The generalised Mooney space for modelling the response of rubber-like materials

Abstract: Soft materials such as rubbers, silicones, gels and biological tissues have a nonlinear response to large deformations, a phenomenon which in principle can be captured by hyperelastic models. The suitability of a candidate hyperelastic strain energy function is then determined by comparing its predicted response to the data gleaned from tests and adjusting the material parameters to get a good fit, an exercise which can be deceptive because of nonlinearity. Here we propose to generalise the approach of Rivlin and Saunders [Phil Trans A 243 (1951) 251-288] who, instead of reporting the data as stress against stretch, manipulated these measures to create the 'Mooney plot', where the Mooney-Rivlin model is expected to produce a linear fit. We show that extending this idea to other models and modes of deformation (tension, shear, torsion, etc.) is advantageous, not only (a) for the fitting procedure, but also to (b) delineate trends in the deformation which are not obvious from the raw data (and may be interpreted in terms of micro-, meso-, and macro-structures) and (c) obtain a bounded condition number \k{appa} over the whole range of deformation; a robustness which is lacking in other plots and spaces.

• The paper presents the Generalised Mooney Space (GMS), a novel framework that recasts hyperelastic parameter estimation into a well-conditioned linear regression problem.
• The work demonstrates GMS using models like Mooney-Rivlin, Yeoh, and Gent, revealing its effectiveness in exposing nonphysical trends and improving model validation.
• The framework extends to multiple deformation modes, ensuring enhanced numerical robustness and streamlined parameter identification across diverse rubber-like materials.

Generalised Mooney Space for Hyperelastic Modelling of Rubber-like Materials

Introduction

The accurate constitutive modelling of rubber-like materials under large deformations hinges on the selection of appropriate hyperelastic strain energy functions and robust parameter identification methodologies. Traditional approaches, often executed in Cauchy or engineering stress-strain spaces, encounter both conceptual and numerical challenges, including non-uniqueness in parameter estimation and instability in the fitting process. The classical Mooney plot, introduced by Rivlin and Saunders, reformulated experimental data for uniaxial tension into a linear relation via the Mooney-Rivlin model, providing not only computational simplification but also deeper insight into the microstructural regime differentiation. However, the generality, extensibility, and physical interpretation of such transformation spaces for arbitrary strain energy functions and deformation modes remained limited.

Theoretical Framework: Generalised Mooney Space

This work introduces the Generalised Mooney Space (GMS), a formal extension of the classical Mooney plot framework, systematically designed for a broad class of strain energy functions and deformation modes. In GMS, experimental data $\{\lambda_i, P_i\}$ (stretch and engineering stress) are transformed into pairs $\{\zeta, \mathscr{M}\}$, where the relationship between transformed variables is constructed to yield a linear, or polynomial, regression problem for a specific model family: $\mathscr{M} = \sum_{j \in A} C_j \zeta^{j}$ where $C_j$ are material parameters and $A \subset \mathbb{Q}$.

This formulation enables, for each strain energy function $W$, the design of model-specific transformations such that the parameter fitting task is recast into a well-conditioned linear regression problem. The key theoretical contribution is the identification and construction of $\zeta$ and $\mathscr{M}$ variables that canonically map the original stress-stretch experimental data into a "Mooney" space appropriate for the functional form of $W$.

A critical technical advantage of GMS is the bounded and typically reduced condition number $\kappa$ for regression, which is analytically shown to be strictly less than or equal to unity across the deformation range for canonical Mooney-Rivlin models. This is in direct contrast to the Cauchy or engineering stress spaces, where $\kappa$ diverges in the vicinity of the unstrained configuration, leading to numerically unreliable parameter estimation.

