In the isotropic Cosserat solid or micropolar continuum, there are six elastic constants, in contrast to the classical elastic solid in which there are two independent elastic constants. The micropolar theory also incorporates a micro-inertia that is pertinent to dynamic deformation at sufficienly high frequency.

In a Cosserat solid [1] with the micro-inertia incorporated in the micropolar formulation, dispersion of shear waves [2] is expected to occur. In particular, the wave speed of plane dilatational waves in an unbounded Cosserat elastic medium is independent of frequency as in the classical case, that is, no dispersion. The speed of shear waves depends on frequency in a Cosserat solid. A new kind of wave associated with the micro- rotation is predicted to occur in Cosserat solids. Wave dispersion also occurs in the Mindlin microstructure theory [3] and in nonlocal solids [4]. A theory similar to Mindlin microstructure was called micromorphic by Eringen. In such theories the points in a continuum may translate as in classical elasticity, rotate as in Cosserat elasticity, and deform as well.

If wave methods are to be used in experiments [5], a difficulty is that wave dispersion also arises from viscoelasticity of materials. Wave methods are therefore most suitable if the wave attenuation due to viscoelasticity is small enough to be neglected. In viscoelastic materials, all material coefficients, including the Cosserat ones, can depend on time or frequency. One must therefore decouple the frequency dependence associated with the material coefficients from the wave dispersion associated with the generalized continuum, e. g. Cosserat, model. This cannot be done in wave methods, since the wavelength which governs the strain gradient cannot be decoupled from the frequency. Moreover, dispersion may be so strong that no transmitted waves are discerned.

A wave method was used by Gauthier [6] to examine the particulate composite consisting of aluminum spheres embedded in an epoxy matrix. This material had appeared classical in the quasi-static size effect studies of Gauthier and Jahsman [7]. The micro-inertial characteristics were determined, and it was found that the coupling number N

1. E. and F. Cosserat, Theorie des Corps Deformables. A. Hermann et Fils, Paris, 1909.

2. A. C. Eringen, Theory of micropolar elasticity. In Fracture (Edited by H. Liebowitz) Vol. 2, pp. 621- 729. Academic Press, New York, 1968.

3. Mindlin, R. D., "Micro-structure in linear elasticity", Arch. Rational Mech. Analy, 16, 51-78, 1964.

4. Eringen, A. C., "Linear theory of nonlocal elasticity and dispersion of plane waves", Int. J. Engng Sci, 10, 425-435, 1972.

5. Lakes, R. S., "Experimental methods for study of Cosserat elastic solids and other generalized continua", in Continuum models for materials with micro-structure, ed. H. Muhlhaus, J. Wiley, N. Y. Ch. 1, p. 1-22, 1995. Review article pdf

6. Gauthier, R. D., "Experimental investigations of micropolar media", In Mechanics of micropolar media, ed. O. Brulin, R. K. T. Hsieh, World Scientific, Singapore, 1982.

7. Gauthier, R. D. and W. E. Jahsman. "A quest for micropolar elastic constants." J. Applied Mechanics, 42, 369-374, 1975.

8. Hlavacek, M., "On the effective moduli of elastic composite materials", Int. J. Solids and Structures, 12, 655-670, 1976.

9. Berglund, K., "Structural models of micropolar media", in Mechanics of Micropolar Media, edited by O. Brulin and R. K. T. Hsieh, World Scientific, Singapore, 1982.

10. Merkel, Tournat, Gusev "Experimental evidence of rotational elastic waves in a granular phononic crystal", Phys. Rev. Lett. 2011.

11. Nowacki, W., Theory of micropolar elasticity, Poznan, 1970.

12. M. Nagawa, K. Arakawa, M. Yamada, Diamond Crystals as Cosserat Continua, phys. stat. sol. (a) 57, 713-718 (1980).

Cosserat equations, using the equations of Eringen for micropolar elasticity are as follows. Other authors use different symbols.

Cosserat equations. Click on image for larger image.

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