Saga of Dr. Kushch, V.I., 1994-2009

Has been since 1994.

Dr. V.Kushch for more than 20 years doing solutions of actually one problem - Low scale local composite media exact analytical elucidation while the effective field for one of the phases - matrix, being assigned. He became a great internationally known specialist on this problem. The problems are stated in such a mode, that the method should be called - Effective Field Method with the One Phase Assigned Nonlocal (Upper) Scale Solution and Assigned the Macrostrain (or other field) Values. Also with the Macroscopic strain value assigned as 1. We show below why this arrangement is helping to go Upstairs to the Second nonlocal Upper scale effective coefficients. Anyway incorrectly assessing them - we discuss that below why.

He started to study and work for me in 1994 as a contractor. He was taught on numerous issues of HSP-VAT as of general concepts and ideas, general theorems and math and applications primarily for Thermal Physics and Fluid Mechanics.

Dr. Kushch signed the Nondisclosure Agreement in 1994 for 10 years. I had assigned to him and disclosed some information that was not available at that time, some of that information, techniques are not available up to now.

We spent a huge number of hours on my visits to Ukraine, Kiev in 1994-1997, and more hours, days, and months for email exchange, stating the problems and discussing the outputs.

He was not too happy with work and assignments, as well as I was not happy with his 3 year long attempts to dethrone, refute the HSP-VAT math and physics.

In doing this he had done with me the very good studies and simulations for problems that were in his second doctorate's agenda in his institution. He had not performed finally the simulations that I had assigned to him for start - the simulation of the two-scale local-nonlocal and second scale nonlocal equations for a few morphologies and calculation of the effective coefficients for the Upper scale HSP-VAT governing equations.

Nevertheless, in 1997 despite signing the NA for ten years he published two papers (of which only I know) - Kushch (1997a, 1997b) where some definitions of HSP-VAT had been used.

And were used incorrectly - the effective coefficients were calculated wrongly - I have done analysis of the papers only in 2001-2002 and some of the texts related to (1997a) was published in -

So, the errors went unanswered and are staying up to now as the incorrect data for comparisons, as the "litmus data"? for other scientists in the CM field.

The second paper review in terms of the alleged macroscopic, averaged characteristics for elasticity in heterogeneous medium is given here in this script.

Finally, in this way Dr. V.Kushch became a conscious gatekeeper for HSP-VAT because in a three year period being allowed to recognize the internal features and mathematical details in implementation of HSP-VAT for Thermal Physics and Fluid Mechanics he was not able to find out any fundamental wrongdoings, errors in HSP-VAT. He did not refute this science.

More on that - Dr. Kushch himself took part in the development of simulation procedures for the HSP-VAT.

Well, after all he chose to be sided with his "homogeneous" book (Golovchan et al., 1993) elasticity mechanics statements and his, might be, personal ego while doing the false science in elasticity mechanics of heterogeneous media and composites since 1994.

That was his choice.

Our choice is to put at last this story before public eyes for educational and pedagogical purposes - as it is said - people are controlled by their agreements and by ethics of them.

Since then, for more than 14 years Dr. V.Kushch has been doomed to be also the Wheel Re-Inventor. Because for more than 40 years the better, more precise, and fundamental tool for Heterogeneous, Scaled and Hierarchical physics than the HSP-VAT has not been invented!

But Dr. Kushch continuing to publish a very good Lower (ground) scale solutions for heterogeneous two-scale problems in Elasticity theory, CDM with the consecutive wrong deliberations for the Upper scale properties.

Involving his co-authors and funding agencies in this conspiracy as well.

To our knowledge, Dr. V.Kushch never publicly delivered decisive "disproove" of HSP-VAT as was doing, for example, for near 10 years Chinese physicist regarding the Boltzmann equation. Even in the online manuscripts as C.Y.Chen does -

Now, it appears that all co-authors of Kushch in the US and other countries should be aware at least of what kind of mistakes they have been doing when consider the composite's elasticity, plasticity and CDM problems as of a prime ground scale - the Lower conventional homogeneous physics problems and later on doing the generalization of simulation calculating the "Upper" (second) scale nonlocal "effective" coefficients and like "estimations."

All those are suffering from mistakes of using inappropriate averaging techniques with the GO theorem.

And these mathematical constructions are incorrect.

To the misfortune of the Conventional one-scale for all (OSFA) Heterogeneous Continuum Mechanics (HtCM) the workers like Dr. V.I.Kushch are of plenty. We will still be taking to the Histology table the most interesting works and disclose to students and sober young researchers the path of mistakes in current HtCM.

Recently, doing my studies I again accidentally came across of writings by Kushch and his co-authors on solid state mechanics, elasticity theory, continuum damage mechanics in heterogeneous, multiphase materials, composites.

That can be counted as of ~ 14-15 years of incorrect studies and influences on the US funded workers.

Well, those researchers themselves make up the problems in that kind of homogeneous pretending to be "heterogeneous" studies. But that is another story.

(Only few outstanding personalities, of hundreds working with the false philosophy regarding a Heterogeneous matter)

Effective Field Method with the One Phase Assigned Nonlocal (Upper) Scale Solution and Assigned the Macrostrain (or other field) Values.

Kushch, V.I., Sevostianov, I., and Mishnaevsky, L.Jr., "Effect of Crack Orientation Statistics on Effective Stiffness of Mircocracked Solid," Int. J. Solids and Structures, Vol. 46, No. 6, pp. 1574-1588, (2009).

Few outstanding thoughts need to be written here regarding this paper deliverables. That is the must thing to follow the developments and remark on them if misfit occurred.

Excerpts from (Kushch et. al (2009)):

In page 1574 in the abstract one can find that:

"This paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics. To do so, the representative unit cell approach has been used. The microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks; a sufficient number is taken to account for the microstructure statistics. "

" The exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields. "

In page 1577 one can find that:

"By adding 2D model, we assume within this framework, the

(a) anti-plain shear (in direction) and

(b) plane strain problems are considered. In the first case, is the only nonzero component of the displacement vector

Also, we have two non-zero components of the strain and stress tensors, they are and , i=1 and 2, respectively, where is the shear modulus of a solid. The stress equilibrium equation in this case reduces to the 2D Laplace equation

It enables finding as where is an analytical function of the complex variable

In the plane strain problem, we have The complex value displacement as well as its derivatives (strain and stress tensors) can be expressed in terms of two complex potentials, and (Muskhelishvili, 1953). We write as

where and is the Poisson ratio of a crack-free solid. Components of the stress tensor are given by

We assume the strain and stress fields to be macroscopically uniform and defined by the constant macroscopic strain tensor components Due to the cell-type periodicity of geometry, the displacement field is the quazi-periodic function of coordinates:

and

In the model, we consider the crack surfaces defined by are traction free, i.e., we assume the cracks to be open. "

Our comments:
here in the BC (2.6) and (2.7) the local variables are connected to the macroscopic variables directly as all of them are the local ones. This is the frivolous scaleblending - the term we are using to call the unjustified action or procedure (often even mathematical) that equals or makes simple linear (most often) or other appropriate at the moment mathematical or verbal dependency between the local (lower scale) and of the Upper (nonlocal, scaled) physical fields.

