Table 2.

Flowchart of the integration scheme.

Step	Brief description
1	Input initial variables ${\mathbf{\sigma }}_k$, ${\mathbf{\epsilon }}_k$, ${\mathbf{c}}_k$, $p_{\text{c}k}$, $R_k$ and the strain increment $\mathrm{\Delta }{\mathbf{\epsilon }}_{k+1}$
2	Set the pseudo time $T=0$, $\mathrm{\Delta }T=1$ and the tolerance error $STOL=1\text{e}-6$
3	Calculate the first trial elasto-plastic stiffness matrix ${\mathbf{E}}_1^{\text{ep}}\left\{{\mathbf{\sigma }}_k,{\mathbf{c}}_k,p_{\text{c}k},R_k\right\}$ by substituting the initial variables into Eq. (34)
4	Compute the subincrement according to $\mathrm{\Delta }{\mathbf{\epsilon }}_{k+1}^{\text{sub}}=\mathrm{\Delta }T\times \mathrm{\Delta }{\mathbf{\epsilon }}_{k+1}$
5	Calculate the first trial stress increment $\mathrm{\Delta }{\mathbf{\sigma }}_1={\mathbf{E}}_1^{\text{ep}}:\mathrm{\Delta }{\mathbf{\epsilon }}_{k+1}^{\text{sub}}$ and update the first trial state variables ${\mathbf{c}}_1$, $p_{\text{c1}}$ and $R_1$ related to plastic deformation
6	Compute the second trial elastoplastic stiffness matrix ${\mathbf{E}}_2^{\text{ep}}\left\{{\mathbf{\sigma }}_1,{\mathbf{c}}_1,p_{\text{c}}1,R_1\right\}$ by substituting the first trial variables into Eq. (34)
7	Calculate the second trial stress increment $\mathrm{\Delta }{\mathbf{\sigma }}_2={\mathbf{E}}_2^{\text{ep}}:\mathrm{\Delta }{\mathbf{\epsilon }}_{k+1}^{\text{sub}}$
8	Calculate the average stress increment $\mathrm{\Delta }{\mathbf{\sigma }}_{k+1}=\frac{1}{2}(\mathrm{\Delta }{\mathbf{\sigma }}_1+\mathrm{\Delta }{\mathbf{\sigma }}_2)$
9	Compute the relative local error $LE=∥\frac{1}{2}(\mathrm{\Delta }{\mathbf{\sigma }}_1-\mathrm{\Delta }{\mathbf{\sigma }}_2)∥/∥{\mathbf{\sigma }}_k+\mathrm{\Delta }{\mathbf{\sigma }}_{k+1}∥$
10	If $LE>STOL$, the size of this substep is too large and a smaller pseudo time needs to be found: $\rho =max\left\{0.9\sqrt{STOL/LE},0.1\right\}$, $\mathrm{\Delta }T=max\left\{\rho \mathrm{\Delta }T,0.001\right\}$, then return to step 4 Else, go to next step
11	Compute the size of next substep $\rho =min\left\{0.9\sqrt{STOL/LE},1.1\right\}$ If the above equation is rejected, then $\rho =min\left\{\rho ,0.9\right\}$ Update the pseudo time $T=T+\mathrm{\Delta }T$, $\mathrm{\Delta }T=\rho \mathrm{\Delta }T$
12	Recalculate the pseudo time to ensure the size of next substep is bigger than that of minimum step and is not bigger than 1, the following conditions must be applied $\mathrm{\Delta }T=max\left\{\mathrm{\Delta }T,0.001\right\}$, $\mathrm{\Delta }T=min\left\{\mathrm{\Delta }T,1-T\right\}$
13	While $T=1$ end this iterative calculation and output the Jacobi stiffness matrix ${\mathbf{C}}_{k+1}$ to the finite element routine for the global equilibrium iterations. The Jacobi stiffness matrix can be derived from the same procedure in solving the elastoplastic matrix58: ${\mathbf{C}}_{k+1}={\mathbf{E}}_{k+1}^{\text{ep}}\left\{{\mathbf{\sigma }}_{k+1},{\mathbf{c}}_{k+1},p_{\text{c}k+1},R_{k+1}\right\}$