## 4 Absolute Nodal Coordinates Formulation
### Generalized Forces of Thin Plate Element  
The generalized nonlinear elastic forces vector \(\mathbf{Q}_{e}\) is obtained in the current configuration concerning the stress-free reference configuration, which may have a curved geometry. The strain energy of the thin plate element is obtained based on the membrane effect and the curvature of the mid-surface:  
\[\mathbf{U}=\frac{1}{2}\int_{V}e^{T}\,\mathbf{D}_{e}\,\epsilon\,|\mathbf{J}_{o} |\,\mathbf{dV}+\frac{1}{2}\int_{V}\kappa^{T}\,\mathbf{D}_{e}\,\kappa\,|\mathbf{ J}_{o}|\,\mathbf{dV} \tag{30}\]  
Figure 5: Various related configurations for ANCF plate.  
\[\mathbf{D}_{\epsilon}=\frac{E}{1-\
u^{2}}\begin{bmatrix}1&\
u&0\\ \
u&1&0\\ 0&0&\frac{1-\
u}{2}\end{bmatrix},\qquad\mathbf{D}_{\kappa}=\frac{h^{2}}{12} \mathbf{D}_{\epsilon} \tag{31}\]  
where \(\mathbf{D}_{\epsilon}\) and \(\mathbf{D}_{\kappa}\) are the matrices of elastic coefficients, \(\mathbf{E}\) is the elastic modulus, \(\mathbf{\
u}\) is Poisson's ratio, and \(\mathbf{h}\) is the plate thickness. The Green-Lagrange strain tensor \(\mathbf{\epsilon}_{s}\) of the plate mid-surface, which guarantees zero strain in the initial reference configuration in terms of the matrices of position vector gradient \(\mathbf{J}\), can be written as  
\[\mathbf{\epsilon}_{s}=\frac{1}{2}\left(\mathbf{J}^{\mathsf{T}}\mathbf{J}-\mathbf{ I}\right)\]  
\[\mathbf{\epsilon}_{s}=\frac{\mathbf{1}}{2}\left(\left(\mathbf{J}_{\mathbf{e}} \mathbf{J}_{\mathbf{o}}^{-\mathbf{1}}\right)^{\mathsf{T}}\left(\mathbf{J}_{ \mathbf{e}}\mathbf{J}_{\mathbf{o}}^{-\mathbf{1}}\right)-\mathbf{I}\right) \tag{32}\]  
With a constant thickness plate, the Green-Lagrange strain tensor is written in the vector form \(\mathbf{\epsilon}\) as  
\[\mathbf{\epsilon}=\begin{bmatrix}\epsilon_{s11}&\epsilon_{s22}&2\epsilon_{s12} \end{bmatrix}^{T}=\begin{bmatrix}\epsilon_{xx}&\epsilon_{yy}&2\epsilon_{xy} \end{bmatrix}^{T} \tag{33}\]  
Here \(\epsilon_{xx}\), \(\epsilon_{yy}\), and \(\epsilon_{xy}\) are the axial, transverse, and shear strains, respectively. The curvature vector \(\mathbf{\kappa}\) can be expressed as the difference between the current \(\mathbf{\kappa}_{c}\) and initial \(\mathbf{\kappa}_{o}\) evaluated for the current and initial configurations:  
\[\mathbf{\kappa}=\mathbf{\kappa}_{c}-\mathbf{\kappa}_{o} \tag{34}\]  
where  
\[\mathbf{\kappa}_{c}=\begin{bmatrix}\kappa_{xx}&\kappa_{yy}&2\kappa_{xy}\end{bmatrix} ^{T} \tag{35}\]  
\[\mathbf{\kappa}_{o}=\begin{bmatrix}\kappa_{xxo}&\kappa_{yyo}&2\kappa_{xyo}\end{bmatrix} ^{T} \tag{36}\]  
such that the curvature components are defined in terms of the position vector gradients as  
\[\kappa_{xx}=\frac{\mathbf{r}_{xx}^{\mathsf{T}}\mathbf{n}}{\|\mathbf{n}\|^{3} }\quad,\quad\kappa_{xxo}=\frac{\mathbf{r}_{xxo}^{\mathsf{T}}\mathbf{n}_{o}}{ \|\mathbf{n}_{o}\|^{3}}\]  
\[\kappa_{yy}=\frac{\mathbf{r}_{yy}^{\mathsf{T}}\mathbf{n}}{\|\mathbf{n}\|^{3} }\quad,\quad\kappa_{yyo}=\frac{\mathbf{r}_{yyo}^{\mathsf{T}}\mathbf{n}_{o}}{ \|\mathbf{n}_{o}\|^{3}}\]  
\[\kappa_{xy}=\frac{\mathbf{r}_{xy}^{\mathsf{T}}\mathbf{n}}{\|\mathbf{n}\|^{3} }\quad,\quad\kappa_{xyo}=\frac{\mathbf{r}_{xyo}^{\mathsf{T}}\mathbf{n}_{o}}{ \|\mathbf{n}_{o}\|^{3}}\]  
The vector \(r_{ij}\) represents the derivative of \(r_{i}\) with respect to \(j\), and the normal vectors \(\mathbf{n}=\mathbf{r}_{x}\times\mathbf{r}_{y}\) and \(\mathbf{n}_{o}=\mathbf{r}_{xo}\times\mathbf{r}_{yo}\) are at the mid-surface for the current and initial configurations.  
The elastic forces \(\mathbf{Q}_{\epsilon}\) can be obtained by differentiating the strain energy \(\mathbf{U}\) by the vector of the element nodal coordinates:  
\[\mathbf{Q}_{\epsilon}=\frac{\partial\mathbf{U}}{\partial\mathbf{e}} \tag{38}\]  
Thus,  
\[\mathbf{Q}_{\epsilon}=\int_{V}\!\left(\frac{\partial\mathbf{\epsilon}}{\partial \mathbf{e}}\right)^{\mathsf{T}}\!\mathbf{D}_{\epsilon}\,\epsilon\,|\mathbf{J}_ {o}|\,dV+\int_{V}\!\left(\frac{\partial\mathbf{\kappa}}{\partial\mathbf{e}} \right)^{\mathsf{T}}\!\mathbf{D}_{\kappa}\,\mathbf{\kappa}\,|\mathbf{J}_{o}|\,dV \tag{39}\]