Yeoh hyperelastic model

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The Yeoh hyperelastic model is a simple, practical way to describe how rubbery materials stretch. It treats rubber as nearly incompressible (volume stays almost the same) and nonlinear, so its stiffness changes as it stretches.

- The key idea: the model uses a strain energy function that describes how much energy is stored when the material is deformed. For incompressible rubbers, this energy mainly depends on the first deformation invariant I1. For compressible rubbers, there can also be a dependence on I3 (which relates to volume change). Because it uses only some invariants, it’s called a reduced polynomial model.

- Original and generalized forms:
- The original Yeoh model uses a cubic function of I1 with material constants. The combination 2C1 acts like the initial shear modulus.
- A more general version adds up to n terms, so the energy is a sum of several powers of I1. When n = 1, the Yeoh model reduces to the neo-Hookean model for incompressible materials.
- The model is built so that it behaves consistently with linear elasticity at small strains (the small-strain shear modulus is recovered).

- Incompressible vs compressible:
- Incompressible Yeoh: the energy depends on I1 only, giving straightforward formulas for common tests like uniaxial, equibiaxial, and planar extensions.
- Compressible Yeoh: I3 is included, using a modified form that introduces parameters related to the material’s initial shear and bulk moduli. When n = 1, it also reduces to neo-Hookean for incompressible materials.

- How it’s used in experiments:
- Uniaxial extension: the model yields simple relations between stretch and stress.
- Equibiaxial extension and planar extension: similar simplifications apply, with appropriate changes to the stretch directions.

- Practical notes:
- The Yeoh model is popular because it fits rubber data well with just a few parameters and remains stable for moderate strains.
- It’s named after Oon Hock Yeoh, who developed the model.

In short, the Yeoh model is a flexible, easy-to-use way to describe how rubber-like materials deform, balancing accuracy with a small number of material constants.