Asymptotic estimates for the torsional rigidity of rods with thinning cross sections

We study the torsional rigidity for rods with thinning cross sections. The main purpose of the paper is to prove rigorous asymptotic formulas for the torsional rigidity as the thickness of the cross section tends to zero. These asymptotic formulas are empirically known and are widely used in the field of Mechanics of Materials. From a more theoretical point of view, thinning domains are considered when studying optimal inequalities for suitable classes of functionals depending on domains. We recall as an example of this kind of inequalities the celebrated Saint Venant inequality stating that, among planar domains with fixed Lebesgue measure, the disk is the cross section corresponding to a maximal torsional rigidity. It is well known that this statement is equivalent to say that disks are maximizer of a suitable functional, see for example (Amato et al. in On the optimal sets in Pólya and Makai type inequalities, 2025) and the references therein. Actually, in the present paper other kinds of functionals are more relevant when considering thinning domains. We refer in particular to the so-called Pólya and Makai functionals, see the papers (Makai in On the principal frequency of a membrane and the torsional rigidity of a beam, Stanford Univ. Press, Stanford, 1962; Pólya, J Indian Math Soc (N.S.) 24(1960):413–419, 1961; Pólya and Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951) for more details and (1.14) in the present paper for the precise definitions.