PSI - Issue 39

Fabrizio Greco et al. / Procedia Structural Integrity 39 (2022) 638–648 Author name / Structural Integrity Procedia 00 (2019) 000–000

641

4

Figure 1. (a) Representation of a regular brick masonry structure; (b) cohesive/volumetric finite element discretization at mortar joints: a schematic of brick/mortar and mortar/mortar interface elements and related notations.

where b C and m C are the elasticity tensors of brick and mortar continuous phases, respectively. Besides, U is the set of kinematically admissible displacement fields compatible with homogeneous Dirichlet boundary conditions on D Γ , ε is the usual infinitesimal strain tensor. In Eq. (1), the third term of the left-hand side represents the interface contribution, in which int int int + − = = − t t t represents the interface traction exchanged across the brick/mortar and mortar/mortar interfaces, while − + = − u u u § ¨ indicates the displacement jump vector across these interfaces. The constitutive behavior of both brick/mortar and mortar/mortar interfaces is described trough a mixed-mode traction-separation cohesive law incorporating a nonlinear softening response. According to this model, cohesive tractions ( int t ) can be expressed in the following matrix form:

n     s t t

n δ δ     s  

  

  

0

0

K

(

)

(2)

t

1 = = −  

d

n

int

0

0

K

s

being 0 s K the normal and tangential stiffness components, and n δ = ⋅ u n § ¨ and s δ = ⋅ u s § ¨ the normal and tangential components of displacement jump vector, respectively. Besides, d represent a single scalar damage, which is described by means of the following linear-exponential evolution law: 0 n K and

         

max δ δ ≤ 0

0

for

m

m

      

      

  

  

max δ δ α δ δ  −  − 0 f m m m 

  

0

1 exp

− −

m

0

δ

0 δ δ δ < ≤ max m m m f

1 = − 

1

for

(3)

d

−

m

( ) α

max

1 exp

δ

− −

m

max δ δ >

f

1

for

m

m

0 m δ ,

max m δ , which are special values of the mixed-mode separation displacement m δ

f m δ , and

Note that d depends on

, defined as:

2

(4)

2

δ

=

δ δ +

m

n

s