What is in use in Continuum Mechanics of Heterogeneous Media as of Through ~1950 - 2005 ?

SECTION 1.4 ANALYSIS (LIMITED AND SELECTIVE) OF "HETEROGENEOUS" THEORIES FOR HETEROGENEOUS MEDIA (MATERIALS) EXISTING UP TO BEGINNING of THE XXI CENTURY IN CONTINUUM MECHANICS.

THAT IS UNBELIEVABLE, PEOPLE WANT TO DO THIS OBSOLETE SCIENCE UP TO THEIR RETIREMENT?

To understand why the Homogeneous GO theorem based studies are insufficient, softly said, or incorrect for the Heterogeneous media problems in Continuum Mechanics we have to study laboriously few prominent, influential probably studies of such kind. The other way around might give an impression that we mostly bluntly criticize what was done by workers in this way of science making. That is not our goal.

We have the sufficient place (and desire or possibility) here for considering these studies thoroughly, because the nature of this way of doing science and business in multiphase heterogeneous media Continuum Mechanics mainly in the same mode of mathematical methodology as in other fields of physics representing a treatment of Heterogeneous Media as a kind of Homogeneous ones, and that we studied many years ago while publishing on the subjects as well as uploading and saying on that in many parts of this website.

Meanwhile, because this field - the Continuum Mechanics is forming the base for many sub-disciplines particular in Solid State mechanics, we ought to discuss on few major developments as on examples of wrongdoing, of false science, that strangling Continuum Mechanics for more than 40 years.

Keeping in mind what was said above in Sections 1.1 - 1.3 we might now considering the tedious comparing and analysis of those homogeneous GO studies using the point of view as of the Heterogeneous WSAM theorem and of this kind other HSP-VAT advancements, found throughout the last 40 something years.

If reader has enough qualifications and patience to read this analysis up to the end, that is my goal to help professionals and students to understand what's wrong in this field and that the way out of this mess with CM actually was found in 90s first through the WSAM rather theorems (HSP-VAT), and then the Heterogeneous (Ht) Continuum Mechanics started sooner to stand on its own. Right now we have done just few basic initial steps.

Meanwhile, don't get me wrong - I am not abolishing the conventional Homogeneous one scale CM, I just saying that it is the part, the inherent native part of the broader entity of Heterogeneous CM, see our "Fundamentals .....". Say, nobody seems right now be so stupid saying that the Ht Thermal Physics or Ht Fluid Mechanics are false subjects? Or somebody in the traditional CM might dare to do this?

Also, there are no personal attacks on someone, as I've heard in the past, read in our -

  • " Announcements"

    This is a healthy critique to force people stop doing silly things and teach students wrong subject. I am not so naive to believe they can do this for a moment. The publicity might get them stop doing this.

    ~~~~~~~~~~~~~~~~~~~~~~~~

    Let's start with the good old publication incorrectly presenting the subject. There have being published hundreds like that.

    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).

    So, Why this the very good and general quality paper did not get much of attention and the method either have not brought after more than 30 years much of results ??

    Beran (1974) stressed the attention (focused ) on the methods of statistical treatment of elliptical second kind partial differential equations. That is common base mathematics for homogeneous matters in many physical disciplines.

    He discusses both an ensemble averaging as well as a volume averaging. Author speaks on effective coefficients, also tells about the problem of the design of composite materials with needed effective coefficients.

    The basic starting field's distribution equation (3)

    MATH

    after averaging has the form

    MATH

    and we note that in this statistical averaging there is no techniques, recognition of the specifics for the left hand side operator's averaging. No difficulties with the GO theorem conditions fulfillment or about the WSAM theorem application if any at all?

    Also, we have to note here that in this kind of problem analysis and evaluation, including a solution, no place is for material's consideration - just a statistics, no place is for the phase (phases) description, no place for interface description, and on the physics of interface elaboration. There are the statistics of morphology and morphology parameters that are not directly tied to the GE above.

    That is out of consideration because there is no tools and power for doing this in the statistical mechanics of one scale.

    There is no talking about the first scale (there is no scale consideration at all) physics and its mathematical representation.

    Taking an ensemble average of equations is no problem at all in these types of statistical theory of heterogeneous materials.

    1) The simple treatment of effective coefficients problem is given in section V as for the equation

    MATH

    that is said needs to be compared to the original equation with the constant coefficient

    MATH

    In this comparison, while actual substitution of one equation by another there are so many issues mixed and out of description. Among them the vital one is the issue of how to justify the connection of the one initial equation to the effective coefficient equation?

    Another natural question - Is this equation with effective coefficient correct? What kind of fields (all of them) is used in these equations, etc.?

    Regarding the very important issue of the effective coefficient bounds we need to recognize that using the homogeneous concepts and theoretical constructions we inevitably developing the same kind of useless characteristics on bounds.

    In section V we can find the text in p. 232: "This is so because as we know from bounds we shall present below that it is impossible to specify $\epsilon ^{\ast }$ from just a knowledge of low-order statistical information."

    While this is the wrong statement generally, we would like to point out to the tools of HSP-VAT.

    The matter is that the techniques of HSP-VAT gives the ability to model, simulate and make calculation practically of an exact nature, as long as we are able to figure out the morphology features (that is now accepted as a solvable task) and willing to invest into the much larger then the usual mathematical physics project that is the two scale (at least) HSP-VAT mathematics simulation.

    I would say - What is the point in finding the bounds of product 2x2 if then saying whether it result will be more than 3 or/and less then 7, if someone can calculate the exact value which is 4!? The similar story is with the bounds for effective coefficients for heterogeneous media.

    2) Thus, the substantial portion of section V given to the detailed analysis of statistical theory using the three-point correlation function for assesments of bounds for efective coefficients.

    These issues have no sence to consider. Thou, efforts in finding the effective coefficients (constants?) and their bounds by homogeneous tools for heterogeneous materials used for more then 40 years by numerous researchers are examples of waste of resources. The most valuable among them are the human resources.

    3) Section VI is devoted to experiment which is meant to be in line with the assessment of the some three-point correlation functions. All of this discussion on the issue of three-point correlation function is actually based on and brought in life by Miller's (1969) "cell model for real materials."?

    This model is quite simplistic and is far beyond of correct construction for averaging, averaged fields, and averaged equations.

    4) In section VII on "Applications" done the text that could be seen as the introductory text to a research on the purpose of all these efforts in materials science, composites.

    We can read in p. 243: "The principal application of statistical ideas to problems of heterogeneous materials will probably be in the design of materials with desired effective constants. The simplest way to illustrate this point is to consider the design of a two-phase composite material that has a maximum effective thermal conductivity.

    For example, high thermal storage-low thermal conductivity materials are sometimes seeded with high-conductivity inclusions to increase their effective conductivity. Since the inclusion material is usually expensive relative to the matrix material, it is important that inclusion shape and packing be optimally chosen."

    Also, in p. 244 we can read on the same: "Unfortunately, at this time, no one has published specifications that will tell the engineer how to place and size-distribute the inclusions in order to obtain the maximum value of MATH "

    Since these 30+ years passed then anyone has heard of the solution of this problem ?

    Not at all. People even stopped to discuss and promise on this subject.

    During the last 40 something years these kind of principal problems are not solved. The reason is the same - for these two- and more scale multiphase problems models used are insufficient in mathematical sence for claiming answers to these questions on optimization.

    See more explanation into our -

  • " HSP-VAT Design of Scaled Devices"

    That is why the final section VII is full of incorrect concluding remarks and wishes.

    ++++

    5)For example, in the chapter by Beran (1974) in pp. 239-242 given the solid piece on 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.

    6) On the Dyson Equation said that:

    Page 226 - "IV. Equations Governing the Mean Field

    In section III, we pointed out that in order to determine MATH we must in principle solve an infinite hierarchy of equations that contain all the statistical information on the MATH field."

    That's the BIG,BIG difference with the VAT procedures, where no infinite series of GE.

    Page 226 - "In this section we wish to show that MATH formaly satisfies an integro-differential equation usually termed a Dyson equation. The kernel of the integral portion of the equation is a function of all statistical information in the MATH field. Since all the statistical information is contained in a single kernel function, this equation may be used phenomenologically in the absence of detailed information about the MATH field.

    The character of the integrodifferential equation is quite different from that of an ordinary differential equation, and we shall point out the need for such an equation near rapidly varying source functions or boundaries."

    While in the stochastic approach so nicely described in works like by Beran (1974) one would note that this kind of approach has some features that we can name as unacceptable, for example, such as:

    st1) There is no volume of statistical averaging, but either all the 3D cartesian (or in another geometry) coordinates or the volume (body) of the problem. This assumption instantly brings the question on correctness of the boundary conditions all-together with the principal issue what kind of averaging theorem being applied - the homogeneous GO theorem, or the heterogeneous WSAM kind of theorems (there are many of them).

    st2) Further, the statistical averaging comes with the necessity to consider "an infinite statistical hierarchy of equations" (Beran, 1974; p. 222).

    While we can read in p. 223 - "There is, however, no way in general to find a finite set without some approximation."

    That means - no dreams about the exact two scale, or even only an Upper scale rigorous solution of heterogeneous problem! That is the disappointing conclusion, isn't it ?

