PSI - Issue 66

Domenico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 396–405 Author name / Structural Integrity Procedia 00 (2025) 000–000

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2. Theoretical Background This section provides the most important theoretical concepts at the base of the proposed modeling approach. In particular, an overview of the Cohesive Phase-Field Model proposed by Wu (Wu, 2017; Wu et al., 2020) is reported.

Fig. 1. Schematic representation of the smearing of an internal sharp crack.

Fig. 1 shows a two-dimensional domain u t   and an internal sharp crack face c  . Tractions t act on t  , while the displacements u are imposed on u  . Under these assumptions, the governing equations for structural mechanics, in the weak form, are:   : d d d 0                  C u u f u t u (1) where C is the isotropic elastic tensor, while  u and f represent the virtual displacements and the body forces (per unit volume) vectors, respectively. In general, the PFM replaces the internal sharp crack face with a smeared interface by introducing a continuous scalar crack phase-field variable c  , which evolves between 0 (intact material) and 1 (fully fractured material). In addition, the PFM defines a crack surface function   c c ,     expresses in terms of c  and its spatial gradient c   , which, according to Wu formulation (Wu, 2017), is evaluated as follows: 2 R  , which has an external discontinuous boundary

0 0 1 1 c l   

  





  c





(2)

, c    c

l

 

  

 

 

0

c

c

In Eq. (2), 0 l is an internal length scale regularizing the cohesive sharp crack, 0 c represents a scaling parameter, and   c   is the crack geometric function describing how the material’s stiffness degrades as the crack evolves. The latter can be defined as:   c 2 c c (1 ) 0          (3)