Computational Shell Mechanics by Helicoidal Modeling, I: Theory

Starting from recently formulated helicoidal modeling in three-dimensional continua, a low-order kinematical model of a solid shell is established. It relies on both the six degrees of freedom (DOFs) on the reference surface, including the drilling DOF, and a dual director - six additional DOFs - that controls the relative rototranslation of the material particles within the thickness. Since the formulation pertains to the framework of the micropolar mechanics, the solid shell mechanical model includes a workless stress variable - the axial vector of the Biot stress tensor, referred to as the Biot-axial - that allows us to handle nonpolar materials. The local Biot-axial is approximated with a linear field across the thickness and relies on two vector parameters. On the reference surface, the dual director is condensed locally together with one Biot-axial parameter, leaving the surface strains and the other Biot - axial parameter as the basic variables governing the two-dimensional internal work functional. The continuum-based shell mechanics are cast in weak incremental form from the beginning. They yield the two-dimensional nonlinear constitutive law of the shell in incremental form, built dynamically along the solution process. Poisson thickness locking, related to the low-order kinematical model, is prevented by a dynamical adaptation of the local constitutive law. No hypotheses are introduced that restrict the amplitudes of displacements, rotations, and strains, so the formulation is suitable for computations with strong geometrical and material nonlinearities, as shown in Part II.