    Meanwhile, with HSP-VAT tools we are able to solve either scale - an Upper or Lower, or both scale problems.

    st3) To solve the problem for averaged fields in statistical averaging techniques (Beran, 1974) suggested the only way is by using the integrodifferential equation of Dyson that is

    MATH

    where $K_{ij}$ is represented by the infinite summation. We read in p. 227: "equation (46) is an integrodifferential equation where the kernel MATH is given by an infinite number of integrodifferential operators. In general MATH is very complex for arbitrary geometry even if Eq. (47) " (with $K_{ij}$) " is truncated at the first term."

    We need to note here that this equation is all about the unkown field MATH which is the gradient of our main sought field MATH So, the problem is not solved yet in the case that MATH might be known.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Brenner, H. and Adler, P.M., "Dispersion Resulting From Flow Through Spatially Periodic

    Porous Media. II. Surface and Intraparticle Transport," Phil. Trans. R. Soc. Lond., Vol. A 307, pp. 149-200, (1982).

    This is the useless large laborious study with the good topic and still is useless. See what I had found as by their using of the GO theorem:

    One can read in page 161 the shocking disclosure of their preemptive ignorence on the control volume (REV) selection and on existing of other than GO theorems:

    "Physically irrelevant mathematical questions of continuity may arise from possible real physical discontinuities across $s'$ which is the interface surface (boundary between the phases) "in the event that the unit cell faces MATH (bounding the unit cells $\QTR{bf}{r}_{0}$ externally) coincide partly or wholly with the phase boundaries $s.$ Such unnecessary mathematical complications can be avoided by reverting to the artifice adopted in I of imagining the unit cell faces to be drawn in such a way as to lie entirely within the continuous phase. This choice of "curvilinear unit cell: insures complete continuity of all functions across the cell faces MATH despite possible discontinuities across the phase boundaries $s.$ From the point of view of the physics, the particular mathematical artifice adopted here is irrelevant since the unit cells and their boundaries are but imaginary mathematical constructions, devoid of physical reality."

    What forced these authors to write such an ignorent passage?

    This particular mathematical artifice is the one which allowed authors to draw their incorrect studies, by the way.

    Reading further we find that: "Indeed, ultimately we shall do away entirely with the concept of unit cells and revert wholly to a scheme based upon the introduction of spatially periodic functions satisfying boundary conditions only on the phase boundaries $s$ (see пїЅ6). In such a scheme the unit cell boundaries play no role whatsoever."??

    Which results down the road in page 163 in this formula of such a "divergence" theorem - "The divergence theorem for any tensor-valued field $A$ applied to the domain MATH possesses the form"

    MATH

    that is the homogeneous media GO theorem actually.

    As the consequence of this derivation finally authors found the numerous irrelevant formulae. For example, on the dispersivity dyadic for a Darcy flow in porous media (5.4), (5.15) in pp. 170-171.

    As of the most frivolous we can found the next one on the layered media effective overall conductivity coefficients - (8.45)-(8.47) - "Equations (8.45) - (8.47) apply for any one-dimensional spatially periodic conductivity field D(z), continuous or discontinuous."?

    We might conclude on this work that: Because the time of this paper was 1980-82 that is many years after 60ths and 70ths paper publications on basics of the VAT, then these authors would blame nobody but themselves that their work (Brenner and Adler, 1982; Brenner, 1980) was useless and incorrect.

    Useless study - INCORRECT - because used the GO theorem!

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Russel, W.B., Saville, D.A., and Schowalter, W.R., Colloidal Dispersions,

    New York, Cambridge, Cambridge University Press, 1991.

    This book came later then the one by Happel, J. and Brenner, H., (1965, 1973), and the content is more advanced, still of not such a tedious and calm story on the subject.

    And I would say that the book came under the editorship by Batchelor, G.K., and as such has a lot of references to his works!!

    The book has content which hide the more complicated sense of the field and describe many things in a simplified manner.

    For example, in pp. 35-37 given the text on assessment of stress and strain in the REV, without the description of the REV and that alone makes possible to write the formulae based on the GO theorem with no explanation of why is that?

    In page 397 given the ensemble average velocities calculation based on the following contemplation - "The integrals over the $A_{i}$ (that is the surface of particles - our notation) account for interactions among particles within the representative volume, while the integral over $S$ (this is the boundary surface of the REV- our notation) represents the effect of particles outside. Strictly speaking, $S$ should deviate from the surface enclosing $V$ in order to bypass particles and remain in the fluid."

    The same old story we have been pointing out so often while reading the kind of the similar derivation in any physical field actually - it is about convenience and safe way to say this while using the GO theorem. But it is grossly incorrect - see in our pieces of text in

  • "Effective Coefficients in Electrodynamics"

    Reading further - "However, in the dilute limit taking $S$ to coincide with the surface of $V$ introduces an error of only $0(\phi ^{2}).$"

    And where they took this estimation? This statement is untrue, because the WSAM theorem should be applied in this case and this alone might give an error even in hundred percent for estimation of average fields and their operators.

    Going further we can read and find in the pages 493-497 in "Macroscopic stresses" that formulae (14.10.2) and other in this subsection that relied heavily upon the Homogeneous divergence theorem are incorrect and thou the estimations are incorrect too.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Truesdell, C., A First Course in Rational Continuum Mechanics, Baltimore, John Hopkins Univ., (1972).

    Firstly I would like to say that this fundamental monograph probably will not serving to students to the up high level as it is written with and was earlier mentioned over in many publications.

    The reason is the same as we argue here - because this is about physics, mathematical physics, that means that author and other public interested in progress in the field need from time to time ask themselves - Any communication to reality we can observe or not? Any comparisons with the real world we have to do or not now ?

    The science of Continuum Mechanics that is the reflection of some concept that the many fields in physics can be worked out with the single cartesian coordinate system - No matter what!

    So - how about an other or two of physically connected fields?

    This book gives no answer to the question and would not, because it is about one scale for all occasions. Even the most complicated physical phenomena described with the mathematics of a one scale. Period.

    There is no need in this book to talk on Heterogeneous media.

    Theorem of divergence, of coarse Homogeneous GO, mentioned only in the mathematical supplement.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Berryman, J.G. "Variational Bounds on Elastic Constants for the

    Penetrable Sphere Model," J. Phys. D: Appl. Phys., Vol. 18, pp. 585-597, (1985).

    (This work done at Lawrence Livermore National Laboratory. P.O. Box 808. L-200. Livermore, CA 94550).

    (Sponsored by - under the auspices of the US DOE under Contract W-7405-ENG-48 and supported specifically by the Base Technology Program of the Earth Sciences Department.).

    This is the example of the waste scientific product - and I use this one as an example while analysing it for it's "scientific" value for the Milton numbers - for spatial correlation functions?

    In page 585 we can read: "Since Hashin and Shtrikman (1963) introduced their variational principle for bounds on elastic constants of composites, many improvements on these bounds have been found (Beran and Molyneux, 1966; Miller, 1969; McCoy, 1970; Silnutzer, 1972; Milton and Phan-Thien, 1982). The improved variational bounds always require statistical and geometrical information about the random composite - usually in the form of spatial correlation functions."

    and further "... The penetrable sphere model is a hypothetical random material composed of spherical particles randomly imbedded in a host material. The locations of the sphere centres are uncorrelated in this model so the particlas do overlap and the spatial correlation functions are relatively easy to calculate. This model plays several important roles in developing the theory......"

    in page 586:

    " The present discussion is limited to the three spatial correlation functions measurable (Corson, 1974; Berryman, 1985) using photographs of cross-sections of the material:


    MATH

    MATH

    where the brackets indicate a volume average over the spatial coordinate $\QTR{bf}{r.}$ The volume fraction of constituent 1 is given by $\varphi .$ In writing equations (2) - (4), it has been assumed that the composite medium is statistically homogeneous so that on the average only the differences in the coordinate values are important."

    in page 589:

    "...various formulas providing estimates of physical constants for composite materials are known (Beran, 1968). Milton (1981, 1982) has recently shown how to simplify the formulas of Beran and Molyneux (1966) and of McCoy (1970) by introducing two parameters ( $\zeta _{1}$ and $\eta _{1}$) which depend on the microgeometry of the random composite through the three-point correlation function $S_{3}.$.."

    in page 590:

    " Three averages of any modulus M can be defined according to

    MATH

    MATH


    MATH

    where MATH and $M_{0}$ and $M_{1}$ are the moduli value in corresponding phase 0 or 1. THAT'S JUST SIMPLE WAY TO FIND OUT THE EFFECTIVE COEFFICIENTS!!

    in page 596: "The main results of this paper are the values of the Milton numbers for the penetrable sphere model which are presented in table 1 and figure 2. These tabulated values may be used to provide estimates of elastic constants in a wide variety of situations assumimg that materials of interest have geometries which do not differ too much from that of the model. Until spatial correlation functions of real material become readily available, this model will provide a useful test case with narrower limits that those for the Hashin-Shtrikman bounds. We would expect this model to be a reasonable approximation to the geometry of a sintered powder or other synthetic materials made by heating and/or pressing aggregates of particles to form a cohesive solid."

    Useless study: because that was done for characteristics that are not shown in governing equations or parameters of interest, but just declared as useful - these bounds of elastic moduli that are not correct themselves. What for are these mathematical correlation dependences estimated??

    In more than twenty years since then I have not heard of implementation - that might happen, that again being useful for scientific and technology points of view.

    All is this happen because authors of this direction (finding the bounds of elastic constants of composites) took the modeling mode for their studies based on Homogeneous GO theorem, nothing less.

    ~~~~~~~~~~~~~~~~~~~~~~~

    Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics,

    Sijthoff and Noordhoff, Alphen aan den Rijn, (1973).