This is incorrect in scaled hierarchical HtCM (Heterogeneous Gontinuum Mechanics).

In page 1578 one can find that:

"We assume the stress field in a solid with cracks is induced by the far field constant strain tensor "

Our comments:
because later on this globally assigned tensor components will be concerned and used in the problem's second scale simulation assessments - it will be used for the macroscale strain computed based on the microscale local solution, that is the incorrect implication of the Upper scale field for the incorrectly simulated Lower scale fields.

And Dr. Kushch knows about this.

In page 1579 one can find that:

"3.3 Effective stiffness tensor

The above analytical solutions provide an accurate evaluation of the local fields at any point of the unit cell as well as the stress intensity factors (SIFs) at the crack tips. Their detailed analyses is a subject of a separate paper: for our purpose, it is important that these fields can be integrated analytically to obtain the exact, closed form expressions for the effective elastic moduli of a cracked solid, defined by the formula

where are the volume-averaged, or macroscopic stresses: In our case, . And, since the components of the mean strain tensor were taken by us as the input (load governing) parameters, all we need to determine the effective stiffness tensor is to integrate the local stress field. Indeed, , where the stress field corresponds to the uniaxial strain

Our comments:
and which is the wrong formula, because the effective characteristics on the Upper scale are determined via the different equations and averaging mathematics.

In page 1580 one can find continuation of the thought:

"Averaging the anti-plane shear stress gives us

The Gauss formula yields

where is the outer boundary of the cell, is the q-th crack surface and are the components of the outward normal unit vector.

Accounting for the displacement field periodicity, we obtain expectedly

As to the integrals over the cracks' surfaces, they can be written as

Our comments: and which is the wrong formula as well.

In (3.37) the matrix averaged value assigned as of the two-phase? In (3.38) cracks should be of the two kinds not one. The WSAM theorems and consequences should be used. Let's explain this with more graphical detail.

The features of the REV by present authors we should discuss in this review are shown in

Well, the trick here is that the REV is taken including all the cracks seen in the figure.

In this case the cracks always to be included within the REV and Kushch and co-authors can use assigned up-front the strain tensor with consequtive elimination of integration over the ANY boundary of matrix phase.

That means also that the matrix bounding surface can be freely drawn.

Our comments: Meanwhile, the real correct REV drawn ought to be intersecting any phase that might occur in the way of drawing

We address this issue with more detail in the review of the following paper - Kushch et al. (2008a).

Summary to this and generally to all Kushch' papers:

1) The lack of heterogeneous media scaled understanding brought out the strange and erroneous statements used throughout the literature on Homogeneous Continuum Mechanics applied for heterogeneous problems, that the assignment of the remote initial values for like "bulk" strain and stress in the composite automatically determines the matrix effective, "bulk" characteristics .

That is the trick, because those properties are not equal. These characteristics are of different media and of different scales even.

Nevertheless, based on this erroneous assumption - the great number of studies using the definition that the only functions they need to simulate are of the inclusions phase, because the matrix phase effective properties are known as assigned?

Which is wrong for heterogeneous, scaled physics and math.

2) We can not write the equality (2.6)

because the mean macroscopic strain tensor can not be defined by this method as via the differential of microscopic displacements vector components.

This is the two scale problem, and effective macroscopic coefficients of this kind are not defined and equal to just differentiated or averaged volumetrically (meaning in most cases also stochastically) microscopic fields.

And Dr. V.Kushch knows about that, this is the most interesting.

3) We can not avoid mentioning an important feature of analytical solution - it is to give a guide and the cornerstone assessment and results for future controlled comparison.

So, it is vitally important to have correct analytical solutions. In problems with the Kushch's solutions for the two-scale local-nonlocal heterogeneous problems we have not gained this advancements, because the Upper scale results, estimates by Kushch and co-authors (but works with V.S.Travkin over the HSP-VAT two-scale heat transfer, fluid mechanics, and the two-scale simulation) are wrong.

4) Taking into account the incorrectness of the Upper scale estimates in this paper (Kushch at el. (2009)), the pages starting from p. 1581 - are not worth to read - the published data is incorrect.

5) In (3.38) (Kushch at el. (2009a)) should be written not

6) In p. 1580 left pane - the text has the hidden fact and mathematical errors that consist of the mathematical treatment of the outer surface that is presented in the Fig. 1 as it is intersected by cracks, but the crack's surfaces in real treatment are considered as if they are not intersected by the outer

This is so important hide that I was forced to write the whole piece of analyses deliberating the truth why this intersection is so important to hide for the conventional homogeneous, one-scale physics, including electrodynamics and, of course, elasticity mechanics, presented in the "Heterogeneous Electrodynamics" section of this website -

particularly in the critics of work - 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 disdain of the REV's features and more of that the declaration of unimportancy of the exact form and concept of drawing the REV was done, our guess, to disregard the HSP-VAT or Averaging Theory quite different conceptual approach to the Scaled, Nonlocal Continuum Mechanics Governing Equations (GE) at that time - ~30 years ago, that was still the new direction needed to be destroyed.

Destroyed, because the Non-invariant GEs for the same problem, for the same phenomena but at different scales have been appeared in the CM, and that was a heresy at that time?

7) That means also the formula (3.38) (in the Kushch at el. (2009a)) is incorrect as well, as soon as it doesn't distinguish the internal cell's cracks and the intersecting boundary cracks.

8) We need to stress this out that it can not be written based on a "periodicity" the formula (3.37)

because no matter what, the "mean strain tensor" was taken as the assigned input (load governing) parameter, and that assigned tensor is not the mean strain tensor within the medium of the problems. Because the mean effective strain tensor within the heterogeneous medium determines with the HSP-VAT mathematics and procedures very differently of what is considered in homogeneous CM presentation followed in this study.

This was explained hundreds times in literature on thermal physics and fluid mechanics.

Just because the simple volume averaging, and incorrect averaging (Dr. V.Kushch has been taught in detail on averaging procedures in 94-97, and knows on the problems with the "mean" and the "averaged," phase averaged and intrinsic phase averaging functions) does not constitute the effective strain tensor. The mean and the effective coefficients are often not the same functions as they are and in this case.

That is - the (3.37) is invalid by definition, so

the macroscopic strain tensor and the microscopic displacement components are connected in a very different way.

9) Regarding the "famous" often simply hidden, but crucial for Kushch's and other researcher's methodology - that the centers of inclusions should not lay down exactly at the boundary of the REV? We need to state that this restriction is not applicable or even reasonable requirement at all, it is quite artificial and can not be maintained as long as the REV can and should be moving over the space for averaging process to be performed. The REV can be placed at any location and serve for any assessment locally within the problem's region.

10) It is worth to note here also, that this kind of problem for composites refined and described in the two-scale local-nonlocal HSP-VAT statements allow analysis and results to be obtained that are unachievable with the kind of 1-scale elasticity formulations as Dr. Kushch does with his co-authors.

See some of the outcomes of that kind in -

Conclusions:

1) The lack of heterogeneous media scaled understanding brought out the strange and erroneous statements used throughout the literature on Homogeneous Continuum Mechanics applied for heterogeneous problems, that the assignment of the remote initial values for like "bulk" strain and stress in the composite automatically determines the matrix effective, "bulk" characteristics

That is strange, because those properties are not equal. These characteristics are of different media and often of different scales even.

