  We show that near a second order phase transition in a two-component elastic
medium of size L in two dimensions, where the local elastic deformation-order
parameter couplings can break the inversion symmetry of the order parameter,
the elastic modulii diverges with the variance of the local displacement
fluctuations scaling as $[ln(L/a_0)]^{2/3}$ and the local displacement
correlation function scaling as $[ln(r/a_0)]^{2/3}$ for weak
inversion-asymmetryThe elastic constants can also vanish for system size
exceeding a non-universal value, making the system unstable for strong
asymmetry, where a 0 is a small-scale cut-off. We show that the elastic
deformation-order parameter couplings can make the phase transition first
order, when the elastic modulii do not diverge, but shows a jump proportional
to the jump in the order parameter, across the transition temperature. For a
bulk system, the elastic stiffness does not diverge for weak asymmetry, but can
vanish across a second order transition giving instability for strong
asymmetry, or displays jumps across a first order transition. In-vitro
experiments on binary fluids embedded in a polymerized network, magnetic
colloidal crystals or magnetic crystals could test these predictions.
