## 4 Shape Functions for Uniaxial Loading and Unloading Curves  
With the evolving elastic potential \(\overline{W}\), the stress limit \(r\) and the Prager modulus \(c\) given in the last section, the elastoplastic \(J_{2}\)-flow model established in Section 2 exactly reproduces the monotone loading curve \(h=f(\tau)\) and any given number of unloading curves, \(h=g_{s}(\tau)\). It may be straightforward to find out suitable forms of such single-variable functions fitting any given data for the monotone loading case and for various unloading cases, without involving usual complicated procedures of treating several nonlinear differential equations incorporating a number of parameters to be identified. The foregoing two kinds of single-variable functions, referred to as the shape functions for loading und unloading curves, may be given directly by usual interpolating functions fitting any given data without error. Alternatively, these shape functions may also be given by other forms of functions. In particular, it is of interest to have simple forms of shape functions with parameters of direct physical meanings. Such forms of shape functions are given below.  
Firstly, below a simple form of the shape function is given to represent each unloading curve:  
\[\tau=\frac{(1-\gamma)E(h-h^{p})}{1-\frac{h-h^{p}}{u}}+\gamma\,E(h-h^{p})\quad \text{for}\quad h-h^{p}\geq 0\,, \tag{32}\]  
and its inverted form is as follows:  
\[h=g(\tau)=h^{p}+\frac{2u\sigma}{\tau+Eu+\sqrt{(\tau+Eu)^{2}-4\gamma^{+}Eu\sigma }}\,. \tag{33}\]  
In the above, \(h^{p}\) is the plastic strain after unloading, \(E\) is the Young's modulus at infinitesimal strain, \(u\) is the strain limit and \(\gamma<1\) is dimensionless parameter characterizing the growth degree of stress as the strain is approaching the strain limit \(u\). In general, each of these parameters may rely on the effective plastic work \(\tilde{\kappa}\), namely,  
\[E=E(\tilde{\kappa}),\quad u=u(\tilde{\kappa}),\quad\gamma=\gamma(\tilde{ \kappa})\,. \tag{34}\]For each unloading curve associated with a value of the effective plastic work, values of the above parameters may be determined by fitting the above shape function to given data. As such, a set of parameter values may be obtained for a family of unloading curves. Then, each of the foregoing \(\tilde{\kappa}\)-dependent parameter may also be given directly by an interpolating function. It may be noted that the parameters introduced directly represent the very features of each elastic unloading curve at issue and, accordingly, are determinable from each unloading curve in a decoupled manner.  
In particular, the unloading curve starting at the initial yield points \((h_{0},\tau_{0})\) is associated with \(\tilde{\kappa}=0\) and coincident exactly with the initial elastic curve from the origin to the initial yield point (cf., Figure 1). This curve may be given simply by a linear relation between the axial stress \(\tau\) and the axial Hencky strain \(h\), namely,  
\[\begin{cases}h=g_{0}(\tau)=\frac{\tau}{E_{0}}\quad\left(0\leq\tau\leq\tau_{0} \right),\\ \tilde{\kappa}=0\,,\end{cases} \tag{35}\]  
where \(E_{0}=E(0)\) is the initial [PERSON]'s modulus.  
On the other hand, the shape function for the monotone loading curve starting at the initial yield points \((h_{0},\tau_{0})\) may be given by (c.f., [46; 47; 48])  
\[\begin{cases}h&=f(\tau)=\frac{\tau_{0}}{E}+\xi_{0}(\tau-\tau_{0})+\\ &\quad+\frac{1}{2}(\xi_{1}(\tau-\tau_{0}-s)+a_{0})\left(\tanh(m(\tau-\tau_{0}- s))+\tanh(ms)\right)\,.\end{cases} \tag{36}\]  
The meanings of the parameters incorporated may be found in previous works [46; 47; 48].  
The above shape functions for the monotone loading curve and the unloading curves of SMPs are new and much simpler, as compared with the counterparts in a most recent study [40]. With these new shape functions, accurate accord with test data may be achieved, as will be shown in Section 6.1.