# T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff-Love Shells
## 3 Formulations for the Large Deformation of Elastoplastic Kirchhoff-Love Shells
### Finite Strain Plasticity
The update of the hardening variable is governed by  
\[e^{p}=\sqrt{\frac{2}{3}}\gamma. \tag{16}\]  
The Kuhn-Tucker relationships are used to calculate the plastic consistency parameter \(\gamma\) under loading/unloading conditions:  
\[\gamma\geq 0,\ \ f(\tau_{dev},e^{p})\leq 0,\ \ \gamma f(\tau_{dev},e^{p})=0. \tag{17}\]  
With the preceding formulas, an algorithm for strain-driven return mapping is presented in Algorithm 1. ```
1:\(\mathbf{F}_{n+1}\) is computed in the time step \(t_{n+1}\), \(\mathbf{C}_{n}^{\text{p}^{-1}}\) and \(e_{n}^{p}\) are saved from the last time step \(t_{n}\)
2:\(\mathbf{F}_{n+1+1}\), \(\mathbf{C}_{n}^{\text{p}^{-1}}\), \(e_{n}^{p}\)
3:\(\mathbf{C}_{n+1}^{\text{p}^{-1}}\), \(\mathbf{S}_{n+1}\)
4: (1) Compute elastic predictor:
5:\(\bar{\mathbf{F}}_{n+1}=\int_{n+1}^{-1/3}\mathbf{F}_{n+1}\quad J_{n+1}=\text{ det}[\mathbf{F}_{n+1}]\quad\bar{\mathbf{b}}_{n+1}^{\text{c},tr}=\bar{\mathbf{F}}_{n+1} \mathbf{C}_{n}^{\text{p}^{-1}}\bar{\mathbf{F}}_{n+1}^{\text{T}}\)
6:\(\mathbf{\tau}_{\text{dev},n+1}^{tr}=\mu\text{dev}[\bar{\mathbf{b}}_{n+1}^{ \text{c},tr}]\quad\mathbf{\tau}_{\text{vol},n+1}^{tr}=\frac{\kappa}{2}(J_{n+ 1}^{2}-1)\mathbf{I}\)
7: (2) Check for yielding:
8:if\(f_{n+1}^{\text{tr}}=\left\|\mathbf{\tau}_{\text{dev},n+1}^{\text{tr}}\right\|- \sqrt{\frac{2}{3}}h\left(e_{n}^{p}\right)\leq 0\) (elastic step) then
9: set\((.)_{n+1}=(. )^{tr}\) and exit. 10:else(plastic step)
11: go to (3)
12:endif
13: (3) The return mapping:
14: Compute \(\Delta\gamma\) according:
15:if\(h\left(e_{n}^{p}\right)=\sigma_{Y}+H\cdot e_{n}^{p}\) (linear hardening law) then
16:\(\Delta\gamma=\frac{f_{n+1}^{\text{tr}}/2\bar{\mu}}{1+H/3\bar{\mu}}\quad\bar{ \mu}=\frac{1}{3}\mu tr\left(\bar{\mathbf{b}}_{n+1}^{\text{c},tr}\right)\)
17:else(nonlinear hardening law)
18:\(\Delta\gamma\) is computed by the Newton iteration
19:endif
20: Compute stress:
21:\(\boldsymbol{\tau}_{\text{dev},n+1}=\boldsymbol{\tau}_{\text{dev},n+1}^{\text{ tr}}-\frac{2}{3}\Delta\gamma\mu tr\left[\bar{\mathbf{b}}_{n+1}^{\text{c},tr} \right]\mathbf{n}\quad\mathbf{n}=\frac{\boldsymbol{\tau}_{\text{dev},n+1}^{ tr}}{|\boldsymbol{\tau}_{\text{dev},n+1}^{2}|}\)
22:\(\boldsymbol{\tau}_{\text{vol},n+1}=\boldsymbol{\tau}_{\text{vol},n+1}^{tr}\)
23:\(\sigma_{n+1}=\frac{1}{\mu_{n+1}}\big{\{}\boldsymbol{\tau}_{\text{vol},n+1}+ \boldsymbol{\tau}_{\text{dev},n+1}\big{\}}\)
24: Update the state variables:
25:\(\mathbf{C}_{n+1}^{\text{p}^{-1}}=\mathbf{C}_{n}^{\text{p}^{-1}}-\frac{2}{3} \Delta\gamma tr\left[\bar{\mathbf{b}}_{n+1}^{\text{c},tr}\right]\mathbf{F}_{n+ 1}^{-1}\mathbf{n}\mathbf{F}_{n+1}^{-T}\)
26:\(\epsilon_{n+1}^{p}=e_{n}^{p}+\sqrt{\frac{2}{3}}\Delta\gamma\)
27:\(\mathbf{S}_{n+1}=J_{n+1}\mathbf{F}_{n+1}^{-1}\boldsymbol{\sigma}_{n+1}\mathbf{F}_ {n+1}^{-T}\)
28: (4) Compute consistent tangent moduli:
29: See Algorithm 2
```  
**Algorithm 1** Return mapping algorithm. In addition to the update of state variables as achieved by the return mapping algorithm in Algorithm 1, we also provide an algorithm to compute consistent elasto-plastic material tangent \(\mathbb{C}\), as listed in Algorithm 2. The tangent moduli can be obtained by taking the derivative of the Kirchhoff stress tensor with respect to the right Cauchy-Green deformation. ```
0: Same as Algorithm 1
0:\(\mathbb{C}\)
1: Spatial elasticity tensor:
2: II is the fourth-order symmetric identity tensor.