Application to Hyperelastic Constitutive Models

The practical efficacy of GMS is demonstrated for several widely adopted hyperelastic models:

• Mooney-Rivlin Model: In GMS, the mapping $\mathscr{M} = P/[2(\lambda - \lambda^{-2})]$, $\zeta = 1/\lambda$ recasts the fitting into a strict linear regression, with the condition number bounded ($0 < \kappa \leq 1$).
• Yeoh and Polynomial Neo-Hookean Models: Transformed as $\mathscr{M} = C_1 + 2C_2\zeta + 3C_3\zeta^2$, with $\zeta = \lambda^2 + 2\lambda^{-1} - 3$, enabling direct high-order polynomial regression in GMS. This representation exposes the regimes (e.g., large stretch $\,\lambda \geq 3$) where the model adequately captures the data and where it systematically fails.
• Gent and Fung-Demiray Models: For these exponentially stiffening or limiting-chain extensibility models, the mappings involve mixed algebraic-logarithmic forms. For instance, with the Gent model: $$\mathscr{M}_G = \mu + (1/J_m)\zeta, \quad \text{where} \quad \zeta = \frac{P(\lambda^2 + 2\lambda^{-1} - 3)}{\lambda - \lambda^{-2}}$$ In all cases, the transformed space reveals non-trivial patterns in the data (e.g., V-shaped trends), allowing for precise identification of linear regimes and model selection or validation pitfalls such as nonphysical negative stiffening parameters.

For highly nonlinear or non-polynomial models (e.g., Ogden-type, logarithmic, or entropic models), GMS identifies cases where such a transformation to linear regression is not possible, marking intrinsic limits of the approach and emphasizing the need for careful model choice.

Generalisation to Multiple Deformation Modes

The GMS framework is systematically extended to a spectrum of classical deformation modes, including:

• Uniaxial and equi-biaxial tension
• Pure shear
• Simple shear
• Simple torsion

For each mode, GMS provides explicit mappings for $\mathscr{M}$ and $\zeta$ tailored to the geometry and kinematics, again enabling model-appropriate linear or polynomial regression forms. This universalization of the Mooney plot concept allows for transparent cross-mode model validation and robust comparative fitting.

Numerical and Practical Implications

A major practical implication is the strict improvement in numerical robustness and reproducibility of parameter identification due to the controlled condition number and transparency in the fitting residuals. The GMS framework standardizes the minimization of relative, rather than absolute, error in the transformed space. This ensures invariance of optimal parameters with respect to chosen stress representation and deformation mode, resolving previous ambiguities in the literature.

The GMS also serves as a diagnostic tool for model inadequacy—regions where no physical or consistent fit is possible (e.g., negative moduli or stiffening parameters) are immediately exposed in the geometry of the GMP, streamlining model selection and falsification.

Theoretical Implications and Future Directions

The systematic construction of Mooney spaces for arbitrary $W$ elucidates structural facets of the Hauptproblem of nonlinear elasticity—parameterization and validation—by delivering a tool to probe and visualize material response regimes inaccessible in raw or classical transformed stress-strain spaces. GMS sharpens the detection of micro- and meso-structural deformation mechanisms via data trends that are obfuscated in traditional plots.

Future research directions could include:

• Development of GMS for anisotropic, viscoelastic, or heterogeneous materials.
• Automated model selection and regime identification algorithms via GMP analysis.
• Integration with Bayesian or machine-learning-based inverse modelling strategies leveraging the advantageous conditioning of GMS.

Conclusion

The generalised Mooney space (GMS) formalism provides a mathematically rigorous and numerically efficient framework for the identification, validation, and critique of hyperelastic constitutive models for rubber-like materials. By unifying diverse models and deformation modes under a canonical transformation that regularizes the fitting process, GMS elevates the interpretability and robustness of nonlinear elasticity modeling. As the field advances towards more complex materials and experimental protocols, the GMS methodology stands as a quantitative tool for navigating the growing model landscape and resolving core inverse problems in nonlinear elastic material characterization.

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Reference:

For the complete technical development, see "The generalised Mooney space for modelling the response of rubber-like materials" (2403.01223).