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In memory of Professor Grigory Isaakovich Barenblatt
УДК 539.3
Development of Barenblatt's scaling approaches in solid mechanics and nanomechanics
F.M. Borodich
School of Engineering, Cardiff University, Cardiff CF24 3AA, United Kingdom
The main focus of the paper is on similarity methods in application to solid mechanics and author's personal development of Barenblatt's scaling approaches in solid mechanics and nanomechanics. It is argued that scaling in nanomechanics and solid mechanics should not be restricted to just the equivalence of dimensionless parameters characterizing the problem under consideration. Many of the techniques discussed were introduced by Professor G.I. Barenblatt. Since 1991 the author was incredibly lucky to have many possibilities to discuss various questions related to scaling during personal meetings with G.I. Barenblatt in Moscow, Cambridge, Berkeley and at various international conferences as well as by exchanging letters and electronic mails. Here some results of these discussions are described and various scaling techniques are demonstrated. The Barenblatt-Botvina model of damage accumulation is reformulated as a formal statistical self-similarity of arrays of discrete points and applied to describe discrete contact between uneven layers of multilayer stacks and wear of carbon-based coatings having roughness at nanoscale. Another question under consideration is mathematical fractals and scaling of fractal measures with application to fracture. Finally it is discussed the concept of parametric-homogeneity that based on the use of group of discrete coordinate dilation. The parametric-homogeneous functions include the fractal Weierstrass-Mandelbrot and smooth log-periodic functions. It is argued that the Liesegang rings are an example of a parametric-homogeneous set.
Keywords: statistical self-similarity, parametric homogeneity, scaling, fractals, contact, wear
DOI 10.24411/1683-805X-2018-16011
Развитие масштабных подходов Бареиблатта в механике деформируемого
твердого тела и наномеханике
Ф.М. Бородин
Кардиффский университет, Кардифф, CF24 3AA, Великобритания
Основное внимание в статье уделено методам подобия в применении к механике деформируемого твердого тела и персональному развитию автором масштабных подходов Баренблатта в механике деформируемого твердого тела и наномеханике. Утверждается, что масштабирование в наномеханике и механике твердого тела не должно ограничиваться только эквивалентностью безразмерныж параметров, характеризующих рассматриваемую проблему. Многие из обсуждаемыж методов быши введены в науку профессором Г.И. Баренблаттом. С 1991 года автору сопутствовала невероятная удача, позволившая ему иметь много возможностей для обсуждения различных вопросов, связанных с масштабированием, во время личныж встреч с Г.И. Баренблаттом в Москве, Кембридже, Беркли и на различных международных конференциях, а также путем обмена письмами и электронной почтой. Здесь описываются некоторые результаты этих обсуждений и демонстрируются различные методы масштабирования. Модель накопления повреждений Баренблатта-Ботвиной переформулирована в виде формальной статистической автомодельности множеств дискретных точек и применена для описания дискретного контакта между неровными слоями многослойных пакетов, а также износа углеродныж покрытий, имеющих шероховатости на наноуровне. Другой рассматриваемый вопрос — это математические фракталы и масштабирование фрактальныж мер с применением к разрушению. В заключение, обсуждается понятие параметрической однородности, основанное на использовании группы дискретного растяжения координат. Параметрически однородные функции включают фрактальные функции Вейерштрасса-Мандельброта и гладкие логарифмически-периодические функции. Утверждается, что кольца Лизеганга являются примером параметрически однородного множества.
Ключевые слова: статистическая автомодельность, параметрическая однородность, масштабирование, фракталы, контакт, износ
1. Introduction
Scaling methods may be applied wherever there is a need in studying a phenomenon across many scales. The rescaling techniques include dimensional analysis, renorma-lization groups, intermediate self-similar asymptotics, in-
complete similarity, fractals, parametric-homogeneity and other techniques. Many of these techniques are described in books by Barenblatt [1-4].
