Tuning Nonlinear Elastic Materials under Small and Large Deformations
Huanyu Chen, Jernej Barbic
TL;DR
This work unifies isotropic hyperelastic modeling by introducing PK1-linearization, which maps any isotropic energy $\psi$ to a Linear Corotational material characterized by two Lamé parameters $\lambda_{Lam\'e}$ and $\mu_{Lam\'e}$ derived from a second-order expansion around the rest shape $\lambda_i=1$. It proves that this linearization captures the small-deformation behavior in a rotation-aware manner and enables normalization across materials so they share the same quadratic (PK1-linear) behavior; it further provides a simple, continuous one-parameter family $\psi_\alpha$ to adjust nonlinearity without altering the small-deformation response. The paper also shows how to mix and match material components to combine desired small-deformation properties with controlled nonlinear responses, enabling independent tuning of stiffness, volume preservation, and nonlinearity. Practical demonstrations on volumetric materials and a nonlinearity-adjustment workflow illustrate how artists can explore a wide range of nonlinear behaviors with a compact parameter set, while preserving a consistent baseline for fair comparisons. The approach offers a versatile framework for material normalization, stiffness tuning, and nonlinear control in computer graphics and engineering simulations.
Abstract
In computer graphics and engineering, nonlinear elastic material properties of 3D volumetric solids are typically adjusted by selecting a material family, such as St. Venant Kirchhoff, Linear Corotational, (Stable) Neo-Hookean, Ogden, etc., and then selecting the values of the specific parameters for that family, such as the Lame parameters, Ogden exponents, or whatever the parameterization of a particular family may be. However, the relationships between those parameter values, and visually intuitive material properties such as object's "stiffness", volume preservation, or the "amount of nonlinearity", are less clear and can be tedious to tune. For an arbitrary isotropic hyperelastic energy density function psi that is not parameterized in terms of the Lame parameters, it is not even clear what the Lame parameters and Young's modulus and Poisson's ratio are. Starting from psi, we first give a concise definition of Lame parameters, and therefore Young's modulus and Poisson's ratio. Second, we give a method to adjust the object's three salient properties, namely two small-deformation properties (overall "stiffness", and amount of volume preservation, prescribed by object's Young's modulus and Poisson's ratio), and one large-deformation property (material nonlinearity). We do this in a manner whereby each of these three properties is decoupled from the other two properties, and can therefore be set independently. This permits a new ability, namely "normalization" of materials: starting from two distinct materials, we can "normalize" them so that they have the same small deformation properties, or the same large-deformation nonlinearity behavior, or both. Furthermore, our analysis produced a useful theoretical result, namely it establishes that Linear Corotational materials (arguably the most widely used materials in computer graphics) are the simplest possible nonlinear materials.