Those treatments of Entropy in continuum mechanics that I've viewed on the web introduce Entropy abruptly, as if it is a fundamental property of matter. For example the current Wikepedia article on continuum mechanics ( https://en.wikipedia.org/wiki/Continuum_mechanics ) says:

Are other approaches to entropy? Can entropy be defined as a function of the more familiar properties of matter -such as position, mass, velocity?

For example, making an analogy between mass density and a probability density function, one aspect of an alternate definition of entropy ##H_a## could be to define ##H_a## as a function that increases as mass density becomes more uniform. Making an analogy with the entropy of thermodynamics , another aspect could be that at a given constant mass density, ##H_a## is higher when balance of matter at locations ( due to inflow and outflow) takes place at a high rate (- high "turnover"). Is there a specific function of the fundamental properties of matter that meets those requirements and is a useful definition of Entropy?

If I understand you correctly, my answer is that in continuum mechanics (which includes thermodynamics) entropy is indeed a postulated property of matter- just like mass, pressure, etc. The relationship between entropy and temperature is a (postulated) constitutive equation. The second law of thermodynamics can be formulated either as a balance equation or as the clausius-duhem inequality.

There are other approaches, which become significant in nonequilibrium conditions- 'temperature' may not have a good value and the entropy may no longer be single-valued. Two decent references I have are Balescu's "Statistical Dynamics" and Jou et. al. "Extended Irreversible Thermodynamics". Neither is a quick read.....

Well, equilibrium (or very close to equlibrium) thermodynamics can be derived from statistical physics, where the statistical entropy can be defined (Boltzmann, Gibbs, von Neumann, Shannon).

Continuum mechanics books usually assume the reader has completed a course on thermodynamics. I think that is reasonable.

In general, I think it is reasonable for any graduate level textbook to gloss over or completely omit the development of concepts covered in most undergraduate curricula. The definitions of entropy are covered in (as far as I have seen) every B.A. and B.S. physics program's requirements for graduation.

However, as far as I have seen, no connection is proven between the entropy that is defined in thermodynamics and the quantity called entropy that is assumed to be a property of matter in continuum mechanics. The entropy of thermodynamics is only defined for equilibrium states. Continuum mechanics deals with dynamic situations.

This appears to be your misunderstanding, and is not a fact.

The thermodynamics text I used as an undergraduate defines the second law of thermodynamics in equilibrium, but defines entropy for any system large enough to obey the statistics of very large numbers.

Is a gas that is not in equilibrium such a regime?

Is your point of view that the treatment of entropy in continuum mechanics defines entropy exactly as it is defined in thermodynamics - and that expositions of continuum mechanics don't bother to state the definition because "the student" is assume to have studied entropy as it is defined in thermodynamics? That would be contrary to my interpretation of post #2.

In continuum mechanics entropy is something you calculate by integration using the third law of thermodynamics (entropy of an ideal crystal at 0 K is zero) and heat capacity data that extends from sufficiently low temperatures to the temperature of the considered system. There's no speculation about the microscopic nature of matter involved in it.

Here's how it's done for aluminum oxide: http://www2.stetson.edu/~wgrubbs/datadriven/entropyaluminumoxide/entropyal2o3wtg.html . It's also explained in most standard physical chemistry textbooks. Note that in tables of thermodynamic quantities, molar entropies are listed as absolute quantities, not values relative to something else, like the enthalpies of formation ##\Delta H_f## are.