## 5 ANCF Continuum Trunk Model  
ANCF gradient-deficient plate finite elements were used to model the geometry and conduct the analysis of the trunk. A dedicated subroutine was developed in MATLAB to automate the construction of the trunk's tubular structure. The subroutine systematically defined the elements in a clockwise direction for each cross-sectional slice, progressing sequentially along the elastic centerline of the trunk. The generation of the nodal points is shown in Figure 12. The generated nodal coordinates were then used to assemble the ANCF finite elements, enabling the simulation of the complex deformations and movements of the trunk. Figure 11: Energy of the curved pendulum. Figure 12: Generation of tubular structure using ANCF thin plate element. The lofted profiles were obtained by transforming the position coordinates of the upper cross-sectional mesh points, see Figure 13, which can be expressed as  
\[P_{u}= \begin{bmatrix}R_{u}\cos(\theta_{1})&0&R_{u}\sin(\theta_{1})\\ R_{u}\cos(\theta_{2})&0&R_{u}\sin(\theta_{2})\\ \vdots&\vdots&\vdots\\ R_{u}\cos(\theta_{n})&0&R_{u}\sin(\theta_{n})\\ 1&1&1\end{bmatrix}  \tag{41}\]  
where \(R_{u}\) represents the upper radius and \(\theta\in[0,2\pi]\) is the angle for the mesh points. This matrix defines the original profile that will be subjected to scaling and translation transformations along the length of the trunk until it reaches the tip, depending on the number of profiles \(n\). Each lofted profile, \(P_{k}\), at a given location, \(k\), along the trunk is generated using the transformation  
\[P_{k}=T_{k}\cdot S_{k}\cdot P_{u} \tag{42}\]  
where  
- \(\mathbf{S_{k}}\) is the homogeneous scaling matrix, defined as  
\[S_{k}= \begin{bmatrix}\frac{R_{k}}{R_{u}}&0&0&0\\ 0&1&0&0\\ 0&0&\frac{R_{k}}{R_{u}}&0\\ 0&0&0&1\end{bmatrix} \]  
- \(\mathbf{R_{k}}\) is the radius at position \(k\), given by  
\[R_{k}=R_{u}-\left(R_{u}-R_{L}\right)\frac{k}{n},\quad k=0,1,2,\ldots,n\]  
Figure 13: Tubular trunk ANCF model generated using thin plate element. - \(\mathbf{T_{k}}\) is the homogeneous translation matrix, defined as  
\[T_{k}= \begin{bmatrix}1&0&0&0\\ 0&1&0&k\frac{L}{n}\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \]  
Here, \(R_{L}\) is the lower radius, and \(L\) is the length of the trunk. **Continuum Trunk Model Analysis:** The dimensions and material properties of the continuum trunk model were based on empirical data from a mature female Asian elephant at the National Zoo in Washington, D.C. [31]. The trunk was 1.5 m long with an upper diameter of 32 cm near the head, tapering to a lower diameter of 8 cm towards the tip. This gradual tapering provided an anatomically accurate representation of an elephant's trunk. The material properties for the model included a density of 1180 kg/m\({}^{3}\), a Young's modulus of \(0.938\times 10^{6}\) N/m\({}^{2}\), and a Poisson's ratio of 0.3. Initially positioned in a horizontal configuration, the trunk was allowed to fall naturally under the action of gravity, simulating real-world behavior over a duration of 0.3 seconds. Figure 14 illustrates this motion, highlighting the trunk's deformation as it transitioned into a downward curve due to gravitational forces. In addition to gravitational loading, external bending moments were applied along the length of the trunk to simulate coiling or bending actions, as shown in Figures 15 and 16. This captured one of the trunk's most vital functions: its ability to bend or coil during activities such as grasping or wrapping around objects. To achieve this, a direct methodology was employed to impose concentrated moments on the ANCF finite elements, leveraging the framework proposed in [10; 13]. These simulations demonstrated the model's ability to realistically replicate the complex mechanics of an elephant's trunk, including large deformations and the intricate interplay between the gravitational forces and applied moments. This validated the effectiveness of the ANCF in modeling biological structures with high degrees of flexibility and adaptability.