Arruda–Boyce

ArrudaBoyce is a stateless finite-strain hyperelastic material model (src/models/ArrudaBoyce.jl). It implements Arruda and Boyce's eight-chain network response [1] with an isochoric network term and 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 Jacobian $J = \det\boldsymbol{F} = \sqrt{\det\boldsymbol{C}} \text{.}$ Its isochoric first invariant is defined as

\[\bar{I}_1 = J^{-2/3}\operatorname{tr}\boldsymbol{C} \text{.}\]

The implemented free energy density is defined as

\[\Psi(\boldsymbol{C}) = \mu \sum_{i=1}^{5}\frac{c_i}{N^{i-1}} \left[\bar{I}_1^i - 3^i\right] + \frac{\kappa}{2}\left[J - 1\right]^2 \text{,}\]

while the truncated inverse-Langevin series coefficients are

\[c_1=\frac{1}{2} \text{,}\quad c_2=\frac{1}{20} \text{,}\quad c_3=\frac{11}{1050} \text{,}\quad c_4=\frac{19}{7000} \text{,}\quad c_5=\frac{519}{673750} \text{.}\]

Here, $\mu$ is the network shear modulus, $\kappa$ the bulk modulus, and $N$ the limiting chain extensibility parameter. It sets the stiffness scale of the eight-chain network; the resulting small-strain shear modulus is $\mu_0 = \mu\left(1 + \frac{3}{5N} + \mathcal{O}(N^{-2})\right)$, which approaches $\mu$ as $N\to\infty$. The subtracted constants $3^i$ make the isochoric part vanish at $\boldsymbol{C}=\boldsymbol{I}$, where $\bar{I}_1 = 3$. The volumetric part vanishes there as well, so $\Psi(\boldsymbol{I})=0$ holds in the undeformed state.

The second Piola-Kirchhoff stress (PK2) $\boldsymbol{S}$ follows from differentiating the free energy density as

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

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

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

while the Cauchy stress follows as

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

Implementation details

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

By default (tangent=:AT), the element assembly uses a hardcoded exact material tangent for ∂S/∂C and maps it to the tangent ∂P/∂F. Using the keyword argument tangent=:AD, the same stress formulation is differentiated by Tensors.gradient over the PK1 response. Both tangent modes are exact. Depending on your case, :AD may compute faster than the constructor default :AT. 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

ArrudaBoyce(μ, κ, N; tangent=:AT)

All three constructor arguments must be positive. The optional tangent keyword must be :AT or :AD.

2D usage

material = PlaneStress(ArrudaBoyce(μ, κ, N))   # plane stress
material = PlaneStrain(ArrudaBoyce(μ, κ, N))   # plane strain

For wrapper mechanics, see Wrappers.

Model parameters

Provenance and acknowledgements

The constitutive equations follow Arruda and Boyce's eight-chain model [1]. The decoupled isochoric–volumetric split follows Flory [2]. 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] Arruda, E. M., & Boyce, M. C. (1993). A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41(2), 389-412. doi:10.1016/0022-5096(93)90013-6

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