# 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  
In our work, the rate-independent plasticity theory and the finite strain plastic model based on deformation gradient decomposition are employed as the theoretical founda  
Figure 5: The Bézier extraction of a T-mesh: (**a**) a cubic T-mesh; (**b**) Bézier mesh associated with the T-mesh in (**a**). tions. The deformation gradient \(\mathbf{F}\) can be decomposed as the product of the elastic and plastic parts:  
\[\mathbf{F}=\mathbf{F}^{e}\cdot\mathbf{F}^{p}. \tag{7}\]  
According to Equation (7), the plastic right Cauchy-Green deformation tensor is defined as  
\[\mathbf{C}^{p}=(\mathbf{F}^{p})^{T}\mathbf{F}^{p}, \tag{8}\]  
and the elastic left Cauchy-Green deformation tensor as  
\[\mathbf{b}^{e}=\mathbf{F}^{e}(\mathbf{F}^{e})^{T}. \tag{9}\]  
Combining Equations (7)-(9), \(\mathbf{b}^{e}\) can be given as  
\[\mathbf{b}^{e}=\mathbf{F}(\mathbf{C}^{p})^{-1}\mathbf{F}^{T}. \tag{10}\]  
The elastic response is evaluated by the stored-energy function \(\Psi^{e}\) of the intermediate stress-free configuration, which can be decomposed into a volumetric and deviatoric part as  
\[\Psi^{e}=\Psi^{e}_{\text{vol}}+\Psi^{e}_{\text{dev}^{e}}, \tag{11}\]  
with  
\[\Psi^{e}_{\text{vol}} =\frac{1}{2}\kappa(\frac{1}{2}((f^{e})^{2}-1)-\ln f^{e}), \tag{12}\] \[\Psi^{e}_{\text{dev}} =\frac{1}{2}\mu(tr[\mathbf{\tilde{b}}^{e}]-3),\]  
where \(J^{e}=\det[\mathbf{F}^{e}]\) and \(\mathbf{\tilde{b}}^{e}=(J^{e})^{-2/3}\mathbf{b}^{e}\), and \(\kappa\) and \(\mu\) are the bulk and the shear modulus. The Cauchy stress \(\sigma\) can be calculated by the stored energy:  
\[ \begin{split}\sigma&=\frac{2}{j}\mathbf{b}^{e} \bigg{(}\frac{\partial\mathbf{\Psi}^{e}}{\partial\mathbf{b}^{e}}\bigg{)}= \frac{2}{j}\mathbf{b}^{e}\bigg{(}\frac{\partial\mathbf{\Psi}^{e}_{\text{vol} }}{\partial\mathbf{b}^{e}}+\frac{\partial\mathbf{\Psi}^{e}_{\text{dev}}}{ \partial\mathbf{b}^{e}}\bigg{)}\\ &=\frac{1}{j}\Big{\{}\frac{\kappa}{2}\Big{(}j^{2}-1\big{)} \mathbf{I}+\mu\text{dev}\big{\{}\mathbf{\tilde{b}}^{e}\big{\}}\Big{\}}=\frac{ 1}{j}\{\tau_{\text{vol}}+\tau_{\text{dev}}\}.\end{split}  \tag{13}\]  
where \(\mathbf{I}\) is the second-order unit tensor; \(J=\det[\mathbf{F}]\); \(\text{dev}(\cdot)=(\cdot)-\frac{1}{3}\text{tr}(\cdot)\mathbf{I}\); and \(\tau_{\text{dev}}\) and \(\tau_{\text{vol}}\) are the deviatoric and volumetric parts of the Kirchhoff stress \(\tau\). In order to compute the material yielding, there are two primary yield criteria. One is the maximum shear stress criterion, and the other is the von Mises yield criterion defined using multidimensional stress. It should be noted that the maximum shear yield surface is a polygon or a polyhedron, while the von Mises criterion yield surface is smooth. The smoothness of the yield surface makes it easy to find the yield point by numerical method. The classical von Mises yield function is computed from the deviatoric part of the Kirchhoff stress \(\tau_{\text{dev}}\) and the hardening function \(h(e^{p})\), written as  
\[f(\tau_{\text{dev}},e^{p})=\|\tau_{\text{dev}}\|-\sqrt{\frac{2}{3}}h(e^{p}) \leq 0, \tag{14}\]  
where \(e^{p}\) is the hardening variable. The associative flow rule is used to describe material behavior during plastic deformation, which can be determined by the principle of maximum plastic dissipation. The rate form of the inverse of the plastic right Cauchy-Green deformation tensor \(\dot{\mathbf{C}}^{p^{-1}}\) can be expressed as  
\[\dot{\mathbf{C}}^{p^{-1}}=-\frac{2}{3}\gamma\text{tr}[\mathbf{b}^{e}]\mathbf{ F}^{-1}\mathbf{n}\mathbf{F}^{-T}. \tag{15}\]where \(\mathbf{n}=\boldsymbol{\tau}_{\text{dev}}/\|\boldsymbol{\tau}_{\text{dev}}\|\) and \(\gamma\) is the consistency parameter.