Nevertheless, based on this erroneous assumption - the great number of studies using the definition that the only functions they need to simulate are of the inclusions phase, because the matrix phase effective properties are known as assigned?

Which is wrong for heterogeneous, scaled physics and math.

2) The great, fundamental fault of all of this kind local-nonlocal, pseudo-Upper scale effective characteristics - "effective" stiffness tensor, "effective" strain tensor, "effective" stress tensor, and other "effective"-like properties is that researchers while taking averaging (mean) functions just cut out them from the real heterogeneous physics and proper averaging mathematics.

For example, while tackling the strain tensor Kushch and co-authors (2009; 2008a; etc.) wrote the volumetric averages without any thought about the Hooke's law separately from the elasticity governing equations or the Upper scale in the respected two or more phases, dropping the part of boundary conditions - tractions at the interfaces, external boundary of the REV (they used the "invented" RVE notion), etc.

That kind of cheating or ignorance give the possibility to write the wrong equality in Kushch et al. (2008a), for example,(as well the same kind of statement in every their study)

where is the assigned macroscopic strain tensor, because generally

By doing this assignment the half-problem has become known (well, in reality for Upper (averaged, non-local) scale we can not do such a simple assignment!) and, at the same time - the problem of the Boundary Conditions (BC) at the Upper macroscopic scale has been created.

That is because the local strain tensor at the boundary from outside can not be easily assigned with using the homogeneous formulae for the BC for the nonlocal fields and for the "averaged" strain.

And Dr. V.Kushch knows about that, which is the most interesting.

Because in 1994-97 we provided the specific studies on the BC between heterogeneous and Homogeneous media to state the Upper scale BCs. Because the great prevailing number of problems are interesting and significant for practice when being stated as the Top-Down heterogeneous problems. As, by the way, might be the problem stated in the Kushch et al. (2009).

Assignment of the Upper scale strain is actually the "half-solution" of the problem, as soon as (3.37) and in all other like (3.37) formulae in reviewed here studies by Kushch, is satisfied because of this assignment!

In this way authors assign (found) the nonlocal Upper scale solution in the matrix just at the beginning.

3) Also, the assignments of the Nonlocal macrostrain means not only assignment of the one phase solution actually - see our comments above, to other papers by Kushch, and elsewhere in this website section on Ht Elasticity, Continuum Mechanics, but also the almost unsurpassed technical difficulties with the BC and this Upper (nonlocal) scale problem - because the nonlocal boundary condition would demand the assignment also of the surficial integrals near the boundary within the heterogeneous medium already.

Then, the assignment of both phases nonlocal HSP-VAT boundary conditions with surficial values means in turn the solution of the both Upper Scale displacement - strain fields problem in need.

Which means, in turn, the general solution of the Upper scale HSP-VAT statement, that Dr. Kushch fiercily avoiding since 1994.

4) This is the danger of ignorant usage of the Upper scale (macroscopic) variables to instantaneously (and not innocent) presenting the Lower scale (microscopic) wrong averaging operations for "finding" the "effective" strain?

That is unacceptable.

5) In this way of doing "multiscale", "macroscopic" - in reality screwed homogeneous one-scale media elasticity theory, continuum damage mechanics, general continuum mechanics of heterogeneous, scaled media (HSM) - authors not only do incorrect the Upper scale (macroscopic) assessments, simulation. And often even the Lower scale simulation is going wrong, because they assume the different scales variables can be taken as equal. This we call the Scaleblending.

They have lost the substantial part of a real science for Heterogeneous Scaled Media. This constitutes a great loss both for science and technologies and the financial loss for wrong, unnecessary studies.

This also makes a wrong design of heterogeneous materials, devices as, for example, in their Kushch et al. (2008b) composite CDM study authors in p. 2759 write that -

"The composites of primary interest for us are the unidirectional glass fiber - epoxy matrix FRC's. The pre-requisite of using these materials in the wind turbine components is extremely high (~ and more cycles) lifetime."

How dare they do this if their Upper and Lower scale models were stated without the Scaleportation presented in any slightest hint?

Well, and they don't have and will not have the proper clue on - How to do the Heterogeneous experiment which can confirm the Homogeneous theoretical results? Because of many inconsistencies we are writing here about.

6) In this way of doing the two-scale problems as the one scale authors are loosing the chance to address the most interesting phenomena in the physics of composite media - the phenomena at the interfaces.

This is not achievable with the one scale local homogeneous mechanics.

And Dr. Kushch has been knowing about this.

7) Dr. Kushch certainly uses the HSP-VAT techniques, but in a silent, hidden way, as we point out here. The few most interesting phenomena occurring at the interface in composites with damages, cracks as in Kushch et al. (2009) become lost for investigation with the wrong use of a homogeneous approach. One more example is that, here in the problem described by Kushch et al. (2009) should be derived the Macroscale nonlocal equation of elasticity equilibrium for the one phase only - for the matrix phase. But it was not done? Meanwhile, the elements of the one phase treatment were presented in the study albeit with more errors than Kushch usually demonstrates. My guess, that in this (2009) study his co-authors tried to get a lead, and failed.

8) While avoiding modeling and simulation formulated in terms of HSP-VAT for the Upper scale fields authors of this group, and others we will be deliberating on, acting to prevent themselves from the Upper scale problem's features:

8.1) They can not formulate, even to know, and investigate most of the Upper scale characteristics of interest: a) both scale surficial characteristics; b) both scale fluctuation characterisics, etc.

I am sure most of Dr. Kushch co-authors never knew about those? But he does.

It is interesting to note, that both surficial and fluctuation characteristics Dr. Kushch himself simulated and studied with me in 1995-96 and we derived at many interesting features on those parameters, some of them are unpublished yet.

8.2) They can not study morphology parameters of Ht Elasticity, Plasticity, Continuum Damage Mechanics, etc. Morphology parameters formulated and sought after as of the Upper scale - not those people usually count for, as - number of cracks, statistical characteristics of cracks, length of the cracks, etc. All these are of the Lower scale parameters not of the Upper one.

8.3) They can not study the properties of the interface - correctly determining the arisen characteristics, because the interface is that thing that connects and communicates the both phases (if really both, the declaration of each inclusion a separate phase mostly serves to the fogging the problem features) and has the own characteristics. Those would be studied later when researchers learn how to formulate the issues.

It is not surprising that Dr. Kushch carefully writes about interface, and actually does not study the interface effects in composites, even in the study on debonding - Kushch et al. (2008b)? Why is that?

9) It can be said that the analyzed above issues are related to the problem of Scaleportation from Lower scale local field solutions to the Upper scale averaged nonlocal, but mostly sought after fields. These fields include also the effective properties of composites, heterogeneous media. As we have depicted above the problem of scaleportation is itself not a trivial task of simulation with the Detailed Micro-Modeling - Direct Numerical Modeling (DMM-DNM) homogeneous physics mathematical statements.

