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

143

6

  t α ( P

α ( P − ) α ( P − )

u α ( P t α ( P

+ ) = u + ) = − t

α = 1 , 2 for 2-D problems

(10.1)

 

f ([ u α ]) = g ( t α ( P

+ ))

α = 1 , 2 for 2-D problems

(10.2)

α ( P − )

+ ) = − t

The functions f ( · ) and g ( · ) in the above equation express the interface relationship between displacement and traction. [ u α ] is defined as the di ff erence in the normal displacements of the P + and P − nodes on opposite sides of the interface, i.e., [ u α ] = u α ( P + ) − u α ( P − ). To adopt the interface relationship presented in Equation (9) by Tan et al. (2005) to BEM, the functions f ( · ) and g ( · ) given in Equation (10.2) must be selected to represent this interface model. Interface behavior is expressed by defining the relationship between displacement and traction on either side of the interface. The relationship between traction and displacement for this interface is given in Equation (11).   T 0 Ik = S k ( U 0 Ik − U k I ) T 0 Ik + T k I = 0 (11) In Equation (11), T 0 Ik ve U 0 Ik are defined as the tractions and the displacements of the nodes on the matrix side for the k th interface. Additionally, T k I and U k I are the tractions and the displacements of the nodes on the inclusion side for the k th interface. The S k matrix is a diagonal matrix consisting of sub-matrices containing the elastic modulus k b on diagonal elements and has a size of 2 × 2. Inserting Equation (11) in the BEM equation system given in Equation (8) and making the simplifications, the equation system is recast into the following form

     

      U 0 T 0 U 0 I 1 U 1 I . . . U 0 In U n I

    - G 0 H 0 G 0

   

0 I 1 - G

0 I 1 S 1 · · · G 0

0 In - G

0 In S n

I 1 S 1 - H - G 1 S 1

In S n - H

1

G 1 S 1 - H

= 0 .

(12)

. .

.

- G n S

n

G n S n - H

n

The variables U 0 and T 0 given in the above equation are the displacement and the stress vectors of the nodes on the outer boundary of the matrix domain. The matrices G and H are the coe ffi cient matrices obtained by the boundary element equations. After defining the classical boundary element equation in this way, the system of equations is finalized using the homogenization method developed for three classical boundary conditions, as discussed in previous sections. Since the modulus ( k int ) of each node pair may change during the softening phase of the cohesive interface model, the coe ffi cient matrix S given in Equation (12) that describes the relationship between the traction and the opening displacement will also change. Thus, the new S matrix may not satisfy the equilibrium condition in that load step. Therefore, an iterative procedure is needed to reach an equilibrium solution for a given loading. The steps that are used to reach equilibrium can be described as follows. • STEP 1 : Initially, the relationship between the interface node pairs is defined as k int = k σ and u r - hist = 0. The maximum number of iterations and the tolerance value are set to N and e 0 , respectively. • STEP2 : The next incremental step of macroscopic strain-driven loading is applied. • STEP 3 : The boundary element equations containing k int = k σ interface modulus values specified for the relevant load step are solved. Then, the di ff erence in the normal displacements between the interface node pairs [ u r ] is calculated.