## 2 Micromechanical Finite Element Model
### Constitutive Response of Ta Particles and BMGs  
So far, several constitutive models are developed to describe the deformation of the BMGs. [PERSON] proposed the shear transformation model using [PERSON]'s insightful theory, in which he assumes that the plastic deformation only occurs in the region called the shear transformation zone [30]. Furthermore, [PERSON] et al. developed a tension transformation zone model to describe the quasi-brittle dilatation deformation in metallic glass [31]. Viewing the basic \"flow event\" as an individual atom jump driven by the shear stress, [PERSON] [32] developed the free volume model to analyze the plastic deformation in the BMGs. Compared other constitutive models, the free volume model has a clear physical meaning, and has been widely used to evaluate the mechanical properties of the BMGs. According to [PERSON], the general free volume model flow equation is written as:  
\[\frac{\partial\gamma^{p}}{\partial t}=2f\exp\left(-\frac{\alpha v^{*}}{ \overline{v}_{f}}\right)\exp\left(-\frac{\Delta G^{m}}{k_{B}T}\right)\text{ sinh}\left(\frac{\tau\,\Omega}{2k_{B}T}\right) \tag{5}\]  
where \(f\) is the frequency of atomic vibration, \(\alpha\) is a geometrical factor, \(v^{*}\) is the critical volume, \(\overline{v}_{f}\) is the average free volume per atom, \(\Delta G^{m}\) is the activation energy, \(\Omega\) is the atomic volume, \(\tau\) is the shear stress, \(k_{B}\) is the Boltzmann constant, and \(T\) is the absolute temperature. Unlike the equivalent plastic strain in the usual metal plastic model, the free volume is used as a parameter to describe plastic deformation. The net rate of the free-volume increase is:  
\[\frac{\partial\overline{v}_{f}}{\partial t}=v^{*}f\exp\left(-\frac{\alpha v^{ *}}{\overline{v}_{f}}\right)\exp\left(-\frac{\Delta G^{m}}{k_{B}T}\right) \left\{\frac{2\alpha k_{B}T}{\overline{v}_{f}C_{eff}}\left(\cosh\left(\frac{ \tau\,\Omega}{2k_{B}T}-1\right)\right)-\frac{1}{n_{D}}\right\} \tag{6}\]  
Here \(n_{D}\) is a constant varying from three to 10. \(C_{eff}=\frac{E}{3(1-v)}\) is the effective elastic module for isotropic materials with [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(v\). Figure 1: A representative cell (RVC) for two-dimensional (2D) particle-reinforced composites. Figure 2: Process of the iteration to calculate the distance between two ellipses on a plane. According to the flow rule, the strain can be decomposed into elastic and plastic parts:  
\[\dot{\varepsilon}_{ij}=\dot{\varepsilon}_{ij}^{e}+\dot{\varepsilon}_{ij}^{p} \tag{7}\]  
Moreover, the elastic and plastic strain rate can be expressed as follows:  
\[ \begin{array}{l}\dot{\varepsilon}_{ij}^{e}=\frac{1+v}{E}\big{(}\dot{\sigma}_ {ij}-\frac{v}{1+v}\dot{\sigma}_{kt}\delta_{ij}\big{)}\\ \dot{\varepsilon}_{ij}^{p}=\exp\left(-\frac{1}{v_{f}}\right)\sinh\left(\frac{ \sigma_{e}}{\sigma_{0}}\right)\frac{\dot{\varepsilon}_{ij}}{\sigma_{e}}\end{array}  \tag{8}\]  
where \(S_{ij}\) is the deviatoric stress tensor, and \(\sigma_{e}\) is the equivalent stress. According to the \(J_{2}\) flow rule, the evolution of plastic strain is a function depending on the deviatoric stress. So, the flow equation can be written as follows:  
\[\dot{v}_{f}=\frac{1}{\alpha}\exp\left(-\frac{1}{v_{f}}\right)\left\{\frac{3(1- v)}{E}\left(\frac{\sigma_{0}}{\beta v_{f}}\right)\left[\cosh\left(\frac{ \sigma_{e}}{\sigma_{0}}\right)-1\right]-\frac{1}{n_{d}}\right\} \tag{9}\]  
in which, \(\sigma_{0}=\frac{2k_{B}T}{\Omega}\) is the reference stress, and \(\beta=\frac{v}{\Omega}\) and \(v_{f}=\frac{v_{f}}{\alpha\sigma^{n}}\) are the normalized free volume. A user-defined material subroutine (UMAT) in ABAQUS code is developed to implement the free-volume model [33]. For reinforcing Ta particles, an exponentiation isotropic hardening relationship according to [PERSON] et al.