Abstract: An introduction to some ideas from Continuum Mechanics is given in the context of Nonlinear Elasticity. We will restrict our discussion to the particular case of hyperelastic materials, which are defined in terms of their "elastic energy density", or stored (free) energy. We show that because of a basic constitutive requirement the stored energy function cannot be convex, and thus the minimization problem in this context cannot be solved using the classical tools of the Calculus of Variations. New ideas, and the development of modern analytical techniques are needed to address this and related issues.