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.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Gibson, L. J., and Ashby, M. F., 1997, Cellular Solids, 2nd ed., Pergamon/Cambridge, Oxford, UK.
Cosserat, E., and Cosserat, F., 1909, Theorie des Corps Deformables, Hermann et Fils, Paris.
Eringen, A. C., 1968, “Theory of Micropolar Elasticity,” Fracture, Vol. 1, H.Liebowitz, ed., Academic Press, New York, pp. 621–729.
Hadjesfandiari, A. R., and Dargush, G. F., 2011, “Couple Stress Theory for Solids,” Int. J. Solids Struct., 48(18), pp. 2496–2510. [CrossRef]
Koiter, W. T., 1964, “Couple-Stresses in the Theory of Elasticity, Parts I and II,” Proc. K. Ned. Akad. Wet., 67(1), pp. 17–44.
Lakes, R. S., 1986, “Experimental Microelasticity of Two Porous Solids,” Int. J. Solids Struct., 22(1), pp. 55–63. [CrossRef]
Anderson, W. B., and Lakes, R. S., 1994, “Size Effects Due to Cosserat Elasticity and Surface Damage in Closed-Cell Polymethacrylimide Foam,” J. Mater. Sci., 29(24), pp. 6413–6419. [CrossRef]
Spadoni, A., and Ruzzene, M., 2012, “Elasto-Static Micropolar Behavior of a Chiral Auxetic Lattice,” J. Mech. Phys. Solids, 60(1), pp. 156–171. [CrossRef]
Adomeit, G., 1967, “Determination of Elastic Constants of a Structured Material,” Mechanics of Generalized Continua, Proceedings of the IUTAM Symposium, Freudenstadt/Stuttgart, Aug. 28–31/Sept. 1–2, E.Kröner, ed., Springer, Berlin, pp. 80–82.
Bigoni, D., and Drugan, W. J., 2007, “Analytical Derivation of Cosserat Moduli Via Homogenization of Heterogeneous Elastic Materials,” ASME J. Appl. Mech., 74(4), pp. 741–753. [CrossRef]
Lakes, R. S., 1995, “On the Torsional Properties of Single Osteons,” J. Biomech., 28(11), pp. 1409–1410. [CrossRef] [PubMed]
Gauthier, R. D., and Jahsman, W. E., 1975, “A Quest for Micropolar Elastic Constants,” ASME J. Appl. Mech., 42(2), pp. 369–374. [CrossRef]
Mora, R., and Waas, A. M., 2000, “Measurement of the Cosserat Constant of Circular Cell Polycarbonate Honeycomb,” Philos. Mag. A, 80(7), pp. 1699–1713. [CrossRef]
Sikon, M., 2009, “Theory and Experimental Verification of Thermal Stresses in Cosserat Medium,” Bull. Pol. Acad. Sci.: Tech. Sci., 57(2), pp. 177–180. [CrossRef]
Krishna Reddy, G. V., and Venkatasubramanian, N. K., 1978, “On the Flexural Rigidity of a Micropolar Elastic Circular Cylinder,” ASME J. Appl. Mech., 45(2), pp. 429–431. [CrossRef]
Park, H. C., and Lakes, R. S., 1987, “Torsion of a Micropolar Elastic Prism of Square Cross Section,” Int. J. Solids, Struct., 23(4), pp. 485–503. [CrossRef]
Park, H. C., and Lakes, R. S., 1986, “Cosserat Micromechanics of Human Bone: Strain Redistribution by a Hydration-Sensitive Constituent,” J. Biomech., 19(5), pp. 385–397. [CrossRef] [PubMed]
Lakes, R. S., Gorman, D., and Bonfield, W., 1985, “Holographic Screening Method for Microelastic Solids,” J. Mater. Sci., 20(8), pp. 2882–2888. [CrossRef]
Anderson, W. B., Lakes, R. S., and Smith, M. C., 1995, “Holographic Evaluation of Warp in the Torsion of a Bar of Cellular Solid,” Cell. Polym., 14, pp. 1–13.
Iesan, D., 1971, “On Saint Venant's Problem in Micropolar Elasticity,” Int. J. Eng. Sci., 9(10), pp. 879–888. [CrossRef]
Iesan, D., 1971, “Bending of Micropolar Elastic Beams by Terminal Couples,” An. St. Univ. Iasi Math., 17, pp. 483–490.
Reissner, E., 1968, “On St. Venant Flexure Including Couple Stresses,” PMM, 32(5), pp. 923–929.
Park, S. K., and Gao, X. L., 2006, “Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory,” J. Micromech. Microeng.16(11), pp. 2355–2359. [CrossRef]
Jemielita, G., 1984, “Bending of a Micropolar Rectangular Prism. I,” Bull. Pol. Acad. Sci.: Tech. Sci., 32(11), pp. 625–632.
Mindlin, R. D., 1965, “Stress Functions for a Cosserat Continuum,” Int. J. Solids Struct., 1(3), pp. 265–271. [CrossRef]
Wang, Y. C., Lakes, R. S., and Butenhoff, A., 2001, “Influence of Cell Size on Re-Entrant Transformation of Negative Poisson's Ratio Reticulated Polyurethane Foams,” Cell. Polym., 20(6), pp. 373–385.
Tauchert, T., 1970, “A Lattice Theory for Representation of Thermoelastic Composite Materials,” Recent Adv. Eng. Sci., 5(1), pp. 325–345.
Ilcewicz, L., Kennedy, T. C., and Shaar, C., 1985, “Experimental Application of a Generalized Continuum Model to Nondestructive Testing,” J. Mater. Sci. Lett., 4(4), pp. 434–438. [CrossRef]
Drugan, W. J., 2003, “Two Exact Micromechanics-Based Nonlocal Constitutive Equations for Random Linear Elastic Composite Materials,” J. Mech. Phys. Solids, 51(9), pp. 1745–1772. [CrossRef]
Eringen, A. C., 1983, “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocations and Surface Waves,” J. Appl. Phys., 54(9), pp. 4703–4710. [CrossRef]
Peddieson, J., Buchanan, G. R., and McNitt, R. P., 2003, “Application of Nonlocal Continuum Models to Nanotechnology,” Int. J. Eng. Sci., 41(3–5), pp. 304–312. [CrossRef]


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




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In