Hyperelastic Material Deformation Calibration Code

Overview

This repository contains code used to calibrate the deformation of hyperelastic materials. The calibration process employs two basic algorithms:

• For models with parameters ≤ 2, only uniaxial tensile tests are used to calibrate and obtain optimized parameters. Experimental data from other tests are used for prediction and validation.
  This approach is implemented in the micromechanical_model/ directory.

• For models with parameters ≥ 3, a single loading condition (e.g., uniaxial tensile) is insufficient to obtain a unique parameter set. Therefore, we calibrate the model using multiple sets of experimental data, known as simultaneous fitting.
  This approach is implemented in the AB/, Ogden/, Hill/, and Yeoh/ directories.

For more detailed discussions on the calibration of different hyperelastic models, please refer to:

• Dal, H., Açıkgöz, K., & Badienia, Y. (2021). On the Performance of Isotropic Hyperelastic Constitutive Models for Rubber-Like Materials: A State of the Art Review. ASME Applied Mechanics Reviews, 73(2), 020802.
• Ogden, R., Saccomandi, G., & Sgura, I. (2004). Fitting hyperelastic models to experimental data. Computational Mechanics, 34, 484–502.
• Xiang, Y., Zhong, D., Rudykh, S., Zhou, H., Qu, S., & Yang, W. (2020). A Review of Physically Based and Thermodynamically Based Constitutive Models for Soft Materials. ASME Journal of Applied Mechanics, 87(11), 110801.

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Author

Chongran Zhao
Southern University of Science and Technology, China
Email: [email protected]
Date: Jan. 21, 2025

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Table of Contents

• Experimental Data
• Continuum Basis
• Calibration Details and Material Models
• Hill's Hyperelastic Model with Generalized Strains
• Evaluation Functions (NMAD, MSD, R²)

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References

Experimental Data

We have collected several sets of experimental data from various research papers, including:

• Treloar, L. R. (1944). Stress-strain data for vulcanized rubber under various types of deformation. Rubber Chemistry and Technology, 17(4), 813-825.
• Kawabata, S., Matsuda, M., Tei, K., & Kawai, H. (1981). Experimental survey of the strain energy density function of isoprene rubber vulcanizate. Macromolecules, 14, 154-162.
• Meunier, L., Chagnon, G., Favier, D., Orgéas, L., & Vacher, P. (2008). Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber. Polymer Testing, 27, 765-777.
• Jones, D. F., & Treloar, L. R. G. (1975). The properties of rubber in pure homogeneous strain. Journal of Physics D: Applied Physics, 8, 1285–1304.
• Kawamura, T., Urayama, K., & Kohjiya, S. (2001). Multiaxial deformations of end-linked poly(dimethylsiloxane) networks. 1. Phenomenological approach to strain energy density function. Macromolecules, 34(23), 8252-8260.
• James, A. G., Green, A., & Simpson, G. M. (1975). Strain energy functions of rubber. I. Characterization of gum vulcanizates. Journal of Applied Polymer Science, 19(7), 2033-2058.
• Katashima, T., Urayama, K., Chung, U. I., & Sakai, T. (2012). Strain energy density function of a near-ideal polymer network estimated by biaxial deformation of Tetra-PEG gel. Soft Matter, 8(31), 8217-8222.

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Continuum Basis

For a comprehensive understanding of the continuum basis, please refer to:

• Holzapfel, G. A. (2002). Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science.

The strain energy is decomposed into isochoric and volumetric parts as follows:

$$ \Psi(\mathbf{C}) = \Psi_\mathrm{ich}(\tilde{\mathbf{C}}) + \Psi_\mathrm{vol}(J), $$

For incompressible materials, a Legendre transformation on ( J ) converts the Helmholtz free energy into Gibbs free energy:

$$ G(\mathbf{C}) = G_\mathrm{ich}(\tilde{\mathbf{C}}) + G_\mathrm{vol}(P). $$

For more details, see:

• Liu, J., & Marsden, A. L. (2018). A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction. Computer Methods in Applied Mechanics and Engineering, 337, 549-597.

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Calibration Details and Material Models

For insights into calibration details and a review of material models, see:

• Dal, H., Açıkgöz, K., & Badienia, Y. (2021). On the Performance of Isotropic Hyperelastic Constitutive Models for Rubber-like Materials: A State of the Art Review. Applied Mechanics Reviews, 73(2), 020802.

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Hill's Hyperelastic Model with Generalized Strains

Detailed information on the Hill's hyperelastic model with generalized strains can be found in:

• Liu, J., Guan, J., Zhao, C., & Luo, J. (2024). A Continuum and Computational Framework for Viscoelastodynamics: III. A Nonlinear Theory. Computer Methods in Applied Mechanics and Engineering, 430, 117248. DOI: 10.1016/j.cma.2024.117248

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A New Micro-Macro Transition for the Micro-Mechanical Model

I have reproduced part of the results from:

• Zhan, L., Wang, S., Qu, S., Steinmann, P., & Xiao, R. (2023). A new micro–macro transition for hyperelastic materials. Journal of the Mechanics and Physics of Solids, 171, 105156.

These results are included in the micromechanical_model/ directory, along with some experimental data. Other models are calibrated simultaneously, such as those in the AB/, Hill/, Ogden/, and Yeoh/ directories.

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Evaluation Functions (NMAD, MSD, R²)

For an explanation of the evaluation functions used (NMAD, MSD, R²), please visit:

• Fitness Functions Used by MCalibration

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Please note that the links provided are for reference only and may require a stable internet connection for access. If you encounter any issues with the links, please verify their legitimacy and try again later.

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