    We need to say more on this book - in terms of COLLECTIVE BEHAVIOR Problems.

    In this book (we may say almost or classical good book) two last chapters - 8th and 9th are devoted to the matter of suspension moving in liquid while considering mainly the very dilute and the more dense slow flow regimes.

    The whole content of these two chapters are to make assessments of dynamics of not a separate particle movement, but rather to describe and to find out the parameters of the "clouds" of particles -actually averaged characteristics of the Upper scale physics.

    At the same time, there is no concepts or features (remind that was only the 60th and beginning of 70th) describing the second (Upper) scale field's properties that are in fact sought in studies described and formulated in these two chapters. These are averaged characteristics of bulk effects of the second dispersive phase as, for example, nonlocal pressure drop $\Delta P_{s}$ in liquid, differential volume element $d\tau $ (as REV), permiability coefficient $K$, filtration velocity $U,$ effective viscosity of suspension $\mu ,$ etc. etc.

    This the one scale handling of problems do not allow to obtain correct analysis and mathematical formulae. For example, in chapter 9 we can find the definition of effective viscosity of suspension in expressions (9.1.1), (9.1.2) that are incorrect as soon as the boundaries should not be considered as homogeneous surfaces, etc. etc.

    We elaborated the two-scale theory for suspension flow in liquid and had gotten the exact (or mathematically most strict) solution and simulations for few such kind of problems with Kushch in 1994-1996. These advancements were not published in open literature up to now, while we hope this take place in a future.

    Commonly, these two chapters in Happel and Brenner (1973) book give a nice, broad analysis of single scale attempts to solve these broadly stated Two-scale heterogeneous problems - the flow of suspensions, and the flow in porous media. The chapters can be referred as for studying what can not be done with the one scale mathematical physics statement while the two scales are natural for problems in this field.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Hashin, Z., Theory of Fiber Reinforced Materials,

    NASA CR-1974 Report, Univ. Pennsylvania, Philadelphia, pp. 1-690, (1972).

    This is the useless 700 hundred report (monograph) because of the GO theorem used.

    Will be commenting on the formulae and developments !

    There were a lot of citations (references) in 80ths and 90ths (I remember myself) to the publications authored and co-authored by Z.Hashin, for example, see (1965a,b, 1970, 1972, 1983). In these studies dated back to the beginning of 60ths there were already proclaimed the theory which supposed to explaine almost about everything in continuum mechanics of composites (Heterogeneous Media) - elasticity, heat transport, electromagnetic fields basics, strength, etc.

    The whole course of developments in these studies is based on artificially (we explain why we use this word) simplified concepts and mathematics of bulk averaging through the Volume - Surface GO theorem, and the Virtual Work Theorem.

    Using the assigned one phase boundary REV see, for example Fig. 2.1.2 in 1972 report for NASA by Hashin (1972) -

    MATH

    which explains why much of the formulae in this report, for example, like (5.3.5), (5.3.21) that are the key formulae among others, are illustrations of erroneous mathematics.

    On page 1 we read: "From the engineering point of view the most important type of composites at the present time are Fiber Reinforced Materials (...FRM)."

    On page 3: " it is an attempt to present a theory of FRM which is reasonably rigorous and is at the same time oriented towards the engineering uses of such materials."

    In page 534 -

    The average of gradient of field is given as "the gradient theorem" in the homogeneous interpretation

    MATH

    We can read in the same page; "The proof is immediate: The volume integral in (5.1.3) is converted to two phase region surface integrals by means of the divergence theorem. The surface integrals on the interface cancel because of (5.1.1c) and the reversal of the interface normal. Then (5.1.5) follows at once."

    While this is INCORRECT for Heterogeneous Materials.

    Now it is the good time of questioning - And where are these materials, texts, research, promises, design hopes and models with their embodiments based on this and like this theories now ? After 30-40 like years period of utilization where are the proofs of their usefulness ? Or correctness?

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Hori, M. and Nemat-Nasser, S., "On Two Micromechanics Theories

    for Determining Micro--Macro Relations in Heterogeneous Solids," Mech. Materials, Vol.31, Issue 10, pp. 667-682, (1999).

    Abstract

    The average-field theory and the homogenization theory are briefly reviewed and compared. These theories are often used to determine the effective moduli of heterogeneous materials from their microscopic structure in such a manner that boundary-value problems for the macroscopic response can be formulated. While these two theories are based on different modeling concepts, it is shown that they can yield essentially the same effective moduli and boundary-value problems. A hybrid micromechanics theory is proposed in view of this correspondence. This theory leads to a more accurate computation of the effective moduli, and applies to a broader class of microstructural models. Hence, the resulting macroscopic boundary-value problem gives better estimates of the macroscopic response of the material. In particular, the hybrid theory can account for the effects of the macrostrain gradient on the macrostress in a natural manner.

    Author Keywords: Micromechanics; Homogenization; Average field theory; Multi-scale anisotropy

    Corresponding author. Tel.:+1-619-534-4772; fax:+1-619-534-2727; e-mail: sia@halebopp.ucsd.edu

    .......................

    Here we need to make a dear note that neither any of authors mentioned in the work as Hill (1963), Mura (1987), nor authors themselves are experts or really made a direction in a correct "average-field" theory.

    To our delight and experience we observed and discussed few years earlier (seems in 2003-2004) with one of the authors (Nemat-Nasser) the work itself and outcome produced in this publication.

    Authors writing deserves to be commented here for clarity of agruments. As to the statement, for example, in page 667 where one can read that: "Average-field theory. This theory is based on the fact that the effective mechanical properties measured in experiments are relations between the volume average of the strain and stress of microscopically heterogeneous samples. Hence, macrofields are defined as the volume averages of the corresponding microfields, and the effective properties are determined as relations between the averaged microfields."

    This definition is incorrect and incomplete. We would not be surprised that using this definition authors came to their conclusions.

    What is incorrect regarding this statement: a) "effective mechanical properties measured in experiments are relations between the volume average of the strain and stress of microscopically heterogeneous samples," - these properties are the result of experimental set-up made for Homogeneous medium, and as such bearing the features of GO homogeneous medium theorem based experimental set-up. Not a Heterogeneous Medium Experiment (HtME) on Elasticity properties. This is the experiment (HtME) where the dependency of stress on a displacement field as of an averaged field usually established, thus this dependency would consist also within the HtME provided with the surficial integrals for the displacement fields over the interfaces within the measured volume, within the REV or one of possible REVs taken as for this curtain experiment. So, This should not be the Homogeneous experiment, if we are determined to deal with the our Heterogeneous medium. Well, in this case we have to obtain the the Heterogeneous dependence between the average displacements and average stress in the whole Heterogeneous medium, in the separate phase(s) of the medium.

    b) The Upper scale elasticity model fields are determined throughout not only the Lower scale microelasticity fields, but also and that is the major constitutive part of Upper scale physics, by the Upper scale Ht Governing equations solution for the problem's domain Heterogeneous medium. Otherwise, if the Upper scale GE would be accepted as of Homogeneous medium, then the Upper scale Effective Coefficients (EC) and Boundary Conditions (BC) would not the conventional Homogeneous medium EC and BC.

    c) In many, most probably, situations the Upper scale statement conditions are of the prevailing importance for the Heterogeneous problem (elasticity in this case) and as such the Upper scale mathematical statement must be stated rigorously, as much strict as the theory allows. That means - the Lower and the Upper scale GEs must be constructed so directly mathematically tied, as they are in physical nature of the problem, usually. That means also - that the Top-Down as well as the Bottom-Up sequances should be used for mathematical formulation of the Two Scale Problem. Also, the Upper scale GEs formulated as the conventional homogeneous statement are incorrect if taken separately from the correct Lower and Upper scale Heterogeneous GEs as in this paper, for example.

    In this paper we would find the incorrect Upper scale averaged governing equation for the displacement field MATH, Eq. (12) in page 670

    MATH

    which was obtained, and authors do stress out on that, with help of using the GO theorem.

    By the way, in this equation the average displacement field MATH is the field averaged over the entire elastic body $B$ !?

    So, in this way there will be no solution, properties and analysis of phase distinguished fields can be done. No interaction between phase depending properties, etc., etc. ? Strange development.

    This governing equation (12) for supposed to be the averaged field MATH is of the same look and actually exactly the same as the governing elasticity equation (5) in page 669, for homogeneous elastic body as we know it

    MATH

    The main conclusion of the study in the page 680 is that:

    "It is shown that the two micromechanics theories, the average-field theory and the homogenization theory, can be related to each other, even though they are based on different concepts. In particular, the first order terms in the expanded strain and stress fields of the homogenization theory correspond to the average-field theory."

    We can remark on that - this is obtained because the initial statements for the so-called "average-field theory" were incorrect, following from this the final output conclusion stated in the paper is incorrect also.

    There is no one published source of information on true averaged scaled physical fields theory (VAT, HSP-VAT) referred in this paper and in the book by the same authors. Referred works by Beran, Hashin and others are the incorrect ones in terms of averaging statements and mathematics.

    Nevertheless, we can draw the one at least useful conclusion regarding this work study which is - This study is a unique probably example of comparison (incorrect anyway) of averaging (incorrect) GE and of the homogenization GE for the same physical task. The results authors obtaine are incorrect in terms of comparison - and this gives us the GOOD EXAMPLE of DIFFERENCE between the averaged correctly (HSP-VAT so far is the only way) elasticity in heterogeneous medium statement and of the Homogenization theory.

