## 3 Stress-Softening Effect at Unloading
### Stress Limit and Prager Modulus  
We next determine the stress limit \(r\) and the Prager modulus \(c\). Let a set of data be given for a monotone stress-strain curve starting at the initial yield point \((h_{0},\tau_{0})\) (cf., Figure 1). Then, a single-variable function representing this curve may also be directly given either by an interpolating function or an other form of function, as shown below:  
\[ \begin{cases}h=f(\tau)\text{ for the loading curve}\,,\\ h_{0}=f(\tau_{0}),\quad\tau\geq\tau_{0}\,,\end{cases}  \tag{23}\]  
For the monotone loading in the uniaxial case, from the \(J_{2}\)-flow model in Section 2 the four reduced equations below may be derived:  
\[\tau-\alpha=r,\]  
\[\dot{h}=\overline{\left(\frac{\partial\overline{w}}{\partial\tau}\right)}+ \frac{4}{9}r^{2}\dot{h}^{-1}\dot{\tau}\,,\]  
\[\dot{\kappa}=\frac{4}{9}r^{3}\dot{h}^{-1}\dot{\tau},\]  
\[\dot{h}=\frac{2}{3}cr^{2}+\frac{4}{9}r^{3}r^{\prime}\,.\]  
In the above, \(\overline{w}=\overline{w}(\tau,\dot{\kappa})\) is the reduced form of the elastic potential Equation (20) in the uniaxial case and given by  
\[\overline{w}=\sum_{s=1}^{M}\overline{w}_{s}(\tau)L_{s}(\bar{\kappa})\,, \tag{24}\]  
where \(\overline{w}_{s}(\tau)\) and \(L_{s}(\bar{\kappa})\) are given by Equations (18), (22) and (23). In the monotone loading case, the axial stress \(\tau\) may be reformulated as a function of the effective plastic work \(\bar{\kappa}\), namely, \(\tau=q(\bar{\kappa})\,.\) As such, from the last three of the foregoing four equations the following two equations may be derived:  
\[ \begin{cases}\tau^{\prime}=r^{\prime}+1.5\dot{\tau}^{\xi}_{\tau}\,,\\ \frac{d}{d\bar{\kappa}}\left(f(\tau)-\frac{\partial\overline{w}}{\partial\tau }\right)=\frac{1}{\tau}\,,\end{cases}  \tag{25}\]  
The stress limit \(r\) may be given as follows [40; 46; 47; 48]:  
\[r=r_{\infty}+(\tau_{0}-r_{\infty})e^{-\beta\bar{\kappa}}\,, \tag{26}\]  
where \(r_{\infty}>0\) is a constant and \(\beta>0\) is a large dimensionless parameter. For a large \(\beta>0\), the exponential function in Equation (26) rapidly goes to vanish for \(\bar{\kappa}>\delta\) and the stress limit above is almost constant, i.e., \(r^{\prime}=r_{\infty}\) for \(\bar{\kappa}>\delta\). Here \(\delta\) is very small. The localized property just mentioned means that, after the initial yielding, the stress limit \(r\) may be taken to be constant and given just by \(r_{\infty}\), as will be done below. By setting \(r=r_{\infty}\), the integration of Equation (25)\({}_{2}\) from the initial yield point \((h_{0},\tau_{0})\) to a point \((h,\tau)\) along the monotone curve in Figure 1 yields  
\[\bar{\kappa}=r_{\infty}\left(f(\tau)-\frac{\partial\overline{w}}{\partial\tau }\right)\,. \tag{27}\]At each unloading point \((h_{s}^{*},\tau_{s}^{*})\), the derivative \(\partial\overline{w}/\partial\tau\) is just given by Equation (15) with \(\tau=\tau_{s}^{*}\). From this and Equation (17) and \(f(\tau_{s}^{*})=g_{s}(\tau_{s}^{*})\) we obtain the effective plastic works at the \(M\) unloading points as follows:  
\[\tilde{\kappa}_{s}=r_{\infty}\,h_{s}^{p},\quad s=1,2,\cdots,M\,, \tag{28}\]  
where \(h_{s}^{p}\) and \(\tilde{\kappa}_{s}\) are the plastic strain and the effective plastic work associated with the \(s\)-th unloading curve. Next, Equation (25)\({}_{2}\) with \(r=r_{\infty}\) may be converted to  
\[\left(\frac{df}{d\tau}-\frac{\partial^{2}\overline{w}}{\partial\tau^{2}} \right)\tau^{\prime}=\frac{\partial^{2}\overline{w}}{\partial\tau\partial \tilde{\kappa}}+\frac{1}{r_{\infty}}\,. \tag{29}\]  
From this and Equation (25)\({}_{1}\), an expression for the Prager modulus \(c\) in the uniaxial case may be obtained in terms of the axial stress \(\tau\) and the \(\tilde{\kappa}\). Replacing \(\tau\) by the invariant \(\varphi\) in Equation (19) we eventually obtain [PERSON]'s modulus for general multi-axial cases as follows:  
\[c=\frac{2}{3}\,\frac{1+r_{\infty}\phi(\varphi,\tilde{\kappa})}{f^{\prime}( \varphi)-\zeta(\varphi,\tilde{\kappa})}\,, \tag{30}\]  
where  
\[\left\{