PSI - Issue 66

Domentico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 350–361 Author name / Structural Integrity Procedia 00 (2025) 000–000

353

4

 ( x )  [0,1], referred to as phase-field or damage field. The extreme values  ( x ) = 0 and  ( x ) = 1 correspond to coordinate points within the band for which the material is intact and completely cracked, respectively. Under the hypotheses mentioned above, the governing equation of the solid mechanics and the phase-field problem for the proposed model can be expressed through the following weak forms:

( ) ε

( , )   ε m



b 









ε

ε

 t u

:

:

0

dV

dV

dA







 



ε

ε











(1)

b

m

t

( , )   ε m



 















0

dV G 

dV         





c





 







m

m

( ) b  ε and

( , ) m   ε represent the strain energies stored inside the brick ( b  ) and mortar joint ( m  )

where

domains, respectively. Besides, G c is the critical energy release rate of the mortar and 

 ,       is the crack

surface density function generally defined as:

1 1 

  

1

   

     d



(2)

( ,   

)  

, l       c

4



0

0

c l 

0

0 0

where, l 0 represents the length scale parameter that governs the width of the diffuse band approximating the crack, while   [0,1]    is the so-called geometric crack function, which determines the distribution of the crack phase field inside the band, and it is expressed as:         2 1 0,1 and 0,2              (3) In the present study, it is assumed  =2, which implies   2 2       and 0 c =  . According to the so-called hybrid formulation, two distinct energy functions   0 ( ( ), ) ( ) ( ) m m g     ε u ε u and   0 ( ( ), ) ( ) ( ) m m g     ε u ε u are introduced to express the stress field and the damage evolution law, respectively, so that it results:   0 0 0 0 , : m m g            σ σ σ C ε ε ε (4) ( , ) ( ) ( ) m m m g Y g g                 ε (5) In Eq. (4), 0 C is the fourth-order elasticity tensor, while g (  ) is the energetic degradation function due to material damage. According to Wu et al. (Wu (2017), Wu and Nguyen (2018)), g (  ) is expressed by the following expression for cohesive cracks:     2 3 1 1 2 1 2 3 1 ( ) 1 p p g a aa aa a             (6) Where, the parameters p ≥ 2, a 1 >0, a 2 , and a 3 control the softening behavior of the material, and their expressions can be found in (Wu et al. (2020)). In Eq. (5), m Y represents the effective crack driving force. Different effective driving forces have been proposed in the literature (see, for instance, Ambati et al. (2015), Wu (2018), Wu and Nguyen (2018)). The effective crack driving force adopted in the proposed model is defined in terms of an equivalent strain eq  expressed using a Drucker-Prager damage model with a compression cap as follows (Vandoren et al. (2013)):