  Elastic constants of zero-temperature amorphous solids are given as the
difference between the Born term, which results from a hypothetical affine
deformation of an amorphous solid, and a correction term which originates from
the fact that the deformation of an amorphous solid due to an applied stress
is, at the microscopic level, non-affine. Both terms are non-negative and thus
it is a priori not obvious that the resulting elastic constants are
non-negative. In particular, theories that approximate the correction term may
spuriously predict negative elastic constants and thus an instability of an
amorphous solid. Here we derive alternative expressions for elastic constants
of zero-temperature amorphous solids that are explicitly non-negative. These
expressions provide a useful blueprint for approximate theories for elastic
constants and sound damping in zero temperature amorphous solids.
