nonlocal law, set as:       0 0 max min 1, eq k eq t t t history eq e D D D                  (9) with   eq   x the equivalent nonlocal strain, evaluated at the point x as:         1 eq eq d d                x x y y x y (10) and eq   the equivalent strain introduced as [11]: