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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and Xu and Needleman (1994)). In the 1990s, the first representation of an interface, in which adhesion and friction were modeled sequentially, was presented by Tvergaard (1990). Finally, in the early 2000s, models that combined adhesion and friction behavior with a smooth transition were presented, RCMM Raous et al. (1999) and Chaboche et al. (2001) are the two examples of such models. In Section 2, the boundary element formulation for linear elasticity, incorporating homogenization through bound ary conditions, is thoroughly explained. Subsequently, we introduce the bilinear interface model and explain its im plementation within the context of the Boundary Element Method (BEM). To assess the e ff ectiveness and accuracy of our approach, we provide validation through two numerical examples: involving single-inclusion and multi-inclusion RVEs. These validation exercises are presented in detail in Section 3, wherein we compare our results to established findings from the existing literature. In Section 4, a brief conclusion of the study is presented.

2. Method

2.1. Boundary Element Formulation

Consider a solid body Ω with the boundary Γ which is decomposed into the parts Γ u and Γ t , where the displacement (Dirichlet) and traction (Neumann) boundary conditions are prescribed, respectively. Note that these parts fulfill the conditions Γ = Γ u ∪ Γ t and Γ u ∩ Γ t = ∅ . Within the domain Ω , the balance of linear momentum under quasi-static conditions ( ρ ¨ u i ≈ 0) can be written as ∂σ i j ∂ x j + b i = 0 in Ω , (1) where b i and σ i j denote the given body forces and the stress tensor, respectively. The Kelvin fundamental solution u ∗ ki for the elastic body is used for weighing Equation (1) over the domain Ω . Ω u ∗ ki ∂σ i j ∂ x j + b i d Ω= 0 . (2) In Equation (2), u ∗ ki is the Kelvin displacement fundamental solution for an infinite elastic body. The application of the divergence theorem to the stress term appearing in the integral and following the derivation presented by Akay (2023), the boundary integral equation for linear elasticity with body forces can be respesented as C ki u i + Γ t ∗ ki u i d Γ= Γ u ∗ ki t i d Γ+ Ω u ∗ ki b i d Ω , (3) where the term t ∗ ki stands for the Kelvin’s traction solution. Moreover, t i and u i are the traction and displacement vectors defined on the boundary nodes, respectively. The Kelvin solutions introduced in to the formulation can be summarized for the 2-D plane strain case as follows u ∗ i j = − 1 8 π (1 − ν ) µ (3 − 4 ν ) ln 1 r δ i j − r , i r , j , t ∗ i j = − 1 4 π (1 − ν ) r (1 − 2 ν ) δ i j + 2 r , i r , j ∂ r ∂ n − (1 − 2 ν )( r , i n j − r , j n i ) . (4) In Equation (4), r : = √ r i r i denotes the distance between the source and the field points of BEM with r i : = y i − x i being the components of r and ∂ r /∂ n : = n i ∂ r /∂ y i is the perpendicular distance between the source and the element containing field points. Moreover, r , i is defined as the partial derivative of r with respect to y i . By setting ν → ν/ ( ν + 1) in Equation (4), the plane stress assumption can be adopted. The way to calculate the matrix C ki is discussed by Mantic (1993). A straightforward way to determine it is by using the concept of rigid body motion. After discretizing the integral equation Equation (3) into N e boundary elements with N n nodes and utilizing the isoparametric shape functions N α on the boundary Γ , the discretized form of Equation (3) can be expressed as

N n α = 1

N n α = 1

N e n = 1

N e n = 1

H n α

n α i

G n α

n α i ,

C ki u i +

ki u

ki t

(5)

=