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Stable Neo Hookean

Stable Neo Hookean constitutions implement a stable version of the classic Neo Hookean material model. This model is particularly well-suited for simulating soft, rubber-like materials due to its ability to handle large deformations while maintaining numerical stability.

There are so many constitutive models called "Neo Hookean", we will distinguish them with UID numbers.

Dynamic Deformables: Implementation and Production Practicalities (Now With Code!)

#10 Stable Neo Hookean

Since v0.0.26, this constitution is Stiff-GIPC's SNK1 (energy, gradient, and analytically SPD-projected Hessian ported verbatim from femEnergy.cu).

Deformation energy density:

\[ E = \frac{1}{2} \mu (I_c - 3) + \frac{1}{2} \lambda (J - 1 - \mu / \lambda)^2, \]

where \(J = \det(F)\) and \(I_c = \|F\|_F^2\).

First Piola-Kirchhoff stress (gradient):

\[ P = \frac{\partial E}{\partial F} = \mu F + (\lambda (J - 1) - \mu)\, \mathrm{cof}(F). \]

The 9×9 energy Hessian is projected to SPD analytically — twist/flip eigensystem from the QR-SVD of \(F\) plus a 3×3 direct eigensolve — instead of a generic eigendecomposition with clamping.

In continuum mechanics, \(F\) is called the deformation gradient, \(\lambda\) and \(\mu\) are the Lamé parameters. \(I_c\) is the first invariant of the right Cauchy-Green deformation tensor \(C = \|F\|_F^2\), \(\|\cdot\|_F\) is the Frobenius norm.

Attributes

On tetrahedra:

  • mu: \(\mu\) in the energy above
  • lambda: \(\lambda\) in the energy above