    The other examples given in the Homogenization theory are actually comparisons of the single scale different notation expanded and of the initial GE and have no interest for Multiscale development.

    Further, we can observe in the Fig. 2 (paper's page 669)

    MATH

    that the boundary surfaces for the REV are placed within the single phase. Of course, this is incorrect. One can guess, that the deposition of second and more phases, defects, etc. within the embracing matrix is taken for purpose, see our analysis on this artificial arrangement in -

  • "Effective Coefficients in Electrodynamics"

    In the introduction in page 667 we can read that: "One major objective of micromechanics of heterogeneous materials is to determine the effective overall properties by certain microscopic considerations. The effective properties are then used to evaluate the response of structural elements which consist of heterogeneous materials. Consider two basic approaches for obtaining the overall response of a heterogeneous medium: (1) the average-field theory (or the mean-field theory) and (2) the homogenization theory. Roughly speaking, these are physics- and mathematics - based theories, respectively. Here, the basic characters of these two theories are interpreted as follows: "

    "...Homogenization theory. This theory establishes mathematical relations between the microfields and the macrofields, using a multi-scale perturbation method. The effective properties then naturally emerge as consequences of these relations, without depending on specific physical measurements."

    " In this paper, we are primarily concerned with establishing a link between the average-field theory and the homogenization theory, clarifying their similarities and differences."

    ++

    In the Page 674 authors say: "There are, however, two major differences between these two theories. The first difference is the modeling of the microstructure: the homogenizatio theory uses a unit cell of the periodic structure, while the average-field theory considers a representative volume element of a statistically homogeneous body.

    The second difference is that the homogenization theory is able to treat the macro-micro relations more regorously, allowing higher-order terms in the singular perturbation expansion."

    Both statements are wrong - in the HSP-VAT one not necessarily uses the "statistically homogeneous body" and this is the great advantage (we spoke few times on that); the HSP-VAT just gives with the proper mathematical statement the Exact Solution on both Scales that is opposite to the Homogenization theory always approximate solutions for the Lower scale.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Nemat-Nasser, S. and Hori, M., Micromechanics: Overall Properties of

    Heterogeneous Materials, 2nd edition, Elsevier Science B.V., Amsterdam, (1999). 786 p.

    .....................

    In this fundamental book given the parts related to averaging procedures, theorems, etc. all aplied toward the Elasticity theory for Heterogeneous materials. But the basics for averaging made of conventional Ostrogradsky-Gauss theorem, not heterogeneous theorems by WSAM (Whitaker-Slattery-Anderson-Marle ), see, for example, pp.59-60.

    Page 11 we see the determination of the REV (RVE in notation by authors) - "An RVE for a material point of a continuum mass is a material volume which is statistically representative of the infinitesimal material neighborhood of that material point. The continuum material point is called a macro-element. The corresponding microconstituents of the RVE are called the micro-elements. An RVE must include a very large number of micro-elements, and be statistically representative of the local continuum properties."

    MATH

    This is only the part of the normal definition of the REV, but in its part it is O.K.

    The missed points are the descriptions and properties of the two scales and how the REV and the Lower scale medium are corresponding one to another?

    For What reason we consider the REV ? For the macro-element, point of the Upper scale, and what more for?

    Page 44 gives the unjustifiable communication of the Boundary Conditions on the boundary of REV ?? to the conventional BC as "The RVE is regarded asstatistically representative of the macroresponse of the continuum material neighborhood, if and only if any arbitrary constant macrostress $\QTR{bf}{\Sigma }$ produces through (2.5.26a) a macrostrain MATH such that when the displacement boundary conditions (2.5.27a) are imposed instead, then the macrostress, MATH , is obtained, where the equality is to hold to a given degree of accuracy."

    To this point we must ask -

    What are the BC on the boundary of REV ? For what aim? Is this the problem's domain as a whole ? Seems authors of the book considering this option.

    The excerpt in the page 471 shows explicitly that authors bearing in mind the averaging procedures in the book as for and the one scale methodology. We read - "The second important difference between the concept of an RVE and the periodic model is that, through the application of a Fourier series representation, the periodic (elasticity) model can be solved essentially exactly in many important cases, whereas in the case of a RVE, only estimates based on specialized models (e.g., the dilute distribution, the self-consistent, and the differential models) are possible."

    Which is stated as such because authors do not believe as for existance of the second scale physics and mathematics, although, this statement is also incorrect, as soon as there are the Upper scale problems solved for many instances. The most clear, simple, and straightforward of them can be seen in our texts even and mostly here in the website.

    We have shown in many places of this website, by means of the Two scale solutions, especially with the exact Two scale solutions of those few common textbooks known problems, see in -

  • "Classical Problems in Fluid Mechanics"

  • "Classical Problems in Thermal Physics"

  • "Globular Morphology Two Scale Electrostatic Exact Solutions"

    that this new kind of Mathematical Physics old problems can be successfully tackled and solved either.

    While also obtained after 2002 the analytical (and numerical) solutions of the following classical problems that have not been solved for many decades by other methods (given in textbooks the Lower Homogeneous scale "solutions" are wrongly attributed to the Upper Heterogeneous scale averaged fields)

  • "When the 2x2 is not going to be 4 - What to do?"

  • "Two Scale EM Wave Propagation in Superlattices - 1D Photonic Crystals"

  • "Two Scale Solution for Acoustic Wave Propagation Through the Multilayer Two-Phase Medium"

  • "Effective Coefficients in Electrodynamics"

    These solutions leave no chances for calculations or comparison with experiment of the Upper scale characteristics using the basis of Homogeneous GO. This has no sense, invalid for Heterogeneous problems.

    +++

    These two notions below have the common ground applicable and to this book, that say that the physics of heterogeneous (composite) materials should comply with:

    A) The physics of each separately taken smallest distinguashable Homogeneous part (grain, crystalline, bead, etc. ) of material. Most often this is the piece of a separate phase in a multiphase material.

    B) The physics of overall properties of the material as the whole Upper scale body (the whole volume of given composite material), or as of a substantial part of it containing within itself the infinite number of REVs.

    One of the ground concepts in the book by Nemat-Nasser and Hori (1999) is that the averaging is done over the REV with the applied boundary conditions to the REV's boundary? One needs to ask then - Either this is the whole problem's volume or there is not a REV averaging that supposed to be used in the "averaging" procedure. All this is done probably to prevent connection of neighboring REVs and finding the correct Upper scale governing equations (GE)?

    We can read with amazement, see p. 38, that - "It is seen that the discontinuity in the field quantities does not influence the average stress, strain, strain rate, and the rate of stress-work, if the tractions, the displacements, and the velocity fields are continuous in V."?

    Obviously, just observing the Fig. 2.4.1 given in page 36

    MATH

    which embraces more than 3 phases of the 3D nature and the cracks within the REV which can be considered as 2D phases it's hard to accept the homogeneous concept of elasticity for heterogeneous materials being developed by authors.

    That is, the mathematics and physics of multiphase media arrived at in the Chapter 1, Section 2 are all wrong as long as the whole deductions were achieved throughout the use of Homogeneous Gauss-Ostrogradsky theorem

    MATH

    where $\partial S_{io}$ is the input-output surface surrounding the volume $\Delta \Omega $.

    For a Heterogeneous medium! That all means no distinguishing of the one phase from other phases in a 3D (and 2D, 1D) cases, by taking into account interplay between the phase's related phenomena, etc. etc.

    Actually, in this way the body of solid state mechanics elasticity theory, plasticity theory and related areas being treated as the ordinary Homogeneous media theory, the one is known to professionals and published intensively about. These mathematical models and treatments do not allow to get even correct results, not saying about searching for the properties related effects, see in our -

  • "When the 2x2 is not going to be 4 - What to do?"

    It is of general surprise that even the products of physical fields declared as the linear systems?! And averaged products macrofields are linear - pp. 65-66 also?.

    The content of the book done in the fashion just mimicking (imitating) the theory, description, modeling and simulation of the solid state heterogeneous media, composites mechanics as in the areas of elasticity theory, plasticity and, of coarse, the strength mechanics. Serving for the false hope and effort to reach the truth.

    +++++

    The second failure after using the GO theorem for heterogeneous media as for homogeneous ones in continuum mechanics(in almost 100% of books) is the acceptance and even sometimes "proves"(?) that the averaged product of fields is equal to the product of averaged fields!?

    MATH

    We can read in p.54 (Nemat-Nasser and Hori, 1999) - "Thus, averages taken over any RVE in B$^{^{\prime }}$ are essentially the same as those taken over B when the RVE is suitably large. For statistically homogeneous B, it therefore follows that

    MATH

    where MATH and MATH are the prescribed uniform farfield strains and stresses."

    This is the false statement.

    This expression assumption allows authors to do in many important issues unbelievable simplifications and stretching.

    For example, in pp. 54-57 (Nemat-Nasser and Hori, 1999) we can read the reasonings ("proves") that the operation of differentiation commutes with the operation of local averaging!

    So, there is no need to incorporate the WSAM theorem for heterogeneous media !!!

    ++

    An Extract taken from the HSP-VAT course on "Heterogeneous Electrodynamics ...":

    ''In accordance with one of the major averaging theorem - theorem of averaging $\nabla $ operator, the WSAM theorem (after Whitaker-Slattery-Anderson-Marle) the averaged operator $\nabla $ becomes

    MATH

    Meanwhile, the foundation for averaging made, for example, by Nemat-Nasser and Hori (1999) (and many others) is based on conventional homogeneous Gauss-Ostrogradsky theorem (see pp.59-60), not of its heterogeneous version as the WSAM theorem.

