# 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
### Arc-Length Method  
In the simulation of large deformation elastoplastic shells, overcritical behavior of the thin-walled structures, e.g., failure or buckling, and instabilities in the plastic material, similar to metals, will cause a nonlinear load-displacement equilibrium path, as shown in Figure 7. In such a case, a certain load may correspond to multiple displacement solutions, which makes the classical Newton-Raphson method in Section 3.4 invalid. For the computation of such an equilibrium path, the arc-length method [56; 57; 58] is a popular choice to apply a constraint equation \(f(\mathbf{u},\lambda)=0\), where \(\lambda\) is the load factor. The residual vector in (37) can be formulated as  
\[\mathbf{R}(\mathbf{u},\lambda)=\lambda\mathbf{F}_{\mathrm{ext}}-\mathbf{F}_{ \mathrm{int}}(\mathbf{u}). \tag{41}\]  
On the equilibrium path, \(\mathbf{R}(\mathbf{u},\lambda)=0\). The incremental governing equations of the Newton-Raphson iterative method are based on a first-order Taylor series expansion about the equilibrium point with incremental displacement \(\Delta\mathbf{u}\):  
\[\mathbf{R}(\mathbf{u},\lambda)+\Delta\lambda\cdot\mathbf{F}_{\mathrm{ext}}- \mathbf{K}^{\mathrm{u}}\Delta\mathbf{u}=0. \tag{42}\]Equation (42) can be expressed as  
\[\Delta\mathbf{u}=\mathbf{u}^{\mathrm{R}}+\Delta\lambda\cdot\mathbf{u}^{\mathrm{F}}, \tag{43}\]  
where \(\mathbf{u}^{\mathrm{R}}=\mathbf{R}(\mathbf{u}_{,}\!\lambda)/\mathbf{K}^{\mathrm{ u}}\) and \(\mathbf{u}^{\mathrm{F}}=\mathbf{F}_{\mathrm{ext}}/\mathbf{K}^{\mathrm{u}}\). Hence, the displacement vector and load factor at the \((j+1)\) iterative step can be updated by  
\[ \begin{bmatrix}\mathbf{u}_{j+1}\\ \lambda_{j+1}\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{j}\\ \lambda_{j}\end{bmatrix} + \begin{bmatrix}\Delta\mathbf{u}_{j+1}\\ \Delta\lambda_{j+1}\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{j}\\ \lambda_{j}\end{bmatrix} + \begin{bmatrix}\mathbf{u}^{\mathrm{R}}\\ 0\end{bmatrix} +\Delta\lambda_{j+1}\cdot \begin{bmatrix}\mathbf{u}^{\mathrm{F}}\\ 1\end{bmatrix}  \tag{44}\]  
Equation (44) can be simplified by variable \(\mathbf{t}\):  
\[\mathbf{t}_{j+1}=\mathbf{t}_{j}+\mathbf{t}^{\mathrm{R}}+\Delta\lambda_{j+1} \cdot\mathbf{t}^{\mathrm{F}}. \tag{45}\]  
Let \(\mathbf{t}_{\mathrm{O}}\) be the variable at the last convergence point \(O\), as shown in Figure 7. Then, the variation of this variable at the \((j+1)\) iterative step can be defined as  
\[\blacktriangle\mathbf{t}_{j+1}= \begin{bmatrix}\blacktriangle\mathbf{u}_{j+1} \\ \blacktriangle\lambda_{j+1}\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{j+1}\\ \lambda_{j+1}\end{bmatrix} - \begin{bmatrix}\mathbf{u}_{\mathrm{O}}\\ \lambda_{O}\end{bmatrix} =\mathbf{t}_{j+1}-\mathbf{t}_{\mathrm{O}}. \tag{46}\]  
Combing this with Equation (45), one gets  
\[\blacktriangle\mathbf{t}_{j+1}=\blacktriangle\mathbf{t}_{j}+\mathbf{t}^{ \mathrm{R}}+\Delta\lambda_{j+1}\cdot\mathbf{t}^{\mathrm{F}}. \tag{47}\]  
Herein, we use [PERSON]'s arc-length method [57; 58] to obtain \(\Delta\lambda_{j+1}\). The constraint equation \(f(\mathbf{u},\lambda)=0\) can be defined as  
\[\blacktriangle\mathbf{t}_{j+1}\cdot\blacktriangle\mathbf{t}_{j+1}-L^{2}=0, \tag{48}\]  
where \(L\) is the arc-length measure. Feeding Equation (47) into (48), one gets  
\[\left(\blacktriangle\mathbf{t}_{j}+\mathbf{t}^{\mathrm{R}}+\Delta\lambda_{j+ 1}\cdot\mathbf{t}^{\mathrm{F}}\right)\cdot\left(\blacktriangle\mathbf{t}_{j }+\mathbf{t}^{\mathrm{R}}+\Delta\lambda_{j+1}\cdot\mathbf{t}^{\mathrm{F}} \right)-L^{2}=0. \tag{49}\]  
Figure 7: Example of the arc-length method. Equation (49) is a quadratic equation in \(\Delta\lambda_{j+1}\):  
\[a\Delta{\lambda_{j+1}}^{2}+b\Delta\lambda_{j+1}+c=0, \tag{50}\]  
with  
\[