PSI - Issue 33

Umberto De Maio et al. / Procedia Structural Integrity 33 (2021) 954–965 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

958

5

max  , max  and

where the

I c G , II c G denote the normal/tangential critical interface stresses and the critical mode I/II

fracture energies, respectively. 2.2. The Embedded Crack Model

The embedded crack model allows the crack propagation across the finite element to be accurately predicted, and, to avoid typical crack locking phenomena, due to a kinematically incompatible crack path topology, incorporates a crack adaptation approach for the computation of the correct crack orientation. Initially, in each finite element the displacement jump is = 0 u and a linear elastic behavior =   C is assumed. According with the classical Rankine’s criterion, when the maximum principal stress exceeds the normal critical stress of the material, a crack discontinuity appears, whose orientation, defined by unit normal vector n , is orthogonal to the corresponding direction. In this way, the crack line cuts the finite element in two parts, and as a consequence, some nodes, called “solitary nodes”, move away from the initial configuration ( see Fig. 2a).

Fig. 2. Schematic representation of the Embedded Crack Model: (a) constant stress element with embedded crack and (b) the adopted softening function.

According with the well-known strong discontinuity approach (Oliver, 1996a, 1996b), the approximated displacement field within the cracked element can be computed as the sum of a continuous part c u on the bulk element, and a discontinuous part d u , corresponding to the displacement jump u , on the cracked plane  d , as follows:

 +    = + = +  −  d c d c H u u u u u

(4)

 d H is the Heaviside function at the discontinuity  d and   + = + =  1 ( ) i

i N x is the sum of the standard

where

interpolation function associated with nodes +  i . As consequence, assuming a spatially constant displacement jump, the resulting strain field is obtained as a continuous part c  plus a Dirac’s function on the crack line. The continuous part, which determines the stress field of the element on both crack faces, is written as follows:

+ = −  s c s   u b u c

(5)

where s  stands for the symmetric part of the gradient operator,  s denotes the symmetric part of the dyadic product, and + b is the gradient of the interpolation functions of the positive subdomain nodes. On the other hand, the crack