## Hyperelastic Materials: A Method for Estimation of the Material Parameters of Hyperelastic Material Models in Relation to the Linear Elastic Material Model

Given a particular form of the strain energy function, we present here a quick method by which the material parameters can be related to the shear modulus and the bulk modulus of a linear elastic material.

### Shear Modulus

The material parameters of a hyperelastic material model can be related to the shear modulus of an elastic material as follows. First, a simple shear state of deformation is assumed:

The engineering shear strain in this case is equal to and . The matrix is equal to:

The first and second invariants of are:

The following are examples of some of the compressible and incompressible material models listed above:

For each of these material models, the corresponding Cauchy stress matrix has the form:

For linear elastic materials, the shear stress component and the engineering shear strain component are related by the relationship: . By investigating the component in the above matrices and by setting , the relationship between the shear modulus and the given material parameters are as follows:

- For material 1:
- For material 2:
- For material 3:
- For material 4:

### Bulk Modulus

The material parameters of a hyperelastic material model can be related to the bulk modulus of an elastic material as follows. First, a spherical state of deformation is assumed:

. The matrix is equal to:

The first and second invariants of are:

The following are examples of some of the compressible material models listed above:

For each of these material models, the Cauchy stress has the form:

For linear elastic materials, the hydrostatic stress component is related to the engineering volumetric strain by the relationship . In the deformation described in this problem, the volumetric strain is equal to , therefore, an estimate of the material constant for the above materials is given as: