ANIME-Rod: Adjustable Nonlinear Isotropic Materials for Elastic Rods

We give a method to simulate large deformations of 3D elastic rods under arbitrary nonlinear isotropic 3D solid materials. Rod elastic energies in existing graphics literature are derived from volumetric models under the small-strain linearization assumptions. While the resulting equations can and are commonly applied to large deformations, the material modeling has been limited to a single material, namely linear Hooke law. Starting from any 3D solid nonlinear isotropic elastic energy density function ψ , we derive our rod elastic energy by subjecting the 3D solid volumetric material to the limit process whereby rod thickness is decreased to zero. This enables us to explain rod stretching, bending and twisting in a unified model. Care must be taken to adequately model cross-sectional in-plane and out-of-plane deformations. Our key insight is to compute the three cross-sectional deformation modes corresponding to bending (in the two directions) and twisting, using linear theory. Then, given any ψ , we use these modes to derive an analytical formula for a 5D "macroscopic" large-deformation rod elastic energy function of the local longitudinal stretch, radial scaling, the two bending curvatures and torsion. Our model matches linear theory for small deformations, including cross-sectional shrinkage due to Poisson's effect, and produces correct bending and torsional constants. Our experiments demonstrate that our energy closely matches volumetric FEM even under large stretches and curvatures, whereas commonly used methods in graphics deviate from it. We also compare to closely related work from mechanics literature; we give an explicit expansion of all energy terms in terms of the rod cross-section diameter, allowing independent adjustment of stretching, bending and twisting. Finally, we observe an inherent limitation in the ability of rod models to control nonlinear bendability and twistability. We propose to "relax" rod physics to more easily control nonlinear bending and twisting in computer graphics applications.