2.1.6. Implicit Integration Algorithm

In this section, the implicit integration algorithm is used for the implementation of the models. The calculation steps are as follows:

*J. Mar. Sci. Eng.* **2019**, *7*, 308

(1) Assume that strain increment ∆ε*n*+<sup>1</sup> is complete elastic increment at incremental steps of *n* + 1. For initial iteration count *k* = 0, these variables are defined as follows:

$$\begin{array}{ccccc} \Lambda\_{n+1}^{(0)} = 0, & \Delta\omega\_{n+1}^{(0)} = 0, & \Delta\varepsilon\_{v,n+1}^{p} & = 0 \\ p & (0) & & \\ \Delta e\_{ij,n+1}^{(0)} = 0, & b\_{n+1}^{(0)} = b\_{n\nu} & p\_{c,n+1}^{(0)} = p\_{c\nu} & \\ \overline{K}\_{p,n+1}^{(0)} = \overline{K}\_{p,n\nu} & \xi\_{ij,n+1}^{(0)} = \xi\_{ij,n\nu} & o\_{ij,n+1}^{(0)} = o\_{ij,n} & \\ \end{array} \tag{21}$$

(2) Non-linear elastic predictor:

$$\begin{pmatrix} p\_{n+1}^{(0)} = p\_n \exp\{ \frac{1+\varepsilon\_0}{\kappa} \Delta \varepsilon\_{\upsilon, n+1} \} \\ s\_{ij, n+1}^{(0)} = s\_{ij, n} + 2G\_{n+1}^{(0)} \Delta e\_{ij, n+1} \end{pmatrix} \tag{22}$$

$$\mathbf{K}\_{n+1}^{(0)} = p\_n \frac{1+\epsilon\_0}{\kappa}, \quad \mathbf{G}\_{n+1}^{(0)} = \frac{\mathbf{3} \mathbf{K}\_{n+1}^{(0)} (1 - \mathbf{2}\nu)}{\mathbf{2} (1+\nu)}. \tag{23}$$

(3) Distinguish the unloading process from the loading event, according to Equation (4). (a) reloading, homological center remains constant. (b) unloading, update the homological center and the bounding surface, according to Equations (5) and (6). Then evaluate the following residuals:

$$r\_{l,n+1}^{(k)} = \begin{cases} p\_{n+1}^{(k)} - p\_n \exp\left[\frac{1+\epsilon\_0}{\kappa} \left(\Delta\varepsilon\_{v,n+1} - \Delta\varepsilon\_{v,n+1}^p \right)^k \right] \\ \Delta\varepsilon\_{v,n+1}^p - \left(\Delta\varepsilon\_{v,n+1}^k - \Delta\varepsilon\_{v,n+1}^p \right) \\ s\_{i,j,n+1}^{(k)} - s\_{i,j,n} - 2G\_{n+1}^{(k)} \left(\Delta\varepsilon\_{i,j,n+1} - \Delta\varepsilon\_{i,j,n+1}^p \right) \\ \Delta\varepsilon\_{v,n+1}^p - \left(\Delta\varepsilon\_{v,n}^k - \Delta\varepsilon\_{v,n+1}^{(k)} \right) \\ p\_{n,n+1}^{(k)} - p\_{n,n} \exp\left(\frac{1+\epsilon\_0}{\lambda-\kappa} \Delta\varepsilon\_{v,n+1} \right) \quad \big|\rho\_{n+1}^{(k)} \\ \alpha\_{n+1}^{(k)} - \alpha\_n \exp\left(-\beta \sqrt{\frac{3}{2}\Delta\varepsilon\_{i,j+1}^p + \lambda \varepsilon\_{i,j+1}^p} \right) \\ b\_{n+1}^{(k)} - \eta\_n - \lambda\_{v,n}^{(k)} \frac{\overline{\nu}\_{n+1}^p - \nu\_{v,n}^{(k)} + \lambda\_{p,n+1}^p}{\lambda\_n^{(k)}} \\ \left(\overline{p}\_{n+1}^{(k)} - \xi\_{p,n+1}^{(k)} - \left(\overline{p}\_{n+1}^{(k)} - \xi\_{p,n+1}^{(k)}\right) p\_{n,n+1}^{(k)} + \frac{\overline{\lambda}}{2\left(\lambda \varepsilon\_{n}^{\Delta} - \alpha\_0^{\Delta}\right)\_{i,j,n+1}^{\delta/$$

where *l* is the number of nonlinear equations. Variable *A* can be presented as:

$$\begin{aligned} A &= p\_c \big( p - o\_p \big) + 2 \big( p - o\_p \big) \big( \xi\_p - o\_p \big) + \frac{3}{\left( M^2 - a\_0^2 \right)} \big[ \left( s\_{ij} - o\_{ij} \big) - \left( p - o\_p \big) \alpha\_{ij}^0 \right] \\ \big[ \left( \xi\_{ij} - o\_{ij} \big) - \left( \xi\_p - o\_p \big) \alpha\_{ij}^0 \big] \big] \end{aligned} \tag{25}$$

If the k*r* (*k*) *l*,*n*+1 <sup>k</sup> <sup>&</sup>lt; tolerance (taken as 10−<sup>8</sup> ), THEN EXIT Else GO TO step 4

(4) Solve the linear equations:

$$
\left(\frac{\partial r}{\partial \mathcal{U}}\right)\_{n+1}^{(k)} \delta \mathcal{U}\_{n+1}^{(k)} = -r\_{l,n+1'}^{(k)}\tag{26}
$$

$$\text{where } \delta \mathcal{U} = \left\{ \delta p\_{\prime} \delta \mathbf{s}\_{ij\prime} \delta \Delta \varepsilon^{p}\_{\upsilon\prime} \delta \Delta \varepsilon^{p}\_{ij\prime} \delta p\_{\sigma\prime} \delta b\_{\prime} \delta \Lambda\_{\prime} \delta \overline{\mathcal{K}}\_{p\prime} \delta \omega \right\}$$

(5) Update stresses and internal variable *U* (*k*+1) *n*+1 = *U* (*k*) *n*+1 + δ*U* (*k*) *n*+1 , *k* = *k* + 1 and GO TO step (3).

.

(6) Satisfy the convergence condition, END.
