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:

;; = ; + 2;;