  An incremental (or tangent) bulk modulus for finite isotropic elasticity is
defined which compares an increment in hydrostatic pressure with the
corresponding increment in relative volume. Its positivity provides a stringent
criterion for physically reasonable response involving the second derivatives
of the strain energy function. Also, an average (or secant) bulk modulus is
defined by comparing the current stress with the relative volume change. The
positivity of this bulk modulus provides a physically reasonable response
criterion less stringent than the former. The concept of incremental bulk
modulus is extended to anisotropic elasticity. For states of uniaxial tension
an incremental Poisson's ratio and an incremental Young's modulus are similarly
defined for nonlinear isotropic elasticity and have properties similar to those
of the incremental bulk modulus. The incremental Poisson's ratios for the
isotropic constraints of incompressibility, Bell, Ericksen, and constant area
are considered. The incremental moduli are all evaluated for a specific example
of the compressible neo-Hookean solid. Bounds on the ground state Lam\'{e}
elastic moduli, assumed positive, are given which are sufficient to guarantee
the positivity of the incremental bulk and Young's moduli for all strains.
However, although the ground state Poisson's ratio is positive we find that the
incremental Poisson's ratio becomes negative for large enough axial extensions.
