  A limit elastic energy for pure traction problem is derived from re-scaled
nonlinear energy of an hyperelastic material body subject to an equilibrated
force field. We show that the strains of minimizing sequences associated to
re-scaled non linear energies weakly converge, up to subsequences, to the
strains of minimizers of a limit energy, provided an additional compatibility
condition is fulfilled by the force field. The limit energy is different from
classical energy of linear elasticity; nevertheless the compatibility condition
entails the coincidence of related minima and minimizers. A strong violation of
this condition provides a limit energy which is unbounded from below, while a
mild violation may produce a limit energy with infinitely many extra minimizers
which are not minimizers of standard linear elastic energy and whose strains
are not uniformly bounded. A relevant consequence of this analysis is that a
rigorous validation of linear elasticity fails for compressive force fields
that do not fulfil such compatibility condition.
