Finite Anticlastic Bending of Hyperelastic Solids and Beams

Abstract

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.

Mechanics of High-Flexible Beams Under Live Loads

“…In [27], it has been shown that, for the points belonging to longitudinal basic line X = 0, Y = 0 and Z = Z, the equilibrium equations are exactly satisfied. For these points, the displacement field is correct being it kinematically compatible and equilibrated.…”

“…The equilibrium problem of inflexed hyperelastic blocks and short beams has been investigated by Lanzoni and Tarantino [27] in the fully nonlinear context of the finite elasticity. Following a semi-inverse approach, a three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal anticlastic deformation of cross-sections, has been formulated.…”

“…This kinematic model is quite different from the others existing in the Literature, where the bending of a solid was essentially formulated for a two-dimensional context, neglecting systematically the deformation of the cross sections [28]. In [27], explicit formulae, which describe the displacement field, stretches and stresses at each point of the beam using both the Lagrangian and Eulerian descriptions, have been obtained.…”

“…The beam model proposed in [27] for the uniform inflexion was then developed to study the general case with variable curvature [1]. Such generalization allowed to derive a completely nonlinear moment-curvature relationship for the beam axis.…”

FE Analyses of Hyperelastic Solids under Large Bending: The Role of the Searle Parameter and Eulerian Slenderness

“…Recently, the Rivlin formulation [20] has been extended to a 3D framework by taking into account also the anticlastic deformation arising in the transverse cross sections of the solid subjected to uniform bending [23]. Later, the analysis has been extended to beams subjected to variable bending moment [24,25].…”