Mechanics of deformation and fracture of nanomaterials and nanotechnology

R.V. Goldstein and N.F. Morozov / Physical Mesomechanics 10 5–6 (2007) 235–246

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Mechanics of deformation and fracture of nanomaterials and nanotechnology R.V. Goldstein and N.F. Morozov1 Institute for Problems in Mechanics RAS, Moscow, 119526, Russia 1 St. Petersburg State University, St. Petersburg, 199034, Russia The paper reviews some results for the development of models, calculation methods and experimental investigation of the mechanical behavior (deformation and fracture) of nanoscale objects and nanoparticle-containing materials (nanocomposites). The application of approaches and methods of solid mechanics to nanotechnology simulation is considered. We discuss discrete-continuous models developed for the description of deformation processes in nanoscale objects (systems containing multiple atomic layers, nanotubes and their systems, nanocoatings) and nanoparticle-reinforced materials. The processes of defect formation in them are studied. The opportunity of calculating and estimating effective strain characteristics of the given objects and materials with regard to their structure is analyzed. The notions, models and experimental results concerning the strength and fracture of nanomaterials and nanotubes are discussed with regard to the influence of structure, presence and generation of defects, scale factor and dislocation mechanisms of plastic deformation as applied to nanomaterials. The atomic models of cracks in deformed solids and their relation to the continuous models of cracks in the mechanics of brittle and quasi-brittle fracture are examined. The experimental determination of strain and strength properties as well as fracture resistance parameters of nanomaterials and nanoobjects is considered.

1. Introduction Further development of technologies for the production of nanomaterials and nanoparticle-filled materials, the design of nano- and microscale structural components and products, their application in various fields of engineering and medicine require that strain, strength properties and fracture resistance of these materials be described. Materials filled with nanosized particles have been known long ago. They include, e.g., carbon black filled rubbers. Silicate nanocomposites have been investigated since the 1950s. However, a considerable rise in the study of nanomaterials and nanoparticles relates to the discovery of carbon nanotubes in 1991. Nanotubes have very high strain and strength characteristics, such as high Young’s modulus (~1 GPa), high elastic (up to 5 %) and ultimate strains (up to 20 %). Such a combination of mechanical properties offers great opportunities of using nanotubes as reinforcing fibers in composites. Nanotubes are also a very interesting object from the mechanical standpoint which combines discrete-continuous properties. On the other hand, in multilayer nanotubes a large role in the interaction between individual layers belongs to the Van der Waals forces. Copyright © 2007 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2007.11.002

To take them into account, we should develop appropriate continuum mechanics models. Of great interest are polymer nanocomposites, particularly nanocomposites with a polymer matrix reinforced by minerals or silicate fillers. Among such fillers we would like to distinguish the natural mineral montmorillonite having the same multilayer and crystalline structure as talc or mica. The montmorillonite lattice consists of thin layers ~1 nm thick. The layers form regularly spaced parallel structures (suites). At montmorillonite dispersion in polymer the interlayer spacings can increase depending on the thermodynamics of the formation process, during which the parallelism of the layers breaks under certain conditions (leading to a more uniform nanocomposite reinforcement). The reinforcement with montmorillonite enhances the rigidity (Young’s modulus) and yield strength of the composite, but reduces the relative elongation at rupture. For example, for the capron/montmorillonite nanocomposite Young’s modulus increases almost twice in comparison to the matrix at a montmorillonite content of ~7 %, while the relative elongation decreases 30 times. The yield strength therewith grows by 20 –30 %.  

