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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8

Iteration number

Iteration number

Iteration number

(a) d = 40 µ m

(b) d = 60 µ m

(c) d = 80 µ m

Fig. 5: Macroscopic stress-strain and iteration-strain curves of RVEs with same volume fraction but di ff erent inclusion diameters. Models involve cohesive interface models and results are obtained for three di ff erent boundary conditions.

3.2. Second Model Problem

The second study is conducted by Inglis et al. (2007), which examines the behavior of RVEs with a cohesive in terface containing one or more reinforcing phases under macroscopic strain loading. The interface model presented in Equation (2) is used. A periodic displacement boundary condition (PBC) is defined on the boundaries of the repre sentative volume elements. In the study of Inglis et al. (2007), the problem was solved with the classical Mori-Tanaka approach and micromechanics-based finite elements solution the so-called the Mathematical Theory of Homogeniza tion based Finite Element (MTHFE) approach. Two square RVEs with one centered circular inclusion (Figure 4) and 18 randomly circular inclusions are used in the study. The diameter of the inclusions used in both single and multi inclusion representative volume elements is 60 µ m and the volume fraction of inclusions is 0.32. The material and the interface parameters are given in Table 1. An equibiaxial macroscopic strain-driven loading, with strain increment ( ∆ ε ) of 0.02, is applied to the RVEs. In this study, the e ff ects of a few interface parameters on the results are investigated. The parameters studied are summarized below. • Comparison study of interface algorithm in which, at each load increment, the initial or the last interface con ditions is used. • Comparison study of interface algorithm in which di ff erent maximum iteration and maximum error criteria are used. 3.2.1. Comparison study based on di ff erent initial values of iteration In this study, at each loading step ∆ ε , two cases where the opening displacement of interface node pairs is taken from the state where it remained in the previous loading step (BEM-hist) and from the first state, which is zero opening (BEM-no hist) in the interface equilibrium iterations are examined. This comparison study is conducted for two RVEs including one centered inclusion, and 18 randomly distributed inclusions. Each inclusion is divided in to 30 elements and outer edges of the RVE is discretized into 50 elements. The maximum error and iteration number for the stopping criteria of the interface equilibrium iterations are taken as 10 − 5 and 400, respectively. The results obtained for both models are presented in Figure 6. The red-colored curves are the results obtained with the MFTHE and Mori-Tanaka approaches presented by Inglis et al. (2007). The black colored curves are the results obtained in this study. It is observed that the results, where the interface condition starts from the first loading state (BEM-no hist), give similar results to the Mori-Tanaka approach. It should be noted that Mori-Tanaka approach is not accounting the di ff erence in the case whether single or multi inclusions are used. The consistent result with the MFTHE approach is obtained with the model in which the interface is renewed in each load step (BEM-hist).