Developing the mechanical models for nanomaterials

Composites: Part A 38 (2007) 1234–1250 www.elsevier.com/locate/compositesa

Developing the mechanical models for nanomaterials I.A. Guz a

a,*

, A.A. Rodger a, A.N. Guz b, J.J. Rushchitsky

b

Centre for Micro- and Nanomechanics, College of Physical Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK b Timoshenko Institute of Mechanics, Nesterov str. 3, 03680 Kiev, Ukraine Received 1 May 2005; accepted 1 April 2006

Abstract This paper revisits some of the well-known models in the mechanics of structurally heterogeneous media for the purpose of analysing their suitability to describe properties of nanomaterials (nanoparticles and nanocomposites) and their mechanical behaviour including statics, dynamics, stability and fracture. The paper gives an overview of existing knowledge on the nanomaterials and nanotechnologies, with the emphasis on their mechanical properties. A number of the macro-, meso- and micromechanical models are then reviewed and new nanomechanical models are suggested based on the knowledge accumulated within the micromechanics of composite materials. New directions of research in nanomechanics of materials are also pointed out. As an example of application of the developed models, the paper contains results of determination of the effective properties for particular nanocomposites.  2006 Elsevier Ltd. All rights reserved. Keywords: A. Nano-structures; B. Mechanical properties; C. Analytical modelling; Micro-mechanics

1. Introduction Nanotechnologies and nanomaterials are arguably the most actively and extensively developing research areas at the end of the last/beginning of this century. The number of publications in scientific periodicals and conference proceedings, fully or partially devoted to nanotechnologies and nanomaterials, rapidly increases. Since fundamental results in the area were obtained by chemists and physicists, the majority of new papers also come from chemistry, physics and materials science. In the same time, mechanics of nanomaterials, or nanomechanics, has received considerably less attention, with papers few and far between, covering very specific aspects of research. Development of nanomechanical models and their application to investigation of mechanical behaviour of nanomaterials in a systematic way is not happening yet. * Corresponding author. Tel.: +44 0 1224 272 808; fax: +44 0 1224 272519. E-mail address: [email protected] (I.A. Guz). URL: http://www.eng.abdn.ac.uk/iguz (I.A. Guz).

1359-835X/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2006.04.012

Any mechanics of materials, including mechanics of nanomaterials, envisages analysis of materials for structural applications, be it on macro-, micro- or nanoscale. It is therefore a logical conclusion to any research on nanomaterials, and should precede the analysis of nanomaterials working in various structures and structural members such as shells, plates, beams etc and various devices. Micro- and nano-structural applications look like the most natural and promising areas of the nanomaterials utilisation. They do not require large industrial production of nanoparticles which are currently rather expensive. The first section of the paper gives a brief historical overview of development of nanomaterials and nanotechnologies, with the emphasis on mechanical properties and specific features. In the second section, we attempt to formulate basic problems of nanomechanics and suggest possible ways how to solve them using the knowledge accumulated within the solid mechanics and, more generally, continuum mechanics. Certainly, the choice of papers for the first section and the formulations and approaches suggested in the second section reflect the authors’ opinion. Since mechanics of

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nanomaterials finds itself at the early stages of development, many points made in this paper will be subject to criticism and even rejection. This is natural for any research area at such stage of development, and the authors welcome all critical remarks and suggestion that can contribute to the development of nanomechanics. 2. Brief historical outline of development of nanotechnologies and nanomaterials This section gives an overview of the existing knowledge on the subject as reflected in publications in scientific periodicals and conference proceedings. First of all, the prefix ‘‘nano‘‘ means one billionth part of something. In Greek, nano means ‘‘dwarf ’’. In nanotechnology and nanomechanics, ‘‘nano’’ refers to the length scale of 1 nm, or 1 · 109 m. Many papers mention Richard Feynman as the first person to predict development of nanotechnology. In his historic lecture entitled ‘‘There’s plenty of room at the bottom’’ [30] delivered in 1959 before the American Physical Society, Feynman formulated the fundamental principle on which nanotechnology relies: ‘‘The principles of physics, as far as I can see, do not speak against the possibility of manoeuvring things atom by atom’’. At the time, tools necessary to analyse the nanostructure of a substance did not exist. Electronic microscopes, the principal tools to analyse nanomaterials, have been invented fairly recently. The first scanning electronic microscope was developed in 1942 and became available in the 60s. The scanning atomic-force microscope and scanning tunnelling microscope, suitable for study of nanomaterials, were invented in the 80s, by Binnig and Rohrer of IBM Zurich in 1981 [10,12,34] and Binnig, Quate and Gerber in 1986 [28,90], respectively. The inventors of both microscopes were awarded the Nobel Prize in Physics in 1986 [12]. The fact that these microscopes allowed researches to look at the surface of a material at a nanometre scale was crucial to the success of many experimental observations of nanomaterials. The second important contribution to the nanotechnology was made by Eric Drexler, who helped to shape the emerging technology. In his book [25] he wrote: ‘‘Nanotechnology is the principle of atom manipulation atom by atom, through control of the structure of matter at the molecular level. It entails the ability to build molecular systems with atom-by-atom precision, yielding a variety of nanomachines’’. Nowadays, the term ‘‘atom-by-atom’’ refers to mainly the molecular nanotechnology. Since then, nanotechnology has captured attention of many national scientific funding bodies. Interestingly, the point of view on what constitutes a nanoobject varies from country to country. In the UK, it is 107–109 m, in Germany it is everything below 106 m, and in the USA it is 106–109 m. National Scientific Foundation (USA) also gives its own definition of the nanotechnology [66].

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2.1. Nanotechnology If the task of any technology is to establish physical, chemical, mechanical and other laws with the purpose to discover and implement the most efficient manufacturing processes, then we have to admit that the nanotechnology is still at the early stages of development. In 2001, the Science magazine named nanotechnology ‘‘The Breakthrough of the Year’’ [46], and indeed, there is a number of very promising findings. In the same time, the Scientific American magazine voiced scepticism, which is always present in science. In the special issue, partially devoted to nanotechnology, Gary Stix, a special projects editor, noted that the word ‘‘nanotechnology’’ sounds more like and science-fiction one that its scientific equivalent ‘‘applied mesoscale materials science’’ [81]. He also advised to ‘‘discard the overheated rhetoric that can derail any big new funding effort.’’ It is also indicative that the nanotechnology is not profitable as yet. However, declaring nanotechnology ‘‘enabling and potentially disruptive’’ should be regarded as a scientific advertising and the reality of the near future. There are various points of view as to what directions the nanotechnology is most likely to take. In [91], five major areas within the nanotechnology were identified: (a) molecular nanotechnology; (b) nanomaterials and nanopowders; (c) nanoelectronics; (d) nanooptics and nanophotonics; (e) nanobiomimetics. In the authors’ opinion, the last three areas deal with the application of nanomaterials. Therefore, the concept of a nanomaterial should be discussed fully and comprehensively. However, this discussion has not taken place yet, at least not in the context of the mechanics of materials. This is due to the fact that the overwhelming majority of studies on nanomaterials is carried out within chemistry and molecular physics, because it is primarily chemists and physicists who had experience in working with substances on atomic, molecular and macromolecular levels. Apart from molecular nanotechnology, there are six well-known manufacturing methods to produce nanomaterials [65,70]: (i) Arc plasma spraying or plasma arcing [68]. This plasma ionisation scheme includes two electrodes to generate potential difference. If the electrodes are in a gas medium, the gas loses electrons and becomes ionised, forming plasma. To produce, for example, a nanotube film, carbon electrodes are used, of which one generates carbon cations and the other receives them to form nanotubes. (ii) Chemical vapour deposition [14,27]. According to this technique, a material is placed in a vacuum, heated up until it vapourises and then deposited on the hard surface. Deposition can be either direct or accompanied by a chemical reaction between the material being deposited and the material of the hard surface. The latter usually leads to formation

