  Usual introductions of the concept of motion are not well adapted to a
subsequent, strictly tensorial, theory of elasticity. The consideration of
arbitrary coordinate systems for the representation of both, the points in the
laboratory, and the material points (comoving coordinates), allows to develop a
simple, old fashioned theory, where only measurable quantities -like the Cauchy
stress- need be introduced. The theory accounts for the possibility of
asymmetric stress (Cosserat elastic media), but, contrary to usual developments
of the theory, the basic variable is not a micro-rotation, but the more
fundamental micro-rotation velocity. The deformation tensor here introduced is
the proper tensorial equivalent of the poorly defined deformation "tensors" of
the usual theory. It is related to the deformation velocity tensor via the
matricant. The strain is the logarithm of the deformation tensor. As the theory
accounts for general Cosserat media, the strain is not necessarily symmetric.
Hooke's law can be properly introduced in the material coordinates (as the
stiffness is a function of the material point). A particularity of the theory
is that the components of the stiffness tensor in the material (comoving)
coordinates are not time-dependent. The configuration space is identified to
the part of the Lie group GL(3)+, that is geodesically connected to the origin
of the group.
