Nonuniform bending theory of hyperelastic beams in finite elasticity

Abstract

This paper deals with the equilibrium problem of slender beams inexed under variable curvature in the framework of fully nonlinear elasticity. For the specic case of uniform exion, the authors have recently proposed a mathematical model. In that analysis, the complete threedimensional kinematics of the beam is taken into account, both deformations and displacements are considered large and a compressible Mooney-Rivlin law is assumed for the stored energy function. In the present paper, the kinematics of the aforementioned model has been reformulated taking into account beams under variable curvature. Subsequently, focusing on the local determination of the curvature, new equilibrium conditions on cross sections are introduced in the mathematical formulation. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically through an iterative procedure. Thus, for the generic class of hyperelastic and isotropic materials, explicit formulae for the displacement eld, the stretches and stresses in every point of the beam, following both Lagrangian and Eulerian descriptions, are derived. The analysis developed allows to study a very wide class of equilibrium problems for nonlinear beams under dierent restraint conditions and subject to generic external load systems. By way of example, the Euler beam has been considered and the formulae obtained have been specialized for a specic neoprene rubber material, the constitutive constants of which are determined experimentally. The shapes assumed by the beam as the load multiplier increases are shown through some graphs. The distributions of stretches and Cauchy stresses are plotted for the most stressed cross section. Some comparisons were made using a Finite Element code. In addition, the accuracy of the solution obtained is estimated by evaluating a posteriori that the equilibrium equations are locally satised.

Nonlinear Free and Forced Vibrations of a Hyperelastic Micro/Nanobeam Considering Strain Stiffening Effect

“…Bacciocchi and Tarantino [44] conducted a finite anticlastic bending analysis of hyperelastic beams using two hyperelastic models, namely Mooney-Rivlin and Saint Venant-Kirchhoff. Lanzoni and coworker [45] studied the nonuniform bending of a beam made of the hyperelastic beam, taking the Mooney-Rivlin into account. The large deformation response of hyperelastic beams was explored by Dadgar-Rad and Firouzi [46].…”