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The application of carbon nanotubes as fillers in polymer nanocomposites significantly improves the strain and strength characteristics of the composite. For example, the compression strength of polycarbonate with a filler consisting of multilayer nanotubes (about 30 %) increases ~1.8 times as compared to the compression strength of the matrix material. The multilayer systems and systems with nanothick coatings attract more and more attention. The development of such systems shows much promise for nanoelectronics. Nanostructured coatings often prove to have extraordinary wear resistance, which draws interest to the development of wear-resistant nanocoatings. The above examples point to the necessity of developing mechanical behavior models, calculation methods and techniques for the experimental study of both the nanoscale objects as well as materials, structural components and products on their basis. In the present paper attention is first of all focused on various aspects of materials nanomechanics. 2. Discrete-continuous simulation of elastic characteristics of nanostructures and nanocomposites 2.1. Dependence of elastic characteristics of a packet of atomic layers on their number. Simulation of the influence of subsurface atomic layers When taking into account the atomic structure of nanoscale objects, a question naturally arises of how strongly the elastic characteristics of the nanoscale structure correlate with the macroscopic elastic characteristics of the given material. The answer to the question requires analysis on the basis of discrete and continuous approaches. Through varying the number of atomic layers in the system, one can see that the elastic moduli gradually change and tend to macroscopic characteristics as the thickness of the packet of atomic layers increases. Such transition is studied in [1–4] by the example of a strip of a single-crystalline material with hexagonal closepacked (triangular) lattice. Assuming the interatomic interaction to be pair and taking into account only the interaction of nearest neighbors, the authors consider tension and compression of an infinitely long strip a few atomic layers N wide. Loading is simulated by applying constant loads to the end sections of the strip. Unlike an infinite two-dimensional crystal with hexagonal close-packed lattice, the considered crystal (strip) is anisotropic. Let E1 , Q1 and E 2 , Q 2 are the Young’s modulus and Poisson’s ratio values in tension along and across the strip, respectively. The calculated dependence of the strip elastic moduli on the number of atomic layers is given in Table 1. The calculation data show that in the chosen model Young’s moduli of the system of atomic layers approach corresponding continuous values from above at a growing number of the layers. Even at 20 atomic layers in the sys-

tem the difference of Young’s moduli comprises 1–5 % of the continuous values. From the viewpoint of the rigidity characteristics, the obtained estimate for the number of atomic layers at which the discrete system becomes continuous agrees well with the calculation results of paper [5]. The latter considers a simple cubic lattice as a model of a multilayer nanostructured material with constant-thickness layers and uses harmonic potentials. The discrepancy between the results of papers [1, 2] and [5] is that for the model of [5] Young’s modulus approaches its continuous value from below as the number of atomic layers in the lattice increases. On the other hand, paper [6] demonstrates in the framework of a discrete model that Young’s modulus can approach the continuous value both from above and from below. The cases when this or that variant can be realized are particularly considered elsewhere [7]. Within molecular statics it is shown that when accounting only for pair interaction Young’s modulus approaches the continuous value from below, but when accounting only for multiple interaction — from above (the calculations were carried out for copper using two approaches: one of them used the Lennard-Jones potential as the interaction potential, and the other used the embedded atom potential). A higher rigidity of a packet consisting of a small number of atomic layers in comparison to the continuum is also substantiated in ab initio calculations. The performed analysis has revealed two competing effects that give rise to opposite tendencies of elastic modulus variation at a varying number of atomic layers. The electron density redistribution in the layered system favors an increase of system rigidity at a decreasing number of layers. An opposite effect can stem from a decrease in the coordination number of atoms on the surface, which could in turn diminish the rigidity of the layered system at a reducing number of layers in it. The nature of the revealed model dependences of elastic moduli of the layered system on the number of atomic layers needs further investigation, though the estimate for the number of layers at which the discrete description is changed for continuous is substantiated by calculations performed with various models. Table 1

Elastic moduli of a single-crystalline material strip versus the number of atomic layers N N

E1max E f

Q2

Q 2 Qf

2

2.00

0.18

0.53

1.06

3

1.50

0.23

0.69

1.04

4

1.33

0.26

0.77

1.03

5

1.25

0.27

0.82

1.02

10

1.11

0.30

0.91

1.01

20

1.05

0.32

0.96

1.01

E2 Ef

50

1.02

0.33

0.98

1.00

100

1.01

0.33

0.99

1.00

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Fig. 1. Copper nanosphere of radius 2 nm

