We consider a 3-dimensional elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density.
In the first part of the talk we write down the general dynamic variational functional of our problem. In doing this we follow the logic of classical linear elasticity with displacements replaced by rotations and strain replaced by torsion. The corresponding system of Euler-Lagrange equations turns out to be nonlinear, with the source of this nonlinearity being purely geometric: unlike displacements, rotations in 3D do not commute.
In the second part of the talk we present a class of explicit solutions of our system of Euler-Lagrange equations. We call these solutions plane waves. We identify two types of plane waves and calculate their velocities.
In the third part of the talk we consider a particular case of our theory when only one of the three rotational elastic moduli, that corresponding to axial torsion, is nonzero. We examine this case in detail and seek solutions which oscillate harmonically in time but depend on the space coordinates in an arbitrary manner, which is a far more general setting than with plane waves. We show [1] that our system of nonlinear second order Euler-Lagrange equations is equivalent to a pair of linear first order massless Dirac equations.
In the process of analysing our model we also establish an abstract result, identifying a class of nonlinear second order partial differential equations which reduce to pairs of linear first order equations.
[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl equation and Cosserat elasticity", to appear in J. Phys. A: Math. Theor. Available as preprint arXiv:1001.4726.