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THEORETICAL AND APPLIED MECHANICS |
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TEORIJSKA I PRIMENJENA MEHANIKA 2003 Vol.30 (1), pp.11-28
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Summary: |
Nicolae Boja |
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On the
trace-manifold generated by the deformations of a
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In this paper, concerned to the study of continuous deformations of
material media using some tools of modern 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, w,
which describes the deformation in a small neighborhood of any point along
the orbits. The non-null components wi,i+1 ,
(i=1,2), of w are assimilated as like
curvatures at each point of an orbit in the planes generated by the pairs
of vectors
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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.