Conditions for a maximum or minimum of Poisson’s ratio of
anisotropic elastic materials are derived. For a uniaxial stress in the
-direction and Poisson’s
ratio
defined by the
contraction in the
-direction,
the following three quantities vanish at a stationary value:
,
and
,
where
are the components of the compliance tensor. Analogous conditions for stationary
values of Young’s modulus and the shear modulus are obtained, along with second
derivatives of the three engineering moduli at the stationary values. The stationary
conditions and the hessian matrices are presented in forms that are independent of
the coordinates, which lead to simple search algorithms for extreme values. In each
case the global extremes can be found by a simple search over the stretch direction
only.
Simplifications for stretch directions in a plane of orthotropic symmetry are also
presented, along with numerical examples for the extreme values of the three
engineering constants in crystals of monoclinic symmetry.