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(iii)

(iv)

(v) (vi)

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of oxides or carbides of metals on the metal surface, if oxygen and carbon are present in it. Electro-deposition or galvanisation [8]. This is a well known and long-established technique. In nanotechnology, between one and several atomic layers need to be deposited. Nanogalvanisation is also used to produce dispersive nanomaterials. For example, polymeric membranes, with pores from 10 to 100 nm in size become conductor materials when filled with atoms of metals. Such nanocomposite exhibits adaptable properties and behave as smart materials. Sol–gel synthesis [4–7,32]. Gels are suspensions of disperse particles, which form spatial lattice in a dispersive liquid. The hold the shape like materials and may form by coalescence of sol particle. If the particles in the suspension are rather small and in Brownian motion, then such suspension is termed a sol. A sol–gel is a shape-holding suspension of colloidal particles in a liquid. Nanoparticles form a sol–gel in four stages, including hydrolysis; condensation and polymerisation of monomers; particle growth; and agglomeration of the particles and formation of a gel. The sol–gel synthesis is used to produce nanofilms. Ball milling, or mechanical fracturing [18,70]. Use of natural nanoparticles [64].

2.2. Nanomaterials A theoretical interpretation of a nanomaterial must be based on some primary concept. We suggest using as such a concept the idea that all materials are composed of granules, which in turn consist of atoms. These granules may be visible or invisible to the naked eye, depending on their size, which can range from hundreds of microns to centimetres. The term ‘‘material’’ usually refers to a substance in a solid state, which can be in either crystalline or glassy phase. Since nanoparticles can form both nanomaterials and nanopowders, it is worth pointing out that in mechanics materials are assumed to have stability of shape, while powders are regarded as granular media. Usually, powders are made into materials by application of special techniques such as compression, sintering, irradiation etc. As already mentioned, the size of granules in nanopowders and nanomaterials ranges from 10 to 100 nm in at least one dimension (normally in all three). Many authors believe that nanomaterials, i.e., materials with internal structure of nanoscale dimension, are hardly something new to science – just it has been discovered only recently that some formations of oxides, metals, ceramics, and other substances are nanomaterials. For example, black carbon was discovered at the beginning of the 19th century. Fumed silica powder, a component of silicon rubber, came into commercial use in 1940. However, it is only recently become clear that these two substances are nanomaterials.

It should be noted that the size of a particle is not the only characteristic of nanoparticle, nanocrystal, or nanomaterial. In authors’ opinion, one quite important and specific property of many nanomaterials is that the majority of their atoms are located on the surface of a particle as opposed to conventional materials, in which atoms are distributed over the volume of a particle. Below we give a brief overview of the existing manufacturing methods. The last of manufacturing techniques, mentioned in Section 2.1 (use of natural nanoparticles), deals with materials with 10 nm pores into which medium-sized molecules can permeate. Zeolites (aluminosilicates) and sheet silicates (phyllosilicates) are examples of natural nanomaterials used in nanotechnology. The crystalline structure of aluminosilicates is formed by tetrahedral molecular fragments of silica (SiO2) and alumina (AlO4). These molecules have common vertices in a three-dimensional skeleton pierced with cavities and channels that contain water molecules, metal cations, etc. Aluminosilicates are used as molecular sieves. Flat phyllosilicate sheets are separated with [Al13O4(OH)24]7+ polycation columns, 1 nm in height, to provide channels between the sheets. Films formed by titanium dioxide (TiO2) nanoparticles are produced by the sol–gel technique and used as a bond to living bone. Various nanostructures of such films are shown in [91]. The paper also gives the examples of silicon nanofilms made by the sol–gel technique. The films have a hexagonal lattice nanostructure with cylindrical pores 2– 3 nm in diameter. For ceramics based on silicon nitride (Si3N4) nanoparticles, the atoms are aligned in rows resembling ropes, spaced at 0.6 nm. Oxides of metals are nanomaterials produced by milling. For example, zirconia (ZrO2), aluminium oxide (AlO4), and aluminium titanate (AlTiO3) are used in ceramics. Carbon particles are a well-studied class of nanoparticles. Science has long been aware of three forms of carbon: diamond, graphite, and amorphous carbon. The highly symmetric molecule of carbon C60 was discovered in 1985. It has a spherical form, resembling a football, with carbon atoms on the surface. It contains 60 atoms in five-atom rings separated by six-atom rings. These molecules were named fullerenes and became the subject of extensive and fruitful studies. The scientists who studied fullerenes were awarded the Nobel Prize in Chemistry in 1996 [22,52,80,81]. Since then the number of different varieties of fullerenes has increased considerably, reaching many thousands to date [43,65,91]. The molecule C60 has got its name from R.B. Fuller (1895–1983), an architect who built a house from pentagons and hexagons. The molecule C70 resembles a rugby ball and includes only six-atom rings, but and also bears the name fullerene. Fig. 1 depicts both molecules [91]. Fullerenes can form crystals called fullerites. These crystals have a face-centred cubic lattice with cavities of two types: tetrahedral and octahedral. Placing ions of potassium, rubidium, or caesium into these cavities produces new nanoparticles with unusual properties.

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Fig. 1. The molecules C60 and C70 (fullerenes) as presented in [91].

Fullerenes may be deposited onto a surface, forming a monolayer. There are also rope-like fullerene formations. But the most important thing is that fullerene molecules can form carbon nanotubes, which may be considered as related to graphite. The molecular structure of graphite looks like a sheet of chicken wire, a tessellation of hexagonal rings of carbon [91]. In graphite, these sheets are stacked one over another and can slide past each other. This explains why graphite is soft and greasy and is often used as a lubricant. When graphite lattices are rolled up into a tube, they form nanotubes – molecules with a very large number of atoms, C10,000  C1,000,000. Nanotubes differ in diameter, length, and the way they are rolled. The internal cavities may also be different, and tubes may have more than one sheet. Atoms form ‘‘hemispherical caps’’ at the ends of a fullerene molecule. Sheets may be rolled differently, forming zig-zag, armchair and chiral structures. Since these structures are quite unusual for mechanics, we will discuss them in detail following [91]. Any cylindrical nanotube can be considered as being a rolled graphite sheet. A sheet is schematically

A

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depicted as a periodic hexagonal structure or lattice consisting of regular hexagons with carbon atoms at vertices (Fig. 2). The procedure of rolling a sheet is as follows: choose two atoms A and B that belong to hexagons spaced far enough from each other (it is clear that a tube cannot be made from three hexagons); connect the points with a straight line; cut out a strip from the sheet by two cuts perpendicular to the AB line; and roll the strip so as to superpose the points A and B. The length of AB will be the width of the rolled-up strip and simultaneously the length of a parallel circle of the resultant circular cylindrical tube. Such an arbitrary tube is called a chiral tube. Chirality, which can be left or right, is a concept borrowed from stereochemistry. It refers to the property of non-identity of an object with its mirror image. Three types of chirality are distinguished: central, axial, and planar, corresponding to three chiral elements, namely centre, axis, and plane. Chirality can be measured with the help of a vector ~ ¼ n~ ~ r ¼ AB e1 þ m~ e2

ðn; m 2 N Þ

ð1Þ

where ~ e1 and ~ e2 are vectors, defining two directions intersecting at A. In general, at any vertex of a hexagon there are three directions, important to the lattice. One of them (type 1) is the direction towards the opposite vertex of the hexagon. The straight line crosses the hexagon and then passes along the common side of neighbouring hexagons, with this sequence repeating periodically. The side of a hexagon with two adjacent sides resembles an armchair. The other two directions (type 2) pass through A and the third (counting from A) vertex either on the left or on the right. These two directions define two straight lines that are specific to the lattice; they are differently directed but otherwise identical. The sides connecting all the three vertices (A, the second and the third ones) resemble a zigzag, and this sequence repeats periodically. The vector ~ e1

type 2

ne→1

me→2

B

type 1 type 2

Fig. 2. A cylindrical nanotube represented as a rolled graphite sheet with a periodic hexagonal structure, or lattice consisting of regular hexagons with carbon atoms at vertices [91].

