Nonlinear Hyperelastic Materials: From Experimental Data to Interactive Material Modeling

Why linear elasticity quickly becomes insufficient

Rubber-like materials can look deceptively simple. Pull an elastomeric component and it stretches; release it and, ideally, it returns to its original shape. Mechanically, however, this behavior is very different from the small-strain elasticity commonly used for metals and many structural materials.

For an elastic metal operating below yield, a relationship such as

may describe the response sufficiently well over a limited strain range. For an elastomer undergoing 50%, 100%, or several hundred percent strain, this assumption is no longer appropriate. Geometry changes substantially, the stress-strain response becomes strongly nonlinear, and deformation in one direction is coupled to deformation in the transverse directions.

This is where hyperelastic material models become useful.

From a stress-strain line to a strain-energy function

A hyperelastic material is described through a strain-energy density function. In terms of the principal stretches:

The stresses are then obtained from derivatives of this stored energy. Rather than defining only one nonlinear stress-strain curve, the constitutive law describes how the material stores energy for a complete deformation state.

This framework is particularly useful for elastomers, rubber seals, soft polymers, polymer foams and other materials capable of large approximately reversible deformation.

Why one tensile test is often not enough

A hyperelastic law must describe more than a single uniaxial loading path. Common calibration modes include uniaxial tension, equibiaxial tension, planar tension / pure shear, simple shear and—when compressibility matters—volumetric deformation.

For compressible materials, the volume ratio J becomes another independent part of the constitutive response.

The constitutive model changes the prediction

There is no universal hyperelastic law that is best for every material and every deformation range. Simpler models require fewer coefficients; richer models can reproduce more complex nonlinear behavior but demand sufficient experimental information to identify their parameters reliably.

Neo-Hookean

A useful baseline with one deviatoric stiffness parameter.

Mooney–Rivlin

Introduces dependence on both the first and second deviatoric strain invariants.

Ogden

For the one-term Abaqus-style normalization used by the interactive tool:

The parameter μ governs the initial shear stiffness while α strongly influences the nonlinear shape.

Compressibility should not be treated as a secondary detail

Dense elastomers are frequently modeled as nearly incompressible:

For many applications this is a good approximation. But it is not appropriate for every rubber-like material. Foams, for example, can undergo substantial volume change because of their cellular structure.

For several conventional Abaqus hyperelastic models, compressibility is introduced through a separate volumetric parameter such as D1:

Ogden Hyperfoam is different

Hyperfoam is inherently compressible and couples volumetric and distortional response in the same potential. For the one-term formulation implemented in the tool:

This makes Hyperfoam especially relevant when large volume changes are physically expected rather than treated as a small numerical correction.

Where hyperelasticity appears in real products

For a sealing problem, the engineering question is rarely just “how much does it move?” Contact pressure, local strain, compressibility, friction, temperature and extrapolation of the constitutive law may all influence the result.

What can the experiment actually identify?

More fitting coefficients do not automatically mean more information about the material. For incompressible Mooney–Rivlin behavior in simple shear:

and therefore

Simple-shear data can identify the sum C10 + C01, but not the two coefficients independently. Planar deformation has a similar limitation. Additional independent deformation modes such as uniaxial or biaxial data are needed to separate them.

Making nonlinear constitutive behavior easier to “feel”

Hyperelastic laws are often presented as equations and coefficient tables. An engineer may receive a parameter set such as

or

without immediately developing an intuitive understanding of how those values change the material response.

That motivated the development of MakeCoTech Hyperelastic Explorer v1.0: an interactive engineering environment in which the constitutive law can be explored before geometry, contact, meshing and solver settings are added to the problem.

Checking the fitting workflow against Abaqus examples

The tool provides two examples with distinct purposes:

The Hyperfoam example uses generated model data. It demonstrates the fitting workflow and requires separate experimental or solver data for external validation.

A good fit is not the end of the engineering work

A parameter set can reproduce measured data closely and still behave poorly when extrapolated. Pure hyperelasticity also does not automatically represent rate dependence, hysteresis, Mullins softening, permanent deformation, temperature dependence or damage.

From equations to engineering intuition

The motivation behind the tool is to move from

toward a more transparent workflow:

When the material law is explored independently, it becomes easier to ask the questions that matter: Why do two fitted models diverge in another loading mode? Is a parameter actually identifiable? How much volume change is predicted? What happens beyond the experimental range?

Benchmark and formulation references