Issue 24

G. Cricrì, Frattura ed Integrità Strutturale, 24 (2013) 161-174; DOI: 10.3221/IGF-ESIS.24.17 163 Figure 1 : Physical vs. equivalent voids distribution (plane representation). The homogenized constitutive law is defined by (tensors are indicated with bold characters):     2 2 2 1 3 3 , , , 2 1 0 2 eq m eq m q f q fcosh q f                          (1) p d d      ε σ (2)   1 : p df f d   ε I (3) (1 ) : p f d d       (4)   p   σ C ε ε (5) 1 0 0 0 0   0 0 N N if and d d otherwise                       (6) Where: Eq. (1) defines the plastic surface; Eq. (2) is the plastic flow rule; Eq. (3) is the void growth rate definition; Eq. (4) imposes the equivalence between micro and macro-mechanical plastic work; Eq. (5) is the global stress-strain relationship; Eq. (6) is the plastic hardening power law for the matrix material. Further, the symbols indicate:  : stress tensor;    p : total and plastic strain tensors; C : constitutive elastic law;  eq : Von Mises equivalent global stress;  m : global mean stress;  : current matrix flow stress (internal variable of the model);  : current matrix equivalent strain (internal variable); f : voids volume fraction (internal variable); N : hardening coefficient;  0   0 : yield equivalent stress and strain; q 1 , q 2 , q 3 : Tvergaard correction coefficients.