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always lies on a type 2 line (horizontal line in Fig. 2), while the vector ~ e2 can be directed arbitrarily. Chiral structures are distinguished by the numbers n and m and denoted by (n, m). If ~ e1 and ~ e2 are co-directional, i.e., B and ~ e1 lie on the same straight line, then we have a zig-zag chiral structure. Its typical feature is that m in the notation (n, m) is always equal to zero. If the point B lies on a straight line of type 1, then ~ e2 lies on the second straight line of type 2 and the type 1 line is the bisector of the angle formed by the type 2 lines. It is an armchair structure and its numbers n and m are always equal. Any other structure is called a chiral structure. Fig. 3 [85] shows an armchair (a) and a zig-zag (b) structure. Fig. 4 [53] gives examples of single-walled nanotubes with different chirality for the particular values of (n, m). The length of nanotubes can be 1000 times (and more) greater than their diameter. They can be single-walled and multi-walled. Multi-walled nanotubes, between 4 and 30 nm in diameter, were discovered by Iijima [42] in 1991.

Fig. 3. Atomic structure of (a) an armchair and (b) a zig-zag nanotube, after [85].

The thinnest single-walled nanotube was described in [82,89]; its wall thickness, 0.4 nm, appears in the titles of both papers; by comparison, the diameter of the carbon atom is equal to 0.15 nm. An assembly of nanotubes may have a diameter of 30 to 50 nm and a length of more than 50 lm [91]. Single-walled nanotubes may form rough structures or ropes. Ropes consist of nanotubes bundled in an ordered manner to form a lattice. Nanotube ropes may have a diameter of 10–20 nm and a length of 100 lm and more [91]. Nanotubes are technologically advantageous over conventional carbon fibres, since they are produced from colloidal solutions at room temperatures, whereas carbon fibres require high temperature environment. Research of nanotubes follows mainly three directions: (i) mechanical and electric properties of nanotubes in polymeric films, (ii) arrangement of nanotubes in polymeric composites, and (iii) properties of the nanotube-matrix transition layer. Fullerenes and others nanoformations have been studied in many research centres, and the scientific papers and books [13–16,23,24,31,32,47,48,54,56,59,61,77,86,88,94] reflect well the dynamics of these studies. According to [74], although nanotubes exhibit ideal characteristics when shrouded within pristine, ultrahigh vacuum environments, samples in more ordinary conditions, where they are exposed to air or water vapour, evince electronic properties that are markedly different. Mechanical properties are likely to show similar sensitivity. A similar observation was made in [54]. The authors pointed out that despite the specific surface structure of nanotubes, the adhesive bond between carbon nanotubes and polymeric matrix in nanocomposites is of about the same nature as in carbon fibres–polymer microcomposites. It is interesting to note that fibre whiskerisation, a technique developed and used to improve interface adhesion in fibre-reinforced composites, has been adopted without much variation for nanotubes. During whiskerisation, microfibres are covered with fibres of even smaller diameter. Difficulties that physicists encounter in studying nanoparticles are, perhaps, most clearly described in [73,74]: According to the author, matter at the mesoscale is often awkward to explore. It contains too many atoms to be easily understood by straightforward application of quantum mechanics (although the fundamental laws still apply). Yet these systems are not so large as to be completely free of quantum effects; thus, they do not simply obey the classical physics governing the macroworld. It is precisely in this intermediate domain, the mesoworld that unforeseen properties of collective systems emerge. 3. Mechanical properties and characteristics of nanoparticles and nanomaterials

Fig. 4. Examples of single-walled nanotubes with different chirality for the particular values of (n, m), after [53]. From top to bottom: armchair, zizzag and chiral nanotubes.

In this section, we will discuss the mechanical properties and characteristics of nanomaterials (nanoparticles and nanocomposites).

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3.1. Overview of the mechanical properties While the level of knowledge on the subject of mechanical properties of nanomaterials can be described as extremely low, for microcomposites quite a good database of mechanical characteristics have been accumulated over the years. Composites have been studied extensively both within the framework of macromechanics and micromechanics. Of many important achievements in the mechanics of composites, the following two are the most relevant to nanomechanics. The first one is that the role of a mechanical model is appreciated and understood. In mechanics of composites, the same composite material can be described by a number of micromechanical models – from complex multi-continuum models to discrete ones in the form of a lattice [19,63,75]. Each model offers its advantages when analysing different aspects of a problem. The abundance of models is indicative of the deep insight achieved in the micromechanics of materials. The second achievement is that a wide variety of mechanical characteristics has been established for composite materials and their constituents. For example, the following properties characterise aramid fibres (Kevlar) [40]: (i) density, (ii) diameter of a single fibre, (iii) equilibrium humidity, (iv) ultimate tensile strength, (v) elongation at rupture, (vi) initial modulus of elasticity, (vii) maximum modulus of elasticity, (viii) modulus of elasticity in bending, (ix) design modulus of elasticity in axial compression, (x) dynamic modulus of elasticity, (xi) ratio of loop strength to ultimate tensile strength, (xii) fatigue properties (the number of bending cycles to failure), (xiii) creep under loading to 90% of ultimate strength, and (xiv) coefficient of friction. By comparison, the data on the mechanical characteristics of nanoparticles are extremely poor, and the exemplary fourteen characteristics of a single fibre should be regarded as a remote goal for nanomechanics to pursue. Interestingly, some composites include reinforcing elements that are so small that according to the modern classification such composites should be attributed to the nanomaterials and their fillers – to the nanoparticles. Among such fillers are crystals of inorganic titanate (commercial name Fybex) and special forms of whiskers – silicon carbide (SiC) whiskers and sapphire whiskers named cobweb [41]. Fybex is a microfibre with a diameter of 100– 160 nm. Silicon carbide whiskers also have the form of long fibres, hundreds of nanometres in diameter. Cobwebs are long twisted fibres approximately 18 nm in diameter [41]. Not surprisingly, some of the nanoparticles have been produced during research of highly technological fillers such as those mentioned above (oxides and whiskers). In particular, it is worth mentioning nanocrystals of cerium (CeO2) [91] and zinc oxide (ZnO). The fourteen mechanical characteristics of a single microfibre confirm the well-known fact that density, stiffness, and strength are the most important characteristics of a material. Today, theses and other related quantitative characteristics of nanomaterials have been obtained in var-

