Pure shear axes and elastic strain energy

It is well known that a state of pure shear has distinct setsof basis vectors or coordinate systems: the principal axes,in which the stress is diagonal, and pure shear bases, in whichdiag = 0. The latter is commonly taken as the definition ofpure shear, although a state of pure shear is more generallydefined by tr = 0. New results are presented that characterizeall possible pure shear bases. A pair of vector functions arederived which generate a set of pure shear basis vectors fromany one member of the triad. The vector functions follow froma compatibility condition for the pure shear basis vectors,and are independent of the principal stress values. The complementarytypes of vector basis have implications for the strain energyof linearly elastic solids with cubic material symmetry: fora given state of stress or strain, the strain energy achievesits extreme values when the material cube axes are aligned withprincipal axes of stress or with a pure shear basis. This impliesthat the optimal orientation for a given state of stress iswith one or the other vector basis, depending as the stressis to be minimized or maximized, which involves the sign ofone material parameter.