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Hyperelastic materials are materials that can undergo large deformations without permanent damage. They are often used to model rubber-like materials, biological tissues, soft robots, and other applications that involve large strains. Hyperelastic materials can be characterized by a strain energy function, which relates the strain to the stored energy in the material. ABAQUS is a finite element software that can simulate the behavior of hyperelastic materials under various loading conditions. In this blog post, we will introduce the basics of hyperelasticity and how to model hyperelastic materials in ABAQUS.

## Hyperelacticity behavior:

Hyperelasticity is a branch of continuum mechanics that deals with the nonlinear response of elastic materials. Unlike linear elasticity, which assumes small strains and linear stress-strain relations, hyperelasticity can account for large strains and nonlinear stress-strain relations. The stress-strain curve of hyperelastic materials is shown in Fig 1.

Hyperelasticity is based on the principle of material frame-indifference, which states that the material response should not depend on the choice of reference configuration. This means that the material properties are invariant under rigid body motions, such as rotations and translations. Hyperelasticity also assumes that the material is isotropic, meaning that its properties are the same in all directions.

One way to describe the behavior of hyperelastic materials is by using a strain energy function (SEF), which is a scalar function of the strain measures. The SEF represents the amount of energy stored in the material due to deformation. The SEF can be derived from experimental data, theoretical models, or molecular simulations.

## Different SEFs in ABAQUS:

### Neo-Hookean:

This is the simplest form of SEF, which assumes a linear relation between the first invariant of the right Cauchy-Green deformation tensor and the SEF. The Neo-Hookean model is suitable for moderately stretched rubber-like materials.

Where *U* is the strain energy per unit of reference volume; C10 and D1 are temperature-dependent material parameters; I_{1} is the first deviatoric strain invariant defined as:

Where the deviatoric stretches λ_{i }= J^{-1/3} λ_{i }; *J* is the total volume ratio; J* ^{el }*is the elastic volume ratio and λ

_{i}are the principal stretches. The initial shear modulus and bulk modulus are given by:

In contrast to materials that exhibit linear elasticity, the stress-strain relationship of a neo-Hookean material is not linear. Instead, the material initially follows a linear relationship between applied stress and strain, but eventually reaches a plateau. The neo-Hookean model does not consider the release of energy as heat during material deformation, and it assumes perfect elasticity at all stages of deformation. The model is not capable of predicting an increase in modulus at high strains and is generally only accurate for strains below 20%. Furthermore, it is not suitable for analyzing stress states in multiple directions and has been replaced by the Mooney-Rivlin model.

The neo-Hookean model offers several advantages. Firstly, it is a simple model with only two input parameters, or even just one if the material is assumed to be incompressible. This simplicity results in a smaller number of required tests for obtaining the necessary parameter. Secondly, the model is compatible as the material parameter obtained from one type of stress-strain curve can be applied to predict other types of deformations, particularly under small to medium strain conditions.

### Mooney-Rivlin:

This is an extension of the Neo-Hookean model that incorporates a additional term that is influenced by the second measure of the right Cauchy-Green deformation tensor. The Mooney-Rivlin model is able to account for the increased stiffness observed at high levels of deformation.

Where U is the strain energy per unit of reference volume; C10, C01 and D1 are temperature-dependent material parameters; I_{1} and I_{2} are the first and second deviatoric strain invariants defined as:

Where the deviatoric stretches ; *J* is the total volume ratio; is the elastic volume ratio and are the principal stretches. The initial shear modulus and bulk modulus are given by:

The Mooney-Rivlin model has gained significant recognition and widespread utilization. It particularly excels in small and medium strain applications, spanning from 0% to 100% in tensile strain and 30% in compression. Within this range, the Mooney-Rivlin model effectively captures and accurately represents the mechanical behavior exhibited by rubber materials.

The Mooney-Rivlin model does have certain limitations that should be considered. It is not appropriate for deformation exceeding 150%. Additionally, the higher-order versions of the Mooney-Rivlin model require a relatively complex process of parameter acquisition, often involving curve fitting of experimental data. Another restriction is that the Mooney-Rivlin model is not suitable for analyzing compressible hyperelastic materials like foams. Furthermore, when the strain or stress exceeds the range of input experimental data, there is a possibility of encountering significant errors.

### Ogden:

This is another generalization of the Neo-Hookean model, which uses a power series expansion of the principal stretches to define the SEF. The Ogden model can accommodate different types of nonlinearities and anisotropies.

Where λi are the deviatoric principal stretches λi = J-1/3 λi ; λi are the principal stretches; *N* is a material parameter; and α_{i}, μ_{i }and D_{i} are temperature-dependent material parameters. The initial shear modulus and bulk modulus for the Ogden form are given by:

The hyperelastic constitutive relations of this model make it highly versatile and applicable to a wide range of scenarios. It demonstrates accurate results for the entire strain range of rubber materials, making it particularly suitable for handling large strain problems. Even at extremely high strains, such as 700%, a desired level of accuracy can be achieved when the model is set to N = 3 or higher. Additionally, this model is capable of describing non-constant shear modulus and slightly compressed material behavior. It effectively captures the rapid increase in stiffness during the later stages of deformation. However, it’s important to note that the material parameters obtained from one type of experiment cannot be reliably used to predict outcomes in another type of deformation. As a precaution, it is not advisable to rely on this model when there is insufficient experimental data, such as when only uniaxial tensile test data is available.

### Yeoh:

This is a polynomial form of SEF, which uses a series expansion of the first invariant of the left Cauchy-Green deformation tensor. The Yeoh model can capture the softening effect at large strains.

