Neo-Hookean

The Neo-Hookean model is the recommended default for general-purpose deformable simulation. SuperDex Physics uses the inversion-robust formulation of Smith et al. (2018).

Enum value: SoftMaterialType::NeoHookean

Formulation

Following Smith et al. (2018), SuperDex Physics implements the strain energy density

$$\Psi(\mathbf{F}) = \frac{\widehat{\mu}}{2}(I_C - 3) + \frac{\widehat{\lambda}}{2}(J - \alpha)^2 - \frac{\widehat{\mu}}{2}\ln(I_C + 1)~,$$

where

$$I_C = \operatorname{tr}(\mathbf{F}^T\mathbf{F})~, \qquad J = \det(\mathbf{F})~,$$

and, in terms of the standard Lamé parameters $\mu$ and $\lambda$,

$$\widehat{\mu} = \frac{4}{3}\mu~, \qquad \widehat{\lambda} = \lambda + \frac{5}{6}\mu~, \qquad \alpha = 1 + \frac{3\widehat{\mu}}{4\widehat{\lambda}}~.$$

This reparameterization ensures that the model's small-strain response corresponds to linear elasticity with the standard Lamé parameters. The first Piola–Kirchhoff stress is

$$\mathbf{P} = \widehat{\mu}\left(1 - \frac{1}{I_C + 1}\right)\mathbf{F} + \widehat{\lambda}(J - \alpha)\operatorname{cof}(\mathbf{F})~.$$

Because $I_C + 1$ is positive for every real deformation gradient, the logarithmic term remains defined through element inversion and regularizes the response as $I_C$ approaches zero. It is not a divergent barrier: the energy remains finite at complete collapse.

Parameters

Parameter	Default	Description
youngsModulus	100,000 Pa	Stiffness
poissonRatio	0.45	Compressibility
psdStrategy	Projection	PSD enforcement strategy

Supported concrete PSD strategies: None, Projection, Fast, AbsEigenProjection. MaterialDefault resolves to the model's default, Projection.

The public parameter type is NeoHookeanMaterialParams, an alias for SmithNeoHookeanMaterialParams.

When to Use

• Recommended for most simulations. The Smith et al. (2018) formulation supports large deformation, extreme compression, and element inversion.
• It is somewhat more expensive than Linear Elastic, but substantially more appropriate for finite rotations and strains.

References

• B. Smith, F. de Goes, and T. Kim, Stable Neo-Hookean Flesh Simulation, 2018.