«Holographic evaluation of warp in the torsion of a bar of cellular solid: effect of Cosserat elasticity Anderson, W. B., Lakes, R. S., and Smith, M. ...»
Holographic evaluation of warp in the torsion of a bar of cellular solid: effect
of Cosserat elasticity
Anderson, W. B., Lakes, R. S., and Smith, M. C., "Holographic evaluation of warp in the torsion of
a bar of cellular solid", Cellular Polymers, 14, 1-13, (1995).
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 Cosseratsolids.
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 specimen to provide torque reaction. Torsion was achieved by advancing the micrometer screws on each rotation stage so that the top and bottom of the specimen rotated in opposite directions. Twist per unit length was 0.071 rad/m giving rise to a predicted classical peak strain of 8 x 10-4. The amount of rotation of the top and bottom rotation stages was chosen so that a region of zero angular displacement of the specimen was imaged near the center of the holographic film. The specimen and the holographic components were installed on a research grade damped table top supported upon four pneumatic isolation mounts ( Newport Corporation ). A light beam from a 15 milliwatt helium-neon laser was divided by a variable beamsplitter into an object beam and a reference beam.
Image plane double exposure transmission holograms were made via a unity magnification configuration using a four element coated lens system. For the purpose of simplifying the calculation, the reference and reconstruction beams were collimated. Moreover the object illumination beam was directed at an oblique angle, 16.6° with respect to the specimen surface by a cube beamsplitter, as shown in Fig. 1. The reason for the oblique illumination is to obtain maximum sensitivity to the warping deformation, which is in the plane of the bar surface, in the direction of the bar axis. Each exposure was about three seconds, achieved using a digitally controlled shutter. Agfa 8E75 holographic film was used. The film was developed in Kodak D19 and bleached in ammonium dichromate bleach to increase the image brightness. The images were viewed and photographed in laser light of 633 nm wavelength, the same as was used for making the holograms.
Since the fringes themselves have some depth, a sufficiently small aperture stop was used in the camera lens to achieve adequate resolution of the fringes. The photographic images were enlarged to 11" by 14", and the fringe deflection associated with warp was measured from the enlargements.
Fringe patterns obtained by double exposure holography represent a map of displacement in a direction specified by experimental conditions. The zero-order fringe method [see, e.g. Schumann and Dubas, 1979] was used for interpretation of the holographic fringe patterns. The relation between the displacement vector u and the fringe order n at a point P on the illuminated surface of the specimen is given by nλ=uλ(k-h) (1) in which λ is the wavelength of the light, k is the unit vector from the object to the observer's eye, and h is the unit vector from the light source to the object. The angle required to determine h was obtained from the length of a shadow cast by a test object upon the specimen. The angle required to determine k was obtained from the change in spacing of black marks made upon the specimen surface.
Fringe patterns were evaluated via existing analytical solutions of the torsion problem for a square section bar in classical elasticity (see, e.g. Sokolnikoff, 1983) and in Cosserat elasticity (Park and Lakes (1987)). The warp does not depend on the shear modulus, but it does depend on the Cosserat characteristic lengths and coupling number as defined below. In a Cosserat solid the warp is reduced in comparison with the classical elastic value. As a result, surface strain does not vanish at the corner of the cross section and peak strain at the center of the lateral surface is reduced. The degree of warp was evaluated from the fringes as shown in Fig. 2, which shows a theoretical fringe pattern for a classical elastic material. Consider the vertical dotted line as the edge of the plane which does not rotate, that is, for which the displacements ux = uy = 0. Fringe deviations from this line are due to warp. The distance ζ is proportional to the slope of the warp curve at the center-line of the bar surface, based on Eq. 1 and the displacement field for torsion. ζ was measured on the fringe photographs, and the ratio of the ζ values for the PMMA and the foam was calculated and compared with the corresponding theoretical ratio for classical and Cosserat theories.
Analysis: Elasticity theories Foams and other materials with micro-structure are not continuous media. However no material in the physical world is a continuum either, since all materials are composed of atoms.
Analysis of material properties for engineering application is facilitated by the use of a continuum theory which allows the details of the micro-structure to be averaged. The continuum theories considered in this study are classical elasticity and Cosserat elasticity.
Classical Elasticity The constitutive equation for classical isotropic elasticity (Sokolnikoff, 1983; Fung, 1968), is as follows, in which there are the two independent elastic constants λ and G, the Lamé constants.
