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

Nicolae Boja

Theoretical and applied mechanics, Tome 30 (2003) no. 1.

Résumé

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", $\boldsymbol M$, associated to the deformations, is constructed. The manifold $\boldsymbol M$ 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, $\omega$, which describes the deformation in a small neighborhood of any point along the orbits. The non-null components $\omega_{i,i+1}$, ($i =1,2$), of ω are assimilated as like curvatures at each point of an orbit in the planes generated by the pairs of vectors ($\tilde{\mathbf e}_i,\tilde{\mathbf e}_{i+1}$) of a moving frame in $\boldsymbol M$ 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.