Outlooks in Saint-Venant theory III: torsion and flexure in sections of variable thickness by formal expansions

We study the Saint-Venant shear stress fields arising in a family of sections we call Bredt-like, i.e. in a set of plane regions D_epsilon whose thickness we scale by a parameter epsilon. For each epsilon we build a coordinate mapping from a fixed plane domain cal D onto D_epsilon. The shear stress field in D_epsilon can be represented by a Prandtl-like stress flow function. This is naturally done in torsion (torsion), while in flexure (flexion inégale) we face a gauge choice whose physical interpretation is uncertain. We then consider the Helmholtz operator in a fixed system of coordinates in cal D and represent the shear stress field in a basis field which is not the covariant basis associated to any coordinate system. Formal epsilon-power series expansions for the shear stress field, the warping, the resultant force and torque and the shear shape factors tensor lead to hierarchies of perturbation problems for their coefficients. We obtain all the technical formulae at the lowest iteration steps and their generalization at higher steps-i.e., for thicker sections. No attempt is made to apply the methods proposed to estimate the distance between the generalized formulae we provide and the true solutions for the Saint-Venant shear stress problem.