PSI - Issue 61

Ahmet Arda Akay et al. / Procedia Structural Integrity 61 (2024) 138–147 A.A. Akay et al. / Structural Integrity Procedia 00 (2024) 000–000

142

5

elastic phase ( | OA | ), the softening phase ( | AC | ), and the dissociation phase ( | CD | ). Here, it should be noted that this model accounts only for the behavior in the normal direction of the interface.

= 

k σ [ u r ] ,

[ u r ] < [ u r − A ]

 1 +

k σ ˜ k σ 

  0 ,

σ max − ˜ k σ [ u r ] , [ u r

σ int

− A ] < [ u r ] < [ u r − C ]

(9)

[ u r − C ] < [ u r ]

Fig. 2: Cohesive traction-separation model Wu et al. (2020)

The relationship between the opening displacement and the traction is given in Equation (9). In Equation (9), σ int is the interface stress in the normal direction, σ max is the maximum allowable interface stress, [ u r ] is the opening displacement in the normal direction, k σ and ˜ k σ are the elastic and softening moduli of the interface, respectively. The interface conditions in each section of the bilinear interface behavior can be summarized as follows: • [ u r ] < [ u r − A ]: The interface is in linear elastic phase ( | OA | ),where k σ denotes the interface modulus. The loading and unloading are on the line | OA | . • [ u r − A ] < [ u r ]: This interface is in the linear softening phase. The point which corresponds to the value of [ u r ] is determined. For example, it is point B 1 in Figure 2. Afterward, the stress corresponding to this point, σ B 1 , is calculated by using Equation (9). The value of the modulus is updated as k B 1 . For the new interface modulus, the problem is solved again by an iterative solution method in the relevant load step until the equilibrium is reached. • [ u r ] < [ u r − B 1 ]: Loading and unloading of the interface are done on the line by using modulus k B 1 (this is the case once [ u r ] = [ u r − B 1 ] previously in the loading history). • [ u r − C ] < [ u r ]: The interface is in the dissociation phase. The interface modulus k σ is taken as zero, resultantly, the stress on the interface is zero.

(a) Sticky

(b) Separation

Fig. 3: Interface nodes Wu et al. (2020)

In the boundary element analysis of a two-phase composite without interface modeling, an equation system is created based on the assumption that the nodes in the matrix and those on the reinforcing side of the interface move together, as shown in Figure 3(a). Between the nodes indicated by P + and P − , the displacement and traction relations given in Equation (10.1) are used. To model the separation of the interface as shown in Figure 3(b), an interface law between the nodes P + ve P − should be inserted into the equation system. The displacements of P + and P − are discontinuous, while tractions are continuous. In addition, the displacements and stresses on either side of the interface must satisfy a certain interface condition, as shown in Equation (10.2).