Onset of Non-Linearity in the Elastic Bending of Blocks

TL;DR

This paper analyzes the non-linear elastic bending of blocks, deriving a moment-angle relationship dependent on a key non-dimensional parameter, and proposes experimental methods to estimate non-linear shear coefficients.

Contribution

It introduces a novel approach to relate bending moment and angle in non-linear elasticity, highlighting the dependence on a single non-dimensional parameter and connecting theoretical coefficients to experimental measurements.

Findings

Derived a general moment-bending angle relationship dependent on one non-dimensional parameter.

Found a Maclaurin series expansion revealing the influence of shear modulus and non-linear shear coefficient.

Presented experimental results for estimating non-linear shear coefficients.

Abstract

The classical flexure problem of non-linear incompressible elasticity is revisited assuming that the bending angle suffered by the block is specified instead of the usual applied moment. The general moment-bending angle relationship is then obtained and is shown to be dependent on only one non-dimensional parameter: the product of the aspect ratio of the block and the bending angle. A Maclaurin series expansion in this parameter is then found. The first-order term is proportional to $\mu$, the shear modulus of linear elasticity; the second-order term is identically zero, because the moment is an odd function of the angle; and the third-order term is proportional to $\mu(4\beta -1)$, where $\beta$ is the non-linear shear coefficient, involving third-order and fourth-order elasticity constants. It follows that bending experiments provide an alternative way of estimating this coefficient, and the results of one such experiment are presented. In passing, the coefficients of Rivlin's expansion in exact non-linear elasticity are connected to those of Landau in weakly (fourth-order) non-linear elasticity.