*3.3. Mie-Grüneisen Formulation*

To govern the hydrodynamic behavior of the materials, an equation of state (eos) was used, which defines the pressure (of a solid at a given temperature) as a function of the density and the internal energy [44]. The Mie–Grüneisen (M–G) eos was applied to model the materials' volumetric strength at the high pressures present in the LIW process. The M–G eos is a function of energy, relating the shock velocity and the particle velocity. Equations (7)–(14) are found in [44]. The M–G eos is given in Equation (7):

$$P - P\_H = \Gamma \rho (E\_m - E\_H) \, , \tag{7}$$

where *P*, *P H*, Γ, ρ, *Em*, and *EH* are pressure, Hugoniot pressure, Grüneisen ratio, current density, internal energy per unit mass, and Hugoniot specific energy per unit mass, respectively. The Grüneisen ratio is given in Equation (8):

$$
\Gamma = \Gamma\_0 \frac{\rho\_0}{\rho},
\tag{8}
$$

where Γ0 is a material constant and ρ0 is the density at the reference point. The Hugoniot parameters are functions of density only and are related by Equation (9):

$$E\_H = \frac{P\_H \eta}{2\rho\_0},\tag{9}$$

where η is the nominal compressive volumetric strain and is given in Equation (10):

$$
\eta = 1 - \frac{\rho\_0}{\rho},
\tag{10}
$$

Replacing the terms in Equation (7) with their definitions in Equations (8)–(10) gives Equation (11):

$$P = P\_H \left( 1 - \frac{\Gamma\_0 \eta}{2} \right) + \Gamma\_0 p\_0 E\_{m\_{\prime}} \tag{11}$$

The Hugoniot pressure is defined through curve fitting to the experimental data and is given in Equation (12):

$$P\_H = \frac{\rho\_0 c\_0^2 \eta}{\left(1 - s\eta\right)^2} \tag{12}$$

where *c*0 and *s* linearly relate the shock velocity (*Us*) and particle velocity (*Up*) by Equation (13):

$$
\mathcal{U}\_s = c\_0 + s\mathcal{U}\_{\mathcal{V}}.\tag{13}
$$

Replacing *PH* in Equation (11) with its definition in Equation (12) gives the linear Hugoniot form of the M–G eos in Equation (14):

$$P = \frac{\rho\_0 c\_0^2 \eta}{\left(1 - s\eta\right)^2} \left(1 - \frac{\Gamma\_0 \eta}{2}\right) + \Gamma\_0 \rho\_0 E\_{m\nu} \tag{14}$$

The M–G eos parameters of the materials used in the simulations are listed in Table 4.

**Table 4.** M–G eos parameters [31,45], with permission from Elsevier, 2018.


The different numerical techniques employed for simulation of the LIW process are described next.
