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Research Papers

# Onset of Nonlinearity in the Elastic Bending of Blocks

[+] Author and Article Information

School of Electrical, Electronic, and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Irelandmichel.destrade@ucd.ie

M. D. Gilchrist

School of Electrical, Electronic, and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Ireland

J. G. Murphy

Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland

1

Corresponding author.

J. Appl. Mech 77(6), 061015 (Sep 01, 2010) (6 pages) doi:10.1115/1.4001282 History: Received March 06, 2009; Revised January 21, 2010; Published September 01, 2010; Online September 01, 2010

## Abstract

The classical flexure problem of nonlinear 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 nondimensional 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 $μ$, 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 $μ(4β−1)$, where $β$ is the nonlinear 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 nonlinear elasticity are connected to those of Landau in weakly (fourth-order) nonlinear elasticity.

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## Figures

Figure 1

Bending of a block of silicone rubber with length L=24 cm and thickness 2A=2 cm. The dimension of each square drawn on the top surface is 1×1 cm2. The picture on the right shows that even for such a “homemade” bending experiment, there exists a region about 3 cm wide around the median where plane strain is respected.

Figure 2

Sketch of Rivlin’s deformation for the bending of a block made of an incompressible isotropic solid with length L and thickness 2A into a circular annular sector with inner and outer radii ra and rb, respectively. The angle α is the bending angle. The deformation is plane strain but makes no assumption about the dimensions of the block or the amount of bending.

Figure 3

Variations in the moment M with ϵ, the product of the aspect ratio by the bending angle, for the large bending of an elastic block modeled by Rivlin’s strain-energy density 53. Exact results in the cases where the nonlinear shear coefficient β is equal to 0.0 (Mooney–Rivlin material), 0.1, 0.5, and 1.0. See Fig. 4 for a comparison with experimental results when β=1.0.

Figure 4

Bending of a strip of polyurethane: variation in the nondimensional measure of the bending moment M/Mm with ϵ, the product of the strip’s aspect ratio by the angle of bending. Circles: experimental data; dashed straight line: linear elasticity, Eq. 77; and full line plot: fitting of third- and fourth-order elasticity effects with the data by adjusting the nonlinear parameter β to the value 1.0 in Eq. 76.

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