Mooney–Rivlin

MooneyRivlin is a stateless finite-strain hyperelastic material model (src/models/MooneyRivlin.jl). It implements the two-parameter Mooney–Rivlin response [1, 2], expressed here in the isochoric–volumetric split with a quadratic volumetric penalty.

Mathematical formulation

The model uses the deformation gradient $\boldsymbol{F}$, the right Cauchy-Green tensor $\boldsymbol{C}=\boldsymbol{F}^{\mathrm{T}}\boldsymbol{F}$, and the volume ratio $J=\det\boldsymbol{F}=\sqrt{\det\boldsymbol{C}}$. The first two invariants of $\boldsymbol{C}$ are

\[I_1 := \operatorname{tr}\boldsymbol{C} \text{,} \qquad I_2 := \frac{1}{2}\left[\left(\operatorname{tr}\boldsymbol{C}\right)^2 - \boldsymbol{C}:\boldsymbol{C}\right] \text{.}\]

Mooney's original theory [1] was expressed in principal stretches; Rivlin [2] later developed the invariant-based form. The same invariants can equivalently be evaluated from the left Cauchy-Green tensor $\boldsymbol{b}=\boldsymbol{F}\boldsymbol{F}^{\mathrm{T}}$, since $\boldsymbol{b}$ and $\boldsymbol{C}$ have identical principal invariants (see e.g., Holzapfel [3], Ch. 6).

The implemented isochoric invariants are

\[\bar{I}_1 = J^{-2/3} I_1 \text{,} \qquad \bar{I}_2 = J^{-4/3} I_2 \text{,}\]

while the implemented free energy density is

\[\Psi(\boldsymbol{C}) := C_{10}\left[\bar{I}_1 - 3\right] + C_{01}\left[\bar{I}_2 - 3\right] + \frac{\kappa}{2}\left[J - 1\right]^2 \text{.}\]

Here, $C_{10}$ and $C_{01}$ control the distortional response and $\kappa$ is the bulk modulus of the volumetric penalty. The small-strain shear modulus at the reference state is $\mu_0 = 2\left[C_{10}+C_{01}\right]$.

Stress evaluation

The second Piola-Kirchhoff stress (PK2) is

\[\boldsymbol{S} := 2\frac{\partial\Psi}{\partial\boldsymbol{C}} \text{.}\]

With $\boldsymbol{I}$ denoting the identity tensor, the implemented closed-form PK2 stress is

\[\boldsymbol{S} = 2 C_{10} J^{-2/3} \left[\boldsymbol{I} - \frac{I_1}{3}\boldsymbol{C}^{-1}\right] + 2 C_{01} J^{-4/3} \left[I_1\boldsymbol{I} - \boldsymbol{C} - \frac{2I_2}{3}\boldsymbol{C}^{-1}\right] + \kappa J\left[J-1\right]\boldsymbol{C}^{-1} \text{.}\]

The first Piola-Kirchhoff stress (PK1) used by the assembler is

\[\boldsymbol{P} = \boldsymbol{F}\boldsymbol{S} \text{,}\]

while the Cauchy stress is

\[\boldsymbol{\sigma} = \frac{1}{J}\boldsymbol{P}\boldsymbol{F}^{\mathrm{T}} \text{.}\]

Implementation details

State variables

MooneyRivlin is rate-independent and stateless (create_state returns NoState()). The dt keyword is ignored.

Tangent formulations

Two exact tangent routes are provided:

tangent=:AD: differentiates the PK1 stress response with Tensors.gradient.
tangent=:AT: uses the hardcoded material tangent ∂S/∂C and maps it to ∂P/∂F.

The constructor default is tangent=:AD; representative stiffness_matrix checks favor this route for this model.

Wrapper paths

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

MooneyRivlin(C10, C01, κ; tangent=:AD)

The optional tangent keyword must be :AD or :AT. The constructor requires finite C10 and C01, a positive sum C10 + C01, and a positive finite κ. Individual negative values of C10 or C01 are accepted when their sum is positive. The constructor check does not guarantee stability over all finite-strains.

2D usage

material = PlaneStress(MooneyRivlin(C10, C01, κ))   # plane stress
material = PlaneStrain(MooneyRivlin(C10, C01, κ))   # plane strain

For wrapper mechanics, see Wrappers.

Model parameters

Constructor argument	Symbol	Description
`C10`	$C_{10}$	Coefficient of the first modified invariant
`C01`	$C_{01}$	Coefficient of the second modified invariant
`κ`	$\kappa$	Bulk modulus
`tangent`	-	Selects the exact tangent route, `:AD` by default or optional `:AT`

Provenance and acknowledgements

The constitutive equations follow the two-invariant form of Mooney [1] and Rivlin [2]. The decoupled isochoric–volumetric split follows Flory [4]. The Julia implementation was written for FerriteSolidMechanics.jl; no external code was translated or adapted.

The implementation is maintained by Lukas Laubert. Sebastian Pfaller is gratefully acknowledged for proofreading this page.

References

[1] Mooney, M. (1940). A theory of large elastic deformation. Journal of Applied Physics, 11(9), 582-592. doi:10.1063/1.1712836

[2] Rivlin, R. S. (1948). Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), 379-397. doi:10.1098/rsta.1948.0024

[3] Holzapfel, G. A. (2000). Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons. (Chapter 6, Hyperelastic Materials).

[4] Flory, P. J. (1961). Thermodynamic relations for high elastic materials. Transactions of the Faraday Society, 57, 829-838. doi:10.1039/TF9615700829