  The mechanical response of solids depends on temperature because the way
atoms and molecules respond collectively to deformation is affected at various
levels by thermal motion. This is a fundamental problem of solid state science
and plays a crucial role in metallurgy, aerospace engineering, energy. In
glasses the vanishing of rigidity upon increasing temperature is the reverse
process of the glass transition. It remains poorly understood due to the
disorder leading to nontrivial (nonaffine) components in the atomic
displacements. Our theory explains the basic mechanism of the melting
transition of amorphous (disordered) solids in terms of the lattice energy lost
to this nonaffine motion, compared to which thermal vibrations turn out to play
only a negligible role. It predicts the square-root vanishing of the shear
modulus $G\sim\sqrt{T_{c}-T}$ at criticality observed in the most recent
numerical simulation study. The theory is also in good agreement with classic
data on melting of amorphous polymers (for which no alternative theory can be
found in the literature) and offers new opportunities in materials science.
