The bending-stretching effect of a nonlinear elastic beam under finite deformation

Abstract

Second-order solutions for the pure bending of a beam under finite deformation are obtained in this paper. The equilibrium equations are solved analytically for the second-order displacements, from which the first Piola–Kirchhoff stresses, the curvatures and a bending-stretching parameter are calculated for a cross section with a transverse axis of symmetry. The displacements are dependent on the square of the inverse of elastic constant, whereas the stresses on the inverse of elastic constant. The curvatures and anticlastic curvatures are gradated in the transverse direction. The longitudinal stress is nonlinearly distributed across the section, and exhibits anomalous characteristics in materials with a negative Poisson's ratio. The shear stresses have vanishing force and moment resultants, and bending-twisting coupling does not occur. The normal stresses give rise to force-couple resultants. The longitudinal force resultant representing the bending-stretching effect is dependent on a reduced elastic parameter ξ. This effect is present as long as there is finite deformation, even if the material is linearly elastic. A small shear modulus μ and/or a large third-order elastic constant l, m or n will result in a large ξ. This study may be applicable to the elasticity design of soft medical and wearable robots, where elastic compatibility with tissues is a primary consideration.