Unorthodox Thoughts about Deformation, Elasticity, and Stress

Abstract

The nature of elastic deformation is examined in the light of the potential theory. The concepts and mathematical treatment of elasticity and the choice o f equilibrium conditions are adopted from the mechanics o f discrete bodies, e. g., celestial mechanics; they are not applicable to a change o f state. By nature, elastic deformation is energetically a Poisson problem since the buildup of an elastic potential implies a change of the energetic state in the sense of thermodynamics. In the Euler-Cauchy theory, elasticity is treated as a Laplace problem, implying that no change of state occurs, and there is no clue in the Euler-Cauchy approach that it was ever considered as one. The Euler-Cauchy theory of stress is incompatible with the potential theory and with the nature of the problem; it is therefore wrong. The key point in the understanding of elasticity is the elastic potential.

On the Systematics of Energetic Terms in Continuum Mechanics, and a Note on Gibbs (1877)

“…A detailed and exhaustive discussion of the EulerCauchy theory is given in Ref. 4, where it is shown that the Euler-Cauchy theory, and thus the entire body of continuum mechanics, is profoundly incompatible with the theory of potentials 8 ; in fact, the latter is entirely unknown in mechanics of solids, although it is the theoretical framework of classical physics. The known theories of elastic deformation are therefore in need of revision.…”

“…3 The continuum mechanics theory has been critically reviewed, and the first steps towards a new approach have been outlined. 4 It is this author's firm opinion that continuum mechanics should have been founded again 150 years ago, right after the discovery of the First Law of Thermodynamics when the systematics of energetic terms became completely known; unfortunately, this has not been done. Instead, the difference between conservative and nonconservative physics, the most profound and fundamental difference among the classes of physical processes, was so thoroughly blurred in material science that up to this day, people have serious difficulties to recognize it -because they have not been trained to see it.…”

An Approach to Deformation Theory Based on Thermodynamic Principles

“…This is not correct in the case of changes of state, however. 11,12 Consider a solid container which is filled with air, and closed by a piston. If the piston is moved inward the pressure increases; but it does not matter if it is the piston or any other part of the walls that move, or if all the walls move inward -the pressure increases on all the walls.…”

“…eddies which rotate in either sense about an axis which is close to x 3 . Since flow has been considered a conservative physical problem, and since elastic deformation has not been properly understood as a change of state, 11,12 irreversible relaxation of the elastically loaded state has so far not been considered in the search for the origin of turbulent flow. However, it should be readily recognized as the cause of "elastic turbulence".…”

“…In the Euler-Cauchy theory of continuum mechanics the shape, including the radius, was lost through the Cauchy continuity approach which, however, violates an existence theorem in potential theory. 11,12 The shape of the region bounded by the surface integral in the divergence theorem is arbitrary only if the surface does not pass through mass. This is not the case in continuum physics; the shape is therefore of profound importance because the surface-volume ratio per unit mass is not arbitrary.…”