Symmetry in Nonlinear Mathematical Physics - 2009

Alexander Yakhno (Universidad de Guadalajara, Mexico)

'Homotopy' of Prandtl and Nadai solutions

The hyperbolic system of plane ideal plasticity equations under the Saint-Venant - Mises' yield criterion is considered. Its characteristics curves are deformed by the action of admitted group of point transformations, that permits to construct a new analytical solutions.

We provide two families of exact solutions for the system of plane ideal plasticity using the concept of homotopy of two functions. These functions are the well-known exact solutions of A. Nadai (for the flow of plastic material through the wedge-shaped converging channel and for the plastic zone around a circular cavity) and solution of L. Prandtl.

The mechanical sense of obtained characteristic fields is discussed. The analysis of the envelopes of corresponding characteristic curves permits to determine the boundaries for obtained solutions, which give the description of the stresses for the channels and cavities of specific forms.

The general algorithm of the relation of solutions of quasilinear hyperbolic system of two homogeneous equations of two independent variables is proposed.

This is joint work with S.I. Senashov and L. Yakhno (Siberian State Aerospace University, Krasnoyarsk, Russia).