An essay on deformation measures in isotropic thin shell theories. Bending versus curvature

It has become commonplace for the stored energy function of any realistic
shell model to align ``within first order" with the classical Koiter
membrane-bending (flexural) shell model. In this paper, we assess whether
certain extended Cosserat shell models are consistent with the classical linear
Koiter model. In doing this, we observe that there are numerous reasons why a
modified version of the classical Koiter model should be considered, a
consensus reached not only by Koiter himself but also by Sanders and Budiansky,
who independently developed the same theory during the same period. To provide
a comprehensive overview of the strain measures employed in our Cosserat shell
models, this paper presents them in a unified manner and compares them with the
strain measures previously utilized in the literature. We show that all our new
strain tensors either generalize (in the case of nonlinear constrained or
unconstrained models) or coincide (in the case of the linear constrained model)
with the strain tensors recognized as the ``best" or those possessing a
well-defined geometric interpretation connected to bending or curvature.