PSI - Issue 66

Domenico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 320–330 Author name / Structural Integrity Procedia 00 (2025) 000–000

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Finally, cohesive finite elements have a mixed-mode Traction-Separation Law (TSL) of anisotropic type, i.e. , depending on the interface orientation n , which represents the softening regime after the occurrence of macrostrain localization associated with microcrack coalescence. Such a constitutive law can be expressed as coh [[ ]]   t K u ,

   K K n being the second-order secant stiffness tensor of the interface. 2.2. Implicit gradient-enhanced damage model at the microscopic scale

In order to describe damage evolution in the continuous matrix, an implicit gradient-based regularized isotropic damage model has been considered at the microscopic scale, following the formulation proposed in Peerlings et al. (1996). To this end, an additional variable is introduced, i.e. , the nonlocal equivalent strain eq,nl  , which can be obtained as the solution of the following additional equation:

2      eq,nl eq c

(2)



eq,nl

where eq  denotes the (local) equivalent strain, and c is the gradient parameter, which plays the role of regularization parameter in the presence of microscopic strain localization. Eq. (2) is accompanied by the natural boundary condition eq,nl 0     n at the boundaries of individual embedded holes as well as by a periodic boundary condition on the boundaries of the Repeating Unit Cell adopted for the microscale problem, as will be shown in Section 2.3. In the present work, the local equivalent strain is defined by the well-known Mazars criterion, whereas the scalar damage variable   0,1 d  has the following evolution law: 2  is the Laplacian operator,





  0



0 1 1       

 

exp       

d

(3)

which is valid for 0    ,  being the history variable evolving according to the well-known Karush-Kuhn-Tucker     , and 0  indicating the damage threshold. Moreover,  governs the residual stress, and  denotes the softening slope, depending on the fracture toughness of the material. 2.3. Cohesive/bulk homogenization scheme This section is devoted to describing the combined cohesive/bulk homogenization scheme used in conjunction with the adopted Diffuse Interface Model at the macroscale. In detail, the macroscopic bulk has nonlinear constitutive behavior, which can be derived through a standard hierarchical first-order homogenization performed on a suitably defined Representative Volume Element (RVE) subjected to periodic boundary conditions, given the periodic nature of the considered 2D composite structure. This means that the RVE coincides with the Repeating Unit Cell (RUC) of the microstructural arrangement. Its region and external boundary are denoted by  and  , respectively. The macroscopic stress and strain tensors are defined in terms of their microscopic counterparts as: conditions for the loading function eq,nl F

1

1

 

 

d    t x d

  





(4)

1

1

 

 

d

d  

u n









s





where t is the boundary traction, u is the boundary displacement,  is the tensor product, x is the position along the RUC boundary, and n is the outward normal at x .