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

Abstract

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.