10) Here, the remark is appropriate on that the option authors usually include in the like by Dr. Kushch's statements, that the number of phases N can be substantially >1, means actually that the problem is stated with the phase dependent coefficients or C_{ijkl}(x). Well,the Upper nonlocal GEs become much more cumbersome and tricky for analysis and solution then in the case of two phases.

11) It is interesting to mention regarding the use of assigned averaged strain and stress values - Whether authors thought about the verification of their solution and simulate the values of effective strain and stress by averaging the Lower scale local displacement fields? And they would find out the nonconservative fields. Which is inexplicable in Homogeneous CM.

12) Not the last and definitely not the least remark is that the assignment of the averaged gradient of the strain field (other physical fields of interest) is followed by the non-uniqueness of the solution for the Upper scale statement!

Followed by the non-uniqueness of solutions for the Lower scale statements!

Followed by the non-unique determination and simulation for the "effective" coefficients!

Authors apparently do not realize this.

*********************

Kushch, V.I., Sevostianov, I., and Mishnaevsky, L.Jr., "Stress Concentration and Effective Stiffness of Aligned Fiber Reinforced Composite with Anisotropic Constituents," Int. Journal of Solids and Structures, Vol. 45, No. 18-19, pp. 5103-5117, (2008a).

Excerpts from (Kushch et. al (2008a)):

In page 5103 in the abstract one can find that:

"The paper addresses the problem of calculation of the local stress field and effective elastic properties of a unidirectional fiber reinforced composite with anisotropic constituents. For this aim, the representative unit cell approach has been utilized. The micro geometry of the composite is modeled by a periodic structure with a unit cell containing multiple circular fibers. "

" Exact expressions for the components of the effective stiffness tensor have been obtained by analytical averaging of the strain and stress fields."

In page 5106 one can find that:

"Recently, an analytical method was developed by Kushch et al., (2005) and an accurate solution has been obtained for a finite array of elliptic fibers embedded into an infinite matrix. The method is based on the properly taken sets of outer and inner potentials and the relevant re-expansion formulae providing an efficient reduction of the initial BVP to the linear algebra problem. As shown below, this method, with some modefications, is appropriate for the problem under consideration."

Our comments 1: This method of final array of elements in composites should be recognized as an incorrect one, when this solution is used for the Upper scale generalization, as it is done usually. Not only in this work.

The reason for this argument is based on the idea that the transfer to the Upper scale fields should be done with the mathematics of HSP-VAT, which is the only correct one at this moment for scaled. heterogeneous local-nonlocal physics.

More of that:

a) the set of inclusions is of the final quantity, meanwhile the results are being expanded to the second scale without concern that the BC are different at the Lower scale than at the Upper one if we consider that the problem is the same, only the scale of physical modeling is different, right?

b) this trick is the one that used in homogeneous CM (HCM) as one of the arguments to avoid consideration of the second Upper scale - mostly we have to talk on the Upper scale modeling and simulation, because the Top-Down sequence in CMH is not the area of apprehension yet.

In page 5107 one can find that:

"The fibers non-overlapping condition is where In the framework of the exposed method, considering the case of polydisperse and multiphase composite is rather straightforward and does not meet any problems. The conditions in the remote points and at the interface are the same as in the previous problem, Eq. (17):

Our comments 2: this is the BCs at the Lower scale variables. Their consideration and effects are almost missing or wrong in the homogeneous treatments of the Ht problem.

In page 5108 one can find that:

"The fiber whose edges are shown in Fig. 1 by dashed line do not belong to the cell while occupying a certain area within it. Thus, geometry of the unit cell is given by its length , height and the coordinates defining the center of q-th fiber, The entire composite is obtained by translating the cell along two orthogonal directions.... "

Our comments 3: As it is one of the critical points in the mathematical statement in HSP-VAT, we should be quite aware, that this action of exclusion of few fibers that are parted between the selected REV and the surrounding media has been done for a purpose. And the aim of this exclusion is to use the Gauss-Ostrogradsky theorem (GOT) instead of the Whitaker-Slattery-Anferson-Marle theorem (WSAMt) while the latter is the only correct way for integration. We show this down the road during our analysis.

We already have shown this trick by Dr. Kushch (apparently) performed in the other science - the Thermal Physics paper (applicable to electrostatics as well) analyses-review can be seen in -

Further in this page one can read that:

"The stress field in the composite bulk is assumed to be macroscopically homogeneous, which means constancy of the volume-averaged, or macroscopic, strain and stress tensors, where and is the cell volume."

Our comments 4: In the HSP-VAT these brackets mean intrinsic phase averaging And again we need to recognize that these variables as defined here, when applied to the two-phase medium under investigation do not determine the real Upper scale (second scale nonlocal) strain and stress because what is assigned in the far away homogeneous medium - apart of the studied two-phase medium domain, as done in the present work and numerous others, does not automatically recognised as equal to the variables, parameters that are just being averaged as the local variables. That is the HSP-VAT conclusion determined long ago.

In page 5111 authors for evaluation of the Upper scale properties use notations as:

"On the other hand, the strain and stress fields given by this solution can be integrated analytically to get the exact closed form expression for the macroscopic, or effective, stiffness tensor defined by

where means averaging over the RVE (REV actually). In the problem we consider, the stress field is macroscopically homogeneous and governed by the strain tensor . Also, RVE coincides with the cell volume and, due to the structure periodicity,

where is a cell volume. In our case, (unit length is assumed in z-direction)."

In page 5112 one can find continuation of that definition:

"To evaluate , we write

where is the volume of the pth fiber and is the matrix volume inside the cell: Applying the Gauss' theorem leads to

where is the cell outer surface and are the components of the unit normal vector. Taking into account the first condition (51) and decomposition we get expectedly

i.e. has a meaning of the macroscopic strain tensor. Hence, all components of the effective stiffness tensor can be determined from (57) as where the stress field corresponds to the macroscopic strain for and

The macroscopic stress tensor can be written in the following form

Therefore, we need to integrate the strains over the fiber volume only. By Gauss' theorem, integrals in (57) are reduced to

which we can obtain analytically."

Our comments 5:

In this the only section of the paper devoted to an attempt to address the averaged (Upper) scale characteristics of the problem we just need to debacle on these formulae (58), (59) that are like cheating on the integration, because used the GOT instead of correct WSAM theorem.

Well, we say it again that the proper usage of the GOT might bring the singled out correct value at the Lower scale and formulae for a special artificial configuration - which does not state averaged solution at the Upper scale in any way, meaning it does not explain or state the Upper scale nonlocal mathematical statements including the governing equations - GEs and solution for the Upper scale phenomena fields.

And Dr. Kushch knows on that difference since 1994.

In the expression is involved the Scaleblending of the different scales variables.

In the formula (60) should be used the both phases strain tensor.

The formula (61) is incorrect one because is the trick done because by doing this with the help of assigned these and other authors making the Upper nonlocal scale stress assessment dependent only on the inclusion phase simulation! And by doing this authors actually forcefully assign the macroscopic features, up to a solution for the matrix scale (mostly).

Doing this along with the hidden definition of the used REV is a cheating on readers and themselves - because by hiding the Upper scale nonlocal concepts and coefficients definitions (well, even if authors, but Kushch himself, are not familiar with this science) deprive authors, readers and the funding agencies the proper treatment of the heterogeneous problems. See our summary and conclusions above for Kushch et al. (2009).

