Weak Nonlinearity and Linear Equalization for Isotropic Hyperelastic Materials

Abstract In computer graphics and engineering, isotropic hyperelastic volumetric materials are commonly chosen from a small set of families (e.g., StVK, linear corotational, Neo‐Hookean, Ogden) and tuned by adjusting family‐specific parameters. This workflow obscures how different material families relate and provides no unified notion of “distance” or “amount and direction of nonlinearity” across families. Furthermore, for a general isotropic 3D hyperelastic energy density function ψ, the relationships between its parameter values, and visually intuitive properties such as an object's “stiffness”, volume preservation, or the “amount of nonlinearity”, are less clear and can be tedious to adjust. Starting with an arbitrary isotropic ψ, we define and compute Lamé parameters (capturing linear behavior: stiffness and volume preservation), using the Taylor expansion of ψ at the rest state. This makes it possible to quantitatively “equalize” the small‐deformation behavior of two materials, even if they come from different material families, or are defined by arbitrary isotropic energy densities ψ. We rigorously state and prove that Linear Corotational materials (arguably the most widely used materials in computer graphics) are the simplest possible nonlinear materials. To compare nonlinear behavior beyond the linearization, we define a distance metric between linearization‐matched materials, using third‐order terms of ψ (“weakly nonlinear behavior”). We further define SCIEN, a directional nonlinearity measure obtained by projecting a material onto the standard one‐parameter Seth‐Hill continuum (which includes Hencky, Linear Corotational, and StVK as special cases), thereby distinguishing stretch‐stiffening from compression‐stiffening and quantifying its strength. As a result, we can equalize and edit materials agnostically from any material family: two materials can be matched in small‐deformation response, in weakly nonlinear behavior, or in both, while keeping these properties decoupled. This provides a unified language and practical tools for comparing, organizing, and adjusting common nonlinear material models.