## 4 Absolute Nodal Coordinates Formulation
### Kinematics of ANCF Thin Plate Element
Integration and differentiation are applied to the straight configuration using the matrix of the position vector gradient \(\mathbf{J}_{o}=\frac{\partial\mathbf{r}_{o}}{\partial\mathbf{x}}\), where \(\mathbf{r}_{o}=\mathbf{S}\mathbf{e}_{o}\) and \(\mathbf{e}_{o}\) is the vector of the element nodal coordinates in the initial reference configuration. Another matrix of the position vector gradient \(\mathbf{J}_{e}=\frac{\partial\mathbf{r}}{\partial\mathbf{x}}\) is utilized to establish the relationship between the current and straight configurations [29], thus  
\[\mathbf{J}=\frac{\partial\mathbf{r}}{\partial\mathbf{r}_{o}}=\frac{\partial \mathbf{r}}{\partial\mathbf{x}}\cdot\frac{\partial\mathbf{x}}{\partial \mathbf{r}_{o}} \tag{24}\]  
\[\mathbf{J}=\mathbf{J}_{e}\,\,\,\mathbf{J_{o}}^{-1} \tag{25}\]  
In the fully parameterized elements, the position vector gradient is a \(3\times 3\) matrix. However, the gradient-deficient element has only two gradient vectors as the deformation along the thickness direction is not considered, and as a consequence, the matrix of the position vector gradients is not square. In order to allow for using the standard procedure as in the general continuum mechanics approach, the unit normal vector \(\mathbf{n}_{m}\) at the mid-surface, which is not a position vector gradient, is inserted to ensure that the gradient of the position vector forms a square matrix [20; 30]. \[\mathbf{n}_{m}=\frac{\mathbf{r}_{x}\times\mathbf{r}_{y}}{\|\mathbf{r}_{x} \times\mathbf{r}_{y}\|},\quad\mathbf{n}_{m0}=\frac{\mathbf{r}_{x0}\times \mathbf{r}_{y0}}{\|\mathbf{r}_{x0}\times\mathbf{r}_{y0}\|} \tag{26}\]  
\[\mathbf{J}_{e}=\Bigl{[}\mathbf{r}_{x}\quad\mathbf{r}_{y}\quad\mathbf{n}_{m} \Bigr{]},\quad\mathbf{J}_{o}=\Bigl{[}\mathbf{r}_{x0}\quad\mathbf{r}_{y0}\quad \mathbf{n}_{m0}\Bigr{]} \tag{27}\]  
where \(\mathbf{J}_{e}\) and \(\mathbf{J}_{o}\) are modified square nonsingular matrices of position vector gradients evaluated at the current and initial nodal coordinates. The volume in the current deformed configuration \(v\) and the volume of the curved reference configuration \(V_{o}\) are related to the volume of the straight configuration \(V\) through the relationship \(dv=|\mathbf{J}_{e}|\,dV\) and \(dV_{o}=|\mathbf{J}_{o}|\,dV\), where \(|\mathbf{J}_{o}|\) is the determinant of the position gradient matrix for the initial configuration, which means that integration over the domain \(V_{o}\) can be transformed to the straight element domain \(V\). The zero strain of the initially curved reference configuration can be obtained by using the inverse of the matrix of the position vector gradients evaluated for the curved reference configuration as \(\mathbf{e}=\mathbf{e}_{o}\) and \(\mathbf{J}_{e}=\mathbf{J}_{o}\). The generalized inertia forces can be obtained by evaluating the kinetic energy T of the element as  
\[T=\frac{1}{2}\int_{V_{o}}\rho\,\dot{\mathbf{r}}^{T}\,\dot{\mathbf{r}}\,dV_{o}= \frac{1}{2}\int_{V}\rho\,\dot{\mathbf{r}}^{T}\,\dot{\mathbf{r}}\,|\mathbf{J}_{ o}|\,dV \tag{28}\]  
where \(\dot{\mathbf{r}}\) is the velocity vector of an arbitrary point on the plate, and \(\rho\) is the element mass density; this equation leads to a constant mass matrix defined as  
\[\mathbf{M}=\int_{V}\rho\,S^{T}\,S\,|\mathbf{J}_{o}|\,dV \tag{29}\]