Professor Grigory Isaakovich Barenblatt (1927-2018) was a remarkable applied mathematician who worked in
© Borodich F.M., 2018
many areas of mechanics, physics, and engineering. I will not list his numerous titles, awards and prices as well as his memberships in various prestigious scientific organizations because such a list would be too long. In 1994 G.I. Barenb-latt told me the following story. When he was considered for the position of G.I. Taylor Professor of Fluid Mechanics at the University of Cambridge, K.L. Johnson said that the position is in the area of fluid mechanics and why this famous expert in solid mechanics was under consideration? Thus, D.G. Crighton had to explain that solid mechanics is just one of the fields of Barenblatt's expertise. His famous fracture criterion [5] will not be discussed here either. There are papers where this criterion is discussed in detail (see, e.g. [6-8]). Here I will discuss several problems related to similarity methods of solid mechanics and nanomechanics where I have got some new results and these results were influenced either by our discussions with G.I. Barenblatt or by the techniques I learnt from his papers and books.
In particular, I consider here the self-similarity techniques developed by Barenblatt and Botvina in application to damage accumulation (see, e.g. [9-11]) and scaling of mathematical fractals [1, 3, 12-17]. I will discuss also the concept of parametric-homogeneity [18-21] and its application to contact problems for smooth and fractal punches and nanomechanics. All the above mentioned problems will be discussed through the prism of my personal reminiscences.
I started to enjoy my studies at Faculty of Mechanics and Mathematics (Mekhmat) of Moscow State University (MSU) in 1974. First I heard about G.I. Barenblatt from my teacher Askold Georgievich Khovanskii in 1975. Because A.G. Khovanskii is a pure mathematician [22], he could not be my official supervisor in Solid Mechanics. Hence, I asked him about researchers who could supervise my studies in this field. A.G. Khovanskii said that he heard only about Prof. Barenblatt as a researcher of very high reputation working in solid mechanics. Unfortunately Prof. Barenblatt did not collaborate with MSU that time. Later being the second year student, I found a comment by Timo-shenko, that the questionable aspect of the infinite stress at the end of the crack "has been removed by Barenblatt, who introduces instead large but finite stress to represent atomic cohesion" [23]. Because our lectures in mechanics of continuum media were delivered by Prof. Yurii Nikolaevich Ra-botnov who liked to say "all of us are pupils of Timoshen-ko", I asked him about this comment. Prof. Rabotnov was very kind and discussed with me the comment in detail. In 1976 I continued my studies at Mechmat with narrow specialization at Theory of Plasticity Division led by Prof. Ra-botnov. Thus, I heard about Prof. Barenblatt from my teachers and only much later I learnt that our beautiful sporty classmate Nadezhda Kochina was the oldest daughter of G.I. Barenblatt.
I had enjoyed by several remarkable seminars delivered by G.I. Barenblatt before I was introduced to him personally
in 1991. Prof. Barenblatt was a remarkable lecturer. Each of his talks in some way was structured like an Agatha Christie mystery novel. The lecture started by showing a mystery of a natural phenomenon (that could often obey a power-law behaviour at intermediate stage of its development). Then he presented a list of suspects (a list of possible approaches to the problem) and showed the investigation process. The denouement was made in the final part of his lecture, when G.I. Barenblatt revealed a proper explanation of the initially mysterious phenomenon. I had also studied some of his papers and models before our first meeting. On the other hand, I knew that in 1987 he and Prof. R.L. Salganik discussed my models of discrete contact between uneven surfaces [24] where I used the mathematical techniques and ideas of the Barenblatt-Botvina damage accumulation model [9-11]. During our first personal meeting, Prof. Barenblatt presented me the Russian version of his book [1] and a reprint of his remarkable paper [12]. Both presents had a great impact on some areas of my studies.
2. Statistical self-similarity of arrays of discrete points
In 1983 I prepared my PhD thesis [25] that I defended a year later at Mekhmat. In this thesis I developed a rather awkward self-similar model of discrete contact between uneven layers of multilayer metallic stacks. Modelling of nonlinear deformation of such stacks subjected by high pressure was important for understanding work of thick multilayer vessels of high pressure. I found that contact between layers exists just in discrete points and the number of these points increases as the compressing pressure. The problems of discrete contact is an active field of research because these problems are very important for tribology [26]. However, the problem I studied differs from the traditional ones because the number of contact spots increases us the pressure grows.
