Theory¶

Continuum Mechanics Background¶

Deformation¶

The deformation gradient \(\mathbf{F}\) maps material points from the reference to the current configuration. The right Cauchy-Green tensor \(\mathbf{C} = \mathbf{F}^T \mathbf{F}\) characterizes the deformation state.

Invariants¶

The principal invariants of \(\mathbf{C}\) are:

\(I_1 = \text{tr}(\mathbf{C})\)
\(I_2 = \frac{1}{2}[(\text{tr}\,\mathbf{C})^2 - \text{tr}(\mathbf{C}^2)]\)
\(I_3 = \det(\mathbf{C}) = J^2\)

Their isochoric (volume-preserving) counterparts:

\(\bar{I}_1 = J^{-2/3} I_1\)
\(\bar{I}_2 = J^{-4/3} I_2\)

For anisotropic materials with fiber direction \(\mathbf{a}_0\):

\(I_4 = \mathbf{a}_0 \cdot \mathbf{C} \mathbf{a}_0\) (fiber stretch squared)
\(I_5 = \mathbf{a}_0 \cdot \mathbf{C}^2 \mathbf{a}_0\)

Strain Energy Function (SEF)¶

Hyperelastic materials are defined by a strain energy function \(W(\mathbf{C})\) from which all stress and tangent quantities derive:

Second Piola-Kirchhoff stress: \(\mathbf{S} = 2 \frac{\partial W}{\partial \mathbf{C}}\)
Cauchy stress: \(\boldsymbol{\sigma} = \frac{1}{J} \mathbf{F} \mathbf{S} \mathbf{F}^T\)
Material tangent: \(\mathbb{C} = 4 \frac{\partial^2 W}{\partial \mathbf{C} \partial \mathbf{C}}\)

Supported Material Models¶

Isotropic Models¶

Model	SEF	Parameters
Neo-Hooke	\(W = C_{10}(\bar{I}_1 - 3) + U(J)\)	\(C_{10}\)
Mooney-Rivlin	\(W = C_{10}(\bar{I}_1 - 3) + C_{01}(\bar{I}_2 - 3) + U(J)\)	\(C_{10}, C_{01}\)
Yeoh	\(W = \sum_{i=1}^{3} C_{i0}(\bar{I}_1 - 3)^i + U(J)\)	\(C_{10}, C_{20}, C_{30}\)
Demiray	\(W = \frac{C_1}{C_2}[\exp(C_2(\bar{I}_1-3)) - 1] + U(J)\)	\(C_1, C_2\)
Ogden	\(W = \sum_p \frac{\mu_p}{\alpha_p}(\bar{\lambda}_1^{\alpha_p} + \bar{\lambda}_2^{\alpha_p} + \bar{\lambda}_3^{\alpha_p} - 3) + U(J)\)	\(\mu_p, \alpha_p\)
Fung	\(W = \frac{c}{2}[\exp(Q) - 1] + U(J)\)	\(c, b_1, b_2\)

Anisotropic Models¶

Model	SEF	Parameters
Holzapfel-Ogden	\(W = \frac{a}{2b}[\exp(b(\bar{I}_1-3))-1] + \frac{a_f}{2b_f}[\exp(b_f(I_4-1)^2)-1] + U(J)\)	\(a, b, a_f, b_f\)
GOH	\(W = \frac{a}{2b}[\exp(b(\bar{I}_1-3))-1] + \frac{a_f}{2b_f}[\exp(b_f \bar{E}^2)-1] + U(J)\)	\(a, b, a_f, b_f, \kappa\)
Guccione	\(W = \frac{C}{2}[\exp(Q)-1] + U(J)\) with fiber-frame Q	\(C, b_f, b_t, b_{fs}\)

where \(\bar{E} = \kappa(\bar{I}_1 - 3) + (1-3\kappa)(I_4 - 1)\) is the GOH generalized strain invariant with dispersion \(\kappa \in [0, 1/3]\).

Volumetric Contribution¶

All models use the standard volumetric penalty:

\[U(J) = \frac{K}{4}(J^2 - 1 - 2\ln J)\]

where \(K\) is the bulk modulus.

Surrogate Pipeline¶

Architecture¶

Material -> DeformationGenerator -> Dataset -> Model -> Trainer -> FortranEmitter -> .f90

Define a Material with its SEF and parameters
Generate synthetic deformation gradients (uniaxial, biaxial, shear, combined)
Compute invariants and stress/energy targets
Train a neural network (MLP, ICNN, PolyconvexICNN, or CANN)
Export to Fortran 90 UMAT subroutine

Neural Network Architectures¶

MLP: Standard feedforward network. Flexible but no physics guarantees.
ICNN: Input-Convex Neural Network. Guarantees convexity of energy w.r.t. inputs via non-negative weights on the z-path.
PolyconvexICNN: Polyconvex ICNN with separate branches for invariant groups.
CANN: Constitutive ANN with interpretable basis functions and non-negative weights. Enables model discovery through sparsification.

Thermodynamically Consistent Training¶

The EnergyStressLoss enforces thermodynamic consistency by jointly minimizing:

\[\mathcal{L} = \alpha \|W_{pred} - W_{true}\|^2 + \beta \left\|\frac{\partial W_{pred}}{\partial \mathbf{I}} - \frac{\partial W_{true}}{\partial \mathbf{I}}\right\|^2\]

where the stress (gradient) term is computed via automatic differentiation through the network.

Experimental Data Integration¶

The ExperimentalData class loads biomechanical test data (uniaxial, biaxial) and converts to deformation gradients for integration with the surrogate pipeline.

The fit_material function uses scipy.optimize.minimize to fit material parameters to experimental data by minimizing the stress residual:

\[\min_{\theta} \sum_i \|\boldsymbol{\sigma}_{model}(\lambda_i; \theta) - \boldsymbol{\sigma}_{exp,i}\|^2\]

References¶

Holzapfel, G.A., Gasser, T.C., Ogden, R.W. (2000). A new constitutive framework for arterial wall mechanics. J. Elasticity, 61, 1-48.
Gasser, T.C., Ogden, R.W., Holzapfel, G.A. (2006). Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface, 3, 15-35.
Linka, K., et al. (2023). A new family of Constitutive Artificial Neural Networks towards automated model discovery. Comput. Methods Appl. Mech. Eng., 403, 115731.
Amos, B., Xu, L., Kolter, J.Z. (2017). Input convex neural networks. ICML.
Guccione, J.M., McCulloch, A.D., Waldman, L.K. (1991). Passive material properties of intact ventricular myocardium determined from a cylindrical model. J. Biomech. Eng., 113, 42-55.