PSI - Issue 39

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

642

5

In particular, f m δ represent the effective displacement jumps at damage initiation and complete decohesion. These variables are defined by means of the following mixed-mode crack initiation and propagation criteria: 0 m δ and

2

2

n t         f c     t t

I G G G G

(5)

1 and

1

s + =

II + =

I

II

c

c

being n t and s t the normal and tangential components of the cohesive traction vector, respectively, t f and c the tensile strength and the cohesion, I G and II G the Mode-I and Mode-II energy release rates (the subscript c indicating their critical value). In addition, in Eq. (3), α is a softening shape parameter that influences the rate of damage evolution of the cohesive interfaces. 2.2. On the cohesive strength values of the embedded interface elements In diffuse interface modeling approaches, the presence of interface elements embedded along all the boundaries of the bulk finite elements forming the computational mesh can negatively affect the stability of the numerical model. Specifically, when multiple interface elements experience softening behavior simultaneously, the numerical model can suffer from bifurcation and localization instabilities, which determine a physically meaningless prediction of the fracture behavior (García et al. (2015), Lazarus et al. (2015)). Moreover, convergence issues can rise when using standard nonlinear solvers. To avoid such drawbacks, a possible remedy consists of introducing a certain form of imperfection in terms of material variability. In the proposed model, a random spatial distribution of the cohesive strength values for the embedded damageable interface elements is used. Specifically, cohesive strengths are distributed in the embedded interfaces according to a Weibull distribution. Hence, the probability that the minimum strength value within an assemblage of cohesive interfaces could be reached in more than one interface element is zero. This strategy implicates that strain localization originates across a single (weakened) interface, thus giving rise to the unique and stable solution with physical meaning. In the proposed model, both tensile strength t f and cohesion c values of all embedded interfaces are expressed as a product of reference values ( , t f c ) and the following imperfection function taken from (Okabe et al. (2008)) and defined as follows: ( ) 1 0 0 1 , , , ln 1 m e e L IF L L m L η η     =    −      (6) being e L the interface element length, 0 L a reference length (here chosen as the characteristic mesh element size mesh L ), and m the Weibull modulus. Further, ( ) , RS η η = x is a random real number, expressed by a function generating a (white noise) uniform spatial distribution of values ranging between [0,1] with mean of 0.5. Note that η depends on the spatial coordinate ( x ) as well as on a random seed ( RS ) that ensures arbitrariness by returning independent random values. 2.3. Numerical implementation The proposed DIM is implemented in a commercially available finite element software, that is Comsol Multiphysics (COMSOL (2018)). This software provides an efficient FE environment to handle complex models in a personalized version (Greco et al. (2013), Bruno et al. (2016), Lonetti and Pascuzzo (2016), Lonetti et al. (2016), Lonetti et al. (2019), Lonetti and Pascuzzo (2020)). In addition, it provides suitable meshing tools that permit generating both regular and irregular mesh configurations based on finite elements of triangular and/or quadrilateral shape. The numerical implementation of the proposed DIM model requires the construction of a finite element mesh and the insertion of cohesive interface elements within a predefined portion of the created mesh. In particular, interface elements are placed along the brick-mortar boundaries and between the bulk finite elements of the mortar joint. Note that, the former are physical interfaces between distinct phases (i.e., bricks and mortar joints), whereas the latter is