  Quasi-static strain-controlled measurements of stress vs strain curves in
macroscopic amorphous solids result in a nonlinear looking curve that ends up
either in mechanical collapse or in a steady-state with fluctuations around a
mean stress that remains constant with increasing strain. It is therefore very
tempting to fit a nonlinear expansion of the stress in powers of the strain. We
argue here that at low temperatures the meaning of such an expansion needs to
be reconsidered. We point out the enormous difference between quenched and
annealed averages of the stress vs. strain curves, and propose that a useful
description of the mechanical response is given by a stress (or strain)
dependent shear modulus for which a theoretical evaluation exists. The elastic
response is piece-wise linear rather than nonlinear.
