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.