The formula (62) is incorrect one because used the GOT instead of correct WSAM theorem.

From page 5113 this paper results can not be trusted because of errors with scaling modeling and simulation.

Summary:

Conclusions:

*********************

Kushch, V.I., Shmegera, S.V., and Mishnaevsky, L.Jr., "Meso Cell Model of Fiber Reinforced Composite: Interface Stress Statistics and Debonding Paths," Int. Journal of Solids and Structures, Vol. 45, No. 9, pp. 2758-2784, (2008b).

Excerpts from Kushch et. al (2008b):

In page 2758 in the abstract one can find that:

"The primary goal of this work is to develop an efficient analytical tool for the computer simulation of progressive damage in the fiber reinforced composite (FRC) materials and thus to provide the micro mechanics-based theoretical framework for a deeper insight into fatigue phenomena in them. An accurate solution has been obtained for the micro stress field in a meso cell model of fibrous composite.

......By averaging over a number of random structure realizations, the statistically meaningful results have been obtained for both the local stress and effective elastic moduli of disordered fibrous composite. A special attention has been paid to the interface stress statistics and the fiber debonding paths development, which appear to correlate well with the experimental observations."

Our comments: As we will see this is the same homogeneous treatment - very good for the Lower scale with that one phase assignment, but is a complete failure when authors seeking the generalization of their Lower scale results into the Upper scale characteristics.

In page 2759 one can find from the review of predecessors work:

"In the comprehensive review by Degrieck and Van Paepegem (2001), the progressive damage models which use one or more damage variables related to measurable manifestations of damage (interface debonding, transverse matrix cracks, delamination size, etc.) have been claimed as the most promising ones because they quantitatively account for the damage accumulation in the composite structure.

The gradual deterioration of a FRC - with a loss of stiffness in the damaged zones - leads to a continuous redistribution of stress and a reduction of stress concentrations inside a structural component (e.g., Allen et al., 1990; Shokrieh and Lessard, 2000; Degrieck and Van Paepegem, 2001). Hence, prediction of the final state of the composite structure requires simulation of the complete path of successive damage states. In order to provide an adequate description of progressive damage with account for the local stress redistribution, one needs to use a complicated structure model, able to reflect both the micro structure statistics and the local damage events."

"....The composites of primary interest for us are the unidirectional glass fiber - epoxy matrix FRC's. The pre-requisite of using these materials in the wind turbine components is extremely high (~ and more cycles) lifetime."

".....The approximate multi-particle effective field method (MEFM) is used which allows to estimate the second statistical moments of stresses in both the constituents and the interfaces between the matrix and fibers and thus predict the effective envelope for failure initiation.

However, MEFM does not provide evaluation of local stress fields in the constituents and at the interfaces."

Our comments: These correct evaluations of the one scale - Lower scale, modeling and simulation techniques can have the basis for present authors, at the same time, do the incorrect evaluation of the second scale properties of the composite.

In page 2760 one can find that on the methods used for the Lower scale solutions:

"Even more numerically efficient version of MEM (Golovchan et al., 1993; Kushch, 1997, among others) utilized the series expansion of displacement vector over a set of appropriate periodic singular solutions. In this case, the matrix coefficients are expressed in terms of the easy-to-calculate sums and the only remaining problem is to find an efficient way to solve the resulting set of linear equations. "

"....Recently, the very efficient solving techniques have been developed based on the fast multipole method (Greengard, 1994; Sangani and Mo, 1996; Kushch et al., 2002; Wang et al., 2005b, among others). In this and similar advanced though rather involved algorithms, a computational effort scales as where is a number of inclusions per cell, which makes it rather efficient for studying the very large models (say, with and more), often used in the fluid suspension mechanics in order to account for the long-range interactions accurately. In the solid composites no such long-range interactions exist, and a number from 100 and 200 fibers per cell was reported by many authors as quite sufficient to provide the statistically meaningful results. "

".....In the present work, an analytical approach by Golovchan et al., (1993) has been further developed and applied for studying the local stress and the effective elastic properties of FRC composite using the meso cell model."

Our comments: Amidst the description of the variety of mathematical approaches used for solution of the Lower scale heterogeneous problem authors writes that "and the effective elastic properties of FRC composite using the meso cell model" - this is really a stretching.

This is the claim that the averaging over the REV would help to find the Upper nonlocal characteristics. That is the way Kushch had assignments for simulating Heterogeneous test problems in the HSP-VAT Thermal Physics and Fluid Mechanics in 94-97. Unfortunately, he does this here with co-authors using the Lower scale homogeneous methods and thus they failed to achieve that goal using the REV defined Lower scale local solution.

In page 2761 one can find that the critical definition of the REV (authors used the "internal" self-invented term RUC):

"The fibers whose edges are shown in Fig. 1 by dashed line do not belong to the cell while occupying a certain area within it. Thus, geometry of the unit cell is given by its length , height the coordinates being the centers of inclusions and their radii The whole composite bulk can be obtained by translating the cell in the two orthogonal directions."

"Number N of the fibers with centers inside the cell can be taken large sufficiently to simulate micro structure of an actual disordered composite. In the considered model, a diameter and the elastic moduli are defined individually for each separate fiber. It provides applicability of this model for studying the multi-component systems and the effect of fiber diameter scattering, which can be quite considerable even in the commercial FRCs, ...."

Our comments: The same vital in consequences definition of the REV shown in the figure and of the real hidden volume as we noted in the above analysis of Kushch et al. (2008a).

In page 2763 one can find the interface BC and averaging conditions that are wrong:

"At the matrix-fiber interfaces, the perfect bonding conditions are prescribed:

where

The stress field in the composite bulk is assumed to be macroscopically homogeneous, which means constansy of the volume-averaged, or macroscopic, strain and stress tensors, where and is the volume of meso cell."

Our comments: Practically the same equations and definitions as in Kushch et al. (2008a).

And the averaging equalities are wrongly written because authors do not use (but Kushch was familiar with HSP-VAT) the correct means for averaged properties in composites.

In page 2764 one can find the BC for the REV:

"Next, it is of common knowledge that under macroscopic stress homogeneity condition periodicity of structure results in periodicity of relevant physical fields. In our case, the periodicity condition

can be alternatively regarded as the cell boundary condition providing continuity of the displacement and stress fields between the adjacent cells."

Our comments: these BCs are incorrect for the two-scale heterogeneous problems (physics), because of the two reasons:

1) is as for the HSP-VAT assessments;

2) because of the 2-nd scale BCs that should be satisfied.

In page 2764 further one can read on the method of solution:

"The theory we apply to study this problem is the Kolosov-Muskhelishvili's method of complex potentials, widely recognized as the powerful analytical technique for a given class of problems. However, unlike the case of a finite array of inclusions (Buryachenko and Kushch, 2006; Mogilevskaya and Grouch, 2001), periodicity of the model implies introducing the appropriate periodic complex potentials. The computational cost-efficient way to build up solution of the model BVP in the class of periodic (rather than doubly periodic) functions has been suggested by Golovchan et al. (1993); a brief summary of the theory of periodic potentials is given in Appendix A."