    The differentiation theorem for intraphase averaged function is

    MATH

    where $\partial S_{w}$ is the inner surface in the REV, MATH is the second-phase, inward-directed differential area in the REV ( MATH = MATH).

    The same kind of operator involving $\QTR{bf}{rot}$ will result in the following averaging theorem

    MATH

    also as its consequence the another theorem for intraphase average of MATH is

    MATH

    ~~~~~~~~~~~~~~~~~~~~~~~~

    Chen, J.-S. and Mehraeen, S., "Multi-scale modelling of heterogeneous materials with fixed and evolving microstructures,"

    Modelling Simul. Mater. Sci. Eng., Vol. 13, pp. 95--121, (2005).

    MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING;

    Jiun-Shyan Chen and Shafigh Mehraeen, Modelling Simul. Mater. Sci. Eng., 13 (2005) pp. 95--121.

    This is the rather thick paper with the claim for Multiscale formulation of the Heterogeneous problem. While the Fig. 10 (p. 106) depicts the desire to describe the two scale solid state mechanics Problem for elasticity, in reality it is the single physical scale Homogeneous description of state using homogenization solution method.

    This is not a Multiscale Modeling - this is the Homogenization modeling using the special method of multigrid numerical Galerkin method.

    Still, this paper and the paper below might be serving as examples of an attempt to develop the moving intergrain boundaries in composites.

    ~~~~~~~~~~~~~~~~~~~~~

    Chen, J.-S. and Mehraeen, S., "Variationally Consistent Multi-Scale Modeling and Homogenization of Stressed Grain Growth," Comput. Methods Appl. Mech. Engrg., Vol. 193, pp. 1825-1848, (2004).

    This is one of numerous claims for multiscale modeling of physical phenomena, in this case using the asymptotic expansion of field variables with the following development of "multi-scale" Euler governing equations along. We will follow the main arguments and concepts referred in this paper in the effort (the next one following many previous) to address the "Multi-scale" treatment issues raised in this paper.

    In the Introduction to the paper given a nice review of studies related to "multiscaling" in continuum mechanics. We can read in p. 1826:

    "Quasicontinuum method [12,22,23] has been proposed for problems requiring the simultaneous resolution of continuum and atomistic length scales in a unified manner. In this approach, the continuum part is furnished by finite element method where mesh adaptivity is employed to provide multi-scale analysis capabilities near lattice and other highly energetic regions, and proper weight distribution is introduced for handshaking regions. A bridging scale scheme has been proposed to separate basis functions of 2 scales in the handshaking region [26]. Based on the bridging scale technique and projection of the MD solution onto the coarse scale shape functions, a coupling of molecular dynamics and continuum mechanics simulations has been proposed [16, 27]. In the context of multi-scale methods, a multi-resolution analysis by utilizing wavelet-like functions in the framework of meshfree method has also been introduced to construct hierarchical coarse-fine decomposition [14,15,30]. "

    The main objection to this kind and to the one specific statement above in the referred paper by Chen and Mehraeen (2004), is that in these simulation techniques they can not describe properly the - "coupling," "multiscaling," "connection of scales," "scale bridging," etc., because they can not properly address and formalize the collective behavior, collective physical subjects phenomena at the neighboring physical scales as well as the surficial phenomena all the said simultaneously in Modeling Governing Equations.

    That is because they use for GE developing the Homogeneous GO theorem! That's simple.

    These are really the MULTI-RESOLUTION methods those using sometimes the different scales to solve the INCORRECT problem's statement.

    Why it is incorrect - BECAUSE FOR MULTISCALE PROBLEM INSTEAD OF MULTISCALE PHYSICAL STATEMENT FORMULATED AS THE MULTISCALE MATHEMATICAL STATEMENT BEING USED AND "SOLVED" THE SINGLE SCALE PHYSICAL AND MATHEMATICAL PROBLEM USING the MULTIRESOLUTION STATEMENTS!

    HAVING NO DIFFERENT SPACES - JUST SINGLE SPACE FOR ANY PHYSICAL PHENOMENA!

    WHERE the DOT for every resolution statement IS THE POINT with no physical subject, physical meaning inside, just the value of the field at this dot.

    AND EVERYTHING BEING ADJUSTED TO THE ONE SCALE EXPERIMENTAL DATA (for any scale experiment) USING THE "COEFFICIENTS", the magic property of coefficients. After all, these coefficients are also of the single scale Homogeneous matter.

    Let's follow the formulae in page 1827 where we can read on the basics of approach:

    "The multi-scale hierarchy from a macro-scale continuum to a meso-scale network of polycrystalline material under consideration is shown in Fig. 1.

    MATH

    Let x-coordinate be the macro-scale coordinate system in the physical domain. A unit cell with domain $\Omega $ and boundary $\Gamma $ of a continuum in the physical domain is mapped to referential domain $\Omega ^{y} $ and boundary $\Gamma ^{y}$ measured by a meso-scale y-coordinate. The macro- and meso-scale coordinate are related through a scaling parameter $\lambda $ by

    MATH

    where $\lambda $ is a very small real number. Thus a given length measured in the y-coordinate is scaled by a factor $1/\lambda $ by its true scale measured in the x-coordinate."

    This means and authors write straight about that - the scale in y-coordinate is not a physical scale, it is the mathematical convenience, the mathematical habitat trick for convenience of the finding the solution. We are not looking into some physical sense while selecting this or numerous other kind of coordinate resolution.

    Going further in p. 1827 we read:

    "Consider a unit cell in the physical domail subjected to a surface traction $\QTR{bf}{h}$ on boundary $\Gamma _{h}$ as shown in Fig. 2.

    Fig. 2.

    The grain boundaries in the unit cell are denoted as $\Gamma _{gb}.$ A variational equation for stresses grain growth based on the principle of virtual power described in the x-coordinate [4] is as follows:

    MATH

    MATH

    MATH

    where $\delta \Pi _{e}$ is the virtual power associated with deformation, $\delta \Pi _{gb}$ is the virtual power associated with driving forces acting on grain boundaries, $\QTR{bf}{\nu }$ is the grain material velocity, MATH is the grain boundary migration velocity, $\overline{v}_{n}$ is th enormal velocity pointing away from the center of curvature of the grain boundary, $\overline{v}_{s}$ is the tangential velocity along the grain boundary, $s$ is the coordinate along the grain boundary, $\gamma $ is the surface tension (the boundary energy per unit area), $R$ is the radius of curvature of the grain boundary, $\mu $ is the mobility representing the ease with which the grain boundary can migrate, $\QTR{bf}{h}$ is the surface traction applied on the traction boundary $\Gamma _{h},$ $\QTR{bf}{b}$ is the body force, MATH are the stress and strain in the grain that gains virtual area MATH (grain A), respectively, and MATH are the stress and strain in the grain located on the other side of the boundary (grain B), respectively, as shown in Fig. 3. The detail discussions can be found in [4]. We assume the stress-strain relation follows an anisotropic creep law:


    MATH

    To describe the multi-scale material behavior, the material velocity $\QTR{bf}{\nu }$ is expressed in the following asymptotic expansion form


    MATH

    where MATH and MATH are macro (coarse)-scale and meso (fine)-scale components of material velocity, respectively, and the superscript $[n]$ denotes the level of scale.

    For a function expressed in both coarse and fine scale coordinates, the following relationship can be obtained,

    MATH

    In the above excerpt the biggest doubts are toward the expressions for a virtual power (2.3), (2.4) - those are just homogeneous decriptions of the possibilities at, along and with the grain boundary surface $\Gamma _{gb}$ and, of coarse, the averaging concerning formulae for averaging of differential operators over the domain $\Omega $.

    These all expressions (2.2) - (2.4) for virtual powers separate grain growth might be seen as useless ones when the real scaled consideration applied.

    For example, when the elasticity laws being aplied at the scale of domain $\Omega ^{y}$ (of a separate grain fine scale), and for the Upper scale of domain $\Omega $ which is based on the x-coordinate system. This analysis on the averaging of the elasiticity homogeneous dependenceies we have made back in the 90s.

    Also importantly to say this for students, that the relationship shown in the page 1828 - (2.7) that tying up the partial derivatives for two system coordinates is actually incorrect in n-coordinate systems, $n\geqslant 2.$ That is, remember, that they need this transform while applying to parts of the differential operators. This kind of errors often workers in Fluid Mechanics - Turbulence specifically, did in the past while doing deduction course.

    This is the transformation that quite often used in the past and is incorrecly used for the conservation equations.

    Summarizing the outlook for this excerpts in pp. 1827-1828, we can say that this kind of methodology, expressed in this work of 2004 and other works of given authors group (2005, etc.), presents itself as the homogenization procedures aimed to deal with the complicated heterogeneous media solid state mechanics problem.

    But this is not even a two-scale method. Not a "multi-scale" physics treatment as authors want to position this work at.

    No interscale dependencies were property set-up and determined in terms of Lower Scale Domain Upper Scale Point, etc. as can be expected from Fig. 1 (page 1827).

    So, this is the mathematical methodology (no good physical grounds) attempting to develop and claim more than the one scale used in the solution procedure. Anyway, it is not a physical and mathematically formulated two- or more physical scale methodology.

