PSI - Issue 66

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

354

5

A C D B A C D B    



if

AI

BJ

J

I







1 2

1,

2,

2,

1,









2   , eq eq

Y E 

(7)

 

0

m

CI  

if

DJ

J

I





1,

2,

2,

1,









where 1, I  and 2, J  are the first invariant of the strain tensor and the second invariant of the deviatoric strain tensor, respectively, that under the assumption of plane stress conditions are defined as:

    

    

  

  

1 2 

1 6

1

1 4













  2    yy xx



  

  

2

2

(8)

2

;

I

J

xx   

xx  

m xx   

m yy   







 



m

1,

2,

yy

yy

xy









2

1





1





m

m

being m  the Poisson's ration of the mortar. The parameters A , B , C , and D are defined in terms of the uniaxial tensile t  , uniaxial compressive c  , and biaxial compressive b  (generally expressed as 1.2 b c    ) as follows:     2 1 3 ; ; ; 3 ; 2 2 b c t b c t c t c t c c b c b c A B C D                         (9) 0    , the effective crack driving force is usually replaced by the history variable H m , which represents the maximum value of m Y ever reached, defined as follows: Because the crack phase-field ( )  x must satisfy the irreversibility condition

0 [0, ] max( ,max ) n m ,

m n T H Ym Y  

(10)

, nm Y the value of

being 0 m Y the initial values of effective crack that denotes the limit of the undamageable state and

the crack driving force reached at time t n within the time interval [ 0,T ]. Based on Eqs. (4), (5), and (10), the weak form assumes the following form:

  

  

l

1

 

     













: σ ε 

:   σ ε

 t u

(11)

( )

0       dV    2

0

dV g 

dV H g 

dV G 

dA

 



0

m

c

c l

c

0 0

0











b

m

m

m

t

2.2 Macro-modeling strategy adopted for the masonry The coarser regions in the proposed multiscale approach are represented by macro elements that behave as linear elastic, thus sufficient for a coarse scale resolution. In such regions, the assumptions of periodicity and separation of scales can be supposed to be valid so that an equivalent homogenized material can replace the microstructure of the masonry. A standard numerical homogenization methodology is used to derive the overall macroscopic constitutive behavior of such a material, i.e., the components of the constitutive tensor hom C . Specifically, the Repeating Cell (RC) depicted in Fig. 1-b is considered, and its linear microscopic structural response ( i.e. , without the crack phase field) due to two uniaxial and shear paths under periodic boundary conditions is analyzed. The results of such analyses expressed in stress-strain relations are expressed in components as hom ij ijhk hk C    (with i,j,h,k = 1,2). Because of the nature of the periodic masonry, the homogenized constitutive tensor hom C is orthotropic, so its overall definitions require only four independent moduli, i.e., hom 1111 C , hom 2222 C , hom hom 1122 2211 C C  , and hom 1212 C , being hom 0 ijhk C i j    .