In page 2770 one can find the models for the second Upper scale:

"3.2 Effective stiffness

The stress field obtained from the above solution can be integrated analytically to get the closed form exact expression of the macroscopic, or effective, stiffness tensor defined by

where means averaging over the RVE (REV actually-our comment ). In the model we consider, RVE coincides with the meso cell and, due to periodicity of structure,

where is a cell volume. In our case, (unit length is assumed in -direction)."

Our comments: wrong definition of as long as authors excluded from the some boundary fibers while, at the same time, any boundary for the matrix phase can be accepted - see the comments, summary and conclusions to the Kushch et al. (2009) above. Nevertheless, the definition of REV's volume should include definition of the both phases exactly.

In page 2770 we see the continuation of basic definitions:

"In the problem we consider, the stress field is macroscopically homogeneous and governed either by the constant macroscopic strain tensor or stress tensor. To evaluate , we write

where is the matrix volume inside the cell and is the volume of the pth fiber: Now, we apply the Gauss' theorem to reduce the volume integrals to surface ones: taking into account also the first of adhesion conditions (2) we get

where is the cell outer surface and are the components of the unit normal vector."

Our comments: Here is the same comments to these formulae - wrong formulae (51) can not be put in here - see the comments to the same formulae in Kushch et al. (2008a) above. The reason to cheat on the boundary specific is might be for reason of not to disclose the existing WSAM theorems regarding this matter and their consequences for the Upper scale averaging.

Also, the REV outlined surface is not specified here?

Meanwhile, this surface consists of the two surfaces. One is the matrix cutting surface - input and output for fields fluxes, displacement field via the matrix; another one is the surface cutting the fibers and the cutting should be involved in any REV because of the nature of the problem.

The two phases interface surfaces of the REV are for fiber phase, where is the within the REV interface of the fibers; and the where is the interface for matrix within the REV.

Both surfaces equal to where is the notation what is used in the HSP-VAT usually for this kind of interface.

In page 2771 we see the continuation of basic definitions:

"Evaluating the macroscopic stress tensor follows the same way:

after some algebra, we came to

Thus, the problem is reduced to evaluating the

Our comments: We have to comment with almost the same remarks as for the previous study by Kushch et al. (2008a). These averaging of strain and stress formulae (59)-(61) are incorrect - see the above notes.

Besides, the REV'S outlined surfaces are not the boundaries where we can have the assigned boundary values!? That is because the macroscale major characteristic as which is being assigned in each problem by Dr. Kushch - to get somehow the values of the field nonlocal displacement and strain need being accounted with GOT or WSAM theorem as the already known ones.

That the numerical simulation of even the Lower scale elasticity problem is incorrect - because of usage the macroscopic bulk composite assigned strain for the matrix averaged quantity without specifics of Heterogeneous fields averaging, and vise versa the Upper scale assessments are incorrect because the science of homogeneous elasticity theory is incorrect when applied to this kind of Heterogeneous two-scale problems seeking the Upper scale properties by just application of the homogeneous media GOT.

From page 2772 this paper results can not be trusted because of errors with scaling modeling and simulation.

Summary:

1) Some of the comments to this publication detail must be considered as the additional summary to those outlined in the Kushch et al. (2009).

Conclusions:

1) We came to the needed point by point analysis with regard of the homogeneous treatment of heterogeneous problems when the scaled (two-scale) structure-properties relations are at stake, and have to be obtained as such, but unreachable for homogeneous community workers.

The result of analysis is unfortunate for that kind of frivolous mixing of the physics and math for HtCM that performing as "strictly justified" by numerous HCM micro-communities including Dr. V.Kushch and his co-authors.

2) We have developed these HSP-VAT physical modeling and mathematics fundamentals for the Heterogeneous media Elasticity theory (HtEt), Continuum Mechanics scaled statements and simulation, that needs to be applied in all Dr. Kushch's studies.

*********************

Kushch, V. I., "Microstresses and Effective Elastic Moduli of a Solid Reinforced by Periodically Distributed Spheroidal Particles," Int. J. Solids and Structures, Vol. 34, Iss. 11, pp. 1353-1366, (1997).

Excerpts from Kushch (1997):

In page 1353 in the abstract one can find that:

"......Analytical averaging of the strain and stress tensors gives the exact expressions for all components of effective elasticity tensor of composite considered. The influence on stress concentration and effective moduli of the structural parameters of composite is investigated and the comparison is made with known approximate solutions."

Our comments: The errors started here in the abstract. As we will see the same homogeneous tools were applied to the Heterogeneous problem with the claim that the averaged Upper scale characteristics were simulated and calculations were done with the high accuracy. This study was the beginning of the Kushch's becoming the gatekeeper for the HSP-VAT and the future Wheel re-inventor for the scaled, local- nonlocal heterogeneous elasticity theory and Continuum Mechanics as a whole since 1994.

Fig. 1, as shown in the publication Kushch (1997b)

In page 1354 one can find that:

"We do not suppose the whole inclusion to be lying completely inside the cell. Intersection of the inclusions with the sides of a parallelepiped is possible; the only requirement is that the centers of inclusions do not lie on the boundary of the cell." ?

Our comments: Here is the certain restriction about the localization of the particles and the boundary of the REV. Nevertheless, this is the full scale REV defined and we should expect even with this artificial restriction the corresponding correct averaging procedures following the WSAM theorem. See our Conclusions to the Kushch et al. (2009) on that issue.

It won't happen. The GO theorem will be used in this work instead.

In page 1355 one can find that:

After presenting the common knowledge Homogeneous elasticity theory GE for each phase - particles and the matrix combined always, author writes:

"We suppose that the stressed state of a composite medium is induced by the remote constant strain tensor prescribed. The displacement vector in each phase of the composite satisfies the Lame's equation

where is the Poisson's ratio; in the matrix, in the particle of the qth kind."

"On interfacial surfaces the continuity conditions of the displacement vector and normal stress tensor

are prescribed

"As it is easy to prove, due to the periodicity of the structure, the solution also has periodicity features. So, the displacement vector in a continuum phase (matrix) can be presented as a sum of the linear far field and the periodic disturbance field caused by the presence of inhomogeneities:

where

"It was shown elsewhere (e.g. Kushch, 1985; Sangani and Lu, 1987) that in this case the tensor has a sense of the average strain tensor of the composite, i.e.

where and is the representative volume of the composite. Taking into account eqns (5) and (6), the periodicity unit (Fig. 1) may be chosen as a representative volume. Also it follows from eqn (6) that the satisfaction of contact conditions (4) for the particles with centers lying inside the structure cell means their equal satisfaction for the rest of inclusions in a medium.

Thus, the problem here is to construct the solution of eqn (2) satisfying the periodicity conditions (5) and (6), and the interfacial conditions (4). "

Our comments: Author uses the known from HSP-VAT term here - "representative volume", and the statement that the remote constant strain tensor is equal to the nonlocal averaged two-phase strain value. That might be as well, but only if the nonlocal averaged two-phase strain is derived and formed correctly, that is not the case for this author treatment. Meanwhile, in the further deliberations one would learn that the assigned strain greatly helped to make the pseudo-averaging over the total medium, which is the real reason to assign the macroscopic strain tensor. And that is also leads to the open box of errors we spoke above.