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ious research areas, using the theoretical and experimental methods accepted within them. The next step is to correlate these quantitative data with the approaches and concepts of the mechanics of materials and its standards for determining corresponding quantities. In mechanics of materials, the analysis of a material is carried out in relation to the phenomena occurring in structural members made of it. Research within the framework of the mechanics of nanomaterials should ultimately lead to the analysis of the behaviour of nanomaterials in structural elements (shells, plates, beams, etc.), which predetermine the use of these materials. Therefore, the densities of nanotubes of different chirality theoretically predicted in [91] – namely, 1.33 g/cm3 for zig-zag nanotubes (17, 0), 1.34 g/cm3 for armchair nanotubes (10, 10), and 1.4 g/cm3 for chiral nanotubes (12, 6) – as well as studies of the diameter-dependence of the density [84], should be regarded only as a probing effort to investigate the mechanical properties of nanoparticles. Also, Young’s moduli suggested by various authors should probably be treated in the same way. One more important lesson that could be learned from the mechanics of composites [19,63] is that the mechanical properties of fillers (granules, fibres, and thin foil), determined with all the rigor of standard tests, mostly appear to be different (and sometimes even very different), when those fillers are being components of composites. The prediction may appear true for the nanoparticles, which as fillers or as assemblies could actually have somewhat different mechanical properties from those calculated for a single particle. To simplify the subsequent comparison, we will present the available data [40] on Young’s modulus and ultimate tensile strength of graphite and carbon microfibres used in the mechanics of composites and quite close in structure and properties to carbon nanotubes. Young’s modulus of graphite fibres depend on the orientation graphite planes. If the degree of orientation is high, it is larger than 345 GPa. If the degree of orientation of fibres is small, Young’s modulus is lower than 345 GPa, and the fibres are termed carbon fibres. Two types of fibres are distinguished: (i) low modulus/high strength (Young’s modulus 210 GPa, the ultimate tensile strength 3.275 GPa) and (ii) high modulus/low strength (Young’s modulus 450 GPa, the ultimate tensile strength 1.62 GPa). Carbon fibres in pitches may have Young’s modulus of 880 GPa and ultimate tensile strength from 1.38 to 2.2 GPa. High-modulus and simultaneously high-strength Thornel fibres have the following Young’s modulus and ultimate tensile strength: 170 GPa and 1.27 GPa for Thornel 25 and 250 GPa and 2.65 GPa for Thornel 75. Today’s estimates for Young’s modulus of nanotubes range from 600 GPa to 1.8 TPa. Theoretical predictions for the above-mentioned nanotubes of different chirality are as follows [91]: 648.43 GPa for zigzag nanotubes, 640.3 and 673.94 GPa (different modifications) for armchair nanotubes, and from 1.22 to 1.26 TPa for chiral nanotubes. The latter depends on chirality: 1.22 TPa for (10, 0) and (6, 6), 1.26 TPa for (20, 0). One

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of the latest publications [54], referring to [56,60], estimates Young’s modulus ranging from 270 GPa to 1 TPa and ultimate strength from 11 to 200 GPa. Many authors have studied intensively Young’s modulus and other characteristics (first of all, tensile strength) of nanoparticles [17,23,24,47,48,54–56,59,61,71,77,86,92,94] and others. Studying the mechanical characteristics of nanocomposites appears less difficult. In this field, a great many results have been obtained and there is a greater variety of approaches [1,13,20,33,44,45,48,53,57,58,60,62,64, 67,72,76,83,87,93] and many others. 3.2. Discussion Based on the review of existing studies of mechanical properties of nanoparticles and nanomaterials, the following brief conclusions can be formulated. It should be noted that by nanocomposites we mean (i) traditional composites reinforced with microfibres or microparticles and containing nanoparticles in the matrix and (ii) nanocomposites reinforced with nanoparticles. 3.2.1. Nanoparticles and nanoformations a) The unifying property of all known nanoparticles is their dimensions, while their internal structure may be quite different. For example, carbon nanotubes are large single molecules, whereas oxide nanoparticles are crystals or crystal formations. b) The integral physical properties of isolated nanoparticles (especially, of the most intensively studied carbon nanoparticles, fullerenes) are substantially different from the physical properties of particles in conventional materials, since most atoms are located at the surface of a nanoparticle. Nanoparticles represent an almost limiting case of surface increase. Substances with a high level of surface localisation have greater chemical, mechanical, optical, and magnetic characteristics. c) High level of surface localisation and peculiarities in the chemical and physical structure of nanoparticles, including their intermediate position between macroworld and the atomic world, manifest themselves as their peculiar mechanical properties. Their mechanical characteristics exceed considerably those of traditional materials. For example, Young’s moduli of such traditional polymeric materials as polystyrene and epoxy resin EPON828 are equal to 4 and 2.7 GPa, respectively; the Young’s moduli of wood (oak) and steel are equal to 80 and 200 GPa, respectively. The same modulus for carbon nanotubes may be larger by one or two orders of magnitude (1.26 TPa). It is worth mentioning, once again, that the mechanical parameters of fillers as constituents of composites are somewhat lower than the maximum values of these parameters for the same fillers (particles, fibres, etc.) as bulk materials.

d) The internal structure of some nanoformations is so complex and irregular that their continuum description will probably involve complex microstructural approaches – see, e.g., [19,63,77]. Nanoformations (usually nanofilms or nanocrystallites) may basically be regarded as a porous material whose skeleton consists of nanoparticles. As opposed to microcomposites, nanofilms do not appear in load-bearing structures and their strength is not so important. e) Present studies of the mechanical behaviour of nanoparticles, nanoformations, and nanomaterials are at the early stage. Only external manifestations of mechanical phenomena are detected, but their mechanisms have not been studied yet.

3.2.2. Nanocomposites a) Reinforcing composites with nanoparticles improves their integral mechanical properties considerably. For example, nanomaterials of the two first types employ the capability of nanoparticles to slide past each other. In particular, nanomaterials used in the aerospace and automotive applications, and made from metals and oxides of silicon and germanium, exhibit superplasticity and elongation to fracture from 100% to 1000%. Nanoparticles exhibit high chemical activity, since the number of molecules or atoms at the surface of these particles is much greater than inside them. Therefore, to maintain the desirable properties, it is sometimes necessary to introduce a stabilizer in order to avoid further reactions. This improves the resistance of all kinds of nanomaterials to friction, erosion, and corrosion. b) The integral properties of insufficient adhesion have already been detected in nanocomposites. They are most likely due to local defects at the interface and initial (manufacturing) stresses. This degradation of properties is especially prominent in nanocomposites with polymer matrix and carbon nanotubes [54]. Therefore, establishing the theoretical and computational models for studying the binding force between the nanotube and the matrix, the stress transfer properties and the whole variety of interfacial phenomena in nanomaterials is essential. c) Along with the traditional granular, fibrous, and laminated nanostructures, there are also unconventional spatial structures that have to be described by more sophisticated models.Finally, it seems pertinent to recall a discussion on mechanical properties of new materials which took place more 40 years ago. In the concluding remarks, Professor J.D. Bernal said [11]: ‘‘Here we must reconsider our objectives. We are talking about new materials but ultimately we are interested, not so much in materials themselves, but in the structures in which they have to function.’’ The authors believe that nanomechanics faces the

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same challenges that micromechanics did 40 years ago, which Professor J.D. Bernal described so eloquently.

4. Levels of modelling in mechanics of materials In this section, we will introduce the concept of the structural mechanics of materials, which comprises macromechanics, mesomechanics, micromechanics, and nanomechanics (the areas within it being distinguished by the dimensions of constituents in the material structure, manufacturing techniques, and mechanical processes studied). 4.1. Structural mechanics of materials In the second half of the 20th century, a new branch of solid mechanics, the mechanics of materials, established itself, along with the mechanics of structural members. Nowadays, a considerable part of studies in solid mechanics is performed within the framework of the mechanics of materials, and as a result of this, solid mechanics is sometimes regarded as a branch of materials science. In mechanics of materials, the internal structure of a material is always taken into consideration in one form or another. In the majority of studies, the internal structure is used only to characterise or identify materials (e.g., by means of tracking changes in the internal structure under loading or during manufacture). In fewer studies, the information about the internal structure is incorporated into the model of a material and is used to formulate constitutive equations. Thus, the structural mechanics of materials can be considered an independent research area within the mechanics of materials. Henceforth by the structural mechanics of materials we will mean mechanics of materials that accounts, both qualitatively and quantitatively, for the internal structure of materials while developing models of materials and solving various problems. Its subject of research is a wide range of modern materials, including reinforced concrete, whose internal structure is defined by the reinforcement; metals, alloys, and ceramics, whose internal structure is defined by the presence of grains and other structural elements; composites, whose internal structure is determined by the presence of particles, fibres, and layers; and nanomaterials (nanocomposites), whose internal structure is determined by the presence of nanoparticles. If understood in this way, structural mechanics of materials incorporates macromechanics, mesomechanics, micromechanics, and nanomechanics, which are well defined and widely used terms. The only necessary common requirement for the four research areas is to take the internal structure of materials into consideration when establishing the mechanical models and solving the corresponding problems. To characterise quantitatively the internal structure of materials, it is expedient to introduce a geometrical param-