Later I read the paper on damage accumulation by Ba-renblatt and Botvina [9, 10] where a usual lemma of dimensional analysis ("the dimension function is always a power-law monomial" [1, 3]) was used in a very unusual and elegant way. In fact, they showed that the growth of damage in fatigue tests is statistically self-similar.
Hence, when one looks at the images of the points of damage at an initial moment t0 and at an arbitrary moment t, we cannot distinguish them statistically. Hence, one can write l(t)/1(t0) = F(t/t0) where Fis a function of the di-mensionless time. It follows from the lemma of dimensional analysis that F(x) = xa, a = F'(1), i.e. it is a power law function. Thus, if t/t0 =X then l(Xt0) =Xal(t0).
I understood that this is an universal approach to statistical self-similarity of discrete sets. Indeed, data in the form of a set of points distributed in an irregular way within a planar region, arise in many disciplines. The Barenblatt-Botvina scaling law means that the transformation of a set
of discrete points is a steady-state process and it transforms with the process time statistically in a self-similar way. In other words, if the distribution of the points of the set normalized by the average distance then the distribution is the same for dimensionless time.
Let us write this in a more formal way. One of popular techniques of statistical analysis of spatial point sets is the so-called distance method or the theory of the nearest neighbour. The method considers a point as the basic sampling unit and the distances to neighbouring points are recorded, i.e. the distances to the first, second, and to the kth nearest point. This technique converts a list of point coordinates to a unique data set relevant to study of the population density. It is known that if the spatial pattern is characterized by some one-dimensional probability distribution function fx (x) for the distances to the nearest point then the distribution function can be completely represented by its mean MX, i.e. its expected value E(X), and its higher central moments
M x = E (X) = J xfx (x)dx, 0
MX) = E[(X - Mx)" ] = J (x- Mx)n fx (x)dx, n = 1, 2,....
0
Using the mean, the higher central moments can be made dimensionless MxVMx. Hence, the statistical properties of the point set can be characterized by a single quantity with the dimension of length, namely the average distance {l(t)) = mx between points, and by an ensemble of dimen-sionless statistical characteristics.
I applied this statistical scaling to describe the evolution of spots of multiple contact between uneven layers in multilayer stacks and vessels loaded by external pressure [24]. Due to imperfections of the layer surfaces, there are gaps between layers. These interlayer gaps and the field of points of interlayer contacts develop statistically in a self-similar manner, and the volume of the gaps VP is described as a power-law function
Vp (P) = Vp ( pi)( p/p)a,
where P is the current pressure, P1 is the initial pressure, and a is the self-similar exponent [24, 25].
The same approach was used to describe the abrasive-ness of modern hard carbon-based coatings such as diamond-like carbon or boron carbide. Indeed, S.J. Harris and his co-workers discovered a remarkably simple power-law relationship observed between the abrasion rate of an initially spherical slider by hard carbon-containing films and the number of sliding cycles n that the film has been subjected. The power-law relationship is valid up to 4 orders of magnitude in n. We modeled this phenomenon by connecting it with nanoroughness of the coatings and explained the phenomenon by a statistically self-similar variation of the pattern of relatively sharp nanometer-scale asperities of the films [27, 28].
Later this approach was combined with classical scaling [29-33] to describe wear and abrasiveness of hard carbon-containing coatings under variation of the load [34]. The pin-on-disk tests in which the disks are coated and the counterparts (sliders) are steel balls, were analyzed. It was assumed that the dominant mechanism for slider wear by these nominally smooth coatings is mechanical abrasion of the slider by nanoscale asperities having relatively large attack angles, i.e. by sharp asperities. We propose a model in which we assume that (i) the abrasiveness of a contact is proportional to the number of asperities in the contact; (ii) the areal contact density is uniform; and (iii) the effect of increasing the load is to enlarge the initial apparent contact region between the ball and the disk. Using this model, the observed dependence of the wear rate on load follows relationships that are similar to Hertzian relationships.
