  Despite of some progresses in investigating the roles of the higher-order
strain gradients on elastic stabilities of solids, the physical nature on the
higher-order elastic instabilities of crystals, especially under extreme strain
rates, is still a mystery. In this work, a generalized elastic instability
criterion for infinite crystals is consistently established at both continuum
and atom level under frameworks of a higher-order phenomenological theory. The
established criterion could consistently reproduce the well-known strain-based
lattice instability criteria, such as modified Born criterion,
{\Lambda}-criterion, as well as a higher-order one proposed by Bardenhagen et
al. Our results show that modified Born criterion is not as precise as the
{\Lambda}-criterion under heterogeneous stress states. Different from the
higher-order criterion, contributions from the third order gradients of
displacements are considered so that the well-known sign paradox in the first
strain-gradient theory could be reproduced. According to microscopic
comprehensions on the higher-order phenomenological theory, the sign paradox is
well clarified. Finally, the established criterion is employed to investigate
the elastic stabilities of single crystalline copper and aluminum. The obtained
results could well explain the singularities of elastic responses of the two
metals under ramp compressions.
