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.
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