The helicoidal modeling in computational finite elasticity. Part I: Variational formulation

The finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields. The rototranslation between two material particles can be described by a single, complex tensorial quantity, which is recognized to be orthogonal. Its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body. After due introduction to dual tensors and rototranslations, the polar description of the continuum is addressed, with particular care to mixed differentiations of the rototranslation field. Then, a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis, and the three-field, two-field and one-field principles are drawn and linearized. The proposed modeling is expected to be profitably exploited in non-linear finite element analyses of solids undergoing finite displacements, rotations and strains.