On the trace-manifold generated by the deformations of a body-manifold

Abstract

In this paper, concerned to the study of continuous deformations of material media using some tools of modem differential geometry, a moving frame of Frenet type along the orbits of an one-parameter group acting on a so-called "trace-manifold", M, associated to the deformations, is constructed. The manifold M is defined as an infinite union of non-disjoint compact manifolds, generated by the consecutive positions in the Euclidean affine 3-space of a body-manifold under deformations in a closed time interval. We put in evidence a skew-symmetric band tensor of second order, ?, which describes the deformation in a small neighborhood of any point along the orbits. The non-null components ?i,i+i, (i =1,2), of ? are assimilated as like curvatures at each point of an orbit in the planes generated by the pairs of vectors (?i,?i+i) of a moving frame in M associated to the orbit in a similar way as the Frenet's frame is. Also a formula for the energy of the orbits is given and its relationship with some stiffness matrices is established.