σkl = λεrr δkl + 2G εkl (2) The Poisson's ratio ν = λ/2(λ + G) is restricted by energy considerations to have values in the range from -1 to 1/2. There is no length scale in classical elasticity.
Cosserat (micropolar) Elasticity The Cosserat theory of elasticity (Cosserat, 1909) incorporates a local rotation of points as well as the translation assumed in classical elasticity; and a couple stress (a torque per unit area) as well as the force stress (force per unit area). The force stress is referred to simply as 'stress' in classical elasticity in which there is no other kind of stress. Several authors developed the theory in the language of modern continuum mechanics (Aero and Kuvshinskii, 1960; Mindlin and Tiersten, 1962; Mindlin, 1965; Eringen, 1868; Nowacki, 1970). Eringen (1968) incorporated micro-inertia and renamed Cosserat elasticity micropolar elasticity. Here we use the terms Cosserat and micropolar interchangeably. In the isotropic Cosserat solid there are six elastic constants, in contrast to the classical elastic solid in which there are two, and the uniconstant material in which there is one. The constitutive equations for a linear isotropic Cosserat elastic solid are, in the symbols of
σkl = λεrr δkl + (2µ + κ )εkl + κeklm (rm-φm) (3) mkl = α φr,r δkl + βφk,l + γφl,k (4) The usual summation convention for repeated indices is used throughout, as is the comma convention representing differentiation with respect to the coordinates. σkl is the force stress, which is a symmetric tensor in equation 2 but it is asymmetric in Eq. 3. mkl is the couple stress, εkl = (uk,l + ul,k )/2 is the small strain, uk is the displacement, and eklm is the permutation symbol. The microrotation φk in Cosserat elasticity is kinematically distinct from the macrorotation rk = (eklm um,l )/2 obtained from the displacement gradient.
In three dimensions, the isotropic Cosserat elastic solid requires six elastic constants λ, µ, α, β, γ, and κ for its description. A comparison of symbols used by various authors was presented by Cowin (1970a). The following technical constants derived from the tensorial constants are
beneficial in terms of physical insight. These are (Eringen, 1968; Gauthier and Jahsman, 1975):
Young's modulus E = (2µ+κ)(3λ+2µ+κ)/(2λ+2µ+κ), shear modulus G = (2µ+κ)/2, Poisson's ratio ν = λ/(2λ+2µ+κ), characteristic length for torsion lt = [(β+γ)/(2µ+κ)]1/2, characteristic length for bending lb = [γ/2(2µ+κ)]1/2, coupling number N = [κ/2(µ+κ)]1/2, and polar ratio Ψ = (β+γ)/(α+β+γ).
When α, β, γ, κ vanish, the solid becomes classically elastic. The case N = 1 (its upper bound) is known as 'couple stress theory' (Mindlin and Tiersten, 1962; Cowin, 1970b). This corresponds to κ → ∞, a situation which is permitted by energetic considerations, as is incompressibility in classical elasticity.
Discrimination among continuum representations Analytical predictions of Cosserat characteristic lengths have been developed for a variety of structures. In cellular solids it may be comparable to the average cell size (Adomeit, 1967). In some cellular materials, the characteristic length can exceed the cell size, in contrast to theory which predicts it should be smaller. In fibrous composites, the characteristic length l may be the on the order of the spacing between fibers (Hlavacek, 1975). Structure, however, does not necessarily lead to Cosserat elastic effects. Composite materials containing elliptic or spherical inclusions are predicted to have a characteristic length of zero (Hlavacek, 1976; Berglund, 1982).
One may determine the appropriate continuum representation for a material by experiment.
Cosserat elasticity manifests itself in size effects in the torsion and bending rigidity as reviewed in the Introduction. It is known that common engineering materials such as steel and aluminum behave classically (Ellis and Smith, 1968). Several materials have been demonstrated to behave as Cosserat solids: several foams (Lakes, 1983, 1986, as well as human bone, a natural composite (Yang and Lakes, 1982; Lakes and Yang, 1983; Park and Lakes, 1987).
Results and Discussion Fringe patterns for PMMA and for the Rohacell foam are shown in Figures 4 and 5 respectively. The patterns disclose less warp in the Rohacell specimen than in the PMMA specimen. A computed fringe pattern for the PMMA based on classical elasticity is shown in Fig.
2. The calculated fringe deviation due to warp agrees with the observed value within 7%, equivalent to the limit of error associated with the breadth of the fringes.