PSI - Issue 61

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I.E U¨ nsal and T. Yalc¸inkaya / Structural Integrity Procedia 00 (2024) 000–000

Izzet Erkin Ünsal et al. / Procedia Structural Integrity 61 (2024) 164–170

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Fig. 1. 2D model with a central imperfection for di ff erent length scales l / w = 0 . 0 (left), l / w = 0 . 5 (middle), l / w = 1 . 0 (right).

plastic hardening behaviour. The internal length scale parameter was varied according to the side length w and the resulting shear bands are plotted. Figure 1, shows the plastic strain in the y-direction. As the internal length scale parameter was varied, the model was able to capture varying degrees of shear band thicknesses. The l / w = 0 . 0 case corresponds to the macroscopic case which is identical to the classical J 2 deformation theory without any size e ff ects. Increasing the l / w ratio corresponds to a smaller specimen size hence the stronger response. The thickness of the shear bands is of particular relevance. The Figure shows that shear bands become more noticeable and thicker shear bands appear as the specimen size decreases. However, as was already indicated in the introduction, experimental studies (Bharathula et al. (2010), Dubach et al. (2009)) have demonstrated that decreasing the size of the specimen both slows and stabilizes the process of shear localization, which increases ductility. This indicates that the compression model lacks this attribute. It is only natural to expect the shear bands to become less prominent as we move towards smaller scales. For the second numerical example, a bending problem is analyzed. In contrast to the compression problem, the structural disorder is ensured by fluctuating local yield stresses at each finite element. This enables us to nucleate many di ff erent shear bands at various locations instead of the previous approach which was a single nucleation point at the center. This is a method followed by Roux and Hansen (1992), Sandfeld et al. (2015) The two-dimensional model has an aspect ratio of 5 : 2 with side length being w , where w is the thickness through the bending direction. The model is discretized with a uniform mesh of 200 × 80 linear quadrilateral elements. The plate is subjected to pure bending boundary conditions for the simulation. The aim here is to simulate the bending of thin foils made up of amorphous material which have been proven to show size-dependent behaviour (Suto et al. (1992)). Moreover, the local yield stresses at each element were varied according to a normal distribution with the average yield stress being σ = 0 . 35 and fluctuations of ∆ = ± 0 . 05. The elasticity modulus and Poisson’s ratio are set to be E = 1 and ν = 0 . 3 respectively. Similar to the previous example, there is no hardening for the plastic region. For both of the specimens, the initial distribution of yield stresses is the same with the length scale parameter being the only di ff erence. It should be noted that making l = 0 renders the conventional J 2 plasticity solution. In Figure 2, the total strain is plotted. Here it can be seen that the smaller specimen has a less prominent shear band formation which in contrast to the previous example, makes a more realistic case. Furthermore, as a result of the loading scenario, shear bands occur mostly in the exterior portions of the specimen, where more severe shear strains are observed. This implies that for the current model, the production of shear bands requires not just variable yield stresses but also high enough strains to initiate them. 3.2. Bending Example