Consider the case of a nonlinear elastic material, and extend Exercise 6.8 for the case where the strain energy depends on three invariants of the strain tensor U = U(I_1e, I_2e, I_3e), where:

Note that this new choice of invariants does not change any fundamental aspects of the problem.
Show that using the form σ_ij = ∂U/∂e_ij and employing the chain rule, yields the nonlinear relation σ_ij = ∅₁δ_ij + ∅₂e_ij + ∅₃e_ike_jk where ∅_i = ∅_i(I_je) = ∂U/∂I_ie. This type of theory is often referred to as physically nonlinear elasticity in that the constitutive form retains the small deformation strain tensor but includes higher order nonlinear behavior in the constitutive law; see Evans and Pister (1966) and Bharatha and Levinson (1977).

Data from exercise 6.8 

In light of Exercise 6.2, consider the formulation where the strain energy is assumed to be a function of the two invariants U = U (I_e, II_e). Show that using the relation σ_ij = ∂U/∂e_ij and employing the chain rule, yields the expected constitutive law (4.2.7).

Equation 4.2 .7

Data from exercise 6.2 

Since the strain energy has physical meaning that is independent of the choice of coordinate axes, it must be invariant to all coordinate transformations. Because U is a quadratic form in the strains or stresses, it cannot depend on the third invariants III_e or I₃, and so it must depend only on the first two invariants of the strain or stress tensors. Show that the strain energy can be written in the following forms:

Transcribed Image Text:

Ile = ekk, 12e 1 ==e 2 ekmekm, 13e 1 3 ekmeknemn