In page 1356 one can find the method of analytical solution:

"The analytical method applied here to solve the boundary-value problem (2) - (6) is, in fact, the modified variant of the approach used by Kushch (1995b) to solve the elasticity problem for a medium containing a finite number of aligned spheroidal inclusions. The essence of the method, in few words, is the representation of the displacement vector in a multiply-connected region by a series of partial vectorial solutions of Lame's equation, and use of additional theorems for these solutions to reduce the boundary-value problem to an infinite set of linear algebraic equations."

Our comments: That means the Lower scale local solution of the two-phase elasticity problem can be done analytically and with a high accuracy.

Meanwhile, we need to mention again - that this solution is the one that is determined by local point fields and with adjusted usage of the Upper scale assigned one of the main characteristics. That the usually meant the homogeneous media averaging is going to produce just the one of infinity of nonlocal fields.

In page 1358 one can find the usual homogeneous method for Averaging of the local fields to obtain the nonlocal properties for this problem:

"The solution obtained is sufficient for the calculation of the four-rank tensor of the effective elastic moduli defined by the relation

where

is the volume of elementary structure cell (Fig. 1). Hence, for determination of it is sufficient to find the expression for and calculate them for the corresponding values of

By the use of Hooke's law for the material of each composite phase we have

where and is the volume of the th phase within the elementary cell, equal in total to the volume of the th inclusion. Thus

is the volume fraction of the th phase. The equality (22) may be rewritten in the following form

Hence, we need to integrate strains over the volume of inclusions only. With the aid of Gauss' theorem the volume integrals are reduced easily to surface ones:

where is the surface and are the Cartesian components of displacement vector and of unit vector normal to this surface, respectively."

Our comments: We have to comment with almost the same remarks as for the analyzed previous studies by Kushch et al. (2009),(2008a), (2008b). These averaging of strain and stress formulae (22)-(24) are incorrect - see the above notes.

That the numerical simulation even of the Lower scale elasticity problem is incorrect - because usage of the macroscopic bulk composite assigned strain which actually meaning the macroscale value assigned for any ratio and values of elasticity moduli at the Lower scale? Top-Down assignment of physical Conditions? While the Upper scale assessments are incorrect because the averaged assessments of homogeneous elasticity theory are incorrect when applied to this kind of Heterogeneous two-scale problems.

These averaging procedures are incorrect not only because used the GOT instead of correct WSAM theorem. They are incorrect also because there were no Upper scale averaged equations that would enforce the different understanding of the averaged governing equations and fields, different treatment and simulation results.

From page 1360 this paper results can not be trusted because of errors with scaling modeling and simulation.

Conclusions:

UCLA MAE Department Professors Who Do Their Research In Heterogeneous, Scaled, Hierarchical Mechanics and Physics. Introducing the Heterogeneous Continuum Mechanics Actively Publishing Gatekeepers and Wheel Re-Inventors from UCLA -

Why from UCLA? This kind of workers spread all over the world. Well, that is just because the 12 years and numerous contacts with them regarding their "wrongdoings" were useless and they knew what is going on in the field - as soon as I was affiliated with the MAE department for 12 years.

The only one professor was demonstrating the full support and helped me with promotion of the scaled, hierarchical agenda at the department - professor I.Catton, very known for his input in the numerous areas of Thermal Physics and Fluid Mechanics.
Since my departure from UCLA MAE department he has been trying to maintain some level of flame in the local hotbed of HSP-VAT, back then simply the VAT as I named the field at the beginning of the 90s. Unfortunately, the educational base at the department for HSP-VAT is absent because the faculty did not want to let me establish the program on scaled physics, mechanical engineering at the department in the 90s and later. That is why, that source of application of HSP-VAT, mostly in mechanical engineering is dying or already dead.

Now I.Catton tries to present himself as the "founder" of HSP-VAT (VAT), without even naming me or consulting with me - Who is the founder of HSP-VAT? We did should respond to this claim as long as I was affiliated with his lab for 12 years and developed a lot of things over there. But saying that he is a "starter", or "founder" - it is too much. My reply to this claim and comments on the history of HSP-VAT are featured at the beginning of - "Announcements. Announcement #2:".

Another professor at MAE department - G.Pomraning, started in the 90s to footstep in the science of Nuclear Physics (Neutron Transport Theory (NTT), etc.) following the HSP-VAT developments in what was known as a linear processes averaging. Pomraning with co-authors started to recognize that the particle (neutron included) transport equations are actually allegedly accepted as averaged and in a few papers in the middle of the 90s they timidly gave another kind of radiative transport equation for a two-phase (biphasic, binary) medium.

In my paper (1999) on this topic where I introduced the VAT averaged, one of the options, for the upper scale phase averaged, radiative transport equation (VARE) - "Radiative Transport", this subject matter was only touched.

Other faculties when making study on heterogeneous, famous now "multiscale" topics - they waste the public money, pretending that they can do heterogeneous, multiscale studies. Or they should admit that their qualifications are not good enough to understand the mechanics and physics of HS media.

Unfortunately for myself and for my co-authors and colleagues at UCLA generally, and of MAE department, I need to say a word on most of "heterogeneous" like research and publications by MAE and other departments professionals. Few other departments also making themselves the targets for future mockery. Through the years we also tried to make professors at MAE department involved, but without success. These professionals, nevertheless, doing also the heterogeneous and fashioned lately "multiscale" media studies are wasting just their time and public money in a real sense.

They do the incorrect research; some publish incorrectly written books, for decades.

I have no other choice as to say and make obvious that speciaslists who could learn personally from me the proper, the only correct science, (sorry for the language, still I have no real obstacles saying this as I do this science for more than 25 years) for treatment of scaled, Multiscale, heterogeneous, hierarchical matter are doing things that are out of a real field. They play their wrong party and are looking for honored retirement. They teach students having the wrong scripts.

And that is bad for students, that is bad for science, that is bad and costly for industries, when people are doing their "string" theory in mechanics, physics for at least this generation, taking out the public money. Well, it is better than the string theorists hoping to do for a next thousand years? Still is too long for our kids being thankful to them for such an alchemistry.

Regarding professors N.Ghoneim and J.-S.Chen of MAE department, I wrote already on their kind of "multiscale" mechanics of heterogeneous media, they are in the web already -

Professors had a time after my departure from MAE, to look at the HSP-VAT and not waste time, money, and students' resources on pseudo-multiscaling, "nanoscaling" structures, that will not deliver something worthwhile.

Anyway, they also doomed to be the Wheel-Reinventors. There is no other fundamentals exist for doing Heterogeneous Scaled, Hierarchical physics than the HSP-VAT for more than 40 years. Correct fundamentals. They will need to use some elements of HSP-VAT and better sooner than later.

We are addressing these and close issues in our sections on Heterogeneous scaled media Continuum Mechanics, Elasticity theory, etc.; while with more detail and educational write-ups in the sub-section -

where we introduce the concepts, present also some correct formulae for problems published by Dr. V.Kushch and co-authors, by other mentioned here authors, as elements of the two-scale Heterogeneous Elasticity theory in the Bottom-Up direction, and give a notice of the Scaleportation for the Ht Elasticity, Viscoelasticity, Viscoplasticity, Heterogeneous Continuum Mechanics.

