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

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where the body forces are neglected. Moreover, by expressing Equation (5) for all the boundary nodes, a linear system of equations can be obtained. In the matrix form, the equation system can be represented as ˜ Hu = Gt . (6) Components of the matrices in Equation (6) can be expressed separately in indicial form as

N n α = 1

N e n = 1

˜ H ki = C ki +

H n α

(7a)

ki ,

N n α = 1

N e n = 1

G n α

(7b)

G ki =

ki .

By considering the boundary conditions of the problem, the known and unknown nodal boundary quantities of Equa tion (6) can be combined on each side, and the resultant system becomes Ax = b . (8) In Equation (8), matrix A is an unsymmetric and densely populated coe ffi cient matrix. The vector b is composed of the unknowns, which include the tractions and displacements of the boundary nodes, while the vector b is known and determined from the boundary conditions. The major drawback of BEM compared to FEM is the unsymmetric and highly populated coe ffi cient matrix. Various numerical methods have been proposed in the literature for e ffi ciently solving Equation (8), see e.g. Aliabadi (2002). In general, when employing BEM to solve problems, the presence of integral singularities can be categorized into five distinct types. These singularities arise due to the characteristics of the kernel and the positioning of collocation points in relation to the elements being integrated. A comprehensive discussion on how to e ff ectively address and manage these singularities can be found in Aliabadi (2002) and Akay et al. (2023). Determination of the homogenized properties of a heterogeneous medium is very significant for multi-scale analy ses. Generally, in multi-scale analysis, a macro-scale finite element model requires the determination of the homoge nized moduli and the stress tensor at each integration point. Therefore, e ff ective and robust determination of homoge nized properties of heterogeneous representative volume elments (RVEs) by using numerical solutions is crucial. For this purpose, a boundary element method-based approach is developed to determine the homogenized properties of heterogeneous RVEs. The homogenization with BEM is achieved by the implementation of the three classical types of boundary conditions. The detailed explanation of implementing these boundary conditions in to the BEM formulation is given in Akay (2023). The three boundary conditions can be summarized as follows: • Uniform Traction (TBC) : This type of boundary condition ensures a constant stress state on every element located on the boundary of the RVE. • Linear Displacement (UBC) : This type of boundary condition ensures a linear deformation state at nodes located on the boundary of the RVE. • Periodic Displacement (PBC) : This type of boundary condition guarantees periodic deformations and anti periodic traction states at the nodes situated on the boundary of the RVE. 2.2. Homogenization with Boundary Element Method

2.3. Incorporating a Cohesive Traction-Separation Interface Model into Boundary Element Method

In this study, a bilinear cohesive traction-separation interface model based on experimental results presented by Tan et al. (2005) is used. The model allows for the separation of the reinforcing particle from the matrix. In this model, the traction between nodes on either side of the interface is continuous, while the separation displacement vector is the di ff erence in displacement values of nodes. As shown in Figure 2, the model consists of three phases: the linear