## 3 Stress-Softening Effect at Unloading
### The Evolving Elastic Potential  
We first present the evolving elastic potential \(\overline{W}\). The main procedures will follow those suggested in a previous study [40]. Details will be omitted below and may be referred to this reference. A single-variable function for the unloading curve starting at the unloading point \((h_{s}^{*},\tau_{s}^{*})\) may be obtained in a sense of accurately fitting any given date set as shown in Equation (12). Indeed, it may be given by an interpolating function, such as a Lagrange interpolant and an other form of function. Let such a function be designated by  
\[h=g_{s}(\tau)\text{ for unloading curves }s=1,\cdots,M\,, \tag{13}\]  
with  
\[ \begin{cases}g_{s}(0)=h_{s}^{p}\,,\\ g_{s}(\tau_{\alpha})=h_{\alpha}\,,\\ \quad\alpha=0,1,\cdots,N_{s}\,.\end{cases} \]  
In the uniaxial case, it may readily be shown that the evolving potential Equation (2) reduces to a function of the axial Kirchhoff stress \(\tau\) and the effective plastic work \(\bar{\kappa}\), denoted  
\[\overline{w}=\overline{w}(\tau,\bar{\kappa})\,. \tag{14}\]For the unloading curve starting at the unloading point \((h_{s}^{*},\tau_{s}^{*})\), the elastic rate Equation (3) with \(\dot{T}=0\) yields  
\[\dot{h}=\overline{\left(\frac{\partial\overline{\dot{w}}}{\partial\tau}\right)}\,.\]  
For the unloading curve at issue, the effective plastic work is held fixed and given by \(\bar{\kappa}_{s}\). Then, the integration of this equation from the point \((h_{s}^{p},0)\) to the point \((h,\tau)\) in an unloading process gives rise to  
\[h-h_{s}^{p}=\frac{d\overline{w}_{s}}{d\tau}\,, \tag{15}\]  
where \(\overline{w}_{s}=\overline{w}(\tau,\bar{\kappa}_{s})\). In deriving Equation (15), the following initial condition is used:  
\[\left.\frac{\partial\overline{w}}{\partial\tau}\right|_{\tau=0}=0\,. \tag{16}\]  
From Equation (15) and \(h=g_{s}(\tau)\) we deduce  
\[\frac{d\overline{w}_{s}}{d\tau}=g_{s}(\tau)-h_{s}^{p}\,. \tag{17}\]  
This produces  
\[\overline{w}_{s}=\overline{w}_{s}(\tau)=\int_{0}^{\tau}\left(g_{s}(\tau)-h_{s }^{p}\right)d\tau\,. \tag{18}\]  
From the above procedures we obtain \(M\) uniaxial potentials \(\overline{w}_{1},\cdots,\overline{w}_{M}\) for the \(M\) unloading curves at issue. However, each of them is restricted to the uniaxial case. The potential for general multiaxial deformations is obtainable from these uniaxial potentials by means of direct procedures. In fact, replacement of the axial Cauchy stress \(\tau\) in each potential \(\overline{w}(\tau)\) with the invariant \(\varphi\) (cf., Equation (10)) multiplied by the mode invariant \(\gamma\) (cf., [49]) yields a multi-axial potential below:  
\[ \begin{cases}\overline{W}_{s}=\overline{w}_{s}\left(\gamma\varphi\right)\,,\\ \gamma=4.5\text{tr}\dot{\tau}^{3}/\varphi^{3}\,.\end{cases}  \tag{19}\]  
It may be verified that, in the uniaxial case, each function \(\overline{W}_{s}\) for general multi-axial deformations, given by Equation (19), exactly supplies its uniaxial counterpart \(\overline{w}_{s}\). We seek for a multi-axial potential \(\overline{W}(\bar{\tau},\bar{\kappa})\) which exactly gives the \(M\) multi-axial potentials \(\overline{W}_{1},\cdots,\overline{W}_{M}\) for \(\bar{\kappa}=\bar{\kappa}_{1},\cdots,\bar{\kappa}_{M}\). Such a potential may be given by an interpolating function with the \(M\) interpolating nodes \((\kappa_{s},\overline{W}_{s})\) for \(s=1,\cdots,M\).