Continuum Mechanics and Empirical Models of Rubber Elasticity

3 Continuum Mechanics and Empirical Models of Rubber Elasticity 

An earlier and different approach from the molecular models of rubber elasticity presented so far is based on a phenomenological model and continuum mechanics. It considers the elastic energy stored in the system. The work done on the system must be stored as elastic energy W and given by 

The argument that continuum mechanics makes is that for an isotropic system, the scalar variable W must depend on scalar variables only (unlike the a/s that depend on directions). The scalar variables are called strain invariants and denoted by //s. The strain energy is then written in terms of three invariants as 

However, = (V7Vu) = 1 for incompressible materials and is not a variable. A general expression that one can then write for W in terms of and 4 can take the form of a power series  

The force is the derivative of the energy, and for the simple case of uniaxial extension where only a, is nonzero and a = al = 1/ai, from symmetry and volume conservation, one obtains 

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