    Following this, we can also say that the developed grain topology transformation equations including the grain boundaries evolution equations were developed incorrectly.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Mehraeen, S. and Chen, J.-S., "Wavelet-Based Multi-Scale Projection Method in Homogenization of Heterogeneous Media,"

    Finite Elements in Analysis and Design, Vol. 40, pp. 1665-1679, (2004).

    We need to talk here about their better Homogenization technique and about the "mirror technique" at the boundaries.

    Abstract: "Standard homogenization of highly heterogeneous media often filters out the fine

    scale information, and as a consequence, it produces acceptable results only for certain type of periodic structures. In this work, a wavelet-besed multi-scale homogenization is introduced for highly heterogeneous materials where the standard asymptotic technique cannot be effectively applied.

    A set of scaling and wavelet functions based on the linear hat function and its corresponding wavelet transformation matrix are constructed. The advantage of this wavelet transformation constructed by hat function compared to that using Haar function are identified. The mirror image technique has been employed to preserve the accurate representation of boundary conditions and to avoid numerical oscillation near the boundaries. This wavelet-based multi-scale transformation hierarchy filters out the high-scale components of the solution, and thus provides an effective framework for the multi-scale selection of the most essential scales of the solution."

    So, they talk themselves about the single, the one solution -single scale. The problem with these NUMEROUS DRUM-NOISING texts and publications with constant reminding of the words "multiscale," "hierarchy," in connection to words "heterogeneous," "media," "materials," "scaling," as in works by these authors and by Oden et al. (1999), Fish and Belsky (1995), among many.

    These words beat out of unexperienced reader the thought, the mean, that these are the really "Multiscale physical problems and their solutions".

    And this is dangerous profanity of the mean of physical multiscaling!!

    The physical multiscaling has become mixed, substituted by just old known tool of mathematical multiscaling, but the worst thing is that the authors of these works often keep thinking and representing their works as about the Physical Multiscaling !

    In page 1666 they write:

    "The first method is homogenized Dirichlet projection method which was developed based on the concept of hierarchical modeling (outlined by us). In this method, the mathematical model at the coarsest level is represented by homogenized material properties. This is referred to as the homogenized problem, and the exclusion of heterogeneity generally makes the homogenized problem computationally inexpensive compared to models of finer scale. Using a posteriori modeling error, the accuracy of the solution capable of representing the homogenized behavior of the original heterogeneous problem is estimated. In regions where the modeling error exceeds a preset tolerance, a finer scale model

    is used and a correction to the homogenized solution is made. This homogenization procedure is continued until the error tolerance is met [2]."

    In this kind of " RAVING " "hierarchical modeling" sorry for the word, but people do not study anything but their own backyard only, the main error is that the coarsest level scale mathematical model is represented as the unreal homogeneous statement GE with the incorrect Upper scale effective coefficients!!!

    This is the grave mistake of Homogenization theory - pretending for the two physical scales description.

    Here again is used the MULTIRESOLUTION (Multigrid) method - because as for the Multiscale problem instead of Multiscale physical statement formulated as the Multiscale Mathematical statement is being used and "solved" the single scale physical and mathematical problem using the MULTIRESOLUTION or Multigrid (as it used sometimes) statements!

    Among references can be found:

    [2] Oden, J.T., Vemaganti, K., and Moes, N., "Hierarchical Modeling of Heterogeneous Solids," Comput. Methods Appl. Mech. Eng., Vol. 172, pp. 3-25, (1999).

    [3] Fish, J. and Belsky, V., " Multi-grid Method for Periodic Heterogeneous Media, Part 1: Convergence Studies for One-Dimensional Case," Comput. Methods Appl. Mech. Eng., Vol. 126, pp. 1-16, (1995).

    ~~~~~~~~~~~~~~~~~~~~~~~~

    The large publication was recently issued by Ghoniem, N. and Kioussis, N. (2005) on the noble now topic "How to reach Multiscaling in Nanotech?" -

    Nasr M. Ghoniem and Nick Kioussis, "Hierarchial Models of Nanomechanics and Micromechanics," in Handbook of Theoretical and Computational Nanotechnology; edits. Michael Rieth and Wolfram Schommers, American Scientific Publisher 1, pp. 1-97, (2005).

    Because the study was positioned as like in between the Continuum Mechanics applications and The Nanotech fields related to Continuum Mechanics traditional writings, we need to address this one more "multiscaling" text. Few words we drop in the Nanotechnology section. I know the person and that is why can even place more lengthy remarks on these publications.

    In p. 2 we can read the good words of intensions -

    "Multiscale interconnected approaches will need to be developed to interpret new and highly specialized nano-/micromechanical tests. One of the advantages of these approaches is that, at each stage, physically meaningful parameters are predicted and used in subsequent models, avoiding the use of empiricism and fitting parameters...."

    "...In this article, we review several components of mechanics that are collectively needed to design new nano and micro materials, and to understand their performance. We first discuss in Section 2 the main aspects of quantum mechanics based methods (ab initio). Concepts of statistical mechanics that permit the interpretation of modern atomistic simulation methods, such as molecular dynamics (MD) and kinetic Monte Carlo (KMC) are introduced in Section 3, together with essential details of empirical interatomic potentials and computational techniques. For a description of the mechanics of materials at length scales where defects and microstructure heterogeneities play important roles, we introduce mesomechanics in Section 4. Finally, applications of nanomechanics and micromechanics are given in Section 5."

    As usual in this kind of text - there are a number of scientific techniques that had been developed during the last 60-40 years AND NOTHING about - How we can transfer our models (with their data and solutions) from one scale into another? They can not do this while using the mathematics of Continuum Mechanics which based on the Homogeneous GO theorem.

    And they need to know this if not yet know - professors and then students.

    What we are writing here in this website and in the "Nanotechnologies" section -

  • "Nanotechnologies"

    For example, in p. 39 we are reading that -

    "4. COMPUTATIONAL MESOMECHANICS

    4.1. Introduction

    "Understanding the collective behavior of defects is important because it provides a fundamental understanding of failure phenomena (e.g., fatigue and fracture). It will also shed light on the physics of self-organization and the behavior of critical-state systems (e.g., avalanches, percolation, etc.)."

    That again are all the correct words on intension, but the implementation is all about the one scale.

    "In an attempt to resolve these observations, two main approaches have been advanced to model the mechanical behavior in this meso length scale. The first is based on statistical mechanics methods [135--142]. In these developments, evolution equations for statistical averages (and possibly for higher moments) are to be solved for a complete description of the deformation problem. The main challenge in this regard is that, unlike the situation encountered in the development of the kinetic theory of gases, the topology of interacting dislocations within the system must be included [141].

    The second approach, commonly known as dislocation dynamics (DD), was initially motivated by the need to understand the origins of heterogeneous plasticity and pattern formation. An early variant of this approach (the cellular automata) was first developed by [143], and that was followed by the proposal of DD [144--147]. In these early efforts, dislocation ensembles were modelled as infinitely long and straight in an isotropic infinite elastic medium. The method was further expanded by a number of researchers [148--152], with applications demonstrating simplified features of deformation microstructure. We will describe the main features of these two approaches in the following."

    How many studies referred to in this piece and still all are based on the same homogeneous techniques.

    In p.40 -

    " Because of the high density of dislocations and the strong nature of their interactions, direct computer simulations of inhomogeneous plastic deformation and dislocation patterns is still unattainable. "

    ???? Well, that is incorrect. Not for HSP-VAT, they can not write about everything, because they don't have qualifications!

    " Even if direct computer simulations are successful in the description of these collective phenomena, it is very desirable to obtain global kinetic or thermodynamic principles for understanding the self-organization associated with plasticity at the microscale. To attain these objectives, and to enable a direct link with continuum deformation theory, the statistical mechanics approach has been advanced [135--142, 153--157]."

    That's wrong, that was not been advanced - as we wrote in the "

  • "Fundamentals of Hierarchical Scaled Description...""

    and that's just being overcome for similar issues many years ago in HSP-VAT - in the few next rooms at the same department! Well, not directly for the Continuum Mechanics of dislocations, but for the same kind of physics disciplines.

    Reading further -

    "The fundamental difficulty here is that dislocations, unlike particles, are linear objects of considerable topological complexity. Hence, when concepts of statistical mechanics and the theory of rate processes are used, some level of phenomenological description is unavoidable."

    Not when you can use the HSP-VAT. If you know or assume the topology then the exact models can be evaluated.

    In p.44 we can read that -

    "In its early versions, the collective behavior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations [11, 143, 144, 146--152, 172--175]. Recently, several research groups extended the DD methodology to the more physical, yet considerably more complex 3D simulations. The method can be traced back to the concepts of internal stress fields and configurational forces.

    The more recent development of 3D lattice dislocation dynamics by Kubin and co-workers has resulted in greater confidence in the ability of DD to simulate more complex deformation microstructure [176--183]. More rigorous formulations of 3D DD have contributed to its rapid development and applications in many systems [184--193]. We can classify the computational methods of DD into the following categories:"

    Look, and what about the summary of these calculation for the really Upper next scale of this "Mesomechanics" ? Not at all - that is not describable in these above and below (averaging) homogeneous theories, but HSP-VAT allows and is inherently based on and provide for the averaging and multiscaling!

    In pp.83-84 one can read -

    " In this section, we investigate three different averaging methods that have been proposed to extract "equivalent" isotropic elastic constants from the full anisotropic values. Two such methods, the Voigt (V) and Reuss (R) averaging procedures, are common in general elastic deformation problems, while the third one, the Scattergood-Bacon (S-B) procedure, has been specially used in the context of determining equilibrium shear loop configurations [286]. We will examine here the extent of agreement between isotropic PDD with the anisotropic counterpart APDD, when these three approximations are used.