The average abrasion rate {A(n)) is defined as the average of the instantaneous abrasion rates A during the first n cycles. Let Mbe the total volume of steel removed and d be the total distance travelled d = 2nRn. Here R is a pin-on-disk radius. Then we have M 1 n {A(n)) = M =1E 4.
2nRn n i=1
The region of initial contact depends on the external load and its size can be found from formulae of the Hertz-type contact. Assuming self-similarity of the Barenblatt-Botvina type and considering as the internal time t the number of cycles t = nT, where the initial time t1 is equal to the period of the cycle T, i.e. t1 = T, we obtain that after each cycle, the number of sharp asperities N(1) within this initial contact region is reduced according to the power-law N(i) = = N(1)i~2a. We will call the region of initial Hertzian contact G(1) as the central part of the slider. The ball cannot move down until the material of the central part has not been worn away, while the material outside this part is worn away by new sharp asperities.
Hence, the total amount of material M,!llder (n) removed from the central part of the slider is
Mcsllder(n) = | G(1) | h(n) = E mN(i) = Mcsllder (1)i"2a, i=1
h(n) =J h( x)dx = h(1)n1_2a, 0
where | G(1) | h(n) is the area of the central part, h(n) is the thickness of the removed slice of the ball after n cycles, and m is the amount of material removed by each sharp asperity. For an initially spherical slider of radius Rb, the total volume of steel M removed during n cycles is
V (h(n)) = nRbh2(n) =nRbh2(1) n2(1-2a). Finally, we observe the power-law of abrasion
{ A(n)) = { A(1)) n2(1-2aVn = { A(1)) n1-4a that agrees with experimental observations.
The above examples showed that scaling in nanomecha-nics and solid mechanics is not restricted to just the equiva-
lence of dimensionless parameters characterizing the problem under consideration. Even self-similarity of Hertz-type contact problems [29-32] cannot be described by a simple dimensional analysis. The same is related to scaling arguments in problems of nanoindentation [33].
3. Fractals
My regular scientific communications with G.I. Barenb-latt started in 1994. This was the last year Prof. Barenblatt worked at Department of Applied Mathematics and Theoretical Physics (DAMTP) of University of Cambridge and I arrived there by an invitation of Prof. John R. Willis who got support for this visit from the Royal Society. I would like to write several words about inspirational surrounding at the old small building of DAMTP in 1994-1995. Twice per day you could meet all famous professors at meetings at tearoom on the ground floor. Sometimes these tea-breaks were a bit noisy and I guess the noise could disturb Professor Stephen Hawking who had an access to his office directly from the tearoom. I benefited enormously by the care shown by J.R. Willis and personal discussions with him. As an example of his remarkable hospitality, I can mention the following. The main focus of my research was on contact problems and sometimes I wanted to discuss contact problems with Prof. K.L. Johnson who had retired from Department of Engineering by that time. To organize each meeting, Prof. Willis phoned to Prof. Johnson; then K.L. Johnson walked to DAMTP at Silver street and J.R. Willis walked to the DAMTP library to provide his office for our meeting. It looks incredible but it is a fact, two famous Fellows of the Royal Society of London cared about a modest young professor from Moscow like he was their adopted child. In such friendly atmosphere, I could start to write an extended paper of fractal approaches to fracture and I had a considerable progress on theory of parametric-homogeneous functions.
One of the topics I have discussed with G.I. Barenblatt was fractals. First, I learnt about fractals in 1983 when my classmate Mikhail Ermakov shared with Dmitry Onishchen-ko and me his impression about seminar on pelagic animals delivered by G.I. Barenblatt at Institute for Problems in Mechanics in Moscow. I learnt that there is a new branch of mathematical analysis of highly irregular objects, however I started to study applications of fractals to mechanics seven years later. In summer 1990, A.B. Mosolov encouraged me to study contact problems for fractal bodies. I borrowed a book "Fractals in Physics" [35] from my classmate Irina Petrova. Three days later she was quite surprised when I returned the book saying that I intend to write a paper. In fact, I modified the Cantor-Liu profile [36] that was infinite to CB profile having bounded size. Sometimes CB profile is called the Cantor set model [37], and the Cantor-Borodich profile, structure or fractal (see, e.g. [3840]). I found two kind of contact problems that can be solved for a punch described by the CB profile. However, that time I was not experienced in fractals and Alexey Mosolov
expressed these models using fractal terminology. We published two papers on this topis [41, 42]. Later D.A. Onishchenko and I have introduced a multilevel hierarchical profile that allowed us to confirm our statement that fractal dimension of a rough surface alone does not characterize contact properties of the surfaces [43, 44].
