The unfortunate fact in the contemporary common knowledge continuum mechanics - The Homogeneous Continuum Mechanics (HCM), is that the treatment of the Heterogeneous Media, Matters being performed as that they are being indistinguishable from the Homogeneous media treatment!
In Continuum Mechanics through the last near 40 years workers do not recognize the other then the accepted in HCM since ~50ths-60ths Homogeneous mathematical modes (not the theories we can name) by Beran, Hashin, Hill, among others and continued later in 70ths -- 2000ths by Mura, Nemat-Nasser and Hori, we name just few (many can be mentioned for their sizeable input) English writing authors, with the substantial influence as seems throughout the publications of others. The same Homogeneous one scale approach for Heterogeneous, Scaled media has been taken by authors and in Russian continuum mechanics literature.
The disastrous influence we can say as long as these and like works, studies have persistently stuck to the idea of using the only tool - the Homogeneous Gauss-Ostrogradsky Theorem (HGOT or GOT) even for the Heterogeneous media. Meanwhile, this is wrong since 1970-80th and I think they know that this is wrong, but personal issues prevail mostly over everything else in science.
The countless studies appeared in this mode and continue to fill out the literature on Heterogeneous Media (composites) Continuum Mechanics, Structural Mechanics, etc. published in scientific journals and books.
We analyze in this section-science related to Continuum Mechanics many studies of workers that know that they are doing something inappropriate, wrong actions in progress, some of them for more then 14-16 years, since they got some knowledge of HSP-VAT, that the heterogeneous scaled description in this field also is possible, and what's more - it started to move forward.
We depict below that few of workers know about the situation and attempt to find out their "own" way of doing "averaging" or claim for "new" theory, "method" of doing the "multiscaling" meaning the Upper scale mathematics. But mathematics is just stricter than physics.
This gives everything down the road in these modeling efforts belonging either to homogeneous matter or to incorrect description of heterogeneous phenomena, matter, and governing equations.
A reader instantly has a decisive and "killing" straightforward question - Well, so why all the conventional GO continuum mechanics disciplines are well done and seems are not in contravention of basic laws and considered as flawless fundamental sciences?
The answer might get through the many pages of text, but in short it is: Everything is looking good and successful, because the "science" of coefficients provides for ends meet. Or, the left and the right parts of physical definitions and their mathematical expressions in the Homogeneous like Heterogeneous issues are well agreed (provided) when approximate adjusting bulk coefficients used in the right hand side of the equations.
These coefficients are derived based on the same wrongly homogeneous, pretending to be heterogeneous, concepts.
Their definitions (of coefficients) are incorrectly used as for homogeneous media properties, and performance used to get closer to the homogeneous definitions. Nevertheless, those are either grossly incorrect or discussion on approximations going on for decades – Which of the methods is “correct” now?
Besides, many heterogeneous, scaled issues studied as homogeneous ones in the conventional CM and physics literature having nothing to compare with while going down the road. Because there are no hard media publications on the Ht Continuum Mechanics, but our few fairly related to HtCM HSP-VAT texts in other sciences, partly in this website. And this is not an oversight.
We discuss and make examples of that in many places of this website.
For example, in the chapter by Beran (1974) in pp. 239-242 given the solid piece of experimental (morphological mainly) study done for the purpose of comparison the statistical theory based results on effective constants in heterogeneous medium. A useless study because the theory as it was asserted in 60-th - 70-th (and later on) is useless also (see our analysis below).
Remember from the history that the science of chemistry did well and also flourished until the periodic table of chemical elements was discovered?
After that, who had paid attention to the chemical (and other) studies done without knowledge of the chemical elements table and practice of using it?
In this section we will submit various evidences that the some foundations into the formulation of Elasticity theory, Plasticity, ViscoElasticity -- as the Heterogeneous Two-Scale VAT Governing Equations and models have been developed through the last 16-18 years. That has been done along with the concepts of Heterogeneous media problems treatment specifically within these fields.
Provided that, we would be glad to publish in this site some analysis of commonly known tasks and their handling while applying the Homogeneous and Ht Elasticity, Ht Plasticity, Ht ViscoElasticity theories. We are talking about the various faces of Homogenization theory considered in HCM as the only possible way of solving the heterogeneous problems, "multiscaling" for Heterogeneous media.
Among interesting problems we contemplate on are the Elastic Solids (Heterogeneous) with Microcavities and Microcracks; Elastic Solids with the Inclusions -- with the second Phase; Elastic Solids with Periodic Microstructures, including the 1D Superlattices; etc.
As everyone knows that the conceptual approaches developed in Continuum Mechanics (we repeat here - within the Homogeneous CM) prevailing for many, many years over the approaches used or "allowed" to use in all the neighboring disciplines. Now we can say that the continued use of the Homogeneous CM for almost any important task formulated on Elasticity, Plasticity, or/and ViscoElasticity for Heterogeneous materials in industries is costing a lot not only in terms of financial loss, but with the loss of possibilities for obtaining a correct treatment or solution, or design of the device.
Who is the most unfortunate that are the students who study this discipline- they having been taught the same stuff for Homogeneous and for Heterogeneous problems to many generations of students by more than two generations of professors! But they are not alone - variations of the same stuff are taught to students through the last 30 years in Multiphase Transport, Materials Science, Composites courses, etc. In Meteorology is the outrageous case when the wrong governing equations and mathematics of the Upper scale equations and models provided at the Environmental Engineering and Meteorology departments as the "state of the art" in the craft, see our - "Modeling and Averaging in Meteorology of Heterogeneous Domains - Follow-up the NATO PST.ASI.980064"
Beran, M.J., "Application of Statistical Theories for the Determination of Thermal, Electrical, and Magnetic Properties of Heterogeneous Materials," in Mechanics of Composite Materials, Vol. 2, Ed. G.P. Sendeckyj, Academic Press, New York, pp. 209-249, (1974).