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Fig. 3. Average energy of subsystems of surface atoms (1), bulk atoms (2), total energy (3), experimental value of cohesive energy (4) versus sphere radius

The above results point to a special role of the surface and subsurface atomic layers in the deformation of nanostructured objects. This issue is studied in [8] using molecular dynamics methods. The influence of surface atoms on the mechanical properties of nanocrystals has been simulated for defect-free copper crystals of spherical shape. Since for fcc structures, among which is copper, the coordination number is 12, the group of surface atoms of the spherical crystals includes atoms with the number being less than twelve. All atoms whose coordination number is 12 refer to the “bulk” atoms. An example is a sphere of radius 2 nm. The dark circles denote bulk atoms, and the white circles stand for surface atoms (Fig. 1). The investigation was conducted for spheres of radius varying from 1 to 7 nm. Figure 2 gives a significant characteristic of nanostructures, namely, the sphere radius dependence of the ratio between the number of atoms on the surface and the total number of atoms. As obvious, even for spheres of radius 7 nm this ratio is close to 0.1, not tending to zero as for macroobjects. The energy characteristics were calculated with the use of the Johnson interatomic interaction potential [9]. The

dependence of the average energy of surface atom and bulk atom subsystems and total energy (energy divided by the number of atoms in the subsystems) on sphere radius is illustrated in Fig. 3. The energy of the surface atom subsystem exceeds the bulk energy, which starting from 3 nm is almost equal to the average energy of the whole crystal. As one can see, with sphere radius growth the energy values approach the experimental value for the cohesive energy and differ from it by 1.2 % for spheres of radius 7 nm. An additional characteristic of nanostructures can be the ratio of the surface atom energy to the total crystal energy and to the bulk atom energy (Fig. 4). For spheres of radius 1 nm the surface atom energy reaches half of the total crystal energy and even exceeds the bulk energy. With radius growth to 7 nm the fraction of the surface atom energy decreases, but comprises 10 % of the total energy. This proves that surface effects in nanostructures of size up to 10 nm cannot be neglected. If we determine the surface energy U sr as a sum of the surface atom energy and binding energy of atoms of a 10 nm volume, the surface tension coefficient can be estimated by the formula:

Fig. 2. The ratio of the number of surface atoms to the total number of atoms in a spherical copper cluster versus sphere radius

Fig. 4. The ratio of the surface atom energy to the total crystal energy (1) and bulk atom energy (2) versus sphere radius

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Fig. 5. Surface tension coefficient of the copper crystal versus external pressure

§ wU sr · ¨ ¸ . © wS ¹T Here, S is the surface area of the considered object, the area variation is induced by the applied external stress. The authors of [8] have calculated the dependence of J on external pressure (Fig. 5). Its value for low pressures, when the dependence is close to linear, coincides with the experimental value of the surface tension coefficient for copper melt. In different papers this value varies from 0.5 to 1.2 J/m2. In the conducted numerical experiment on nanosphere compression we have found the value of the bulk compression coefficient: § wP · K V ¨ ¸ © wV ¹ T for the whole sphere and internal region including only bulk atoms. The dependence of K on applied pressure has been constructed. The value for the bulk part for low pressures coincides with the experimental value found for macroobjects. On the other hand, the calculation of K for the sphere as a single object gives a 10 % lower value, which points to a strong influence of the surface. J

