Falk H. Koenemann

  In my introductory class in deformation theory in 1980 at the University of California Davis I asked the following questions right there in class, simply out of confusion:

  1. Why do you use an equation of motion and not an equation of state?

  2. Why do you use Newton's equilibrium condition and not the thermodynamic equilibrium condition which distinguishes system and surrounding?

  3. There are bonds in solids, but there's no mention of bonds in this theory. Aren't bonds important for the understanding of a solid?

  4. Newton defined a normal and a rotating force as being parallel or perpendicular to the radius r of a body. Here you define a shear force as being perpendicular to the normal of a planar element, which is a unit vector n. Newton said nothing about plane orientations. These definitions are incompatible with one another because the radius vector is a lever whereas the unit vector is no such thing. Is Newton's definition wrong? Is the lever unimportant?

  5. If you deform a body, say, a sphere, by shortening it in Z, work is done upon the body in Z. Let's say this is negative work. But the body will expand in X and Y, so positive work is done in X and Y. If the volume remains constant, all material paths must cancel. Since force times path = work, no work is done in a volume-constant deformation. Isn't this impossible?

I did not receive an answer - neither then in class, nor in the following 45 years since, 35 of which I spent at or near RWTH Aachen. The professors simply remained silent - for hours, weeks, years.
             These five questions contain everything that's wrong with the Euler-Cauchy theory. If I could ask them in the very first moment I came in touch with the topic, they cannot be so hard. Frankly, I cannot imagine that I was the first student to ask these questions.

Answers:

  1. The energetic state of a system is invariant in Newton's mechanics, as is stated by the energy conservation law. A change of state requires the modern energetic thinking in physics, which arose only from 1840 on. The First Law of thermodynamics was discovered in 1842 by Mayer, in 1845 by Joule, and in 1847 by Helmholtz, triggering the "golden age of classical physics".

    I have not seen a book on continuum mechanics in which the First Law of thermodynamics is given in correct form. It is always "reformulated" to be subordinate to the conservative energy conservation law of Newtonian mechanics. This is the denial of the First Law.

  2. The thermodynamic equilibrium condition of system vs. surrounding developed from the experiments with the steam engine (J. Watt in the 1760s), i.e. from handling a gas. There the concept of the system that interacts with the surrounding, comes natural. Continuum mechanics uses the Newtonian definition of pressure P = f/A to this day; but the thermodynamic definition P = dU/dV is the more fundamental one. They are not at all identical: f/A is taken to imply the Newtonian equilibrium condition of force and counterforce; P = dU/dV requires the thermodynamic equilibrium condition of system and surrounding. They are not at all identical, they are conceptually independent, and they must both hold in the case of equilibrium. The Cauchy stress tensor and the thermodynamic system are mutually incompatible concepts. Proof that the Cauchy stress tensor does not exist because its derivation is flawed, is straightforward.

  3. Bonds were concluded to exist by Maxwell in 1849. It is simply impossible to understand the nature of solids without considering bonds. Bonds have the effect that equilibrium exists by definition inside a solid below the elastic limit. A distance in freespace is merely the difference between two points; a distance within a bonded solid is a lever. This changes the thinking about solids and elasticity in the most profound way.

    There is not a single book on the foundations of continuum mechanics that I have seen from 1980 to this day that mentions bonds or the lever.

  4. Newton is right. Euler's concepts are not compatible with Newton's mechanics, nor with potential theory, nor with the modern definition of Euclidean vector space. Euler's convention for the notation of planes in space is from 1776; the one that we learn at age 12 in high school is known as Hesse convention, is from 1863. They are mutually exclusive, they cannot be transformed into one another. Euler did not know yet the concept of a vector field in the sense of Ax = b. Vector fields arose with the discoveries of Maxwell 1860 to Gibbs 1901.

    Continuum mechanics does not use vector field theory. It uses a tensor, strain. The displacement is a vector field. Just how strain and displacement are related, is never clearly stated in the textbooks. Plainly, there is no relation.

  5. Work as a physical core concept was discovered by Coriolis 1829 for Newtonian mechanics of discrete bodies in freespace. There the total work must indeed cancel since the total energy of the kinetic system is invariant: this is work done within a system. Joule 1845 defined thermodynamic work, which is the work done upon a system, the work done in a change of state. For a gas this is PdV. Only since then we know

              - the principle of least work,
              - that a perpetual motion machine cannot exist,
              - if a theory predicts that a physical process does not cost physical work, it is proof of error.

    The definition of work requires a distance term times a force term. Euler and Cauchy thought about deformation before physical work was discovered. Their concepts are incompatible with work because of the missing lever. The question as to what sort of work is done by shear forces has never been asked in continuum mechanics, it cannot be asked. Shear forces have been completely ignored.

Continuum mechanics does not have a valid work term. All definitions of deformation work I have seen look very scientific if written in Greek letters, but numerically they are all zero - without exception.

There are thick books on the foundations of continuum mechanics which do not mention work one single time. This consistent omission throughout the literature is beyond error; these authors know what they are doing, – they consciously mislead their readers.

Two examples:
- Antman SS (2005) Nonlinear problems of elasticity. Second Edition. 835pp. Springer
- Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, 694pp

The zero sum was actually a desired result for Euler because it appeared to him to observe the energy conservation law of conservative (Newtonian) mechanics
         Ekin + Epot = const
which his teacher Johann Bernoulli had discovered in 1735, and conservation of mass simultaneously. The law says that for processes governed by this law, the mass in a kinetic system and its energy are proportional.

The idea that the energy of a system can be a variable, was far beyond Euler. His thinking was, from today's point of view, pre-modern. The First Law of thermodynamics
         dU = dw + dq,
the energy conservation law for non-conservative physics, was discovered 60 years after Euler's death.

Newtonian mechanics is conservative in the sense of the law given above, because the total energy of the kinetic system is conserved, i.e. constant. Processes that change the internal energy of a system are therefore nonconservative, and their energy conservation law is the First Law of thermodynamics. Mixing up conservative and nonconservative physics is the silliest and most ridiculous error that can be made in classical physics since 1850. This worst of all errors is perpetuated in continuum mechanics to this day - for 180 years since the discovery of the First Law, and for 110 years since the math for this sort of physics was fully known in 1901.

Elastic deformation is by nature a change of the energetic state of a material since the elastic potential is stored within the system. Changes of state may be reversible or irreversible. Elastic deformation is by nature reversible. Plastic and viscous flow are irreversible, and entropy is produced.

Professors of the natural sciences who treat their field like a religion, who expect faith instead of understanding, who refuse to explain their concepts properly, and who defend their errors by excommunicating the questioners, are dangerous and unprofessional.

Proofs of errors are published in "Unorthodox thoughts (2001)", "Systematics of energetic terms (2008)", and "Cauchy's stress theory in a modern light (2014)".
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