## 2 Materials and Methods
### Formulation of the Discrete Element Model
#### 2.1.2 The Bonded Particle Model  
The bonded particle model, based on a parallel bond [9], is implemented to numerically simulate the uniaxial vertical compression of NiAl samples. The parallel bond can be envisioned as a set of elastic springs uniformly distributed over the circular cross-section in 3D, lying on the contact plane and centered on the contact point. Thus, parallel bonds can transmit both the force and the moment between the bonded particles. When the parallel bond is formed between the particles \(i\) and \(j\), the contact force and the moment are initialized to zero. Each subsequent increment of relative displacement and relative rotational velocity produces the increments of the force and the moment that are added to the current values. Thus, in each time step \(l+1\), the normal bond force \(F^{b,l+1}_{ij,n}\) and the tangential bond force \(F^{b,l+1}_{ij,t}\) can be obtained, adding the increments of the normal bond force \(\Delta F^{b,l+1}_{ij,n}\) and the tangential bond force \(\Delta F^{b,l+1}_{ij,t}\) to the normal contact force \(F^{b,l}_{ij,n}\) and the tangential contact force \(F^{b,l}_{ij,t}\) of the previous time step \(l\), respectively:  
\[F^{b,l+1}_{ij,n}=F^{b,l}_{ij,n}+\Delta F^{b,l+1}_{ij,n}=F^{b,l}_{ij,n}+k^{b}_{ ij,n}A\Delta u^{l+1}_{ij,n}, \tag{6}\]  
\[F^{b,l+1}_{ij,t}=F^{b,l}_{ij,t}+\Delta F^{b,l+1}_{ij,t}=F^{b,l}_{ij,t}-k^{b}_{ ij,t}A\Delta u^{l+1}_{ij,t}, \tag{7}\]  
where \(A\) is the cross-sectional area of the bond, and \(k^{b}_{ij,n}\) and \(k^{b}_{ij,t}\) denote the normal and the tangential stiffness of the bond, while \(\Delta u^{l+1}_{ij,n}\) and \(\Delta u^{l+1}_{ij,t}\) are the increments of the relative normal displacement and the relative tangential displacement during the time step \(l+1\), respectively. In each time step \(l+1\), the normal moments \(M^{b,l+1}_{ij,n}\) and the tangential moments \(M^{b,l+1}_{ij,t}\), as well as their increments \(\Delta M^{b,l+1}_{ij,n}\) and \(\Delta M^{b,l+1}_{ij,t}\), are computed as follows:  
\[M^{b,l+1}_{ij,n}=M^{b,l}_{ij,n}+\Delta M^{b,l+1}_{ij,n}=M^{b,l}_{ij,n}-k^{b}_{ ij,t}J\Delta\theta^{l+1}_{ij,n}, \tag{8}\]  
\[M^{b,l+1}_{ij,t}=M^{b,l}_{ij,t}+\Delta M^{b,l+1}_{ij,t}=M^{b,l}_{ij,t}-k^{b}_{ ij,n}I\Delta\theta^{l+1}_{ij,t}, \tag{9}\]where \(\mathbf{M}^{b,l}_{ij,t}\) and \(\mathbf{M}^{b,l}_{ij,t}\) are the current values of the normal and tangential components of the moment at the previous time step \(l\), respectively, and \(\Delta\theta^{l+1}_{ij,n}\) and \(\Delta\theta^{l+1}_{ij,t}\) are the increments of the normal and tangential components of the relative rotational velocity at the time step \(l+1\), respectively, while \(l\) and \(J\) are the moment of inertia and the polar moment of inertia of the parallel bond's cross-section, respectively. The parallel bonds break instantaneously when the maximum tensile stress \(\sigma^{\max}_{ij}\) exceeds the tensile strength \(\sigma_{c}\) or the maximum shear stress \(\tau^{\max}_{ij}\) exceeds the shear strength \(\tau_{c}\), leading to microcrack formation between the two particles. The expressions of the maximum tensile stress and the maximum shear stress, acting on the bond, are derived from the beam bending theory, which gives the criteria of the bond failure formulated as  
\[\sigma_{c} <\sigma^{\max}_{ij}=\frac{-F^{b}_{ij,n}}{A}+\frac{\left|\mathbf{M}^{b}_ {ij,t}\right|}{I}R, \tag{10}\] \[\tau_{c} <\tau^{\max}_{ij}=\frac{\left|\mathbf{F}^{b}_{ij,t}\right|}{A}+\frac{ \left|\mathbf{M}^{b}_{ij,n}\right|}{J}R, \tag{11}\]  
where \(R\) is the bond radius and \(F^{b}_{ij,n}\) is the scalar value of the normal component of the bond force, which can cause bond breakage only in the case of tension. The bond is broken such that a microscopic crack is created to represent the failure of the material at this particular bond.