Merlini T., Morandini M. (2011)

Computational shell mechanics by helicoidal modeling, I: Theory

**Abstract** - 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.

**Key words** -
Nonlinear shell theory.
Micropolar shell variational mechanics. Helicoidal modeling. Geometric
invariance. Shell constitutive equations. Finite rotations and rototranslations.
Dual tensor algebra.

Journal
of Mechanics of Materials and Structures 6/5, 659-692.