PSI - Issue 61

Izzet Erkin Ünsal et al. / Procedia Structural Integrity 61 (2024) 164–170 I.E U¨ nsal and T. Yalc¸inkaya / Structural Integrity Procedia 00 (2024) 000–000

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2

BMGs exhibit negligible macroscopic plastic strain before the formation of localized shear bands, ultimately leading to catastrophic shear failure (Ashby and Greer (2006)). Various modelling approaches have been o ff ered in the literature, however understanding the interplay between the microscale and macroscale behavior remains an ongoing challenge, multiscale modelling approaches o ff er a viable option (see, e.g. Yalc¸inkaya et al. (2022); Aydiner et al. (2024)). On the atomic length scale, deformations of BMGs have been thoroughly studied in terms of molecular dynamics (MD) simulations (Srolovitz et al. (1983), Kobayashi et al. (1980)). However, molecular dynamic simulations are severely constrained in terms of size and time scale, especially when compared to actual results on greater length and time scales, as well as realistic strain rates. Moreover, recent experimental findings (Chen et al. (2013)) indicate that the shear localization process in amorphous materials may be delayed or even suppressed by reducing the sample size (Volkert et al. (2008)), highlighting a size-dependent deformation characteristic. Therefore, it is necessary to use a size-dependent, continuum-based approach to sustain the connection with the macroscale and to fully uncover the underlying micromechanical phenomenon. To this end, a strain-gradient model is used in this study to incorporate size e ff ects in a continuum scale. In literature, strain-gradient (SG) theories have been separated into two classes, regarding the type of stress and boundary condition descriptions (Niordson and Hutchinson (2003)). Higher order SG theories, initially proposed by Aifantis (1984), use higher order stress terms as work conjugate to the strain gradient (see, e.g. Yalcinkaya et al. (2011); Yalc¸inkaya et al. (2012)). As a result, equilibrium equations and the boundary conditions must be changed. In contrast, lower order SG theories, initially proposed by Acharya and Beaudoin (2000), retains almost all of the features of the classical J 2 theory, and requires no extra boundary conditions. Lower order models just include the influence of strain gradient in the flow stress calculation. As a result, they do not necessitate any changes to the equilibrium equations because only the hardening function of the material is altered. This feature enables lower-order theories to be easily implemented in a commercial finite element code, however, the lack of higher-order stress terms and extra boundary conditions may result in inaccuracies with the boundary layer phenomenon being a notable example (Gao et al. (1999)). In the current work, Conventional Mechanism-Based Strain Gradient (CMSG) plasticity theory is employed to numerically analyze and discuss the size e ff ect on microstructure evolution in metallic glasses. CMSG theory (Huang et al. (2004)) is the lower-order equivalent of the higher-order Mechanism-Based Strain Gradient (MSG) theory pro posed by (Gao et al. (1999)). CMSG theory has been widely used for modelling size e ff ects on various di ff erent problems (Zhao et al. (2020), U¨ nsal et al. (2023)). For the current study, this theory is implemented via a User Mate rial Subroutine (UMAT), in the commercial finite element program ABAQUS to model shear band formations under di ff erent sized specimens. Further, the findings are compared with the classical plasticity approaches. For the mod elling of amourphous material, two di ff erent methods are adopted. For the first aproach, a central defect is introduced into an otherwise uniform specimen. Similar to non-local damage models, the defect serves as a nucleation point to first initiate shear localization which is then followed by shear band formation (Peerlings et al. (1996)). For an alter native approach, the methodology introduced by Roux and Hansen (1992), Burnley (2013), and Sandfeld et al. (2015) is followed, wherein the material behavior is considered as ideally plastic and the disorder which causes the strain pat terning formation is introduced through random local variations in the local yield stress. We explore the significance of di ff erent loading cases, examining the resulting deformation patterns. The paper is organized as follows. The constitutive model is explained in Section 2. The numerical approach and the obtained results are presented in Section 3, which is followed by concluding remarks and outlook in Section 4.

2. Constitutive Model

The constitutive relations in CMSG resemble those of conventional plasticty almost exactly, since no higher-order terms are involved. Therefore, the volumetric and deviatoric strain definitions can be expressed in rate form,

˙ σ kk 3 K ˙ σ ′ i j 2 µ

˙ ε kk =

,

(1)

3˙ ε p 2 σ e

˙ ε ′

˙ σ ′

i j .

i j =

+

Where K and µ is the bulk and shear modulus respectively and σ e denotes Von-Mises equivalent stress and ε p is the equivalent plastic strain.