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eter, h. For reinforced concrete, h is the average minimum cross-sectional diameter of metal reinforcement. For metals, alloys, and ceramics, h is the average minimum size of cells, grains, and other structural heterogeneities. For composites with polymer or metal matrix, h is the average minimum diameter of particles in granular materials, the average minimum cross-sectional diameters of fibres in fibrous materials, and the average minimum thickness of layers in laminated materials. For nanomaterials (nanocomposites), h is the average minimum diameter of nanoparticles. The average minimum cross-sectional diameter of metal reinforcement in reinforced concrete may be on the order of centimetres (102 m), whereas the average minimum diameter of nanoparticles in nanomaterials, as follows from the previous section, may reach 0.4 nm (0.4 · 109 m). Thus, the structural mechanics of materials can be said to investigate materials with parameter h that varies within the following limits: 0:4  109 6 h 6 1:0  102 m

ð2Þ

With such a wide range of h, it may be expediently subdivided into several levels for convenience of further analysis, though such a division is rather conditional. Sih and Lui [78] suggested considering three levels: Macro : 104 –105 m;

Meso : 105 –107 m;

Micro : 107 –108 m

ð3Þ

In view of Eq. (2), the introduction of the fourth level seems pertinent. Also, in the authors’ opinion, the levels do not have clear borders and are overlapping. The modelling levels are demonstrated in Fig. 5. Since the atomic level, as defined by the interatomic distance in a crystal lat˚ (1010 m), the nanotice, has an order of one to several A level in Fig. 5 is conditionally restricted to 109 m. For the definition sake, the levels can be conditionally narrowed to the following values: Macro : 102 –105 m;

Meso : 105 –107 m;

Micro : 107 –108 m; Nano : 108 –109 m

ð4Þ

The research areas associated with the specific materials within the structural mechanics of materials are defined not only by the above levels, but also by the manufacturing techniques and mechanical processes in those materials. We will illustrate this point in the next section while discussing branch of the structural mechanics of materials. 4.2. Branches of the structural mechanics of materials We begin with those areas within the structural mechanics of materials, which have already achieved a certain level of development. They include: fracture mesomechanics of cracked materials; physical mesomechanics of metals, alloys, and ceramics; and the micromechanics of polymer and metal matrix composites.

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10 nm

10 μm

1 cm

meso

macro 10-2

10-3

10-4

10-5

10-6

10-7

10-8

m

10-9

micro 100 μm

nano 100 nm

1nm

Fig. 5. Schematic of the modelling levels.

4.2.1. Fracture mesomechanics of cracked materials The basic concepts of fracture mesomechanics are given in [78]. It should be emphasised here that classical Griffith– Irwin fracture mechanics does not account for the internal structure of the material at the crack tip. With the exception of the Neuber model, which isolates a grain at the crack tip to average local stresses within it, the classical fracture mechanics was developed for homogeneous solids. This also includes the case when a crack is located at, or is extending onto the interface between dissimilar homogeneous solids. Sih and Lui [78] concluded that the next stage in the development of fracture mechanics will account for the internal structure at the crack tip. Fracture mechanics that considers the internal structure at the crack tip at any structural level is termed fracture mesomechanics [78]. According to [78], the internal structure of materials (metals, alloys, and ceramics) can be schematically represented as shown in Fig. 6. The ‘‘cluster of grains’’ means a group of grains as an individual formation in the internal structure. Fig. 6(a) shows a crystal lattice and various disloca-

tions in it. Elements of the internal structure at various levels interact at the crack tip. Also, Sih and Lui [78] distinguish several zones at the tip of a stationary crack of length 2a shown in Fig. 7. Thus, if we exclude from consideration dimensions commensurable with the interatomic distance in a crystal lattice (approximately 1010 m), it follows from Fig. 6 that fracture mesomechanics studies cracked materials with h varying within the following limits: 108 6 h 6 105 m

ð5Þ

This interval includes several levels defined by Eq. (3) or (4). 4.2.2. Physical mesomechanics of metals, alloys, and ceramics The main concepts, approaches, and results of the physical mesomechanics of metals, alloys, and ceramics are given in [69]. Since the majority of results in physical mesomechanics were obtained in [69] for metals and alloys, the emphasis of [78] was placed on plastic deformations that lead to fracture. To analyze plastic deformations,

Fig. 6. A representation of the internal structure of materials (metals, alloys, and ceramics), after [78].

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is dispersion particles ranging in size from 0.01 lm = 108 m = 10 nm to 0.1 lm = 107 m = 100 nm, with the volume fraction ranging from 1% to 15%. The large-particle composites are composites whose filler is particles with the diameter bigger that 1 lm = 106 m and the volume fraction of more than 25%. The fibre-reinforced composites are composites whose filler is fibres (from 0.1 lm = 107 m = 100 nm to hundreds of microns in size) with a volume fraction of almost 70%. Therefore, for polymer- and metal-based composites the parameter h varies within the following limits:

Fig. 7. Zones at the tip of a stationary crack according to [78].

108 6 h 6 104 m methods of solid mechanics and the continuum theory of dislocations were applied, focusing on the analysis of mechanisms and their interaction at various structural levels. In order to gain more insight, the concept of mesoscopic structural elements was introduced. According to [69], mesoscopic structural elements are cells, grains, strip structures, precipitated phases, structural heterogeneities, or fragments of structures. Comparing this definition and Fig. 6, it is obvious that physical mesomechanics [69] addresses a somewhat wider or detailed group of structures than fracture mesomechanics [78]. To determine the range of the parameter h, which characterises the internal structure in physical mesomechanics, let us analyze the data on the size of grains in various metals and alloys used in [69]. In lead, the grains can be as large as 103 m; in ultradispersed structures, the grain size is between 0.05 · 106 and 1 · 106 m (0.05 lm = 50 nm = 5 · 108 m). The grain size in other metals and alloys is somewhere between 103 and 5 · 108 m. Thus, the physical mesomechanics of metals and alloys takes into consideration the internal structure of materials with parameter h that varies within the following limits: 5  108 6 h 6 103 m

ð6Þ

According to [69], under a shock-wave load, some grains of special ceramics based on ZrQ2 powder may become 3 nm = 3 · 109 m in size. Thus, an extended interval of variation for h may be considered in the physical mesomechanics of metals, alloys, and ceramics: 3  109 6 h 6 103 m

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ð7Þ

The interval defined by Eq. (6) or the extended interval defined by Eq. (7) encompass several levels defined by Eqs. (3) and (4). 4.2.3. Micromechanics of polymer- and metal-based composites Micromechanics of polymer and metal matrix composites is arguably one of the most developed branches of the structural mechanics of materials. Following [19], composites can be classified according by their internal structure as follows: dispersion-strengthened composites, large-particle composites, and fibre-reinforced composites. The dispersion-strengthened composites are composites whose filler

