# 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
### Kinematics  
The lowercase letters \(\{\mathbf{x},\mathbf{a}_{\alpha},a_{\beta\beta},\text{etc. }\}\) are used to indicate the deformed shell, and the uppercase letters \(\{\mathbf{X},\mathbf{A}_{\alpha\beta},A_{\alpha\beta},\text{etc. }\}\) indicate the undeformed shell. For the shell structures under finite strains, the right Cauchy-Green deformation tensor \(\mathbf{C}\) and the deformation gradient \(\mathbf{F}\) are represented as  
\[\mathbf{F} =\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{X}}=\mathbf{g}_{i} \otimes\mathbf{G}^{i}\mathbf{=}\mathbf{G}_{i}\cdot\mathbf{g}\mathbf{G}^{i} \otimes\mathbf{G}^{j}F_{ij}=\mathbf{G}_{i}\cdot\mathbf{g}_{j}, \tag{24}\] \[\mathbf{C} =\mathbf{F}^{\mathbf{T}}\mathbf{F}=\mathbf{g}_{i}\cdot\mathbf{g} _{j}\mathbf{G}^{i}\otimes\mathbf{G}^{j}=g_{ij}\mathbf{G}^{i}\otimes\mathbf{G} ^{j},C_{ij}=g_{ij}.\]  
The base vectors \(\mathbf{g}\) and \(\mathbf{G}\) are expressed in Equation (21), respectively defined in the deformed shell and the undeformed shell. Due to the fact that the deformation cannot be neglected in finite deformation, the thickness stretch \(\lambda_{3}\) is defined by \(\lambda_{3}\mathbf{=}\sqrt{C_{33}}\), and the thickness base vector can be corrected as \(\mathbf{g}_{3}=\lambda_{3}\mathbf{a}_{3}\). Thus, Equation (24) can be modified by  
\[F_{i\beta}=\mathbf{G}_{i}\cdot\mathbf{g}_{\beta},F_{i3}=\lambda_{3}\mathbf{G} _{i}\cdot\mathbf{a}_{3}. \tag{25}\]  
\(C_{33}\) is defined by the deformation of thickness, which cannot be directly computed from the deformation of the midsurface. Normally, the plane stress condition (the second Piola-Kirchhoff stress component \(S^{33}=0\)) is used to determine the unknown parameter \(C_{33}\):  
\[S^{33}+\frac{\partial S^{33}}{\partial C_{33}}\Delta C_{33}=S^{33}+\frac{1}{2 }\mathbb{C}^{3333}\Delta C_{33}=0. \tag{26}\]  
In order to solve the plane stress constraint, an iterative calculation method is employed to update \(C_{33}\), as described in Algorithm 3. The Green-Lagrange strain tensor is defined as \(\mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})\). The plane Green-Lagrange strain components \(E_{\alpha\beta}\) consist of membrane strain components \(\varepsilon_{\alpha\beta}\) and bending strain components \(\kappa_{\alpha\beta}\):  
\[\mathbf{E}=E_{ij}\mathbf{G}^{i}\otimes\mathbf{G}^{j},\quad E_{\alpha\beta}= \varepsilon_{\alpha\beta}+\xi^{3}\kappa_{\alpha\beta}, \tag{27}\]  
Figure 6: Description of a KL shell for the different configurations. where strains are computed by the covariant metric coefficients \(a_{\alpha\beta}\), \(A_{\alpha\beta}\) and the covariant curvature coefficients \(b_{\alpha\beta}\), \(B_{\alpha\beta}\) defined in Equation (23):  
\[\varepsilon_{\alpha\beta}=\frac{1}{2}\big{(}a_{\alpha\beta}-A_{\alpha\beta} \big{)},\kappa_{\alpha\beta}=B_{\alpha\beta}-b_{\alpha\beta}. \tag{28}\]  
```
0:\(\mathbb{C}\), \(\mathbb{S}\)
0:F
1:while\(S^{33}\
eq 0\)do
2:\(\mathbb{C}_{33}\) is corrected by \(\Delta\mathbb{C}_{33}\), where \(I\) indicates the iteration step:
3:\(\Delta\mathbb{C}_{33}=-2\frac{\xi^{33}}{\xi^{333}}\)
4:\(\mathbb{C}_{33}^{(I+1)}=\mathbb{C}_{33}^{(I)}+\Delta\mathbb{C}_{33}^{(I)}\)
5: The thickness stretch \(\lambda_{3}\), and the deformation gradient \(F_{ij}\)
6: are updated by \(\Delta\mathbb{C}_{33}^{(I)}\):
7:\(\lambda_{3}^{(I+1)}=\sqrt{\mathbb{C}_{33}^{(I+1)}}\)
8:\(F_{ij}^{(I+1)}{=}F_{ij}^{(I)},F_{ij}^{(I+1)}=\lambda_{3}^{(I+1)}\mathbb{G}_{i }\cdot\mathbf{a}_{3}\)
9: With the updated \(\mathbb{F}\), the algorithms in Section 3.1 could be used to
10: calculate new stress \(\mathbb{S}\) and new material tangent tensor \(\mathbb{C}\). 11:endwhile
```  
**Algorithm 3** Iterative update for \(\mathbb{C}_{33}\).