Kirchhoff Rod Bending

Kirchhoff Rod Bending is a constitutive model for simulating the bending behavior of thin rod structures. This model captures the curvature-based bending energy of rod elements defined by three consecutive vertices.

Reference:

• Codimensional Incremental Potential Contact (C-IPC) Section 8

#15 Kirchhoff Rod Bending

For a rod bending element defined by three consecutive vertices at positions \(\mathbf{x}_0\), \(\mathbf{x}_1\), and \(\mathbf{x}_2\), we define:

Edge Vectors:

\[ \begin{aligned} \mathbf{e}_0 &= \mathbf{x}_1 - \mathbf{x}_0\\ \mathbf{e}_1 &= \mathbf{x}_2 - \mathbf{x}_1 \end{aligned} \]

Curvature Vector:

The curvature vector \(\boldsymbol{\kappa}\) is computed as:

\[ \boldsymbol{\kappa} = \frac{2 \mathbf{e}_0 \times \mathbf{e}_1}{\sqrt{\mathbf{e}_0 \cdot \mathbf{e}_0 \cdot \mathbf{e}_1 \cdot \mathbf{e}_1} + \mathbf{e}_0 \cdot \mathbf{e}_1} \]

Rest Length:

The rest length \(L_0\) is the sum of the lengths of the two edges in the reference configuration:

\[ L_0 = \|\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_0\|_2 + \|\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_1\|_2 \]

where \(\bar{\mathbf{x}}_0\), \(\bar{\mathbf{x}}_1\), and \(\bar{\mathbf{x}}_2\) are the rest positions of the three vertices.

Bending Energy

The bending energy of the Kirchhoff rod element is given by:

\[ E = \frac{\alpha \|\boldsymbol{\kappa}\|^2}{L_0} \]

where:

\[ \alpha = \frac{k r^4 \pi}{4} \]

and:

• \(k\) is the bending stiffness parameter
• \(r\) is the radius of the rod cross-section
• \(\pi\) is the mathematical constant
• \(L_0\) is the rest length of the two edges
• \(\boldsymbol{\kappa}\) is the curvature vector

The energy penalizes deviations from the straight configuration, with the stiffness scaling with the fourth power of the rod radius, consistent with classical beam theory.