Issue 17

E. Benvenuti et alii, Frattura ed Integrità Strutturale, 17 (2011) 23-31 ; DOI: 10.3221/IGF-ESIS.17.03 25   1 1 ( ) ( ) ( ) ( ) ( )       n n ρ i i i ρ i i i u x N x v x N x H x a x (2) where: i N represents the i-th standard polynomial shape function of the usual finite element approximation, i v is the i-th standard degree of freedom representative of the continuous part of the displacement field. Finally, i a denotes the i-th enriched degree of freedom. Figure 2 : Regularized Heaviside function and corresponding gradient for various  . Figure 3 : Micro-cracks density function   obtained by differentiation of H  . After collecting the shape functions into the matrix N and the degrees of freedom i v and i a into vectors V and A , the displacement field writes   ( ) ( ) ( )   ρ ρ u x N x V N x H x A (3) The compatible strain is calculated by differentiation of displacement ρ u in the following way ( ) ( ) ( ) ( )( ( ) )        ρ ρ ρ u N x V N x H x A δ x n N x A (4) where H      n and ( ) ( | ( ) | / )   ρ δ x exp s x ρ