Nonholonomic dynamical systems, Elasticity -- Mathematical models, Continuum mechanics
Motivated by the theory of uniform elastic structures we try to determine the conditions for the local flatness of locally integrable connections on non-holonomic frame bundles of order 2. Utilizing the results of Yuen as well as our results for the holonomic case, we show that the locally integrable non-holonomic 2-connection is locally flat if, and only if, its projection to the bundle of linear frames is symmetric and the so-called inhomogeneity tensor vanishes. In the last section of this short paper we show how these results can be interpreted in the framework of the theory of uniformity of simple elastic materials with microstructure.
Elźanowski, M. and Preston, S. (1996). On Non-Holonomic Second Order Connections, with Applications to Continua with Microstructure, Extracta Mathematicae 11(1) 51-58.