# High-Precision Elastoplastic Four-Node Quadrilateral Shell Element
## 3 Assumed Strain Rate Interpolation  
The four-node quadrilateral shell element can transform the node coordinates, node velocities, and angular velocities expressed in the global coordinate system into the unit coordinate system using the coordinate transformation matrix.  
As shown in Figure 2, the origin of the cell coordinate system is located in the center of the cell \((x_{c},y_{x},z_{c})\), and the vectors \(\mathbf{g}_{1}\) and \(\mathbf{g}_{2}\) are equal to  
\[\mathbf{g}_{1} =(\frac{\partial\mathbf{x}}{\partial\xi^{*}},\frac{\partial y}{ \partial\xi^{*}},\frac{\partial z}{\partial\xi^{*}}) \tag{11}\] \[\mathbf{g}_{2} =(\frac{\partial\mathbf{x}}{\partial\eta},\frac{\partial y}{ \partial\eta},\frac{\partial z}{\partial\eta}) \tag{12}\]  
The base vectors \(e_{1}\), \(e_{2}\), and \(e_{3}\) of the element coordinate system are determined as follows:  
\[\mathbf{e}_{3}=\frac{\mathbf{g}_{1}\times\mathbf{g}_{2}}{|\mathbf{g}_{1} \times\mathbf{g}_{2}|}\ \ \tilde{\mathbf{e}}_{2}=\frac{\mathbf{e}_{3}\times\mathbf{g}_{1}}{|\mathbf{e}_{ 3}\times\mathbf{g}_{1}|}\ \ \mathbf{e}_{2}=\frac{\tilde{\mathbf{e}}_{2}+\mathbf{g}_{2}/|\mathbf{g}_{2}|}{ \left|\tilde{\mathbf{e}}_{2}+\mathbf{g}_{2}/|\mathbf{g}_{2}|\right|}\ \ \mathbf{e}_{1}=\frac{ \mathbf{e}_{2}\times\mathbf{e}_{3}}{|\mathbf{e}_{2}\times\mathbf{e}_{3}|} \tag{13}\]  
Therefore, the vectors \(\mathbf{e}_{1}\), \(\mathbf{e}_{2}\), \(\mathbf{g}_{1}\), and \(\mathbf{g}_{2}\) are coplanar, and the angle between \(\mathbf{e}_{1}\) and \(\mathbf{g}_{1}\) is equal to the angle between \(\mathbf{e}_{2}\) and \(\mathbf{g}_{2}\). The coordinate transformation matrix \(\boldsymbol{\lambda}\ =\ \begin{bmatrix}\mathbf{e}_{1}&\mathbf{e}_{2}&\mathbf{e}_{3} \end{bmatrix}\) is then applied. Subsequently, the element node coordinates, node velocity, and angular velocity expressed in the global coordinate system are converted to the unit coordinate system.  
\[\begin{bmatrix}\hat{x}_{I}\\ \hat{g}_{I}\end{bmatrix}=\boldsymbol{\lambda}^{\mathrm{T}}\begin{bmatrix}x_{ I}-x_{C}\\ y_{I}-y_{C}\\ z_{I}-z_{C}\end{bmatrix}\begin{bmatrix}\hat{u}_{I}\\ \hat{v}_{I}\end{bmatrix}=\boldsymbol{\lambda}^{\mathrm{T}}\begin{bmatrix}u_{ I}\\ v_{I}\\ w_{I}\end{bmatrix}\begin{bmatrix}\hat{\omega}_{xl}\\ \hat{\omega}_{yl}\\ \hat{\omega}_{zl}\end{bmatrix}=\boldsymbol{\lambda}^{\mathrm{T}}\begin{bmatrix} \omega_{xl}\\ \omega_{yl}\\ \omega_{zl}\end{bmatrix} \tag{14}\]  
where \(\begin{bmatrix}x_{I}-x_{c}&y_{I}-y_{c}&z_{I}-z_{c}\end{bmatrix}^{T}\) represents the relative coordinates relative to the center of the unit, \(\begin{bmatrix}\hat{x}_{I}&\hat{y}_{I}&z_{I}\end{bmatrix}^{T}\) represents the node coordinates in the unit coordinate system, \(\begin{bmatrix}\hat{u}_{I}&\hat{v}_{I}&\hat{v}_{I}\end{bmatrix}^{T}\) represents the node velocity in the unit coordinate system, and \(\begin{bmatrix}\hat{\omega}_{xl}&\hat{\omega}_{yl}&\hat{\omega}_{zl}\end{bmatrix} ^{T}\) represents the node angular velocity in the unit coordinate system.  
Symbols such as \(x_{I}\), \(u_{I}\), and \(\omega_{lx}\) indicate node coordinates, velocities, and angular velocities that have been transformed to the unit coordinate system.  
The element coordinate system moves and rotates with the unit itself, so the element coordinate system is also a corotational coordinate system, and the objective stress rate is equivalent to the stress rate given by [PERSON] [38]. When calculating the element, the strain rate can be divided into in-plane components and transverse shear components.  
Figure 2: Four-node quadrilateral shell element coordinate system.