  It is well known that a state of pure shear has distinct sets of basis
vectors or coordinate systems: the principal axes, in which the stress is
diagonal, and pure shear bases, in which diag(stress)=0. The latter is commonly
taken as the definition of pure shear, although a state of pure shear is more
generally defined by tr(stress)=0. New results are presented that characterize
all possible pure shear bases. A pair of vector functions are derived which
generate a set of pure shear basis vectors from any one member of the triad.
The vector functions follow from compatibility condition for the pure shear
basis vectors, and are independent of the principal stress values. The
complementary types of vector basis have implications for the strain energy of
linearly elastic solids with cubic material symmetry: for a given state of
stress or strain, the strain energy achieves its extreme values when the
material cube axes are aligned with principal axes of stress or with a pure
shear basis. This implies that the optimal orientation for a given state of
stress is with one or the other vector basis, depending as the stress is to be
minimized or maximized, which involves the sign of one material parameter.
