A skecth about strain history effects*


The determination of the influence of strain history on material constitutive laws may be simplified by the definition of the so-called Deborah number.

I begin by paraphrasing a text of Clifford. A. Truesdell (in "Sketch for a History of Constitutive Relations") relating a conversation with Markus Reiner.

"After the victory over the Philistines, Deborah sang : "The mountains flowed before the Lord". Every thing flows, the mountains flow. But mountains flow before the Lord and not before man. The man in his short lifetime cannot seen them flowing, while the time of observation of God is infinite.

By this song, Deborah defines implicitely a non-dimensional number D

D = time relaxation/time of observation.

The difference between solids and fluids is then defined by the magnitude of D. The greater the Deborah number, the more solid the material; the smaller the Deborah number, the more fluid it is.

*See in Sketch for a history of constitutive relations, Rheology volume 1: principles, Plenum Press, New York and London.

What about soft biological tisssues?*

By extension, there are in fact several and even infinite Deborah number in a material, there are an infinite number of relaxation times.

A typical example would be biological soft tisssue. From experimental point of view, mechanical tests performed on biological tissues illustrate the the stress depends on the strain rate and that the stress decreases when specimens are subjected to a constant strain.

These dependences can be discriminated according to the time scale of their effects. Indeed, any internal process within the tissue has in principle a natural time which is defined as the measure of the time needed for their internal process to move to a new equilibrium after a change of macroscopic external loading.

The various mechanical behaviours can then be classified as a function of their natural time scale, or their Deborah number D. The tissue is elastic if D very large. An elastic tissue remembers entirely its previous state. The tissue has a short time memory (sometimes anelastic or more correctly finite time memory) if D is around unity, or more correctly a finite number. The tissue has a long time memory if it never returns to its original reference meaning that the time of observation is infinitely greater than any time scale of all internal processes, Deborah number D is near zero.

Alternative way to include time relaxation without explicitly defining a "time caracteristic variable" is to introduce an "spatial internal variable"for playing the role of an internal clock. Choice of internal variable seems quite arbitrary, a geometric approach, our preference, would take the affine connection as internal variable. But it was a long time history ...

Nevertheless, it seems curious that the geometric tool for modeling the length scales (space) may also helps to understand the effects of time scales.

*See in D Pioletti, L. Rakotomanana, Non-linear viscoelastic laws for soft biological tissues, European Journal of mechanics, A/Solids 19, 749-759, 2000.

Brief history of connection*

Here a very short history of affine connection in relation with continuum mechanics and physics.

Implicitly, the notion of connection emerged in the work of Newton (1687) when he introduced the transport (parallel) of instantaneous velocity of a material point before calculating the instantaneous acceleration. Any two instantaneous velocities of a same material point at different times do not have the same origin (spatially) but have two different spatial points. Thus any operation cannot be done before the parallel transport from one point to the other. A second step of connection theory development could be attributed to the work of Gauss (1827) when he studied the geodetic curves on a surface and particularly the influence of the curvature on the properties of triangles embedded in the surface. However, he did not explicitly proposed the method of comparing the tangent planes at different locations (edges of the triangles).

Few years later, Riemann extended to multi-dimensional space the work of Gauss on surface (1854). He introduced namely the notion of metric field on a manifold. The change of the metric along an embedded curve on the manifold could be expressed in terms of curvature. Considering the formulations of metric field at any two neighborhood points on the manifold, Christoffel (1869) found the mathematical condition the two metrics should verify to be transformed each other (smoothly). He then proposed the celebrated Christoffel’s symbols. This was the modern starting point of connection theory. Necessary conditions for the equivalence of the two metrics was used to extract the definition of the curvature tensor. Ricci and Levi-Civita laid down the basic concept of connection by considering the invariance of the Laplace-Beltrami operator during a change of frame. They formally derive the concept of connection (1888). They indeed developed the concept of covariant equations and by the way the covariant derivative. Levi-Civita in 1917 and few years later related the connection to the affine connection which was formally derived by Weyl (1918) and which was used to formulate the basic equations of gravitation and electromagnetism. Elie Cartan was among the first who emphasized the need of separation between the metric and the connection concepts, neglected by his predecessors, the connection exists independently on the metric, (1923, 1924) and Cartan proposed the notion of holonomy group.

The affine connection plays role in the fibre bundle concept as proposed by Esherman (1950) and in the physics gauge theory by Yang and Mills (1954). Nevertheless, the precise relation between the affine connection and the physics theory proposed by these authors was only pointed out recently (1980). Another application of the affine connection in field theory in physics was proposed by Weinberg and Salam (1980) and now constitutes the modern development of the concept. The starting point of the application of affine connection in continuum mechanics could be attributed to the work of Noll (1958), particularly on the theory of non homogeneous bodies, a class of continuum with dislocations. Since then, numerous authors have contributed to lay down the continuum theory with various types of singularity.

*Appendix in the monograph  L. Rakotomanana " A geometric Approach to Thermomechanics of Dissipating Continua", Progress in Mathematical Physics, Birkhauser, Boston, 2004.