Pure shear axes and elastic strain energy

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 σ is diagonal, and pure shear bases, in which diag σ = 0. The latter is commonly taken as the definition of pure shear, although a state of pure shear is more generally defined by tr σ = 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 a 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.