(2) When the stress path does not change its direction:

$$\begin{aligned} \xi\_{p,n+1}^{(m+1)} &= o\_p^{(m+1)} + (\xi\_{p,n}^{(m+1)} - o\_p^{(m+1)}) \frac{p\_{c,n+1}^{(m+1)}}{p\_{c,n}^{(m+1)}} \\ \xi\_{ij,n+1}^{(m+1)} &= o\_{ij}^{(m+1)} + (\xi\_{ij,n}^{(m+1)} - o\_{ij}^{(m+1)}) \frac{p\_{c,n+1}^{(m+1)}}{p\_{c,n}^{(m+1)}} \end{aligned} \tag{6}$$

in which ξ (*m*+1) *p* , ξ (*m*+1) *ij* , ξ (*m*+1) *p*,*n* , ξ (*m*+1) *ij*,*n* , and ξ (*m*+1) *p*,*n*+1 , ξ (*m*+1) *ij*,*n*+1 represent the coordinates of endpoints at incremental steps of *0*, *n*, and (*n* + 1) in the (*m* + 1)th loading events, respectively. To describe the degradation of soft clay under cyclic loading conditions, a damage parameter is introduced into the isotropic hardening rule. The evolution of the size of the boundary surface depends on plastic volumetric strains and the damage parameter, which can be defined as:

$$p\_{c,n+1} = p\_{c,\mathbb{M}} \exp(\frac{1+e\_0}{\lambda-\kappa} d\boldsymbol{\varepsilon}\_{v,n+1}^p) \boldsymbol{\omega}\_{n+1\prime} \tag{7}$$

$$
\omega\_{n+1} = \exp(-\beta \,\varepsilon\_A),
\tag{8}
$$

where

$$\varepsilon\_A = \int \sqrt{2d\_{ij,n+1}^p d\_{ij,n+1}^p / 3},\tag{9}$$

in which λ and κ denote the compression index and swelling index in the *e* − *lnp* space, respectively; *e*<sup>0</sup> represents the void ratio after consolidation under *p* = *pc*. The state variable ω is a function of the deviatoric plastic strain *e p ij*, which is decreasing with the accumulated *e p ij*. The model parameter β controls the rate of damage accumulation. The decrease in ω represents the degradation in stiffness of the clay structure.
