Table 1. Summary of potential functions...

Table 1.

Summary of potential functions defining different versions of the MCC model

Model version	Free energy potential	Force potential
Hyper-plastic MCC (equivalent to viscoplastic with n = 1)	$f = κ p_{a} \exp [\frac{ε_{v} - ε_{v}^{p} + (3 / 2) g {(ε_{s} - ε_{s}^{p})}^{2}}{κ}]$	$z = p_{0} [\frac{{\dot{ε}}_{v}^{p} + \sqrt{{({\dot{ε}}_{v}^{p})}^{2} + {(M {\dot{ε}}_{s}^{p})}^{2}}}{2}]$
Hyper-viscoplastic MCC		$z = \frac{r p_{0}}{n} {[\frac{{\dot{ε}}_{v}^{p} + \sqrt{{({\dot{ε}}_{v}^{p})}^{2} + {(M {\dot{ε}}_{s}^{p})}^{2}}}{2 r}]}^{n}$
Hyper-plastic bounding surface MCC (equivalent to viscoplastic with n = 1)	$\begin{aligned} f = & κ p_{a} \exp [\frac{ε_{v} - \sum_{i = 1}^{N} ε_{v i}^{p} + (3 / 2) g {(ε_{s} - \sum_{i = 1}^{N} ε_{s i}^{p})}^{2}}{κ}] \\ + \sum_{i = 1}^{N} (\frac{p_{a}}{k_{p}}) \exp \{k_{p} [H_{i} ε_{v i}^{p} + \frac{3}{2} g_{p} {(H_{i} ε_{s i}^{p})}^{2}]\} \end{aligned},$ $H_{i} = 1 - \frac{i}{N}$	$z = p_{0} \{\frac{{\dot{ε}}_{v N}^{p} + \sqrt{\sum_{i = 1}^{N} [{(K_{i} {\dot{ε}}_{v i}^{p})}^{2} + {(M K_{i} {\dot{ε}}_{s i}^{p})}^{2}]}}{2}\}$ ⁠, $K_{i} = \frac{i}{N}$
Hyper-viscoplastic bounding surface MCC		$z = \frac{r p_{0}}{n} {\{\frac{{\dot{ε}}_{v N}^{p} + \sqrt{\sum_{i = 1}^{N} [{(K_{i} {\dot{ε}}_{v i}^{p})}^{2} + {(M K_{i} {\dot{ε}}_{s i}^{p})}^{2}]}}{2 r}\}}^{n},$ $K_{i} = \frac{i}{N}$

Model version	Free energy potential	Force potential
Hyper-plastic MCC (equivalent to viscoplastic with n = 1)	$f = κ p_{a} \exp [\frac{ε_{v} - ε_{v}^{p} + (3 / 2) g {(ε_{s} - ε_{s}^{p})}^{2}}{κ}]$	$z = p_{0} [\frac{{\dot{ε}}_{v}^{p} + \sqrt{{({\dot{ε}}_{v}^{p})}^{2} + {(M {\dot{ε}}_{s}^{p})}^{2}}}{2}]$
Hyper-viscoplastic MCC		$z = \frac{r p_{0}}{n} {[\frac{{\dot{ε}}_{v}^{p} + \sqrt{{({\dot{ε}}_{v}^{p})}^{2} + {(M {\dot{ε}}_{s}^{p})}^{2}}}{2 r}]}^{n}$
Hyper-plastic bounding surface MCC (equivalent to viscoplastic with n = 1)	$\begin{aligned} f = & κ p_{a} \exp [\frac{ε_{v} - \sum_{i = 1}^{N} ε_{v i}^{p} + (3 / 2) g {(ε_{s} - \sum_{i = 1}^{N} ε_{s i}^{p})}^{2}}{κ}] \\ + \sum_{i = 1}^{N} (\frac{p_{a}}{k_{p}}) \exp \{k_{p} [H_{i} ε_{v i}^{p} + \frac{3}{2} g_{p} {(H_{i} ε_{s i}^{p})}^{2}]\} \end{aligned},$ $H_{i} = 1 - \frac{i}{N}$	$z = p_{0} \{\frac{{\dot{ε}}_{v N}^{p} + \sqrt{\sum_{i = 1}^{N} [{(K_{i} {\dot{ε}}_{v i}^{p})}^{2} + {(M K_{i} {\dot{ε}}_{s i}^{p})}^{2}]}}{2}\}$ , $K_{i} = \frac{i}{N}$
Hyper-viscoplastic bounding surface MCC		$z = \frac{r p_{0}}{n} {\{\frac{{\dot{ε}}_{v N}^{p} + \sqrt{\sum_{i = 1}^{N} [{(K_{i} {\dot{ε}}_{v i}^{p})}^{2} + {(M K_{i} {\dot{ε}}_{s i}^{p})}^{2}]}}{2 r}\}}^{n},$ $K_{i} = \frac{i}{N}$