Keywords: large strain, manifold of metric states, Euclidean tensor, constitutive laws.

We call metric-state the state of deformation at time t of the vicinity of a material point X of a continuous medium (the state itself, considered independently of any reference or initial state). One of the rare undisputed results of large strain theory is that the symmetrical part D of the gradient of the Eulerian velocity field is the good Eulerian representative of the rate of evolution of this metric-state. As a consequence, the Eulerian representative of stress is the Kirchhoff tensor , equal to the Cauchy stress divided by the mass density ( D: is the specific internal power).

These variables lie in the linear tangent spaces which approximate the movement in the vicinity of the considered time t. As a consequence, in the linearised theory of small displacement, the representative of the current metric- state is the symmetrical part of the gradient of the displacement field, the linearised deformation tensor, because its time derivative is equal to D.

In large displacement theory, such a "material time integration" of D, leading to a good representative m for the metric-state, is not at all easy. Note that this problem is closely related to the problem of finding a good material time derivative for tensorial Eulerian variables. And the diversity of the proposed solutions for both problems, with no recognized winner, proves their failure. In [1], we gave a review of the known proposals for m, pointing out their defects. Then, we briefly recalled our own solution for this problem, fully described in [2], considered as "exact" because it corrects these defects. Finally, we analysed the approximations which, starting from this "exact" proposal, lead to the known defective proposals.

In this paper we come back to our solution, still largely unknown, and we focus on several other aspects. The usual identification of a Euclidean linear space with its dual space is performed with success in the case of a rigid solid because it is done by a constant metric tensor. But in the case of deformable bodies, the local material metric tensor is time-dependent: it is the basic metric-state parameter [3]. And with a time-dependent identification, one may anticipate serious difficulties in the time derivations and integrations. The refusal of this identification is thus the starting point of our approach. A precise mathematical definition of Euclidean tensors is then needed and proposed, and the material time integration of D becomes easy.

Secondly, we present the material mathematical world generated by the obtained metric-state variable m. The set M of all possible metric-states for the vicinity of the considered material point is the curved space in which we are moving when considering a strain process of this vicinity. M is a Riemannian manifold, with additional structure because in each point m the linear tangent space is more complex than a linear Euclidean space: it is indeed the space of symmetrical Euclidean tensors on the Euclidean linear space of material segments issued from X in the metric-state m. With each point m of the manifold M are associated all the linear spaces of m-Euclidean tensors of all orders. Thus, the manifold M is the basis of related fibre bundles in which may be stated the constitutive laws. Working above m in these fibre bundles in this intrinsic Lagrangian approach, and working on the current position in the Eulerian approach, are two consistent approaches.

Lastly, we present the first steps, first techniques and first obtained models of this material intrinsic statement of constitutive laws, which remains a largely open field. The covariant derivative technique leads to a natural material time derivative law whose Eulerian counterpart is the Jaumann derivative (associated to the skew symmetric part of the gradient of the Eulerian velocity field). Its application to the study of uniform strain process is very satisfying. This "material covariant time derivative" provides natural rate needed in plasticity or hypo-elasticity constitutive equations. The covariant derivative of tensorial sections above M provides the way to define metric-state independent operators also useful in the statement of laws. This approach does not contain any non-intrinsic ingredient allowing a wrong unintentional physics. For example, with continuity and mass density, which are the only physical properties taken into account at the beginning of our analysis, only very simple elastic laws may be stated: perfect gas and barotrop perfect fluids. The simplicity and the obviousness with which these laws are obtained prove the quality of our approach. In order to make more complex physics, the mathematical tools have to be consciously introduced. For example, introducing a constant relaxed metric state, the geometry of M is enriched and this allows the statement of isotropic and anisotropic elasticity. The new logarithmic rotation is also interpreted.

1

Rougée, P., "Exact and approximated kinematics frames for the statement of constitutive laws in large strain" in "Formulations and constitutive laws for very large strains", J. Plesek editor, Institute of Thermomechanics, AS CR Prague, ISBN 80-85918-74-9.

2

Rougée, P., "Mécanique des grandes transformations". Mathématiques et Application 25, Springer Verlag, 1997.

3

Noll, N., "A New Mathematical Theory of Simple Material" Arch. Rat. Mech. Anal., 48 1-50

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