I define fractals as sets with non-integer fractal dimension and emphasize that we need to split the term in two: mathematical and physical fractals (see, e.g., [14, 15]). Confusion of these two kinds of fractals led often to various erroneous or at least unjustified conclusions (see discussion in [15]). The former term may be fixed for sets with non-integer mathematical fractal dimension. While the latter term is related to irregular physical objects which being covered by some small units (balls, cubes, yardsticks and so on) obey the power law relation between the number of covering units N and the scale of consideration 8. The main distinction between these two meanings is that the power law of natural objects (physical or empirical fractals) is observed on a bounded region of scales only, while mathematical fractals consider limits when the scale of consideration goes to zero.
Modelling of physical objects by means of mathematical fractal sets encounters various difficulties. For example, it is not a simple task to include scaling properties of a fractal object in its mathematical model.
In 1992 thinking about modelling fracture surface as a mathematical fractal surface, I formulated the following paradox: the Griffith criterion in its classical formulation leads to the conclusion that no fractal cracking is possible [13, 14]. Indeed, during the crack propagation, the amount of the released elastic energy U is finite and the area of a mathematical fractal surface is infinite. Then the use of the Griffith surface energy y leads to the above paradox. Therefore, we should introduce some new concepts to consider fractal cracks. I resolved the paradox using the following Barenblatt idea.
Barenblatt (see, e.g. [1, 3]) stated that the surface of the respiration organ of pelagic animals is a fractal and, therefore, the specific absorbing capacity of the organ has to be related not to its area, which is infinite, but to its Hausdorff measure. I developed the Barenblatt idea and suggested to apply this approach not only to sets with the non-integer Hausdorff dimension but also to fractal sets with other definitions of dimensions. I introduced the s-measure ms of a set that is an alternative to the Hausdorff measure. I took into account that the Hausdorff s-measure not always has a bounded positive value for s = D where D is the Hausdorff dimension of a fractal and introduced the concept of D-measurable sets, i.e. sets whose s-measure has a finite positive value mD for s equal to the fractal dimension D. Finally, I suggested to employ explicitly the scaling properties of ms. Bazant and Yavari [45] suggested to name this approach as the Barenblatt-Borodich idea.
Note that in the general case the s-measure ms does not satisfy all restrictions of a mathematical measure. Feng
and I [17] presented the concepts of upper and lower box-counting quasi-measures that should be used in models employing sets with box-counting fractal dimension. The scaling properties of the physical objects should be reflected through the scaling properties of the fractal measures. Roughly speaking it has been proposed to refer various physical quantities to a unit of the fractal measure of a fractal object.
Reading [1, 12] presented to me in 1991, I proposed to extend the Barenblatt idea and to introduction of the so-called "specific energy absorbing capacity" of a fractal surface P(D ). Physically, P(D ) gives the amount of elastic energy spent on forming a unit of the fractal measure mD. Here D is the fractional part of the fractal dimension, e.g. for a fractal dimension of a line is D = 1 + D*. The physical dimension of P(D*) is [P(D*)] = Fl/L2+d", where F and L are the dimensions of force and length respectively. Bazant in his books [46, 47] named P(D ) as the fractal fracture energy.
Because the length of a fractal curve is infinite, we consider its projection on the x-axis. Let us consider a fractal crack of projected length l and its advance a. It is assumed that for some fixed value of crack advance of the projection length L0, the surface energy n(Lj) is bounded and we can write n(L0) = 2tP(D*)mD(Lq) < where t is the thickness of a sample. Then for the surface energy of an advance a of a fractal crack, we obtain
n(a) = 2tp( D>d (Lo)(a/ Lq)d
< <.
