I took my introductory class in deformation theory in 1980 at the University of California Davis. I asked the following questions right there, simply out of confusion, as they developed in my mind while the lecturer explained the topic to us:
I did not receive an answer - neither then in class, nor in the following 28 years since, 18 of which I spent at or near RWTH Aachen. The professors simply remained silent - for hours, weeks, years.
Why do you use an equation of motion and not an equation of state?
Why do you use Newton's equilibrium condition and not the thermodynamic equilibrium condition which distinguishes system and surrounding?
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?
Newton defined a rotating force as being 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 an unit vector n. These definitions are incompatible with one another because the magnitude of the radius vector can vary with direction whereas the unit vector cannot. Why do you believe that Newton's definition is wrong?
If you deform a body, say, a sphere, by stretching it in X, work is done upon the body in X. Let's say this is negative work. But the body will contract in Y and Z, so positive work is done in Y and Z. If the volume remains constant, the negative work in X and the positive work in Y and Z must balance, therefore no net work is done. Isn't this impossible?
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. There must have been a couple 1000 students since 1850 who were all silenced by grade pressure.
People who bang their head on the bible are bad enough; but I can ignore them. However, 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 far more dangerous.
These questions, and others, are now published in my papers "Unorthodox thoughts (2001)" and "Systematics of energetic terms (2008)".
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(1) and (2) are thermodynamic concepts which were unknown in the 18th century; Euler (1707-1783) had only Newton's theory which he could adapt.
In modern view, a change of state - in order to produce a loaded state, a.k.a. stress - can only be understood through an equation of state, separation of system and surrounding, and the equilibrium between them.
(3) Euler did not know yet about bonds, so he could not include them.
Bonds cause a solid to have a finite volume in a vacuum, which leads to the concept of the "internal pressure" of a solid. That concept is meaningful only in thermodynamics; it has no place in any theory based on Newton's mechanics which, after all, does not distinguish between system and surrounding.
(4) Euler only knew Newton's definition of pressure, but he could not know that thermodynamic pressure is internal energy per volume (that concept also requires separation of system and surrounding).
Euler's simplification of Newton's concept - taking n instead of r - is simply wrong. Euler did not know yet the concept of a vector field - in the sense of Ax = b - which was invented by Lagrange in 1784, a few months after Euler had passed on.
(5) 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 state of a system can be a variable, i.e. that energy and mass are not by definition, or not by nature proportional, 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 much later, in 1847.
It is easy to blame Euler; I don't. We are all children of our own time. Euler was a genius on a par with Gauss and Einstein; he just lived too early. The real error was done by those who never reconsidered Euler's concepts when the full outline of classical physics had become common knowledge. Clausius showed the way; continuum mechanics has been out of touch with reality since 1870 at least.