  We derive expressions for the lowest nonlinear elastic constants of amorphous
solids in athermal conditions (up to third order), in terms of the interaction
potential between the constituent particles. The effect of these constants
cannot be disregarded when amorphous solids undergo instabilities like plastic
flow or fracture in the athermal limit; in such situations the elastic response
increases enormously, bringing the system much beyond the linear regime. We
demonstrate that the existing theory of thermal nonlinear elastic constants
converges to our expressions in the limit of zero temperature. We motivate the
calculation by discussing two examples in which these nonlinear elastic
constants play a crucial role in the context of elasto-plasticity of amorphous
solids. The first example is the plasticity-induced memory that is typical to
amorphous solids (giving rise to the Bauschinger effect). The second example is
how to predict the next plastic event from knowledge of the nonlinear elastic
constants. Using the results of this paper we derive a simple differential
equation for the lowest eigenvalue of the Hessian matrix in the external strain
near mechanical instabilities; this equation predicts how the eigenvalue
vanishes at the mechanical instability and the value of the strain where the
mechanical instability takes place.