(1)
If we rewrite the Griffith criterion using P(D ) concept, then the critical stress ac is [14]
<X. =
N( D-1)/2
a |
2p(D)DmD (Lq)E
k1L0(l + a) I L0 I
/2p(D*)(1 + D )mD (Lo)Ei (
or
|
D*/2
Lq
k1L0l (1 + a/l)
Here E1 is an elastic modulus of material and k1 is a di-mensionless coefficient. We see that initially the fractal crack propagated in a perfect brittle solid is stable. Indeed, ac grows with a. Note that the above formula is valid until the upper cutoff of the fractal crack is not reached.
The term fractal was introduced by B.B. Mandelbrot who published a book concerning fractals in 1975 [48]. Surprisingly, one could not find any definition of the term in the book. He gave numerous examples of sets which are more irregular than sets considered in common textbooks on Euclidean geometry. I met rather often an opinion that "the self-similar sets were almost forgotten and just a very small number of mathematicians knew about them. So, the interest in these sets was resurrected only in 70th". Obviously, this is not true. Starting my studies of fractal sets, I realise soon that I met these sets for the first time in 1969 reading the second edition of a popular book about sets written for school children by Vilenkin [49]. Of course, that time the main term was not yet introduced.
After I studied fractals in detail, I obtained new results for both mathematical and physical fractals. However, I was rather disappointed by these techniques. Mandelbrot
[50] noted that natural objects do not have pure shapes of classical mathematical objects and said that coastlines are not circles, clouds are not spheres, and mountains are not cones. D. Onishchenko and I added [44] "roughness of real bodies is not a mathematical fractal". We argued that all these geometrical objects: spheres, cones, circles as well as fractals are only mathematical idealizations of complex shapes of real physical bodies. Usually mathematical fractals possess such properties that do not allow researchers to use classical tools of investigation. For example, fractal surfaces are nowhere differentiable and, therefore there is no normal vector to such surfaces. Hence, even to give a rigorous formulation of the Gauss-Ostrogradskii theorem or the divergence theorem for fractal surfaces one needs to use rather complicated mathematical tools (for details see
[51]). Many statements about fractals are either not justified mathematically or simply wrong. Mathematical fractals are often not appropriate tool for studying physical phenomena, while physical fractals suffer from the lack of proper mathematical justification of the approach (see, for details [52]). Barenblatt and I had discussed this question many times because he disagreed with my opinion that Vilenkin's book for children is more serious than Mandelbrot's books on fractals.
4. Parametric homogeneity
In 1992 I introduced a class of parametric-homogeneous functions and started to study them. Eventually, the concept of parametric homogeneity was developed. This topic studies parametric-homogeneous (PH) and parametric quasi-homogeneous (PQH) functions, PH- and PQH-sets, and corresponding transformations [18-21]. I started to develop the concept under influence of a paper on the Weierstrass-Mandelbrot function [53]. I thought that the concept and the theory had been developed for a long period of time. However, my guess was wrong. I found in literature only particular cases of PH-functions. In 1994 I started to discuss the concept with G.I. Barenblatt, however he noted that Zababakhin introduced a kind of such sets and he and Zeldovich discussed this case of self-similarity in a couple of review papers.
