Holographic evaluation of warp in the torsion of a bar of cellular solid: effect of Cosserat elasticity

Holographic methods are utilized to examine deviations from classical elasticity in a cellular solid, polymethacrylamide closed cell foam. A square cross section bar is subjected to static torsional deformation. The warp deformation is observed to be less in a foam bar than in a homogeneous polymeric bar used as a control. The homogeneous bar obeys the predictions of classical elasticity. Behavior of the foam bar is consistent with Cosserat elasticity. In a Cosserat solid, points in the continuum to rotate as well as translate, and the material supports couple per unit area as well as force per unit area. Cosserat effects can lead to enhanced toughness. Introduction Cellular solids are two phase composite materials in which one phase is solid and the other is a fluid, most often air. If the size scale of the micro-structure becomes large enough, the material may no longer be assumed to be continuous. Some researchers have found that classical elasticity theory, currently used in engineering analyses of deformable objects at small strain, does not always adequately describe the behavior of cellular materials. Other continuum theories for linear isotropic materials are available. Some have more freedom than classical elasticity. The various continuum theories are all mathematically self consistent. Therefore a discrimination among them is to be made by experiment. In classical elasticity, points in the continuum can undergo translation, and the continuum supports a force per unit area, or stress. The rigidity of circular cylindrical bars of diameter d in tension goes as d2; in bending and torsion, the rigidity goes as d4. There is no length scale in classical elasticity. In Cosserat elasticity, there is a rotational degree of freedom of points in the continuum, and the possibility of a couple stress, or moment per unit area. In Cosserat solids, a size-effect is predicted in the torsion of circular cylinders of Cosserat elastic materials. Slender cylinders in torsion appear more stiff then expected classically (Gauthier and Jahsman, 1975); a similar size effect is also predicted in the bending of plates, and of beams (Krishna Reddy and Venkatasubramanian, 1978) No size effect is predicted in tension. The stress concentration factor for a circular hole, is smaller than the classical value, and small holes exhibit less stress concentration than larger ones (Mindlin, 1963). This phenomenon of stress redistribution can be expected to confer a measure of toughness upon Cosserat solids. Physically, the couple stresses in Cosserat and microstructure elasticity represent spatial averages of distributed moments per unit area, just as the ordinary (force) stress represents a spatial average of force per unit area. In cellular solids moments may be transmitted through the cell ribs or walls. The Cosserat characteristic lengths are expected to be on the order of the size scales in the microstructure. In foams they may be sufficiently large to observe experimentally. This study is directed at the deformation field in the case of torsion of a square cross section bar, and how it depends on material micro-structure. Interpretation of the results is aided by generalized continuum theory: Cosserat elasticity. Materials and Methods Torsion experiments were conducted upon square cross section bars of Rohacell® polymethacrylimide closed cell foam (grade WF 300, ρ = 0.38 g/cm3, Cyro Industries) and amorphous polymethyl methacrylate (PMMA) at room temperature. Both the Rohacell bar and the PMMA specimen were 16.6 mm in square cross section; the Rohacell was 135 mm long and the PMMA was 140 mm long. The PMMA, which is transparent, was painted white to obtain a diffusely reflecting surface. Each end of each specimen was cemented firmly on a precision rotation stage (Newport Corporation). The rotation stages were connected by steel rods parallel to the