Neo–Hookean

The compressible Neo–Hookean hyperelastic material model NeoHooke (src/models/NeoHooke.jl) is parameterized by Young's modulus E and Poisson's ratio ν.

Mathematical formulation

The material's behavior is derived from its free energy density function $\Psi(\boldsymbol{C})$ as

\[\Psi(\boldsymbol{C}) = \frac{1}{2}\mu \left[I_1 - 3 - 2\ln J\right] + \frac{1}{2}\lambda \left[J - 1\right]^2 \text{,}\]

where $\boldsymbol{F}$ is the deformation gradient, $\boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F}$ the right Cauchy–Green tensor, $I_1 = \mathrm{tr}(\boldsymbol{C})$ its first invariant, $J = \det(\boldsymbol{F}) = \sqrt{\det(\boldsymbol{C})}$ the volume ratio, and $\mu, \lambda$ the Lamé parameters.

The parameters $\mu$ (shear modulus) and $\lambda$ (first Lamé parameter) are computed from the input Young's modulus $E$ and Poisson's ratio $\nu$ as

\[\mu = \frac{E}{2\left[1 + \nu\right]}, \qquad \lambda = \frac{E\nu}{\left[1 + \nu\right]\left[1 - 2\nu\right]} \text{.}\]

Implementation details

NeoHooke is rate-independent and stateless (create_state returns NoState()). The tangent is obtained analytically via Tensors.hessian over the free energy Ψ(C), yielding both the second Piola-Kirchhoff stress (PK2) S = 2 ∂Ψ/∂C and the material derivative ∂S/∂C in a single call. The generic 2D wrappers use the model's compute_PK1_3D and update_state_from_3D! hooks. Since the material has no history, update_state_from_3D! is a no-op.

Constructor

NeoHooke(E, ν)

2D usage

material = PlaneStress(NeoHooke(E, ν))   # plane stress
material = PlaneStrain(NeoHooke(E, ν))   # plane strain

For wrapper mechanics, see Wrappers.

Model parameters

Provenance

The constitutive energy, stress/tangent derivation, element-level linearization, and related documentation are adapted from the Ferrite.jl hyperelasticity tutorial [1]. The upstream copyright/license notice is retained in the repository LICENSE.

References

[1] Ferrite.jl documentation. Hyperelasticity tutorial. https://ferrite-fem.github.io/Ferrite.jl/stable/tutorials/hyperelasticity/