Finite bending of non-slender beams and the limitations of the Elastica theory. (15th May 2022)

• Record Type: Journal Article
• Authors: Falope, Federico Oyedeji; Lanzoni, Luca; Tarantino, Angelo Marcello
• Is Part Of: International journal of mechanical sciences. Volume 222(2022)
• Publication Date: 2022-05-15
• Subjects: Pure bending -- Moment–curvature relation -- Finite elasticity -- Elastica theory -- Neutral loci; Mechanical engineering -- Periodicals; Génie mécanique -- Périodiques; Mechanical engineering; Maschinenbau; Mechanik; Zeitschrift; Periodicals; 621.05
• DOI: 10.1016/j.ijmecsci.2022.107187
• Languages: English
• ISSNs: 0020-7403

Abstract

Abstract: The problem of slender solids under finite bending has been addressed recently in Lanzoni and Tarantino (2018). In the present work, such a model is extended to short solids by improving the background formulation. In particular, the model is refined by imposing the vanishing of the axial force over the cross sections. The geometrical neutral loci, corresponding to unstretched and unstressed surfaces, are provided in closed form. Two approximations of the models are obtained linearising both the kinematics and the constitutive law. It is shown that the approximations of the model, corresponding to the Euler Elastica formulation, can lead to significant values of the axial stress resultants despite pure bending conditions. For a generic form of compressible energy function, a nonlinear moment–curvature relation accounting for both material and geometric nonlinearities is provided and then specialised for a Mooney–Rivlin material. The obtained results are compared with simulations of 3D finite element models finding good agreement. The normalisation of the moment–curvature relation provides the dimensionless bending moment as a function of the Eulerian slenderness of the solid. This dimensionless relation is shown to be valid for any aspect ratio of the bent solid and, in turn, it highlights the limitations of the Elastica arising in case of large deformations. Graphical abstract: Highlights: Analytical model of 3D bending of a compressible hyperelastic solids is refined. Bending moment–curvature relation for compressible hyperelastic solids is obtained. Dimensionless bending moment vs Eulerian slenderness curve is valid for any solid. Linearisation of the analytic model provides the Euler Elastica formulation. Limitations of Euler Elastica are shown for the cases of fully nonlinear bending.