Fundamentals of Continuum Mechanics

Exercises

1. Prove that as a consequence of restriction (8.64), I₃ = det B = 1. (Hint: Start with B = FF^T and use the properties (2.47) of the determinant.)

2. Show that for the special case of an incompressible, isotropic, nonlinear elastic solid, the Lagrangian conservation laws (8.61) become (8.75a) and (8.75b).

8.3.3 Incompressible strain energy models

In this section, we present some representative strain energy models for incompressible, isotropic, rubbery materials. We do so in the context of the mechanical (or isothermal) variant of the theory presented in Section 8.3.2, which is

$\text{det}\mathbf{F}=1,\overline{\rho }\stackrel{..}{\mathbf{x}}=\text{Div}\mathbf{P}+\overline{\rho }\mathbf{b},$

and

$\mathbf{T}=-\overline{p}\mathbf{I}+2\frac{\partial \overline{W}}{\partial I_1}\mathbf{B}-2\frac{\partial \overline{W}}{\partial I_2}{\mathbf{B}}^{-}$