Non-linear equations of the dynamics of an elastic micropolar medium

Abstract

The non-linear equations for a continuous elastic medium with three additional degrees of freedom associated with local rotation, are considered. Such an elastic medium is called micropolar /1/. The existence of an elastic potential is proposed for it; thermal effects are neglected. The purpose of this paper is to study certain qualtiative properties of the equations that are closely associated with the concept of hyperbolicity. The complete set of equations is represented as a system of local conservation laws, closed by finite relationships yielding the rheology of the material. The possibility of such a representation is based /2, 3/ on the fact that the gradients of the particle displacment and angle of rotation are used as a measure of the deformation. Local conservation laws for the compatibility of the strain and velocity fields of fairly simple structure are formulated. The velocities of propgation of characteristic surfaces are studied for the dynamic equations for the general case of the material under consideration. The existence of real velocities, the necessary condition for hyperbolicity, results in a constraint on the form of the elastic potential function, which is an analog of the SE-inequality /4/ in the classical theory of non-linearly elastic media. The system of non-linear equations being studied is reduced to symmetric form by replacing the vector of the solution. The necessary condition for such a transformation /5/ is the existence of an additional energy conservation law that follows from the system under consideration. The symmetric form of the equations enables us to formulate the sufficient condition for hyperbolicity — the condition of convexity of the elastic potential in its arguments. An estimate is obtained for the growth of the solutions of the Cauchy problem and the ensuing uniqueness theorem. The presence of the symmetric form of the system enables a general form to be obtained for the transport equation that governs the rate of change of a weak discontinuity along a bicharacteristic.