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.