## 4 Absolute Nodal Coordinates Formulation
### Kinematics of ANCF Thin Plate Element  
Over the years, several ANCF elements have been introduced, evolving from 2D to 3D formulations, including beams, plates, shells, and cables, further expanding the versatility of the ANCF approach [24; 25; 26]. In the ANCF, position and gradient vectors are used as nodal coordinates; this formulation leads to a constant mass matrix and, as a consequence, the centrifugal and Coriolis inertia forces are identically equal to zero, which has a great advantage for dynamic analysis. In this paper, the ANCF thin plate element is employed for modeling the continuum model of an elephant trunk. The ANCF thin plate element assumes that the plate has a constant thickness and does not consider the deformation along the thickness direction. The four corner points of the mid-surface are used as element nodes, with each node having a position vector and two gradient vectors as nodal coordinates with nine degrees of freedom, leading to a reduced-order element with 36 degrees of freedom [27]. The global position vector of an arbitrary point, P, on the mid-surface of the thin plate element can be defined as  
\[\mathbf{r}(\mathbf{x},\mathbf{y},\mathbf{t})=\mathbf{S}(\mathbf{x},\mathbf{y})\cdot\mathbf{e}(\mathbf{t}) \tag{18}\]  
where \(\mathbf{S}\) is the element shape function matrix, and \(\mathbf{e}\) is the vector of the nodal coordinates. This vector at node i of an element can be defined as  
\[\mathbf{e}_{i}= \begin{bmatrix}(\mathbf{r}^{i})^{T}&(\mathbf{r}^{i}_{x})^{T}&(\mathbf{r}^{i}_{ y})^{T}\end{bmatrix} ^{T},\quad i=1,2,3,4 \tag{19}\]  
The nodal coordinates of one element can be expressed by the vector \(\mathbf{e}\), which includes the nodal coordinates of the four nodes, as  
\[\mathbf{e}= \begin{bmatrix}\mathbf{e}_{1}^{T}&\mathbf{e}_{2}^{T}&\mathbf{e}_{3}^{T}&\mathbf{e}_{4} ^{T}\end{bmatrix} ^{T} \tag{20}\]  
where \(\mathbf{r}^{i}\) is the global position vector, \(\mathbf{r}^{i}_{x}\) and \(\mathbf{r}^{i}_{y}\) define the gradient vectors at the node \(i\), and the element shape function matrix \(\mathbf{S}\) for a thin plate element can be obtained using incomplete third-order polynomial interpolation functions where the basis \(\mathbf{B}\) can be obtained from [PERSON]'s triangle [28]. \[\mathbf{B}= \begin{bmatrix}1&x&y&x^{2}&xy&y^{2}&x^{3}&x^{2}y&xy^{2}&y^{3}&x^{3}y&y^{ 3}x\end{bmatrix}  \tag{21}\]  
The shape function matrix \(\mathbf{S}\) can be expressed as  
\[\mathbf{S}= \begin{bmatrix}S_{1}\;\mathbf{I}&S_{2}\;\mathbf{I}&S_{3}\;\mathbf{I}&S_{4}\;\mathbf{I}& \cdots&S_{12}\;\mathbf{I}\end{bmatrix}  \tag{22}\]where \(\mathbf{I}\) is a third-order identity matrix and shape functions can be defined as  
\[\left. \begin{aligned} S_{1}&=(\xi-1)(\eta-1) \Bigl{(}-2\,\xi^{2}-2\,\eta^{2}+\xi+\eta+1\Bigr{)}\\ S_{2}&=-a\,\xi(\xi-1)^{2}(\eta-1)\\ S_{3}&=-b\,\eta(\xi-1)(\eta-1)^{2}\\ S_{4}&=-\xi(\eta-1)\Bigl{(}-2\,\xi^{2}-2\,\eta^{2}+3 \,\xi+\eta\Bigr{)}\\ S_{5}&=-a\,\xi^{2}(\xi-1)(\eta-1)\\ S_{6}&=b\,\xi\,\eta(\eta-1)^{2}\\ S_{7}&=-\eta(\xi-1)\Bigl{(}-2\,\xi^{2}-2\,\eta^{2}+ \xi+3\,\eta\Bigr{)}\\ S_{8}&=a\,\xi\,\eta(\xi-1)^{2}\\ S_{9}&=-b\,\eta^{2}(\xi-1)(\eta-1)\\ S_{10}&=-\xi\,\eta\Bigl{(}2\,\xi^{2}+2\,\eta^{2}-3 \,\xi-3\,\eta+1\Bigr{)}\\ S_{11}&=a\,\xi^{2}\eta(\xi-1)\\ S_{12}&=b\,\xi\,\eta^{2}(\eta-1)\end{aligned} \right\} \tag{23}\]  
where \(\xi=x/a\) and \(\eta=y/b\) are the local normalized coordinates and \(a\) and \(b\) are the length and width of the finite element in the reference configuration. Based on the general continuum mechanics theory, three configurations can be involved in describing the deformation of the ANCF element. Figure 5 illustrates the straight configuration \(\mathbf{x}\), the initial reference configuration (stress-free) \(\mathbf{r}_{o}\), and the current deformed configuration \(\mathbf{r}\).