PH-functions and PQH-functions are natural generalizations of concepts of homogeneous and quasi-homogeneous functions when the discrete (discontinuous) group of coordinate dilations (PH-transformation) (r p„k)
rp„k x = (pka1 X1,..., pka-xn), p > 0, k e Z,
is considered instead of the continuous group of coordinate dilations x ^ Xx. The function Bd: Rl ^ R is called a parametric quasi-homogeneous function of degree d and parameterp with weights a = (a1;..., a;) if there exists a positive parameter p, p + 1 such that it satisfies the fol-
lowing identity
Bd (rpakx; p) = Bd (pka1 x,..., pkaix; p) = = pMBd (x; p), k e Z,
and the parameter is unique in some neighbourhood. Here Z is the set of integer numbers. PH-functions are a particular case of PQH-functions when a1 =... = al. To avoid a non-unique definition, the least p: p > 1 is taken as the parameter. The graphs of these functions can be both continuous and discontinuous, they can also be smooth, piece-wise smooth, with singular points of growth, fractal, non-fractal nowhere differentiable (see for details [20]). Smooth sinusoidal log-periodic functions and fractal Weierstrass type functions are examples of PH-functions [18, 20]
b0(x; p) = Acos(2nlnx/lnp + 0) or
bp(x; p) = E p-nh(pnx),
n=-<*>
where A and O are arbitrary constants and h is an arbitrary function. In particular, we can write the so called Weier-strass-Mandelbrot function
C(x; p) = E p(D-2)n (1 - cospnx), 1 < D < 2,
n=-<*>
where D is equal to the box dimension of the graph. One can see that the PH-functions may have the same global trend, while they have different local behaviours.
It is easy to check that the PH-functions near any point x0 are repeated in scaling form near all points pkx0, k e Z. For example, if bd (x0; p) = Ax¡j sin(2nlnx0/lnp + 2nk), then for bd (pkx0; p) we have
A(pkx0)d sin(2nln(pkx0)/lnp) =
= pkdAxd sin(2nlnx0jlnp + 2nk) = pkdbd(x0; p). Contrary to classical scaling when X is arbitrary positive number, rescaling parameterp of the PH-transformation is fixed.
Thus, the PH-sets and PH-scaling can arise in systems having a fixed scaling parameter p. If the fundamental domain is somehow filled, then one can obtain the whole set by applying a PH-transformation to the fundamental domain. If the filling is fractal then the whole set is also fractal.
The concept of PH-tansformatios was applied to contact problems [18, 21]. Borodich and Galanov [54] presented numerical simulations of contact stresses between a PH-punch and an elastic half-space in the case when the profile of the punch was described by a smooth log-periodic function. It was found that the Hertz type contact problems have some features of chaotic systems: the trend of P-h curve (the global characteristic of the solution) is independent of fine distinctions between functions describing roughness, while the stress field (the local characteristic) is sensitive to small perturbations of the indenter shape. Fractal dimension of roughness alone does not characterise the properties of the contact problems.
As G.I. Barenblatt mentioned, the first physical model, where sine log-periodic functions arose, was a model of shock waves in layered systems constructed by Zababakhin in 1965 [55]. The nested self-similar model of turbulent flow constructed by Novikov [56, 57] contains log-periodic modulations as well. Barenblatt and Zeldovich [58, 59] considered the Zababakhin model as a very interesting development of self-similarity. Then numerous authors who studied scaling at critical points of phenomena noted that in many cases the critical behaviour is modulated: instead of observing as a leading singularity a pure power law, one finds a power law multiplied by a periodic function of the logarithm of the distance to the critical point (see, e.g., [60, 61]). I believe PH-functions are very useful for studies of self-similar phenomena exhibiting threshold behaviour. Clearly, real natural phenomena do not exhibit the pure mathematical PH-properties. However, PH-features can be exhibited by some processes on their intermediate stage when the behaviour of the processes has ceased to depend on the details of the boundary conditions or initial conditions. The self-similar approximation theory developed by Yukalov (see, e.g. [62]) is based on the use of log-periodic functions. In particular, he described the oscillatory behaviour of the energy release in the vicinity of fracture using log-periodic models. However, he noted that he could use also other parametric homogeneous functions.
The idea, that various processes possess an intermediate self-similar stage of their development, was successfully used in the study of continuous self-similarity [1, 3]. We can expect that some self-organized processes possess the PH-features. Indeed, log-periodicity, which is a particular kind of parametric homogeneity, was considered in various papers.