**********************************************************************

What is the False Continuum Mechanics of Heterogeneous Media - Continuation of Reviews on Heterogeneous, Multiscale Treatment:

Pindera, M.J., Khatam, H., Drago, A.S., and Bansal, Y., "Micromechanics of Spatially Uniform Heterogeneous Media: A Critical Review and Emerging Approaches," Composites Part B: Engineering, Vol. 40, Iss. 5, pp. 349-378, (2009).

Well, this is pretty fresh and obviously good sample (slice, if we can say this) of the knowledge base and the fundamental tools used in the US, and not only in the US, industries (aerospace particularly) and academies for the Heterogeneous Scaled problems in Elasticity and CDM of composites particularly right now in 2000-2010.

The review refers to substantial number of publications and researches - all in the same mode of Homogeneous approach for the Heterogeneous problems. We almost done, better to say we've already done with the script of analytical notes on this work and preparing part of it for this website to be uploaded soon.

Here we read from research people in the industry. Some authors are affiliated with "Boeing", "Sikorsky Aircraft Corp." That means we get to know how academia and largest corporations support the ignorance in Heterogeneous Continuum Mechanics. In the long list of references - 164 refs., there are

Excerpts from Pindera et al. (2009):

*********************

Guz, A. N., Rushchitsky, J. J., and Guz, I. A., "Establishing Fundamentals of the Mechanics of Nanocomposites," Int. Applied Mech., Vol. 43, No. 3, pp. 247-271, (2007).

The first author is the outstanding professional in Ukraine's national continuum mechanics field and simultaneously having a position of high status official in the Ukraine academic science bureaucracy (he is Director of a large Institution).

And the title of the paper is like they are the first to say the most important words of the groundbreaking knowledge on the subject?

So, we can expect to find the discussion from the Ukrainian science high rank official on issues of composites, nanocomposites and multiscaling, scale "bridging" for composites modeling, the analytical approaches to the scaling in composites, etc.

Excerpts from Guz et al. (2007):

In page ... in the abstract one can find that:

*********************

REFERENCES:

Cohen, I., Bergman, D.J., "Effective Elastic Properties of Periodic Composite Medium," J. Mechanics and Physics of Solids, Vol.51, pp. 1433-1457, (2003)

Drago, A., Pindera, M.J., " Micro-macromechanical Analysis of Heterogeneous Materials: Macroscopically Homogeneous vs Periodic Microstructures," Composites Science and Technology, Vol. 67, Iss. 6, pp. 1243-1263, (2007)

Eshelby, J.D., "The Elastic Field Outside the Ellipsoidal Inclusion," Proc. Roy. Soc. London, A252, pp. 561-569, (1959)

Gusev, A.A., "Representative Volume Element Size for Elastic Composites: a Numerical Study," J. of Mechanics and Physics of Solids, Vol. 45, pp. 1449-1459, (2002)

Guz, A. N., Rushchitsky, J. J., and Guz, I. A., "Establishing Fundamentals of the Mechanics of Nanocomposites," Int. Applied Mech., Vol. 43, No. 3, pp. 247-271, (2007)

...

Kushch, V.I., "Conductivity of a periodic particle composite with transversely isotropic phases," Proc. R. Soc. Lond., A 453, pp. 65-76, (1997a)

Kushch, V. I., "Microstresses and effective elastic moduli of a solid reinforced by periodically distributed spheroidal particles," Int. J. Solids and Structures, Vol. 34, Iss. 11, pp. 1353-1366, (1997b)

Golovchan, V.T., Guz', A.N., Kokhanenko, Yu.V., and Kushch, V.I., Mechanics of Composites. T.1. Statics of Materials, Kiev, Naukova Dumka, 1993, 455pgs, (in Russian)

Kushch, V.I. and Artemenko, O.G., "Determination of Temperature Field in a Granular Layer," Doklady AN Ukr. SSR, No. 3, pp.74-77, (1983), (in Ukrainian)

Kushch, V.I., "Stressed State and Elastic Moduli of a Periodic Composite Reinforced By Coated Spherical Particles," Mechanics of Composite Materials, Vol. 29, No. 6, pp. 816-822, (1993), (in Russian)

Kushch, V.I., "Heat Conduction in a Regular Composite With Transversely Isotropic Matrix," Doklady AN Ukr. SSR, No.1, pp.23-27, (1991), (in Ukrainian)

Kushch, V.I., "Thermal Conductivity of Composite Material Reinforced by Periodically Distributed Spheroidal Particles," Engng.-Phys. Journal, Vol. 66, 497, (1994), (in Russian)

Kushch, V.I, Sangani, A.S., "Stress Intensity Factors and Effective Stiffness of a Solid Containing Aligned Penny-Shaped Cracks," Int. Journal of Solids and Structures, Vol. 37, pp. 6555-6570, (2000a)

Kushch, V.I, Sangani, A.S., "Conductivity of Composite Containing Uniformly Oriented Penny-Shaped Cracks or Perfectly Conducting Disks ," Proc. Roy. Soc. London, A456, pp. 683-699, (2000b)

Kushch, V.I, Sangani, A.S., Spelt, P.D.M., and Koch, D.L., "Finite Weber Number Motion of Bubbles Through a Nearly Inviscid Liquid," J. Fluid Mech., Vol. 460, pp. 241 - 280, (2002)

Kushch, V.I., Sevostianov, I., and Mishnaevsky, L.Jr., "Stress Concentration and Effective Stiffness of Aligned Fiber Reinforced Composite with Anisotropic Constituents," Int. Journal of Solids and Structures, Vol. 45, No. 18-19, pp. 5103-5117, (2008a)

Kushch, V.I., Shmegera, S.V., and Mishnaevsky, L.Jr., "Meso Cell Model of Fiber Reinforced Composite: Interface Stress Statistics and Debonding Paths," Int. Journal of Solids and Structures, Vol. 45, No. 9, pp. 2758-2784, (2008b)

Kushch, V.I., Sevostianov, I., and Mishnaevsky, L.Jr., "Effect of Crack Orientation Statistics on Effective Stiffness of Mircocracked Solid," Int. J Solids and Structures, Vol. 46, No. 6, pp. 1574-1588, (2009)

Pindera, M.J., Khatam, H., Drago, A.S., and Bansal, Y., "Micromechanics of Spatially Uniform Heterogeneous Media: A Critical Review and Emerging Approaches," Composites Part B: Engineering, Vol. 40, Iss. 5, pp. 349-378, (2009)

Shen, H. and Brinson, L.C., "A Numerical Investigation of the Effect of Boundary Conditions and Representative Volume Element Size for Porous Titanium," J. Mechanics of Materials and Structures, Vol. 1, pp. 1179-1204, (2006)

Sangani, A.S. and Behl, S., "The Planar Singular Solutions of Stokes and Laplace Equations and Their Application to Transport Processes Near Porous Surfaces," Phys. Fluids A, Vol. 1, No. 1, pp. 21-37, (1989)

Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York, (2002)

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