Issue 70

F. Greco et alii, Frattura ed Integrità Strutturale, 70 (2024) 210-226; DOI: 10.3221/IGF-ESIS.70.12

Under the common assumptions of small displacements, quasi-static loading, and negligible volume forces, the mechanical problem of such a composite system can be mathematically expressed by a nonlinear boundary value problem (BVP) stated in the following weak form, i.e., find u such that:                      t d int int int \ : d [[ ]], [[ ]] d d  u t u u t u u (1) In Eqn. (1), u is the unknown displacement field and  u its arbitrary kinematically admissible variation,   int \ collectively denotes the subdomains corresponding to the discontinuous phase, i.e. , the units,   s : C u  is the stress tensor, being related to the infinitesimal strain tensor  s u  through the linearly elastic constitutive tensor C , t is the surface traction applied over  t ,   int int t t is the cohesive traction vector, being defined over the negative side of the embedded interfaces,     [[ ]] u u u is the displacement jump between the two sides of the interfaces and  [[ ]] u is its arbitrary variation, and    d 0,1 denotes a (scalar) history variable, which physically represent the damage state of the cohesive interfaces. In particular, the following coupled mixed-mode traction-separation law is assumed for the cohesive interfaces:       d d int [[ ]], 1 [[ ]] t u K u (2) where K is the second-order interface stiffness tensor, representing the initially elastic behavior of undamaged interfaces. The constitutive law expressed by means of Eqn. (2) can be rewritten in the following matrix form, after a suitable local coordinate system attached to each interface is defined:

  n s      

    n s t t

K

0

 





n

d

1

(3)

    

K

0

s

where the subscripts s and n refer to local components with respect to the tangential and normal directions, respectively. In particular,   s t int t s and   n t int t n , and, accordingly,    s [[ ]] s u and    n [[ ]] n u , where s and   n n are the tangent and outer normal unit vectors referring to the negative side of the interface. Moreover, in the absence of coupling stiffness terms between these directions, n K and s K are the normal and tangential interface stiffness coefficients, which can be obtained from the elastic properties of masonry components ( i.e. , Young's moduli u E , j E and shear moduli u G , j G of units and joints) as well as from the joint thickness j h , according to the relations adopted in REF. [1]:

E E

GG

u j

u j





K

K

,

(4)









n

s

 h E E

 h G G

j

u

j

j

u

j

The damage variable d is defined by the following expression:

   max 0

0

if

 

   

  



d

 0

   max 0

   

(5)

   max 0



 exp 2

1

if

t G f

2



  

f

max

0

and strictly depends on the largest value  max ever attained by the equivalent displacement jump  , the latter being defined based on the following interface version of a capped Drucker-Prager criterion, originally proposed in [6]:

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