# High-Precision Elastoplastic Four-Node Quadrilateral Shell Element
## 3 Assumed Strain Rate Interpolation
### Transverse Shear-Velocity-Strain Interpolation  
According to [PERSON] [23], the nodes in the quadrilateral are interpolated within the natural coordinate system, with the covariant components represented by tensors. The covariant components in the unit coordinate system are derived following a transformation. The transverse shear strain rate is expressed through the covariant component in the element coordinate system. In the rectangular coordinate system (xyz), the relationship between the covariant component of the velocity gradient \(L_{ij}\) in the xyz system and the covariant component \(\widetilde{L}_{ij}\) in the \(\xi\eta\zeta\) system is established:  
\[\widetilde{L}_{ij}=\beta_{i}^{m}\beta_{j}^{n}L_{mn}=\frac{\partial x^{m}}{ \partial\xi^{i}}\frac{\partial x^{n}}{\partial\xi^{j}}\frac{\partial w_{m}}{ \partial x^{n}}=\frac{\partial x^{m}}{\partial\xi^{i}}\frac{\partial w_{m}}{ \partial\xi^{j}} \tag{40}\]  
where \(v_{i}= \begin{bmatrix}u&v&w\end{bmatrix} ,\;x^{i}= \begin{bmatrix}x&y&z\end{bmatrix} , \;\dot{\xi}^{i}= \begin{bmatrix}\xi&\eta&\zeta\end{bmatrix} \). In the \(\xi\eta\zeta\) coordinate system, the velocity gradient components \(\widetilde{L}_{\xi\xi}\), \(\widetilde{L}_{\zeta\xi}\), \(\widetilde{L}_{\eta\zeta}\), and \(\widetilde{L}_{\zeta\eta}\) are  
\[ \begin{array}{l}\widetilde{L}_{\xi\xi}=\frac{\partial x^{\mu}}{\partial x} \frac{\partial\omega_{\eta\eta}}{\partial x}=J_{11}\frac{\partial u}{\partial x }+J_{12}\frac{\partial v}{\partial x}+J_{13}\frac{\partial w}{\partial x}\\ \widetilde{L}_{\xi\xi}=\frac{\partial x^{\mu}}{\partial x}\frac{\partial w}{ \partial x}=J_{13}\frac{\partial u}{\partial x}+J_{32}\frac{\partial v}{ \partial x}+J_{33}\frac{\partial w}{\partial x}\\ \widetilde{L}_{\eta\xi}=\frac{\partial x^{\mu}}{\partial y}\frac{\partial w}{ \partial y}=J_{21}\frac{\partial u}{\partial x}+J_{22}\frac{\partial v}{ \partial x}+J_{23}\frac{\partial v}{\partial x}\\ \widetilde{L}_{\zeta\eta}=\frac{\partial x^{\mu}}{\partial x}\frac{\partial w} {\partial y}=J_{31}\frac{\partial u}{\partial y}+J_{32}\frac{\partial v}{ \partial y}+J_{33}\frac{\partial w}{\partial y}\end{array}  \tag{41}\]  
where \(J_{ij}=\frac{\partial x^{i}}{\partial\xi}\) is a term in the \(3\times 3\) Jacobian matrix. We ignore terms containing \(J_{13}\),\(J_{23}\), \(J_{31}\), and \(J_{32}\), and take \(J_{33}\cong\frac{\pi}{4}\). Substituting Equations (14) and (15) into Equation (41), we obtain  
\[ \begin{array}{l}\widetilde{L}_{\xi\xi}=0.5a[J_{11}N_{I}\omega_{yI}-J_{12}N_ {I}\omega_{xI}]\widetilde{L}_{\xi\xi}=0.5 aN_{I,\xi}w_{I}\\ \widetilde{L}_{\eta\xi}=0.5a[J_{21}N_{I}\omega_{yI}-J_{22}N_{I}\omega_{xI}] \widetilde{L}_{\zeta\eta}=0.5 aN_{I,\eta}w_{I}\end{array}  \tag{42}\]  
In Figure 3, the \(A\) point coordinates in the natural coordinate system of \((0,1,0)\) and the \(C\) point coordinates in the natural coordinate system of \((0,-1,0)\) are shown. In this case, we obtain  
\[ \begin{array}{l}\widetilde{L}_{\xi\xi}^{A}=0.5a[J_{11}^{A}N_{I}^{A}\omega_{ yI}-J_{12}^{A}N_{I}^{A}\omega_{xI}]\\ \widetilde{L}_{\xi\xi}^{A}=0.5 aN_{I,\xi}^{A}w_{I}\end{array}  \tag{43}\]  
\[ \begin{array}{l}\widetilde{L}_{\eta\xi}^{C}=0.5a[J_{11}^{C}N_{I}^{C}\omega_{ yI}-J_{12}^{C}N_{I}^{C}\omega_{xI}]\\ \widetilde{L}_{\zeta\eta}^{C}=0.5 aN_{I,\eta}^{C}w_{I}\end{array}  \tag{44}\]  
The distribution of \(\widetilde{L}_{\xi\xi}\) and \(\widetilde{L}_{\zeta\xi}\) in the element is  
\[\widetilde{L}_{\xi\xi}=\frac{1}{2}\widetilde{L}_{\xi\xi}^{A}(1+\eta)+\frac{1}{ 2}\widetilde{L}_{\xi\xi}^{C}(1-\eta)=0.5 aS_{12}^{AC}\omega_{xI}+0.5 lS_{11}^{AC} \omega_{yI} \tag{45}\]  
\[ \begin{array}{l}\widetilde{L}_{\zeta\xi}=\frac{1}{2}\widetilde{L}_{\zeta \xi}^{A}(1+\eta)+\frac{1}{2}\widetilde{L}_{\xi\xi}^{C}(1-\eta)=0.5 lS_{\xi}^{AC} w_{I}\end{array}  \tag{46}\]  
where  
\[ \begin{array}{l}S_{12}^{AC}=-0.5J_{12}^{A}N_{I}^{A}(1+\eta)-0.5J_{12}^{C}N_{I }^{C}(1-\eta)\\ S_{11}^{AC}=0.5J_{11}^{A}N_{I}^{A}(1+\eta)+0.5J_{11}^{C}N_{I}^{C}(1-\eta)\\ S_{\rho}^{AC}=0.5N_{I,\xi}^{A}(1+\eta)+0.5N_{I,\xi}^{C}(1-\eta)\end{array}