On natural strain measures of the non-linear micropolar continuum

We discuss three different ways of defining the strain measures in the non-linear micropolar continuum: (a) by a direct geometric approach, (b) considering the strain measures as the fields required by the structure of local equilibrium conditions, and (c) requiring the strain energy density of the polar-elastic body to satisfy the principle of invariance under superposed rigid-body deformations. The geometric approach (a) generates several two-point deformation measures as well as some Lagrangian and Eulerian strain measures. The ways (b) and (c) allow one to choose those Lagrangian strain measures which satisfy the additional mechanical requirements. These uniquely selected relative strain measures are called the natural ones. All the strain measures discussed here are formulated in the general coordinate-free form. They are valid for unrestricted translations, stretches and changes of orientations of the micropolar body, and are required to identically vanish in the absence of deformation. The relation of the Lagrangian stretch and wryness tensors derived here to the ones proposed in the literature is thoroughly discussed.