Consequences of different definitions of bending curvature on nonlinear dynamics of beams

Abstract

Beam theories may be grouped in two broad categories, namely induced or intrinsic theories. In the former, beam models are obtained as exact consequences of three-dimensional theory, making use either of asymptotic expansions in a slenderness parameter or projections of three-dimensional elasticity on certain function spaces, while beams are inherently one-dimensional bodies in the latter category. Although induced theories show a clear connection between three- and one-dimensional representations, they are often more demanding with respect to intrinsic ones, in which a finite number of strain parameters, depending on just one space variable, characterizes the motion of beams in an internally consistent way and without a direct linkage to three-dimensional material properties. Hence, as a consequence, intrinsic theories do not provide any structure for constitutive equations and, at least in principle, different choices can be allowed. A typical example of this fact is represented by the one-dimensional relationship between the bending moment and the beam curvature, since for this latter two notions are admissible. Indeed, both are adopted in the literature and, apparently, preferring one to the other is only related to the predictive capability of the ensuing model. The arising question is about possible differences in both static and dynamic responses of beams, when one or the other definition of curvature is selected.