    In the Voigt scheme, the averaging is performed over the elastic constant tensor Cijkl, while in the Reuss scheme, it is done over elastic compliances Sijkl. Results of such averaging and relevant parameters for some anisotropic crystals are listed in [225]. All calculations reported in this section are for the case of single crystal copper (A = 3.21)."

    In the p. 86 we start to read on the multilayred materials elastic one scale properties solution

    "5.4.5. Nanolayered Materials

    A variational form of the governing equation of motion for a dislocation loop has been developed for over-damped dislocation dynamics, where the work exerted on dislocation loop expansion is balanced by viscous dissipation [192], "

    And if in this classical application on Multilayer Materials authors can not put in the truly Two Scale models and their solutions, then we would advise to turn to our -

  • "When the 2x2 is not going to be 4 - What to do?"

  • "Two Scale Solution for Acoustic Wave Propagation Through the Multilayer Two-Phase Medium"

    to find out about the two-scale solutions which can be brought to the problem of elasticity in layered materials (superlattice).

    Well, we would cite more and more useless studies and analytical texts and find out the same 1 scale Homogeneous Mechanics tools - see the next subsection talking on the continuing falsification in Continuum Mechanics of Heterogeneous (Ht) Media:

  • - "Who Are in the Continuum Mechanics Continuing to Dwell in an Ivory Tower? Who Tries to Re-Invent the Wheel? What Are the Damage and Financial Loss? "

    ~~~~~~~~~~~~~~~~~~~~~~~~

    For more balance, we would dismantle for the same purpose the monograph in Russian, (published in English as well) that is taught to students heavily and almost replaced old ones without compassion.

    Nigmatulin, R.I., Dynamics of Multiphase Media, Part I, Moscow, Nauka, 1987, 464 p., (in Russian).
    Nigmatulin, R.I., Dynamics of Multiphase Media, Vol. I, Revised and Augmented Edition, English edit. J.C.Friedly, New York, Hemisphere Publish. Corp., (1991). 507 p.

    This is the pretty much known and cited in Russian literature as a "bible" of heterogeneous mixture treatments book by Nigmatulin (1987-Russian edition; 1991- English edition).

    The Gauss-Ostrogradsky theorem (named formula) is given in the book in the second page of chapter 1 as

    MATH

    with the consequtive reference to this expression while doing derivation for many equations.

    Meanwhile, this is one of the two books in Russian, at least as I am aware of, that gives some place to the WSAM theorem and the place to talk about the Heterogeneous Surface - Volume transform theorem. Also, used the consequences of the theorem for developments. The second book in Russian is the book by Luikov (1971) that is the first of the kind, probably, mentioning the WSAM theorem and deliberating on that theorem in Russian.

    1) Author, along with the solid recognition and work on the interface phase (Z-phase, p. 43 in Russian) do not appreciate the fact that these phenomena are from the even lower scale than the 1-st (Low) scale physics in the problem. By doing this - having taken as the phenomena intact - really included, we might improve the accuracy of phenomena accounting, etc., in the Upper scales.

    2) While establishing the general forms of Governing equations for averaged fields (density of i-th phase, velocity, energy and moment of phase impulse) in equations (1.2.33), (1.2.36) with more detailed appearance in (1.2.47) - equation of impulse conservation, and (1.2.52), (1.2.54) - equation of energy transfer, author involves the phenomena and their mathematical interpretations of (actually) the Upper scale into the governing equations of the Lower (homogeneous) scale.

    Thus, he does this in p.55 bottom (hypothesis of small scale impulse pulsation influence), in (1.2.44) - (1.2.46) for the form of pressure distribution within the REV (he does not recognize the importance of the REV and of it's specifications), in (1.2.48) - (1.2.49), (1.2.51) - the expression for the Upper scale pulsation kinetic energy and energy exchange along the REV's boundary?

    3) Strictly speaking, author does not average the initial homogeneous medium transport equations (1.2.1) in (1.2.25) - that is itself incorrect as long as we need to average over the phase within the REV, not over the phase $i-$th only (which is the intrinsic averaging), and the further equations.

    Author hypothesises on the topic - How would be looking the one time averaged over the $i-$th phase governing equations for intraphase homogeneous phenomena.

    Surprisingly, when author comes to the formulation of applications we can see that he writes, for example, in (1.3.3), (1.3.38), (1.3.45), (1.3.52), (1.3.63), 1.3.66) the just pseudo-averaged, ad-hoc averaged equations.

    These and further equations in paragraphs 4 and 5, 6,7,8,9 (the governing equations for porous media are just unacceptable), and paragraph 10 of chapter one are the mostly well known and used homogeneous type equations for heterogeneous media.

    Thus, unfortunately, we observe the huge gap between the mostly correct concepts and their mathematical formulation in the paragraphs 1,2 and their miss-implementations in paragraphs 3 -10 in chapter one. The same non-implementation of the Upper scale averaging theory reader can find in chapters 4,5 in book 1 and in other chapters of book 2 (Nigmatulin, 1987b; 1991b - English edition).

    4) Saying this, we would be not surprized any more why the same author (Nigmatulin R.I.), in the handling of the sonoluminescense phenomena in the attempt to reach the nuclear fusion using the acoustical implosion of gas bubbles in a small reservoir, see in our subsection with comments in -

  • "Sonofusion - What's Up? Thermonuclear:"

    author and his co-authors used the one only scale hydrodynamic model for obviously multiscale acoustical collapse of mutiple bubbles in a liquid.

    5) The hypothesis' in (1.2.16) (author himself calls them as such) have meaning that author suggests the free transform from the local to an Upper scale integrated fields (nonlocal) with preserving the properties of them. One of hypothesis is (for an arbitrary volume)


    MATH

    which can be re-written further as


    MATH

    as long as according to averaging rules (1.2.13)


    MATH

    or the average interface of scalar function MATH in phase $i$ is

    MATH

    6) The equation given in p. 51 (1987a, Russian edition) under (1.2.23) is actually the WSAM theorem while for that is not given credit or references to authors of that theorem whatsoever in the book

    MATH

    where the term

    MATH

    in VAT notations is no less than the very much known expression for the interface surface integrated instant value of field's function MATH that is weighted by volume of the REV.

    Author used to write this equation as derived via the GO theorem exploration - that is not clear how?

    Averaged over the interface $\delta S_{12}$ instant field MATH in the phase $i$ is

    MATH

    The whole this (1.2.23) expression by Nigmatulin (1987b) means in HSP-VAT notations the same WSAM theorem

    MATH

    7) Few important concluding notes would be helpful to students and professionals for appraisal and comparison of these books by Nigmatulin (1987a,b; 1991a,b):

    7.1) Author can not involve the second time averaging into averaged equations - the second time averaging of stress - for example, in (1.2.47) and the second time averaging of fluxes in (1.2.52), (1.2.54).

    7.2) Author does not recognize that the left hand side of some averaged equations are wrong - (1.2.37), (1.2.47), (1.2.54).

    7.3) Author does not include the turbulent phenomena and the Lower scale turbulent initial governing equations into his description of the concepts.

    7.4) Also, there is no engagement of nonlinear phenomena in the right hand side divergence terms in homogeneous governing equations for the averaging process.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Weiler, F.C., "Fully Coupled Thermo-Poro-Elasto Governing Equations,"

    Proc. Computational Mechanics of Porous Materials and Their Thermal Decomposition, ASME 1992, AMD- Vol. 136, pp. 1-28, (1992).

    (author was at that time with the Lockheed Missiles and Space Company, Palo Alto, CA )

    The same city, year, and conference when I firstly spoke to American scientific audience in San Diego on the very part of this topic also talking on Heterogeneous Mechanics problems while the HSP-VAT is being applied.

    In abstract is said: "The objective of the present paper is to outline the development of a set of thermo-poro-elasto governing equations which represents the physical theory upon which new advanced rocket nozzle analysis computer programs will be based. The three relevant areas of interest are (a) nonlinear thermo-poro-structural analysis, (b) nonlinear pyrolysis gas flow, and (c) nonlinear heat transfer, all of which pertain to rocket nozzle ablative materials such as carbon-phenolics."

    In spite the careful construction of the Representative Control Volume (RCV) the analysis derived regarding the averaged stress, strain, and other functions, fields is done as for the homogeneous medium and finally, of course, the governing equations were done as for homogeneous statements usually performed and known from textbooks. That did not add any value to the program of such a scale as developing the multiphysics, multiphase transient continuum mechanics problem.

    As we said above, we can analyze like this many other "studies" and find out the same apparatus of Homogeneous media treatments used for Heterogeneous problems! These works above were selected based on might be personal experiences, nevertheless, they are pretty representative ones. Readers and students can pick up in many other places of this website the similar things said about the "pseudo-averaging" and "multiscaling." That's might be enough for older professionals.

    The subsequent contemporary innovative and more expensive Gatekeepers for Continuum Mechanics are described in the next subsection:

  • - "Who Are in the Continuum Mechanics Continuing to Dwell in an Ivory Tower? Who Tries to Re-Invent the Wheel? What Are the Damage and Financial Loss? "

    REFERENCES

    Bakhvalov, N.S. and Panasenko, G.P., Homogenization: Averaging Processes in Periodoc Media. Mathematical Problems in the Mechanics of Composite Materials, Kluwer Acad. Publishers, Dordrecht (1989). 366 p.

