Application of the Invariant Approach

In this section, we give examples of applications using the approach to the strain energy function listed as (1) in Section 3.4.2. We shall restrict the discussion to incompressible materials, so that only the invariants 7i and h will be of concern. 

Here C is a material constant and the 3 ensures that the strain energy is zero at zero strain. This form is known as the neo-Hookean or Gaussian model. It can be derived for rubbers using a physical argument based on thermodynamics [9], and will be discussed further in Chapter 4. In many cases, a more complex approach is required. 

In a classic work of Rivlin and Saunders [10] on vulcanised rubber, a strain energy function of the form U h, h) was introduced. Experiments were performed on sheets of material in plane stress with the in direction normal to the plane. Homogeneous deformations were applied using a biaxial stretching machine so that the principal extension ratios 7-1 and kii were under experimental control. Incompressibility applies and Equations (3.55) give the principal stresses 

The incompressibility condition (3.30) leads to a simplified expression for h derived from Equation (3.57)  

From this equation and Equation (3.56), expressions for the derivatives are obtained  

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