ð8Þ

This interval corresponds to several levels defined by Eqs. (3) and (4). The above classification of composites is common in the materials science. In the micromechanics of composites, composites are subdivided into layered, fibrous, and particulate composites. Layered composites are composites whose reinforcing elements are much smaller (by several orders of magnitude) in one direction than in the other two, mutually perpendicular, directions. Fibrous composites are composites whose reinforcing elements are similar in size in two mutually perpendicular directions and are much smaller (by several orders of magnitude) in the third direction. Unidirectional fibrous composites are arguably the most widely used type of composite materials. Glass, steel, ceramic, boron, carbon, and other fibres and whiskers are used as reinforcement in fibrous composites. The diameter of superthin basalt fibres ranges from 0.2 to 0.4 lm (2– 4 · 107 m), the diameter of Fybex fibres ranges from 0.10 to 0.16 lm (1–1.6 · 107 m), and TKF sapphire cobweb whiskers have the diameter of 18 nm = 0.018 lm (1.8 · 108 m) [41]. Particulate composites are composites whose reinforcing elements are of similar size in all the three mutually perpendicular directions. Layered, fibrous, and particulate composites can also be distinguished by the nature of their internal structure, which can be either deterministic (ordered) or random. The micromechanics of layered, fibrous, and particulate composites has received the comprehensive treatment in the 12-volume edition of Mechanics of Composites [63] and the numerous other books – see the reviews [2,3,9,35,37,39,49,50,75,79]. In the majority of these studies, the model of piecewise-homogeneous medium was used, and equations of three-dimensional solid mechanics were solved for each constituent of the composite. In some publications (see the review [39]), the methods of linearised solid mechanics are considered which allowed us to investigate various phenomena in composites in the presence of interfacial cracks. The piecewise-homogeneous medium model is one of the two general approaches used within the micromechanics of composites. It employs the three-dimensional equations of solid mechanics for both the filler and the matrix. The second approach uses homogenisation methods and various microstructural models, of which

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we mention the homogeneous anisotropic body model with effective constants calculated in one way or another. Certainly, the first approach is more accurate and rigorous. The second approach takes into account the internal structure of a composite via the effective constants, which depend on the physical and mechanical properties, geometry, and volume fraction of the filler and the matrix. Both approaches were successfully used to study a variety of problems in statics, dynamics, stability theory and fracture mechanics as applied to linear and non-linear elastic and plastic materials. 4.2.4. Additional remarks The prefixes ‘‘meso’’ and ‘‘micro’’ in the words ‘‘mesomechanics’’ (Section 4.2.2) and ‘‘micromechanics’’ (Section 4.2.3) do not at all imply that these research areas study materials, whose internal structure is defined by Eq. (3) or (4). As a rule, an isolated research area covers several structural levels. For example, the physical mesomechanics of metals and alloys [69] covers completely the meso- and microlevels and partially the macro- and nanolevels. The micromechanics of polymer- and metal-based composites covers completely the meso- and microlevels and partially the macro- and nanolevels, see, for instance, [2,3,9,35– 37,39,50,63,75,79] and other publications. A group of materials studied by a specific branch of the structural mechanics of materials is by no means defined solely by the interval of variation for parameter h. For example, the physical mesomechanics and the micromechanics of composites have almost equal intervals for h. A group of materials studied by a specific branch of the structural mechanics of materials is also defined by the similarity in manufacturing processes, and in physical and mechanical properties of these materials. For example, the restructuring and hardening techniques for materials studied in physical mesomechanics [69] are quite similar. One more example is the possibility to produce the materials addressed in the micromechanics of composites in such a way that they would be suitable for certain external loads. Composite materials are also united by the possibility of manufacturing both a material and a structural element in a single concurrent technological process. In some cases, subdivision of the structural mechanics of materials into macro- and micromechanics has only a methodological meaning. To solve specific problems, macromechanics and micromechanics may employ the same formulations, and the same methods of solving the problems, though the phenomena that are being analyzed can be at different structural levels. To illustrate this point, let us consider a composite laminate made of unidirectional plies, with fibres in neighbouring layers located at different angles to each other. The reduced properties of a single unidirectional ply can be determined within the micromechanics of composite materials using various homogenisation techniques. The effective properties of a multi-directional laminate can be determined using homogenisation techniques within the macromechanics of composites. Here,

the same formulations and the same methods of solution are applied at various structural levels (macro and micro). As already mentioned, both, the physical mesomechanics and the micromechanics of composites, as defined by Eqs. (6) and (8), partially cover the nanolevel. Thus, some issues of nanomechanics (the mechanics of nanomaterials) associated with the upper part of the nanolevel may have been already studied within the framework of physical mesomechanics and the micromechanics of composites. The limits of applicability of the micromechanics of composites may actually be wider that than those defined by Eq. (8), since the limits reflect only already existing composites. For instance, the one and only limitation for applying the micromechanics of composites developed in [2,3,9,35,39,49,50,63,75,79] is the possibility of using the solid mechanics formulations in order to describe the stress-strain relations in each reinforcing element. Whether the micromechanics of composites can be extended to some classes of nanomechanical problems will be discussed in the next section. 5. On nanomechanics In this section, we will also consider some trends and approaches in the mechanics of nanomaterials (nanomechanics). Since the nanomechanics is still a developing science, the discussion below is by no means exhaustive and reflects only the authors’ point of view. 5.1. Subjects of research in nanomechanics Since various authors define the nanolevel somewhat differently, we will use the definition due to the National Scientific Foundation (USA) [66]. According to this definition, the nanolevel is the following interval of variation of the structural parameter h: 109 6 h 6 107

ð9Þ

Nanoparticles may reach from 200 to 300 nm in size or constitute tenths of a nanometre. Thus, nanomechanics as a branch of the structural mechanics of materials describes mechanical processes in materials with the internal structure defined by Eq. (9). We assume that the nanomechanics of materials comprises the mechanics of nanoparticles and the mechanics of nanocomposites (i.e., composites made from a matrix and nanoparticles as a filler, or only from nanoformations). However, it should be noted that, as in the other branches of the structural mechanics of materials, nanomaterials are a group of specific materials by no means solely defined by the interval, Eq. (9). We may consider nanomaterials as materials grouped according to the similarity between their manufacturing processes, physical and mechanical properties, and structural peculiarities. A typical technological aspect of nanomaterials (nanoparticles) is the possibility of manufacturing them at a molecular level, ‘‘atom by atom’’ [25,26], which leads to strongly altered and improved properties or new quanti-

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tative and qualitative properties. This improvement of properties or acquisition of qualitatively new properties of nanoparticles does necessarily lead to the improvement of the properties of nanocomposites with the nanoparticles as a filler. Certainly, the properties of nanocomposites can also be improved through reinforcements optimised for the expected loads on structural members. Apparently, this is the basic difference between nanocomposites, as studied in the nanomechanics of materials, and polymer- and metal-based composites, as studied in the micromechanics of composites. The properties of polymer- and metal-based composites can be improved, mainly, through optimal reinforcement.

