Yeoh (hyperelastic model)

The Yeoh hyperelastic material model is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber. The model is based on Ronald Rivlin's observation that the elastic properties of rubber may be described using a strain energy density function which is a power series in the strain invariants formula_1 of the Cauchy-Green deformation tensors. The Yeoh model for incompressible rubber is a function only of formula_2. For compressible rubbers, a dependence on formula_3 is added on. Since a polynomial form of the strain energy density function is used but all the three invariants of the left Cauchy-Green deformation tensor are not, the Yeoh model is also called the reduced polynomial model.
Yeoh model for incompressible rubbers.
Strain energy density function.
The original model proposed by Yeoh had a cubic form with only formula_2 dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as
where formula_6 are material constants. The quantity formula_7 can be interpreted as the initial shear modulus.
Today a slightly more generalized version of the Yeoh model is used. This model includes formula_8 terms and is written as
When formula_10 the Yeoh model reduces to the neo-Hookean model for incompressible materials.
For consistency with linear elasticity the Yeoh model has to satisfy the condition
where formula_12 is the shear modulus of the material.
Now, at formula_13, 
Therefore, the consistency condition for the Yeoh model is
Stress-deformation relations.
The Cauchy stress for the incompressible Yeoh model is given by
Uniaxial extension.
For uniaxial extension in the formula_17-direction, the principal stretches are formula_18. From incompressibility formula_19. Hence formula_20. 
Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
Since formula_24, we have
Therefore,
The engineering strain is formula_27. The engineering stress is 
Equibiaxial extension.
For equibiaxial extension in the formula_17 and formula_30 directions, the principal stretches are formula_31. From incompressibility formula_19. Hence formula_33. 
Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
Since formula_37, we have
Therefore,
The engineering strain is formula_27. The engineering stress is 
Planar extension.
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the formula_17 directions with the formula_43 direction constrained, the principal stretches are formula_44. From incompressibility formula_19. Hence formula_46. 
Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
Since formula_50, we have
Therefore,
The engineering strain is formula_27. The engineering stress is 
Yeoh model for compressible rubbers.
A version of the Yeoh model that includes formula_55 dependence is used for compressible rubbers. The strain energy density function for this model is written as
where formula_57, and formula_58 are material constants. The quantity formula_59 is interpreted as half the initial shear modulus, while formula_60 is interpreted as half the initial bulk modulus.
When formula_10 the compressible Yeoh model reduces to the neo-Hookean model for incompressible materials.