  A general nonlinear theory for the elasticity of pre-stressed single crystals
is presented. Various types of elastic moduli are defined, their importance is
determined, and relationships between them are presented. In particular, B
moduli are present in the relationship between the Jaumann objective time
derivative of the Cauchy stress and deformation rate and are broadly used in
computational algorithms in various finite-element codes. Possible applications
to simplified linear solutions for complex nonlinear elasticity problems are
outlined and illustrated for a superdislocation. The effect of finite rotations
is fully taken into account and analyzed. Different types of the bulk and shear
moduli under different constraints are defined and connected to the effective
properties of polycrystalline aggregates. Expressions for elastic energy and
stress-strain relationships for small distortions with respect to a
pre-stressed configuration are derived in detail. Under initial hydrostatic
load, general consistency conditions for elastic moduli and compliances are
derived that follow from the existence of the generalized tensorial equation of
state under hydrostatic loading obtained from single crystal or polycrystal. It
is shown that B moduli can be found from the expression for the Gibbs energy.
However, higher-order elastic moduli defined from the Gibbs energy do not have
any meaning since they do not directly participate in any known equations, like
stress-strain relationships and wave propagation equation. The deviatoric
projection of B can also be found from the expression for the elastic energy
for isochoric small strain increments, and the missing components of B can be
found from the consistency conditions. Numerous inconsistencies and errors in
the known works are analyzed.