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parameter, L, to characterise those mechanical processes. The parameter L is the minimum distance at which the stress and strain fields (in the case of statics), the wavelength (in the case of dynamics), and the buckling mode wavelength (in the case of stability theory) change substantially. The restriction on the mechanical processes under study can be represented in the form L>H

ð11Þ

Thus, the continuum solid mechanics can be applied to description of a mechanical process in a material with the internal structure, if the geometrical parameters that characterise the process, the minimum volume of the material, and its internal structure satisfy the following inequality:

5.2. Limits of applicability of continuum solid mechanics

L > H > 10h

First it should be noted that the nanolevel interval, Eq. (9), is fairly wide. Its upper part coincides with the lower part of the microlevel interval, Eq. (8), while in the lower part of the nanolevel, the parameter h may reach several interatomic distances. Most nanomechanical problems for nanomaterials with the parameter h in the upper part of Eq. (9) can be formulated and solved using approaches and methods of the micromechanics of polymer- and metal-based composites. In turn, all main results in the micromechanics of polymer- and metal-based composites [9,35,37,39,49,50,63,75,79] (Section 4.2.3) were obtained by applying the relations of continuum solid mechanics individually to the matrix and to each element of the filler (each reinforcing element). Application of these relations is in fact the only restriction on the above-mentioned theory. Since all nanoparticles, as illustrated in the examples of the first section, consist of discrete elements (molecules and atoms), the question naturally arises about the limits of applicability of the continuum solid mechanics to the description of mechanical processes in nanoparticles. Answering this question will automatically answer the question about the applicability limits for the micromechanics of composites. Certainly, the rigorous and complete mathematical solution of this problem is quite difficult. Therefore, we will discuss here an approximate solution. Along with the parameter characterising the average minimum size of particles in the internal structure, we introduce two geometrical parameters, h* and H. The parameter h* is the average center-to-center distance between the particles in the internal structure, and the parameter H is the minimum volume within which the material may be modelled by a homogeneous continuum. Many experts believe that H must be larger than h* by more than one order of magnitude. Therefore, to perform an approximate qualitative analysis, we assume that

Let us consider some examples of applying the above condition to a number of materials.

H > 10h

ð10Þ

This geometrical condition does not fully define the applicability of continuum solid mechanics because the nature of mechanical processes under study is not taken into account. In this connection, we introduce a geometrical

ð12Þ

Example 1. Consider nanomaterials (nanoparticles) with crystalline structure. In this case, the parameter h* in Eq. (12) is the average distance between the neighbouring atomic planes. Since the average interatomic distance in a crystal lattice comprises several Angstrems, it follows from Eq. (12) that H must be of magnitude of several nanometres. Therefore, to analyze nanomaterials (nanoparticles) with crystalline structure, we can apply the tools of continuum solid mechanics (i.e., the homogeneous continuum model) within the minimum volume H of the material if the mechanical processes being studied satisfy condition Eq. (11). Example 2. Consider nanomaterials whose atoms form granules. Examples of such nanomaterials with granules up to 100 nm in size are presented in Section 2. From this value and Eq. (12), it follows that the mechanical processes in granules to be studied within the framework of the homogeneous continuum model must satisfy the condition L > 10 nm. Example 3. The above conclusion is even more optimistic toward nanocomposites reinforced with nanoparticles. 5.3. Three main research directions in nanomechanics Bearing in mind the information about nanomaterials presented in Section 2 and the discussion of some aspects of nanomechanics in the previous subsections, let us point out three main directions of research in the mechanics of nanomaterials, as it is envisaged by the authors. The first direction comprises the studies of materials and processes, for which the applicability conditions, Eq. (12), are satisfied. It covers a wide range of static, dynamic, and stability problems for nanoparticles and nanocomposites and those fracture problems that can be solved within the framework of solid mechanics. Note that if the applicability conditions, Eq. (12), are satisfied for nanoparticles,

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then they are automatically satisfied for nanocomposites in which these nanoparticles are used as the filler. One of the well-known approaches within this direction is based on the principle of homogenisation. The second research direction comprises the studies of materials and processes, for which the applicability conditions, Eq. (12), are not generally satisfied, but an approximate approach (which may be named the principle of continualisation) can be applied. By continualisation we mean constructing approximate continuum theories that describe changes of at least integral characteristics of discrete structures. The principle of continualisation is widely used in various branches of physics. A classical example is the continuum theory of dislocation, expounded, for example, in [29,51]. Dislocations are, according to [21], discrete defects in a discrete system, a crystal lattice. Nevertheless, a continuum theory that describes the laws of propagation of dislocations was developed for such a discrete system. Certainly, the mechanics of nanoparticles requires applying the principle of continualisation; after that, problems for nanocomposites can be treated within the framework of continuum representations. Due to the application of the principle of continualisation, new formulations might be expected to appear based on specific models of continuum solid mechanics. The third research direction comprises the studies of particularly discrete systems, for which the applicability conditions, Eq. (12), are not satisfied. It seems obvious that the third research direction includes only investigations of the mechanics of nanoparticles. Here, it is difficult to obtain specific results for a wide range of structures and mechanical phenomena. When using such results in the studies on nanocomposites with nanoparticles as the filler, it is necessary to interpret them in continuum terms. Subdivision into three research direction proposed here, although conditional, might be useful in analysing approaches to the study of phenomena and in interpreting results. 5.4. Discussion We discussed the three possible research directions mainly with reference to the mechanics of nanoparticles. Let us now point out some issues relevant to the mechanics of nanocomposites. To analyse mechanical phenomena in nanocomposites at the final stage of studies along the first and second research direction, we can use continuum representations of the mechanics of materials with internal structure. The mechanics of nanocomposites may employ formulations, approaches, and techniques from the micromechanics of polymer- and metal-based composites. As already mentioned earlier, the micromechanics of polymer- and metal-based composites is comprehensively developed. With such an approach, it is possible to analyze a wide range of static, dynamic, and stability problems for nanocomposites and those fracture mechanics problems that can be solved within the framework of solid mechanics.

Moreover, the intervals associated with the micromechanics of composites and nanomechanics of composites (defined by Eqs. (8) and (9), respectively), overlap significantly. The term ‘‘nanocomposite,’’ when used in nanomechanics, may imply two different types of nanomaterials. The first type is a material consisting of a matrix reinforced with nanoparticles. Such a structure fully complies with the terminology adopted in the micromechanics of polymer- and metal-based composites. These composites can be studied within the first and second research direction using approaches and methods of the micromechanics of composites. As each nanoparticle or nanoformation has a rather complex internal structure, it can be considered as a nanocomposite itself with its own internal structure. Such a composite (in the form of a nanoparticle or nanoformation) can be studied using approaches from all the three research direction, the approaches of micromechanics being inapplicable within the third research direction. In this context, the term ‘‘nanocomposite’’ cannot be applied in the micromechanics of polymer- and metal-based composites, because here each element of the filler is always modelled within the framework of solid mechanics. The detailed examples of application of micro- and macromechanical models to nanocomposites were given in [38]. In addition to the above discussion, let us point out one fundamental difference between the model representations of the micromechanics of composites and the nanomechanics of nanoparticles. The micromechanics of composites studies composite materials consisting of a matrix (polymer, metals, etc.) and reinforcement (particles, fibres, etc.). Therefore, reinforcing elements interact with one another through a material medium (matrix). Modelling the matrix within the continuum solid mechanics is a logical, conventional, and undisputed practice, provided that the applicability interval for the micromechanics of composites, Eq. (8), is taken into account. Nanoparticles consisting of atoms combined into molecules have to be studied within the framework of the nanomechanics of materials. In the case of nanotubes, for instance, we have to deal with structures whose atoms are located at lattice sites. In ceramics based on silicon nitride (Si3N4) nanoparticles, atoms are aligned into ropes, spaced at 0.6 nm. In both cases, there is no material medium (similar to the matrix in the micromechanics of composites) between atoms, and atoms interact only through interatomic forces. Thus, models of continuum mechanics cannot be directly applied in the mechanics of nanoparticles to describe the interaction between isolated elements (atoms). However, describing a particle in continuum terms seems a preferred way if the particle will then be studied within the framework of the mechanics of materials and is a component of the internal structure of the composite. The experience of many researchers indicates that the mechanics of materials employs exclusively continuum representations; i.e., even in this case constructing various continuum models has considerable promise. The continuum theory of dislocations [29,51] can be mentioned here as an example. The

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fundamental difference between the micromechanics of composites and the nanomechanics of nanoparticles is, apparently, due to the difference between the structural levels. The structural level in the micromechanics of composites is defined by Eq. (8), whereas the structural level in the nanomechanics of nanoparticles is the interatomic distance. 6. On calculation of the effective elastic properties of nanocomposites

