  Strain-based theory on elastic instabilities is being widely employed for
studying onset of plasticity, phase transition or melting in crystals. And size
effects, observed in nano-materials or solids under dynamic loadings, needs to
account for contributions from strain gradient. However, the strain gradient
based higher order elastic theories on the elastic instabilities are not well
established to enable one to predict high order instabilities of solids
directly at atom level. In present work, a general continuum theory for higher
order elastic instabilities is established and justified by developing an
equivalent description at atom level. Our results show that mechanical
instability of solids, triggered by either strain or strain gradient, is
determined by a simple stability condition consisting of strain or strain
gradient related elastic constants. With the atom-level description of the
higher order elasticity, the strain-gradient elastic constants could be
directly obtained by a molecular statics procedure and then serve as inputs of
the stability condition. In this way, mechanical instabilities of three metals,
i.e., copper, aluminum and iron, are predicted. Alternatively, ramp compression
technique by nonequilibrium molecular dynamics (NEMD) simulations is employed
to study the higher order instabilities of the three metals. The predicted
critical strains at onset of instabilities agree well with the results from the
NEMD simulations for all the metals. Since the only inputs for the established
higher order elastic theory are the same as atomic simulations, i.e., atomic
potentials and structures of solids, the established theory is completely
equivalent to empirical-potential based atomic simulations methods, at least,
for crystals.
