# High-Precision Elastoplastic Four-Node Quadrilateral Shell Element
## 3 Assumed Strain Rate Interpolation
### In-Plame Strainrate Interpolation  
The membrane (in-plane) components of the rate-of-deformation tensor are given by  
\[\dot{\epsilon}_{ij}=\frac{1}{2}\left(\frac{\partial V_{i}}{\partial x_{j}}+\frac{ \partial V_{j}}{\partial x_{i}}\right) \tag{15}\]  
Substituting Equation (10) into Equation (15), we obtain  
\[\dot{\epsilon}_{xx}=\frac{\partial u}{\partial x}=N_{I,x}(u_{I}+\frac{a}{2} \zeta\omega_{yI})+N_{I,x}^{u}\omega_{zI} \tag{16}\]  
\[\dot{\epsilon}_{yy}=\frac{\partial v}{\partial y}=N_{I,y}(v_{I}-\frac{a}{2} \zeta\omega_{xI})+N_{I,y}^{v}\omega_{zI} \tag{17}\]  
\[\dot{\epsilon}_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x }=N_{I,y}(u_{I}+\frac{a}{2}\zeta\omega_{yI})+N_{I,x}(v_{I}-\frac{a}{2}\zeta \omega_{xI})+N_{I,y}^{u}\omega_{zI}+N_{I,x}^{v}\omega_{zI} \tag{18}\]  
Writing Equations (16)-(18) in matrix form yields the following:  
\[\dot{\epsilon}= \begin{bmatrix}\dot{\epsilon}_{xx}\\ \dot{\epsilon}_{yy}\\ \dot{\epsilon}_{xy}\end{bmatrix} = \begin{bmatrix}N_{I,x}&0&0&0&\frac{a}{2} \zeta N_{I,x}&N_{I,x}^{u}\\ 0&N_{I,y}&0&-\frac{a}{2}\zeta N_{I,y}&0&N_{I,y}^{v}\\ N_{I,y}&N_{I,x}&0&-\frac{a}{2}\zeta N_{I,x}&\frac{a}{2}\zeta N_{I,y}&N_{I,y}^{ u}+N_{I,x}^{v}\\ \omega_{yI}\\ \omega_{zI}\end{bmatrix}\begin{bmatrix}u_{I}\\ v_{I}\\ w_{I}\\ \omega_{xI}\\ \omega_{yI}\\ \omega_{zI}\end{bmatrix}  \tag{19}\]  
The rate of deformation in Equation (19) can be written as the sum of the membrane and bending parts:  
\[\dot{\epsilon}=\dot{\epsilon}_{\mathbf{m}}+\dot{\epsilon}_{\mathbf{b}}=( \mathbf{B}_{I}^{m}+\mathbf{B}_{I}^{b})\mathbf{U} \tag{20}\]  
where \(\mathbf{v}\) is a nodal velocity (translation and rotation) vector, \(\mathbf{B}_{I}^{m}\) is a submatrix (block) of the membrane part of the gradient matrix, and \(\mathbf{B}_{I}^{b}\) is the bending part:  
\[\mathbf{B}_{I}^{m}= \begin{bmatrix}N_{I,x}&0&0&0&0&N_{I,x}^{u}\\ 0&N_{I,y}&0&0&0&N_{I,y}^{v}\\ N_{I,y}&N_{I,x}&0&0&0&N_{I,y}^{u}+N_{I,x}^{v}\end{bmatrix}  \tag{21}\]  
\[\mathbf{B}_{I}^{b}=\frac{a}{2}\zeta \begin{bmatrix}0&0&0&0&N_{I,x}&0\\ 0&0&0&-N_{I,y}&0&0\\ 0&0&0&-N_{I,x}&N_{I,y}&0\end{bmatrix}  \tag{22}\]  
The derivatives of the shape functions with respect to the referential coordinates are  
\[N_{I,\xi}=\frac{1}{4}(\xi_{I}+h_{I}\eta),\ N_{I,\eta}=\frac{1}{4}(\eta_{I}+h_ {I}\xi) \tag{23}\]  
where  
\[\dot{\epsilon}= \begin{bmatrix}1&-1&-1&1\end{bmatrix}  \tag{24}\]  
\[\eta= \begin{bmatrix}1&1&-1&-1\end{bmatrix}  \tag{25}\]  
Substituting the finite element interpolants in Equation (23) into the \(2\times 2\) Jacobian matrix yields  
\[\mathbf{J}= \begin{bmatrix}\frac{\partial x}{\partial\xi}&\frac{\partial y}{ \partial\xi}\\ \frac{\partial x}{\partial\eta}&\frac{\partial y}{\partial y}\end{bmatrix} = \begin{bmatrix}\frac{1}{4}(\xi_{I}+h_{I}\eta)x_{I}&\frac{1}{4}(\xi_{I}+h_{I} \eta)y_{I}\\ \frac{1}{4}(\eta_{I}+h_{I}\xi)x_{I}&\frac{1}{4}(\eta_{I}+h_{I}\xi)y_{I}\end{bmatrix}  \tag{26}\]  
The determinant \(J\) of the \(2\times 2\) Jacobian matrix can be expressed as  
\[J=|\mathbf{J}|=J_{0}+J_{1}\xi+J_{2}\eta \tag{27}\]where  
\[ \begin{array}{l}J_{0}=\frac{1}{16}\big{[}(\xi_{1}x_{1})\big{(}\eta_{1}y_{1}\big{)} -(\eta_{1}x_{1})\big{(}\xi_{j}y_{1}\big{)}\big{]}\\ J_{1}=\frac{1}{16}\big{[}(\xi_{1}x_{1})\big{(}\eta_{1}y_{1}\big{)}-(\eta_{1}x_{1 })\big{(}\xi_{j}y_{1}\big{)}\big{]}\\ J_{2}=\frac{1}{16}\big{[}(h_{1}x_{1})\big{(}\eta_{1}y_{1}\big{)}-(\eta_{1}x_{1 })\big{(}h_{1}y_{1}\big{)}\big{]}\end{array}  \tag{28}\]  
It is useful to write the shape functions in an equivalent manner:  
\[\mathbf{N_{I}}=\boldsymbol{\Delta}+x\mathbf{b}_{x}+y\mathbf{b}_{y}+h\boldsymbol {\gamma} \tag{29}\]  
where  
\[\boldsymbol{\Delta}_{I}=\frac{1}{4}\big{[}t_{I}-(t_{J}x_{J})b_{xI}-(t_{J}y_{J} )b_{yI}\big{]},\;b_{xI}=\frac{1}{2A}[\vartheta_{24}\quad\vartheta_{31}\quad \vartheta_{42}\quad\vartheta_{13}] \tag{30}\]  
\[b_{yl}=\frac{1}{2A}[\hat{x}_{42}\quad\hat{x}_{13}\quad\hat{x}_{24}\quad\hat{x }_{31}],\;\gamma_{I}=\frac{1}{4}\big{[}h_{I}-(h_{J}x_{I})b_{xI}-(h_{I}y_{I})b_ {yI}\big{]}, \tag{31}\]  
\[x_{IJ}=x_{I}-x_{J},\;y_{IJ}=y_{I}-y_{I} \tag{32}\]  
\[t_{I}=