Research Papers

Bending of a Cosserat Elastic Bar of Square Cross Section: Theory and Experiment

[+] Author and Article Information
Roderic Lakes

Department of Engineering Physics,
Engineering Mechanics Program,
Department of Materials Science,
Rheology Research Center,
University of Wisconsin,
1500 Engineering Drive,
Madison, WI 53706-1687
e-mail: lakes@engr.wisc.edu

W. J. Drugan

Department of Engineering Physics,
Engineering Mechanics Program,
University of Wisconsin,
1500 Engineering Drive,
Madison, WI 53706-1687
e-mail: drugan@engr.wisc.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 23, 2015; final manuscript received May 12, 2015; published online June 16, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(9), 091002 (Sep 01, 2015) (8 pages) Paper No: JAM-15-1106; doi: 10.1115/1.4030626 History: Received February 23, 2015; Revised May 12, 2015; Online June 16, 2015

Pure bending experiments on prismatic bars of square cross section composed of reticulated polymer foam exhibit deformation behavior not captured by classical elasticity theory. Perhaps the clearest example of this is the observed sigmoidal deformation of the bars' lateral surfaces, which are predicted by classical elasticity theory to tilt but remain planar upon pure moment application. Such foams have a non-negligible length scale compared to the bars' cross-sectional dimensions, whereas classical elasticity theory contains no inherent length scale. All these facts raise the intriguing question: is there a richer, physically sensible, yet still continuum and relatively simple elasticity theory capable of modeling the observed phenomenon in these materials? This paper reports our exploration of the hypothesis that Cosserat elasticity can. We employ the principle of minimum potential energy for homogeneous isotropic Cosserat linear elastic material in which the microrotation vector is taken to be independent of the macrorotation vector (as prior experiments indicate that it should be in general to model such materials) to obtain an approximate three-dimensional solution to pure bending of a prismatic bar having a square cross section. We show that this solution, and hence Cosserat elasticity, captures the experimentally observed nonclassical deformation feature, both qualitatively and quantitatively, for reasonable values of the Cosserat moduli. A further interesting conclusion is that a single experiment—the pure bending one—suffices to reveal directly, via the observation of surface deformation, the presence of nonclassical elastic effects describable by Cosserat elasticity.

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Grahic Jump Location
Fig. 1

Open-cell polyurethane foam. Scale bar: 5 mm.

Grahic Jump Location
Fig. 2

Portion of the bent 50 mm wide test beam of 0.4 mm cell size foam viewed at an oblique angle

Grahic Jump Location
Fig. 3

Deformation uy versus position, excluding tilt: (a) larger cell foam, cell size 1.2 mm and (b) size 0.4 mm. Comparison of the two figures shows that the sigmoidal displacement is increased for larger pore size, implying a local length scale effect which suggests modeling via Cosserat theory.

Grahic Jump Location
Fig. 4

Deformed shape of a bar of initially square cross section experiencing pure bending by moments at ends acting in the positive-y direction, showing anticlastic curvature and tilt of lateral surfaces

Grahic Jump Location
Fig. 5

Approximate Cosserat analytical solution for lateral side displacement uy (with tilt portion removed) versus position for foam with 0.4 mm cell size



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