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.

Attributes

On vertices:

• bending_stiffness: \(k\) in the energy above