Testing for Hyperelastic Material Models in CAE
Hyperelastic materials are highly stretchable and deformable substances that return to their original shape after the removal of applied forces. Unlike materials that undergo permanent deformation, hyperelastic materials — commonly rubbers and elastomers — exhibit large elastic strains without yielding or failure. Because they display strong coupling between deformation modes due to near-incompressibility (Poisson's ratio ≈ 0.5), their mechanical behavior must be understood and modeled across multiple modes of deformation.
To accurately simulate hyperelastic materials in finite element analysis (FEA), mathematical models such as Mooney-Rivlin, Ogden, and Arruda-Boyce are employed. These models are constructed by fitting experimental data from various deformation modes. Selecting the correct model and appropriate test data is essential to ensure realistic simulations, especially for products that undergo complex or large deformations.
To learn more regarding material testing and characterization in support of material models for CAE, see our TestPaks or contact us to talk with our materials testing experts.
Assumptions in Hyperelastic Modeling
To fit a hyperelastic model accurately:
- The material must not exhibit yielding, which would make it an elastomer rather than a true hyperelastic.
- It should be isotropic, so behavior does not depend on direction.
- The Mullins effect (work softening due to cyclic loading) must be considered in the testing strategy.
- Materials are typically treated as incompressible, though compressible models are discussed later.
Limitations of Hyperelastic Models
Hyperelastic models are only as good as their input data. These models tend to fit data well only in the strain ranges used during curve fitting. If the model is extrapolated beyond those ranges (especially into high strain), accuracy suffers, and low-strain fidelity may be compromised. Additionally, multiple deformation modes must be tested and fitted to develop robust models.
Implementation Strategies for Testing
When planning hyperelastic testing and modeling, several factors must be defined early:
- Is the simulation capturing initial loading or repeated cyclic behavior?
- What is the expected deformation magnitude?
- What type(s) of mechanical loading will the product experience?
These answers dictate the test types, data requirements, and model fitting strategy.
Modes of Deformation
The complex nature of rubber deformation demands that the full strain energy potential (i.e., energy stored per unit volume) be characterized through testing in multiple modes:
- Uniaxial tension and compression
- Planar tension
- Equibiaxial tension (optional)
Tests must be combined to capture the full 3D behavior of hyperelastic materials. For instance, a rubber block under compression will simultaneously experience shear and tension in other regions.
Magnitude of Deformation
If deformation is small (e.g., vibration damping), simple models like neo-Hookean may suffice. These models are computationally efficient and adequate for low-strain predictions. For large deformation, higher-order models like Mooney-Rivlin (polynomial), Ogden, or Arruda-Boyce are more appropriate.
To guide model selection and test planning, early-stage simulations (even linear elastic) can estimate expected strains and help tailor the testing program.
The Mullins Effect
The Mullins effect describes stress-softening seen in filled rubbers under cyclic loading. Unlike metals, which can exhibit hardening, rubbers typically become weaker after initial deformation due to internal damage (e.g., broken filler-polymer bonds).
- Simulations for cyclic loads must use data from preconditioned samples — rubber specimens cycled repeatedly to the expected strain until the stress-strain curve stabilizes.
- If simulations involve first-time deformation, preconditioned data are not appropriate — the initial, first-cycle response is needed instead.
- In cases of varying loads, neither single-curve nor conventional models are sufficient. A piecewise approach or advanced visco-hyperelastic models may be required.
Hyperelastic Testing Techniques
Hyperelastic testing requires specialized specimen shapes, non-contact strain measurement, and carefully managed clamping. The most common tests include:
Tensile Tests
- Performed on an electromechanical universal testing machine (UTM).
- Specimens often follow ASTM D412 or ISO 37 standards.
- Video or laser extensometry is used due to large strains and potential artefacts from contact devices.
- Wide grips help prevent slippage and local necking.
Planar Tension Tests
- Large width-to-length ratio specimens generate pure shear at 45° angles.
