A continuum-mechanics interpretation of Reissner's non-linear shear-deformable beam theory

This article deals with the non-linear modelling of beams that are bent, sheared and stretched by external forces and moments. In the following, we restrict to plane-deformations and static conditions. Our task is to present a continuum mechanics-based interpretation of the celebrated large displacement finite deformation structural mechanics theory, which was presented by Eric Reissner [On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. The latter formulation was restricted to the notions of structural mechanics and thus did not use the notions of stress and strain, which are fundamental for continuum mechanics. Thus, the common continuum mechanics-based constitutive modelling at the stress–strain level cannot be utilized in connection with Reissner's original theory. Instead, Reissner suggested that constitutive relations between certain generalized strains (bending, shear and axial force strains) and generalized static entities (bending moments, shear and normal forces) should be evaluated from physical experiments. This means that the beam to be studied must be first built up, and the experiments must be performed for the real beam as a whole. Although such physical experiments are indeed to be performed in practice for safety reasons in sensible cases, for example, bridge decks or aircraft wings, it is nevertheless felt to be a drawback that the results of simple standardized stress–strain experiments concerning the constitutive behaviour of the materials, from which the beam is built up, cannot be used. Moreover, relying only on physical experiments on the whole beam means that computations (virtual experiments) can be made only after the beam has been built up. To overcome this problem, we subsequently present a continuum mechanics-based interpretation of Reissner's structural mechanics modelling, by attaching a proper continuum mechanics-based meaning to both the generalized static entities and the generalized strains in Reissner's theory [E. Reissner, On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. Consequently, these generalized static entities can be related to the generalized strains on the basis of a constitutive modelling on the stress–strain level. We show this in some detail in this contribution for a hyperelastic material proposed by Simo and Hughes [Computational Inelasticity, Springer, New York, 1998]. An illustrative numerical example is given which shows the results of large bending and axial deformation behaviour for different constitutive relations. This article represents an extended version of a preliminary work published in [H. Irschik and J. Gerstmayr, A hyperelastic Reissner-type model for non-linear shear deformable beams, Proceedings of the Mathmod 09 Vienna, I. Troch and F. Breitenecker, eds., 2009, pp. 1–7].