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were obtained after some rearrangement based on the general approaches expounded in [63]. The formulas give a good correlation with experimental data for fibrous microcomposites with small volume fractions of reinforcement (see [63]): E L ¼ cm E m þ cf E f þ GL ¼ lm

4lm cm cf ðmf  mm Þ2 ; ½1  cf ð1  2vm Þ þ cm ð1  2vm Þlm =lf

cm þ ð1 þ cf Þlm =lf ; cm  ð1 þ cf Þlm =lf 2

Here we will illustrate how the effective properties of nanocomposites can be calculated (the first research direction of nanomechanics, Section 5.3). First, we have to ensure that the applicability conditions, Eq. (12), giving the relationship between the size of the reinforcement, the minimal volume, and the geometrical parameter of the investigated phenomena are met. Four fibrous composite materials used in the numerical examples of this section have the same epoxy matrix (EPON-828) and differ in the type of carbon reinforcement. The following reinforcements are considered: R1 – commercial carbon microfibre Thornel T-300 with the diameter of 8 lm; R2 – graphite microwhiskers with the diameter 1 lm; R3 – zig-zag carbon nanotubes with the average tube diameter of 10 nm; R4 – chiral carbon nanotubes with the average diameter of 10 nm. Physical properties of the matrix and four carbon reinforcements are given in Table 1. We assume that the resulting composite materials are unidirectional, with the fibre volume fraction of between 1% and 10%. For microscale fillers, these values are very small. For nanofibres, they can be considered as high enough due to the enormous surface area provided by nanofillers. Usually, such materials are treated as transversally-isotropic continua, which, if deforming elastically, are characterised by five independent elastic constants. Alternatively, the following engineering constants are used: Young’s modulus EL, shear modulus GL and the Poisson’s ratio mL in the longitudinal direction, and Young’s modulus EL, shear modulus G and the Poisson’s ratio mT in the transverse direction. At that, the relationship holds mL =GL ¼ mT =GT

ð13Þ

These engineering constants can be calculated using the following formulae for the effective elastic constants, which

mL ¼ mm 

ð3  4mm Þ þ cm þ cf lm =lf ; ð3  4mm Þcm þ ½1  cf ð3  4mm Þlm =lf  1 mL 1  2mm ¼ þ ET EL 2lm ( 2 þ ð1  mf Þlm =lf  ½1  cf ð1  2vm Þ þ cm ð1  2vm Þlm =lf ) 2cf ½1  lm =lf  ET þ 1 ; mT ¼ ð3  4vm Þ þ cf þ cm lm =lf 2GT

GT ¼ lm

ð14Þ Here Em, lm, mm, cm are the Young’s modulus, shear modulus, Poisson’s ratio and volume fraction for the matrix, and Ef, lf, mf, cf are the Young’s modulus, shear modulus, Poisson’s ratio and volume fraction for the reinforcement. Table 2 shows the effective properties of the three composites materials for different fibre volume fractions.

Table 2 Effective values of the elastic constants Fibre volume fraction

Epoxy resin (EPON-828) [40] R1 [40] R2 [41] R3 [91] R4 [91]

Density, q Young’s Shear modulus, Poisson’s (kPa s2/m2) modulus, l (GPa) ratio, m E (GPa) 1.21 1.75 2.25 1.33 1.40

2.68 228 1000 648 1240

0.96 88 385

0.4 0.3 0.3

Fibre type R1

R3

R4

0.03 0.08 0.09 0.10

Longitudinal Young’s modulus 2.20 · 1010 0.94 · 1010 2.07 · 1010 5.43 · 1010 10 2.30 · 10 6.08 · 1010 2.52 · 1010 6.72 · 1010

0.03 0.08 0.09 0.10

Transverse Young’s modulus ET (Pa) 0.406 · 1010 0.448 · 1010 0.465 · 1010 0.456 · 1010 0.481 · 1010 0.489 · 1010 10 10 0.461 · 10 0.485 · 10 0.493 · 1010 0.466 · 1010 0.489 · 1010 0.495 · 1010

0.03 0.08 0.09 0.10

Shear modulus 0.9606 · 109 0.9618 · 109 0.9620 · 109 0.9623 · 109

GL (Pa) 0.9601 · 109 0.9603 · 109 0.9604 · 109 0.9604 · 109

0.9601 · 109 0.9603 · 109 0.9604 · 109 0.9604 · 109

0.03 0.08 0.09 0.10

Shear modulus 1.000 · 109 1.096 · 109 1.115 · 109 1.135 · 109

GT (Pa) 1.01 · 109 1.10 · 109 1.12 · 109 1.14 · 109

1.01 · 109 1.10 · 109 1.12 · 109 1.14 · 109

Table 1 Physical properties of the constituent materials Constituent materials

2cm ð1  mm Þðmf  mm Þ ; ½1  cf ð1  2vm Þ þ cm ð1  2vm Þlm =lf

EL (Pa) 4.0 · 1010 1.02 · 1011 1.14 · 1011 1.26 · 1011

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Reinforcement with chiral carbon nanotubes (R4) leads to increase by a factor of 5 in the longitudinal Young’s modulus value as compared to carbon fibres (R1). The transverse Young’s modulus and all shear moduli are less sensitive to the type of reinforcement, but this is likely to change for higher volume fractions of the reinforcement. 7. Conclusions The mechanics of nanomaterials (nanomechanics) is still at the early stage of development. In this paper, an attempt is made to formulate the general approaches in the mechanics of nanomaterials and to position it within the structural mechanics of materials. It is suggested that continuum solid mechanics can be applied to describe a mechanical process in a nanomaterial, if the geometrical parameters that characterise the process, the minimum volume of the material, and its internal structure, satisfy a certain applicability condition. Three main directions of research in the mechanics of nanomaterials are also pointed out, depending on whether the applicability condition is satisfied in full or just partially. Since mechanics of nanomaterials finds itself at the early stages of development, many points made in this paper will be subject to criticism and even rejection. This is natural for any research area at this stage, and the authors welcome all critical remarks and suggestion that can contribute to the development of nanomechanics. Acknowledgements The authors would like express their gratitude to Professors L.P. Khoroshun, N.A. Shul’ga, I.Yu. Babich, V.N. Chekhov, A.P. Zhuk, Yu.V. Kokhanenko, and Dr. V.A. Dekret (Timoshenko Institute of Mechanics, Ukraine), and Dr. M. Kashtalyan (University of Aberdeen, UK), for the helpful discussions and suggestions. Financial support of the part of this research by the Royal Society (UK) within the International Joint Project Grant Programme is gratefully acknowledged. References [1] Ajayan PM, Stephan O, Colliex C, Trauth D. Alighned carbon nano tube arrays formed by cutting a polymer resin – nanotubes composites. Science 1994;265:1211–4. [2] Akbarov SD, Guz AN. Continuum approaches in the mechanics of curved composites and related problems for members of constructions. Int Appl Mech 2002;38(11):1285–308. [3] Akbarov SD, Guz AN. Mechanics of curved composites (the piecewise homogeneous body models). Int Appl Mech 2002;38(12): 1415–39. [4] Attard GS, Glyde JC, Goltner C. Liquid-crystalline phases as templates for the synthesis of mesoporous silica. Nature 1995; 378:366–8. [5] Attard GS, Edgar M, Emsley J, Goltner C. The true liquid crystal approach to mesoporous silica. Proc Symp Mater Res Soc 1996;425:179–89. [6] Attard GS, Edgar M, Goltner C. Inorganic nanostructures from lyotropic liquid crystal phases. Acta Mater 1998;46:751–8.

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