2.1.2. The Evolution of the Boundary Surface

The generalized isotropic hardening rule is applied in the present model, which assumes that the boundary surface isotropically hardens around the discrete mapping center in the stress space. When the loading path changes its direction, the boundary surface should translate along the direction of stress reversal point to the image stress point. The mapping center can be expressed as:

$$\begin{pmatrix} o\_{p,n+1}, o\_{lj,n+1} \end{pmatrix} = \begin{cases} \begin{pmatrix} o\_{p,n\nu}, o\_{lj,n} \end{pmatrix} & \frac{\partial F}{\partial \overline{\sigma}\_{ij,n}} d\sigma\_{ilj,n+1} \ge 0\\ \begin{pmatrix} p\_{n\nu}, s\_{lj,n} \end{pmatrix} & \frac{\partial F}{\partial \overline{\sigma}\_{ij,n}} d\sigma\_{lj,n+1} < 0 \end{cases} \tag{4}$$

in which *op*,*n*, *oij*,*<sup>n</sup>* and *op*,*n*+1, *oij*,*n*+<sup>1</sup> are the coordinates of the mapping center at incremental steps of n and n + 1, respectively. The location of the boundary surface depends on the stress path direction.

(1) When the stress path changes its direction:

$$\begin{array}{l} \xi\_p^{(m+1)} = \xi\_p^{(m)} (p - \overline{p}) \\ \xi\_{ij}^{(m+1)} = \xi\_{ij}^{(m)} (s\_{ij} - \overline{s}\_{ij}) \end{array} \tag{5}$$