    Bakhvalov, N.S. and Panasenko, G.P., Averaging Processes in Periodoc Media. Mathematical Problems of Mechanics of Composite Materials, Nauka, Moscow, (1984). 352 p. (in Russian).

    Beran, M.J., Statistical Continuum Theories, Wiley-Interscience, New York, (1968).

    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).

    Berryman, J.G. "Variational Bounds on Elastic Constants for the Penetrable Sphere Model," J. Phys. D: Appl. Phys., Vol. 18, pp. 585-597, (1985).

    (This work done at Lawrence Livermore National Laboratory. P.O. Box 808. L-200. Livermore, CA 94550).

    (Sponsored by - under the auspices of the US DOE under Contract W-7405-ENG-48 and supported specifically by the Base Technology Program of the Earth Sciences Department.).

    Borisenko, A.I. and Tarapov, I. E., Vector Analysis and Principles of Tensor Analysis with Applications, Visshaya Shkola, Moscow, (1966) (Russian publication, 3rd edition).

    Borisenko, A.I. and Tarapov, I. E., Vector and Tensor Analysis with Applications, Prentice-Hall, Inc., Englewood Cliffs, (1968) (English translation).

    Brenner, H., "Dispersion Resulting From Flow Through Spatially Periodic Porous Media," Phil. Trans. R. Soc. Lond., Vol. 297, pp. 81-133, (1980).

    Brenner, H. and Adler, P.M., "Dispersion Resulting From Flow Through Spatially Periodic Porous Media. II. Surface and Intraparticle Transport," Phil. Trans. R. Soc. Lond., Vol. A 307, pp. 149-200, (1982).

    Brenner, H. and Leal, L.G., "Conservation and Constitutive Equations for Absorbed Species Undergoing Surface Diffusion and Convection at a Fluid-Fluid Interface," J. Colloid Interface Sci., Vol. 88, pp. 136-184, (1982).

    Chen, J.-S. and Mehraeen, S., "Multi-scale modelling of heterogeneous materials with fixed and evolving microstructures," Modelling Simul. Mater. Sci. Eng., Vol. 13, pp. 95--121, (2005).

    Chen, J.-S. and Mehraeen, S., "Variationally Consistent Multi-Scale Modeling and Homogenization of Stressed Grain Growth," Comput. Methods Appl. Mech. Engrg., Vol. 193, pp. 1825-1848, (2004).

    Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics, Sijthoff and Noordhoff, Alphen aan den Rijn, (1973).

    Fish, J. and Belsky, V., " Multi-grid Method for Periodic Heterogeneous Media, Part 1: Convergence Studies for One-Dimensional Case," Comput. Methods Appl. Mech. Eng., Vol. 126, pp. 1-16, (1995).

    Fish, J. and Shek, K., "Multi-scale Analysis of Composite Materials and Structures," Compos. Sci. Technol., Vol. 60, pp, 2547-2556, (2000).

    Ghoniem, N.M. and Kioussis, N., "Hierarchial Models of Nanomechanics and Micromechanics," in Handbook of Theoretical and Computational Nanotechnology; edits. M. Rieth and W. Schommers, American Scientific Publisher 1, pp. 1--97, (2005).

    Noronha, S.J., Huang, J., and Ghoniem, N.M., "Multiscale Modeling of the Brittle-to-Ductile-Transition," J. Nucl. Mater. 329--333, 1180--1184 (2004).

    Hashin, Z., Elasticity of Random Media, Trans. Soc. Rheol., Vol. 9, pp. 381-406, (1965a).

    Hashin, Z., On Elastic Behaviour of Fiber Reinforced Materials of Arbitrary Transverse Phase Geometry, J. Mech. Phys. Solids, Vol. 13, pp. 119-134, (1965b).

    Hashin, Z., Theory of Fiber Reinforced Materials, NASA CR-1974 Report, Univ. Pennsylvania, Philadelphia, pp. 1-690, (1972).

    Hashin, Z., Analysis of Composite Materials - A Survey, J. Appl. Mech., Vol. 50, pp.481-505, (1983).

    Hill, R., "Elastic Properties of Reinforced Solids: Some Theoretical Principles," J. Mech. Phys. Solids, Vol. 11, pp. 357-372, (1963).

    Hill, R., A Self-consistent Mechanics of Composite Materias, J. Mech. Phys. Solids, Vol. 13, pp. 213-222, (1965).

    Hori, M. and Yonezawa, F., Statistical Theory of Effective Electrical, Thermal, and Magnetic Properties of Random Heterogeneous Materials. III. Perturbation Treatment of the Effective Permittivity in Completely Random Heterogeneous Materials, J. Math. Phys., Vol. 15, pp. 2177-2185, (1974).

    Hori, M. and Yonezawa, F., Statistical Theory of Effective Electrical, Thermal, and Magnetic Properties of Random Heterogeneous Materials. IV. Effective-Medium Theory and Cumulant Expansion Method, J. Math. Phys., Vol. 16, pp. 352-364, (1975).

    Hori, M. and Nemat-Nasser, S., "On Two Micromechanics Theories for Determining Micro--Macro Relations in Heterogeneous Solids," Mech. Materials, Vol.31, Issue 10, pp. 667-682, (1999).

    Korn, G.A. and Korn, T.M., Mathematical Handbook for Scientists and Engineers, (2nd edn.), 599p., McGraw-Hill, New York, (1968).

    Luikov, A.V., Heat- and Mass Transfer (Textbook), Moscow, Energiya, (1971). (in Russian). 560 p.

    Mehraeen, S. and Chen, J.-S., "Wavelet-Based Multi-Scale Projection Method in Homogenization of Heterogeneous Media," Finite Elements in Analysis and Design, Vol. 40, pp. 1665-1679, (2004).

    Mura, T., Micromechanics of Defects in Solids, Martinus Nijhoff, New York, 1987.

    Nayfeh, A.H., Perturbation Methods, J. Wiley & Sons, New York, (1973).

    Nayfeh, A.H., Perturbation Methods, Mir, Moscow, (1976), (in Russian).

    Nemat-Nasser, S. and Hori, M., Micromechanics: Overall Properties of Heterogeneous Elasticl Solids, Lecture Notes Initiated at UCSD, (1989).

    Nemat-Nasser, S. and Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials, 2nd edition, Elsevier Science B.V., Amsterdam, (1999).

    Nigmatulin, R.I., Fundamentals of Mechanics of Heterogeneous Media, Moscow, Nauka, 1978, (in Russian). 336p.

    Nigmatulin, R.I., Dynamics of Multiphase Media, Part I, Moscow, Nauka, 1987, 464 p., (in Russian). Nigmatulin, R.I., Dynamics of Multiphase Media, Vol. I, Revised and Augmented Edition, English edit. J.C.Friedly, New York, Hemisphere Publish. Corp., (1991). 507 p.

    Nigmatulin, R.I., Dynamics of Multiphase Media, Part II, Moscow, Nauka, 1987, 360p., (in Russian).

    Nigmatulin, R.I., Dynamics of Multiphase Media, Vol. 2, Revised and Augmented Edition, English edit. J.C.Friedly, New York, Hemisphere Publish. Corp., (1991). 371 p.

    Oden, J.T., Vemaganti, K., and Moes, N., "Hierarchical Modeling of Heterogeneous Solids," Comput. Methods Appl. Mech. Eng., Vol. 172, pp. 3-25, (1999).

    G.Papanicolaou, ed., Wave Propagation in Complex Media, Springer-Verlag, New-York, 1998. 287 p.

    Russel, W.B., Saville, D.A., and Schowalter, W.R., Colloidal Dispersions, New York, Cambridge, Cambridge University Press, 1991.

    Shenderova, O.A., Brenner, D.W., Nazarov, A.A., Romanov, A.E., and Yang, L.H., " Multi-scale Modeling Approach for Calculating Grain Boundary Energies From First Principle," Phys. Rev. B, Vol. 57, (1998).

    Truesdell, C., A First Course in Rational Continuum Mechanics, Baltimore, John Hopkins Univ., (1972).

    Truesdell, C., A First Course in Rational Continuum Mechanics, Moscow, Mir, (1975), (in Russian).

    Vaithyanathan, V., Wolverton, C., and Chen, L.Q., "Multi-scale Modeling of Precipitate Microstructure Evolution," Phys. Rev. Lett., Vol. 88, (2002).

    Wagner, G.J., and Liu, W.K., "Hierarchical Enrichment for Bridging Scales and Meshfree Boundary Conditions," Int. J. Numer. Methods Engng., Vol. 50, pp. 507-524, (2000).

    Wagner, G.J., and Liu, W.K., "Coupling of Atomic and Continum Simulations using a Bridging Scale Decomposition," J. Comput. Phys., Vol. 190, pp. 249-274, (2003).

    Weiler, F.C., "Fully Coupled Thermo-Poro-Elasto Governing Equations," Proc. Computational Mechanics of Porous Materials and Their Thermal Decomposition, ASME 1992, AMD- Vol. 136, pp. 1-28, (1992).

    Yu, Q. and Fish, J. , "Multi-scale Asymptotic Homogenization for Multi-physics Problems with Multiple Spatial and Temporal Scales: a Coupled Thermo-Viscoelastic Example Problem," Int. J. Solid Struct., Vol. 39, pp. 6429-6452, (2002).

    Yau, W.W. and Kirkland, J.J., and Bly, D.D., Modern Size-Exclusion Liquid Chromatography, Wiley, New York, (1979).


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