  Quasicrystals can carry, in addition to the classical phonon displacement
field, a phason displacement field, which requires a generalized theory of
elasticity. In this paper, the third-order strain invariants (including phason
strain) of icosahedral quasicrystals are determined. They are connected with 20
independent third-order elastic constants. By means of non-linear elasticity,
phason strains with icosahedral irreducible Gamma^4-symmetry can be obtained by
phonon stress, which is impossible in linear elasticity.
