2.1.5. Hardening Modulus

*K<sup>p</sup>* and *K<sup>p</sup>* are related to the actual plastic modulus and the bounding plastic modulus at the image stress *sij*, respectively. According to the consistency condition, the following equation can be obtained:

$$
\frac{\partial \mathbf{F}}{\partial \overline{p}} d\overline{p} + \frac{\partial \mathbf{F}}{\partial \overline{s}\_{lj}} d\overline{s}\_{lj} + \frac{\partial \mathbf{F}}{\partial \varepsilon^p\_v} d\varepsilon^p\_v + \frac{\partial \mathbf{F}}{\partial \omega} d\omega = \mathbf{0}.\tag{17}
$$

Substituting the hardening rules, the loading index and the equation of the bounding surface into the consistency conditions, the bounding plastic modulus at the image stress can be written as:

$$\overline{K}\_p = (\overline{p} - \xi\_p) p\_c \overline{n}\_p (\chi\_0 - \frac{2\beta}{M^2 - \alpha\_0^2} \frac{\overline{q}}{\overline{n}\_p}) \, , \tag{18a}$$

in which:

$$\chi\_0 = \frac{1+e\_0}{\lambda-\kappa'} \tag{18b}$$

$$
\overline{q} = \sqrt{\frac{3}{2} \hat{s}\_{ij} \hat{s}\_{ij\prime}}\tag{18c}
$$

$$
\overline{n}\_p = \frac{\partial F}{\partial \overline{p}} \,\tag{18d}
$$

$$\mathfrak{s}\_{ij} = \overline{\mathfrak{s}}\_{ij} - \xi\_{ij} - (\overline{p} - \xi\_p). \tag{18e}$$

Here, the interpolation is adopted to calculate the plastic-hardening modulus at the current stress state.

$$K\_p = \overline{K}\_p + H(\overline{p}, \overline{q}, \varepsilon\_{\upsilon^\prime}^p \omega)(\frac{\delta}{\delta\_0 - \delta})^r,\tag{19}$$

in which *H*(*p*, *q*, ε *p v* , ω is the shape hardening function. Different shape hardening functions are adopted in the first loading, reloading, and unloading stages, respectively.

$$H(\overline{p}, \overline{q}, \epsilon^{p}\_{\upsilon}, \omega) = \begin{cases} \left| \mathcal{K}\_{m} - \overline{\mathcal{K}}\_{p} \right| & \text{for first} \quad \text{loading} \\ \left| \zeta\_{r} \mathcal{K}\_{m} - \overline{\mathcal{K}}\_{p} \right| & \text{for} \quad \text{reloading} \\ \left| \zeta\_{u} \mathcal{K}\_{m} - \overline{\mathcal{K}}\_{p} \right| & \text{for} \quad \text{unloading} \end{cases} \tag{20a}$$

in which:

$$
\zeta\_{\iota} = \left(1 + \frac{\overline{\partial}\mathcal{F}}{\overline{\partial}\overline{p}} / \eta\right) \zeta\_{r\iota} \tag{20b}
$$

$$K\_m = 8\chi\_0 \left(p\_c\right)^3,\tag{20c}$$

where γ and η are model parameters. The material parameter ζ*<sup>r</sup>* controls the reloading events, of which a detailed account of physical meaning can be found in the paper by Hu et al. [26].
