PSI - Issue 61

166 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 3 In order to include the gradient e ff ects without the higher order stress terms, the definition of ˙ ε p is replaced by a viscoplastic term which expresses the plastic strain gradient in terms of e ff ective stress σ e rather than its rate ˙ σ e (for more details see Huang et al. (2004)),

= ˙ ε   

σ e 2 ( ε p ) + l η p    m

σ e σ f low 

= ˙ ε 

m

˙ ε p

(2)

σ ref  f

Where, l is the intrinsic lenght scale and η p is the e ff ective plastic strain gradient. Note that, the flow stress σ f low is expressed as a function of σ ref , f ( ε p ) and the product of the intrinsic lenght scale and the e ff ective plastic strain gradient l η p . In this relation, σ ref is the reference stress value and f ( ε p ) is a nondimentional hardening curve taken for a pure uniaxial stress-strain curve. By rewriting the flow stress in this way, it is possible to include gradient terms without getting into the higher order terms. Also, setting the lenght scale as zero, the strain gradient term vanishes and the classical J 2 relation is recovered. Stress-rate in terms of the strain rates can be obtained as per first substituting Eq. 2 into the Eqs. 1 and then inverting the relation: ˙ σ i j = K ˙ ε kk δ i j + 2 µ    ˙ ε ′ i j − 3˙ ε p 2 σ e    σ e σ ref  f 2 ( ε p ) + l η p    m ˙ σ ′ i j    (3) Constitutive equation given above relates e ff ective plastic strain gradient with the stress. Since, η p term only comes into play for the calculation of the stress, it fits within the lower-order strain gradient plasticity frameworks (Acharya and Beaudoin). In literature, Zr- or Ti-based bulk metallic glasses have been shown to have an almost ideally plastic hardening behaviour (Wu et al. (2014), Pan et al. (2020)), therefore throughout this study an ideally plastic hardening behaviour is employed. This is a valid assumption for the simulation of bulk metallic glasses (Roux and Hansen (1992)). Therefore, the only hardening observed is due to the plastic strain gradient terms. The definition of e ff ective plastic strain gradient η p is the same as Gao et al. (1999), and given by; η p = ε p ik , j + ε p jk , i − ε p i j , k (4) In the above expression, the comma represents in terms of which variable the derivative is taken as. For more explicit details considering the calculation of the strain gradient and how it is implemented in a finite element setting, the reader is re ff ered to the paper (U¨ nsal et al. (2023)).

3. Numerical Examples

For the numerical examples, three di ff erent loading cases are examined. The aim is to compare and discuss the two di ff erent means of introducing disorder within the amorphous material. For the first example, an imperfection is added in compression to trigger shear bands whereas for the bending and tensile examples, fluctuating local yield stresses are used to obtain the shear bands.

3.1. Compression Example

In this example, a central imperfection is incorporated into an initially uniform specimen. This specimen is then subjected to compressive load to initiate strain patterning, resulting in the emergence of two perpendicular shear bands stemming from the central imperfection. Similar to non-local damage models, this initial defect serves as a nucleation site to start the shear localization process. This example serves as a simple model for the creation of a single stable shear band and is useful for studying strain patterns. The model is a square plate with side lengths w × w . It is discretized with 100 × 100 linear quadrilateral finite elements. The bottom face of the plate is fixed in the y-direction with the bottom left corner node being also fixed in the x-direction to prevent rigid body motion. A displacement of u y is applied from the top face in negative y-direction with the rest of the remaining surfaces being traction-free. Young’s modulus is set to be unity E = 1 with Poisson’s ratio ν = 0 . 3. The yield stress is set to be σ y = 1 . 0 with an ideally