- Proper clamping is crucial to avoid compressive artefacts.
- Non-contact extensometry ensures accurate measurements.
Biaxial Tension Tests
- Stretch sheet specimens in two perpendicular directions using specialized equipment.
- Alternative method: bubble inflation, where a circular sheet is deformed from the underside with air to induce biaxial stress. This method simplifies setup but complicates data analysis.
- The Miller method offers another approach using a circular specimen pulled radially to create a central biaxial stress state.
Uniaxial Compression Tests
- Sometimes substituted for biaxial tension in practical settings.
- Must be done with lubricated platens to minimize friction and ensure homogenous compression.
- Video monitoring can track deformation to avoid artefacts like buckling or barreling.
While hyperelastic theory suggests equibiaxial tension and uniaxial compression responses should be symmetric, experimental friction and boundary effects can disrupt this equivalence. Thus, comparison between compressive and biaxial tension data is crucial.
Fitting Material Models to Data
To develop a reliable hyperelastic model, the following should be noted:
- Plot and compare all test modes (tension, compression, planar, biaxial) on the same graph.
- Data must cover and slightly exceed the strain range of interest.
- Curves must follow a logical order of stiffness: equibiaxial > planar > uniaxial.
- Compression should match tension at low strains.
Model fitting is usually done in one of two ways:
Software-Integrated Fitting
- FEA programs accept raw data files (strain vs. stress or stretch ratio).
- The software internally fits models like Mooney-Rivlin, Ogden, etc.
- Input must follow format and unit requirements exactly.
External Curve Fitting
- Data is fitted manually using nonlinear regression tools.
- Models are expressed in strain energy potential functions, often involving invariants of the deformation tensor.
- Each deformation mode corresponds to a different mathematical form.
- After fitting, model coefficients are entered into the simulation software.
Best practice: start with a low-order model and increase complexity only if needed. Dropping deformation modes or switching models may be necessary if a particular test curve does not fit.
Model selection guidance:
- Neo-Hookean: small strain (<30%)
- Mooney-Rivlin: intermediate strain (<100%)
- Ogden: large strain (>100%)
Choose the lowest-order model that fits the data over the strain range of interest to reduce numerical instability and computational cost.
Failure Modeling in Hyperelastic Materials
Rubbers often fail through tearing. Standardized tests like ASTM D624 Type C (bow-tie shaped specimens) are used to assess tear strength.
- These tests provide controlled crack initiation and propagation.
- The stress state is well-defined, allowing for direct comparison between experiment and simulation.
Modeling crack growth in hyperelastic materials remains a highly nonlinear challenge due to simultaneous deformation and damage processes. Techniques like the penalty method, developed by Deihl and expanded by Nair, have shown promise in simulating these behaviors accurately.
Conclusion
Modeling hyperelastic materials involves far more complexity than conventional linear materials. Due to their nonlinear, multi-directional, and large-strain behavior, hyperelastic modeling requires:
- Accurate data from multiple deformation modes
- Consideration of effects like the Mullins effect
- Specialized testing setups and non-contact extensometry
- Careful curve fitting using models like neo-Hookean, Mooney-Rivlin, or Ogden
- Tailored modeling strategies based on the expected strain range and loading type
For accurate simulations, at least two to three deformation modes should be tested and used in fitting. Advanced use cases such as failure modeling and volume compressibility require even more specialized testing and modeling techniques.
Ultimately, the goal is to use the simplest possible model that captures the necessary behavior within the relevant strain range. Knowing the simulation’s scope in advance is key to specifying the appropriate testing and modeling effort — helping to avoid errors, reduce costs, and ensure realistic simulation results.
To explore the topics discussed on this page further, see Hubert Lobo (Founder, DatapointLabs) and Brian Croop (CEO, DatapointLabs), Determination and Use of Material Properties for Finite Element Analysis (NAFEMS, 2016), Ch.6.
