Issue 17

E. Benvenuti et alii, Frattura ed Integrità Strutturale, 17 (2011) 23-31 ; DOI: 10.3221/IGF-ESIS.17.03 26 In particular, the form of the function   , which is displayed in Fig. 4, reproduces the distribution of micro-cracks within the process zone which is often recognized in quasi-brittle materials [16]. The presence of a regularization parameter  introduces a characteristic length within the present XFEM formulation. After introduction of the compatibility operator   B N and N such that ( ( (x) ))   n N A N A , three distinct strain fields ( ) ( )  ε x B x V (5.1) ( ) ( ) ( )  ρ ρ ε x H x B x A (5.2) ( ) ( ) ( )  c ρ ρ ε x δ x N x A (5.3) can be introduced. They are kept distinguished because they play a different mechanical role within the model. In particular, th e stress field ε is the standard strain field in the absence of the discontinuity. The strain field ρ ε is related to the spatial variation of the jump intensity, while c ρ ε represents the localized strain field resulting from the m icro-cracks distributed within the process zone. Independent stress fields are associated with each strain field by means of the following constitutive laws ( ) (1 ) ( )   σ x D E B x V (6.1) ( ) ( )(1 ) ( )   ρ ρ σ x H x D E B x A (6.2) ( ) (1 ) ( )   c ρ c c σ x D E N x A (6.3) Figura 4 : Microcrack density distribution according to Mihashi and Nomura [16]. where D and D c are isotropic damage scalar parameters which govern the degradation of the elasticity matrices E and c E . A detailed explanation of the transition strategy from a continuum damage description to the regularized discontinuous description can be found in Benvenuti (2011). The key point of the transition procedure is that, as soon as the bulk damage parameter D has reached a critical value, cr D , the damage c D is activated. In particular, for a loading history described by the pseudo-time parameter t, the evolution of the damage parameter c D is governed by the following loading unloading conditions , 0, 0, 0         eff c c φ σ r φ D φD (7) The effective stress scalar eff  selects the positive value of the maximum principal stress max  of the stress field c ρ σ  eff max σ σ (8)