Saint Venant–Kirchhoff (StVK)

Saint Venant–Kirchhoff extends linear elasticity to finite deformation using the Green–Lagrange strain. The constitutive relation is linear in that nonlinear strain measure, making the model frame-indifferent but unreliable under large compression.

Enum value: SoftMaterialType::StVenantKirchhoff

Formulation

The Green–Lagrange strain is

$$\mathbf{E} = \frac{1}{2}\left(\mathbf{F}^T\mathbf{F} - \mathbf{1}\right)~.$$

The strain energy density is

$$\Psi(\mathbf{F}) = \mu\|\mathbf{E}\|^2 + \frac{\lambda}{2}\left(\operatorname{tr}\mathbf{E}\right)^2~,$$

and the first Piola–Kirchhoff stress is

$$\mathbf{P} = \mathbf{F}\left[2\mu\mathbf{E} + \lambda\left(\operatorname{tr}\mathbf{E}\right)\mathbf{1}\right]~.$$

Parameters

The public parameter type is StVenantKirchhoffMaterialParams.

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.

warning

For auxetic materials ($\nu < 0$), Fast does not guarantee a positive-semidefinite tangent. Use Projection or AbsEigenProjection when that guarantee is required.

When to Use

• Problems with large rigid rotations but only moderate elastic strain.
• Educational or comparative work requiring a simple frame-indifferent hyperelastic model.

caution

StVK has no infinite compression barrier. Its energy remains finite at $\mathbf{F}=\mathbf{0}$, and the first Piola–Kirchhoff stress vanishes there, so complete collapse is a pathological stationary state. For robust large-deformation simulation, use Neo-Hookean instead.