Where *U* is the strain energy per unit of reference volume; C_{i0} and D_{i} are temperature-dependent material parameters; I_{1} is the first deviatoric strain invariant defined as:

Where the deviatoric stretches λ_{i }= J^{-1/3} λ_{i }; *J* is the total volume ratio; J* ^{el }*is the elastic volume ratio and λ

_{i}are the principal stretches. The initial shear modulus and bulk modulus are given by:

Yeoh and Mooney-Rivlin are both part of the Polynomial model family and have similarities. Yeoh, however, is considered simpler than Mooney-Rivlin since it does not take into account the second invariant I_{2}. Nonetheless, the volumetric aspects of Yeoh are more intricate than those of Mooney-Rivlin.

The Yeoh model has several advantages. It is simple and only requires a few parameters, making it easy for the user to obtain reasonable numerical results with minimal experimental data, specifically from a tensile experiment. Additionally, the Yeoh model can accurately describe a wide range of deformation, including large uniaxial stretching and simple shear deformation. It is also capable of representing an inverse S-shape stress-strain curve and simulating the sharp increase in material stiffness during later stages of deformation, making it suitable for modeling carbon black filled rubber with non-constant shear modulus.

However, there are some drawbacks to consider when using the Yeoh model. It may show deviations when predicting the deformation under biaxial tension, potentially affecting the accuracy of the results. Additionally, when dealing with complex deformations that involve different types of strains, the Yeoh model may also exhibit inaccuracies. Furthermore, when applied to small strain deformations, it is important to carefully choose the model parameters as there may be a deviation between the Yeoh model and experimental data. However, in finite element analysis, this deviations is not significant as the stress in the small strain region is low, resulting in a small absolute error even if the relative error is relatively large.

## Thermal expansion:

Only isotropic thermal expansion is permitted with the hyperelastic material model. The elastic volume ratio, J^{el}, relates the total volume ratio, *J*, and the thermal volume ratio, J^{th}:

J^{th} is given by

To model hyperelastic materials in ABAQUS, we need to define the material properties and assign them to a section. ABAQUS provides several built-in SEF models, such as Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh, etc. We can also define our own SEF using a user subroutine UHYPER. To define a hyperelastic material in ABAQUS, we need to specify:

– The type of SEF (e.g., NEO HOOKE, MOONEY RIVLIN, OGDEN, YEOH, etc.)

– The material constants (e.g., C10, C01, D1, D2, etc.)

– The reference temperature (optional)

– The density (optional)

## Comparison of Mooney-Rivlin and Neo-Hookean models under uniaxial tensile conditions:

For the case of an incompressible Mooney–Rivlin material under uniaxial elongation, **λ _{1 }= λ_{2}** and

**λ**. Then the true stress (Cauchy stress) differences can be calculated as:

_{2}= λ_{3}= 1/√λIn the case of simple tension, **σ _{22 }= σ_{33 }= 0**, then we can write:

In alternative notation, where the Cauchy stress is written as **T** and the stretch as **α** we can write:

And also, for incompressible neo-hookean materials under uniaxial extension, **λ _{1 }= λ_{2}** and

**λ**, therefore:

_{2}= λ_{3}= 1/√λAssuming no traction on the sides, **σ _{22 }= σ_{33 }= 0**, so we can write:

Where **ε**_{11}**=λ-1** is the engineering strain. This equation is often written in alternative notation as:

To compare the above models and formulations, fig.2 could be considered.

## Defining hyperelastic behavior using test data:

One way to define a hyperelastic material conveniently in Abaqus is by providing experimental data. By utilizing a least squares method, Abaqus can calculate the necessary constants. Abaqus is capable of fitting data from various experimental tests, such as:

- Uniaxial tension and compression
- Equibiaxial tension and compression
- Planar tension and compression (pure shear)
- Volumetric tension and compression

Unlike plasticity data, the required test data for hyperelastic materials in Abaqus should be provided as nominal stress and nominal strain values. Regarding volumetric compression data, it is only necessary to input this information if the material’s compressibility plays a crucial role. Typically, in Abaqus/Standard, the default behavior assumes complete incompressibility. However, in Abaqus/Explicit, a slight level of compressibility is assumed if volumetric test data is not specifically provided.

The accuracy and effectiveness of simulation results using hyperelastic materials in Abaqus largely rely on the quality of the material test data provided. To ensure the best possible calculation of material parameters, there are several measures you can take.

It is recommended to acquire experimental test data from multiple deformation states whenever possible. This enables Abaqus to create a more precise and stable material model. For incompressible materials, the following tests are considered equivalent:

- Uniaxial tension ↔ Equibiaxial compression
- Uniaxial compression ↔ Equibiaxial tension
- Planar tension ↔ planar compression

If you have obtained data from one of these tests that models a specific deformation mode, there is no need to include data from another test. Additionally, it is recommended to gather test data specific to the deformation modes that will occur in your simulation. For instance, if your component experiences compression, it is important to include compressive test data rather than focusing solely on tensile loading. Both tension and compression data should be considered, with compressive stresses and strains represented as negative values. Whenever possible, use compression or tension data that directly corresponds to your application, as trying to fit a single material model to both tensile and compressive data may result in less accurate results compared to individual tests.

Incorporating test data from planar tests measuring shear behavior can greatly improve the accuracy of your hyperelastic material model.

To further enhance your model, ensure that you provide ample test data at strain levels that your material is expected to experience during the simulation. For example, if your material will only undergo small tensile strains below 50%, it is unnecessary to provide extensive test data at higher strain values over 100%.

Utilize the material evaluation functionality available in Abaqus/CAE to conduct simulations of the experimental tests and compare the calculated results with the actual experimental data. If the computational results prove to be poor for a particular deformation mode that is critical for your application, it is advised to obtain additional experimental data specific to that particular deformation mode.