As an example, let us consider Liesegang rings. The Liesegang ring consists of diffusion of silver nitrate which is placed as a drop of aqueous solution onto a gel matrix containing potassium bichromate [63]. During the experiment there arises a fragmented pattern consisting of bands (rings) of different widths. The formation of the rhythmic bands (the Liesegang rings) are explained by self-organization (internal rhythms) and since the discovery of the Liesegang rhythmic patterns, numerous experimental studies have been done. It was found that the Liesegang ring patterns possess PH-features, namely the distances bn between the successive nth and (n + 1)th Liesegang rings increase following the law of geometric series or Jablczynski law [64]
bn+1/bn = p, P >1 Zeldovich, Barenblatt and Salganik [65] developed a rather sophisticated mathematical model of the formation of the Liesegang rings considering nontrivial diffusion equations. However, if we suppose that the solution u(x, t) of the equations is a PQH-function of degree d and parameter p with weights a1, a2 and consider the classical diffusion equation
d , N d2 . . —u( x,t) =—T u( x,t) at dx
as a leading term of the model, then we obtain
a2( n+1)
xn+1 = p x = tt
„ a1n p '
xn p 1 xq
t «20+1) t tn+1 _ p t0
(2)
na2nt p t0
= pa2, a2 = 2a1.
Here x0 and t0 are some values of the arguments for a pattern point, xn and tn are the equivalent values of the arguments. This means that we obtained both the Jabl-czynski law and the so-called time law xn ~-fa.
5. Conclusions
The paper presented a very personal review of application of modern scaling techniques to solid mechnaics and nanomechanics. Using Barenblatt-Botvina techniques, I described scaling of increasing number of discrete points of contact between uneven layers subjected to external pressure; and scaling of decreasing number of discrete sharp nanoscale asperities in pin-on-disk wear tests for some modern carbon-based coatings. Then I discussed Barenblatt idea of attributing scalar physical quantities to fractal measure. This idea provide a proper mathematical tool for studies scaling of mathematical fractals. I developed this idea and demonstrated it in application to fracture mechanics. Finally, I described some aspects of theory of parametric-homogeneous functions. I believe that it can be useful for studying discrete self-similarity and phenomena having oscillatory behaviour on their intermediate stage. I have also discussed briefly the Liesegang rings arguing that this is a PH-phenomenon.
All these ideas were described through the prism of my personal reminiscences. Currently my favorite topic is nano-mechanics, in particular mechanics of adhesive contact. I had many discussions with G.I. Barenblatt on nanome-chanics. He agreed with my definition of nanomechanics as a scientific discipline that studies (i) application of mechanical laws and solutions of problems of mechanics to objects of nanotechnology; (ii) interactions between physical objects using mechanical equations adjusted to the specific character of the interactions at nanometer length scale; and (iii) influence of nanometer scale objects and processes on meso/macroscale phenomena. We discussed some results of B.V. Derjaguin whom I consider as one the founding fathers of nanoscience. He agreed with me that molecular adhesion is a crucial feature of nanoscale interactions. I explained him why I consider the classic Johnson-Kendall-Roberts (JKR) theory of adhesive contact as a very beautiful theory and told him about my extensions of the JKR theory [29]. In turn, G.I. Barenblatt argued that not only the chemical reactions take place at the Angstrom length scale, but this scale has a fundamental physical meaning in nanomechanics and should be included in the list of
governing parameters when the scaling laws are derived. He sent to me a draft of his paper on this topic that was later published [66]. After one of the discussions at University of California, Berkeley, in 2007, the following sentence was born: "Mechanics is like a phoenix. Many times it was declared as dying. However, we see its rebirth again and again. Nanomechanics is one of its new beautiful reincarnations".
G.I. Barenblatt knew that I like very much his ideas. I discussed them in many papers and I applied his ideas to various problems. I am very proud that Z.P. Bazant put our names together [45] as a recognition of my modest contribution to development of one of Barenblatt's brilliant ideas. Certainly, the results of G.I. Barenblatt have earned a place in the Hall of Fame of modern mechanics and applied mathematics.
Acknowledgments
I am very grateful to Professors Lyudmila R. Botvina and Nikita F. Morozov for inviting me to contribute to this Special Issue.
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Received November 15, 2018, revised November 15, 2018, accepted November 22, 2018
Cardiff University, UK, [email protected]