2.2. Discrete-continuous simulation of carbon nanotubes and their systems Like for systems consisting of a small number of atomic layers, the opportunities and applicability limits of continuous description of mechanical properties are also important issues for nanotubes and their systems. In this connection, a discrete-continuous model for calculating strain characteristics of nanotubes and their systems has been developed [10]. The model is based on the possibility of changing a set of atoms that make up a nanotube for an equivalent model of elastic isotropic rods with regard to the regular atomic structure of the nanotubes. The system of rods is then changed for a continuous model, which allows one to pass on to the continuous model of the nanotube depending on the considered problem. As the continuous model either the cylinder or shell model can be used. When developing the equivalent rod model of the nanotube, the energy approach is used which allows selecting such rod model parameters that provide the coincidence of

the rod and atomic model of the nanotube. The interaction of atoms in the molecular structure is quantitatively described by molecular dynamics methods. The attractive and repulsive forces for each pair of atoms depend on the relative position of the atoms and are described by the chosen force field. These forces contribute to the total potential energy related with molecular system oscillations, which is in turn equal to the strain energy of a macroscopic body of equivalent geometry. From the computational standpoint, the rod model significantly reduces the computational time as compared to the molecular dynamics approaches. The potentials for carbon atoms are chosen with the aid of the Morse and Lennard-Jones potentials for describing covalent and noncovalent interactions, respectively. Out of many force fields for describing parameters of aromatic carbon molecules we choose the MM3 force field, because it permits the behavior of various carbon-containing molecules to be described most completely. The nanotube structure can be represented in the following way: a part of the hexagonal plane composed of carbon atoms (graphene plane) is rolled into a nanotube. Every atom in this plane covalently interacts with its three neighbors. A characteristic element of the hexagonal plane consists of four bound atoms within a hexagonal cell. In the given system all degrees of freedom contribute separately to strain energy (owing to stretching energy, bond angle variation and nonforce interaction). In the general form, the total potential energy of the nanostructured system (material) can be represented as consisting of five terms:

E

¦ E U  ¦ E T  ¦ E tor  ¦ E Z  ¦ E VdW.

Here, E U is the bond stretching energy, E T is the energy of angle variation between neighboring atomic bonds, E VdW is the noncovalent interaction energy, E tor is the energy of bond coming out of the plane accompanied by bond torsion, and E Z is the energy of plane bending due to S-electron density variation. In nanotube simulation, in the majority of cases, only two first degrees of freedom are important. The torsion energy is rather low and can thus be neglected. The sum of E Z becomes considerable only in problems with high bending strains. The covalent bond stretching energy E U is equal to the strain energy of the rod connecting two atoms: E U K U u u (r a  R a ) 2 (harmonic potential), where R a is the initial interatomic distance and r a is the distance in the deformed state for the covalent bond; E T K T (T  4) 2, where 4 is the initial angle value and T is the angle value in the deformed state. Taking into account that in the rod model the strain energy depends solely on the strains of the rods themselves, the last expression can be transformed as: E T 4 K T ( r b  R b ) 2 ( R b ) 2, where R b is the initial length and r b is the length of the rod in the deformed state which connects two atoms covalently bound with the third one.

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The elastic strain energy of an arbitrary rod system can be written as follows:

¦¦

E

i

n

Ai Y i i (rn  R i ) 2, 2R i

where summation is taken over all rods of each type, n is the rod number, rni is the rod length in the deformed state, and A i is the cross-section of the rod of the i-th type. Based on the discrete atomic model, we can turn to the rod model if all considerable pair interactions in the atomic structure are taken into account and changed for their equivalent rods. We can thus obtain a rod system equivalent to the atomic model under the same loading conditions. This approach was applied to nanoscale structures elsewhere [11, 12]. For the case of nanotubes and graphene planes, we can obtain the following estimates of the elastic modulus for the rods of the equivalent rod model. Let us distinguish two types of rods (a and b) to which there correspond energies E U and E T, then their Young’s moduli are

2K U R a 3K T b Y Y and . Aa 2 R b Ab The interatomic distances for carbon in graphite R a and b R are known. The constants K U and K T are also the known force field parameters [11]: K U = 46 900 kkal/(mole ˜ nm2), K T = 63 kkal/(mole ˜ rad2). If the rod radius is assumed to be 0.01 nm (which is quite small compared to the interatomic bond length), we can estimate Young’s moduli: Y a = 295 GPa and Y b = 9 GPa. The estimates show that the third-type rods modeling the interaction of opposite atoms in a hexagonal cell can be set aside, because the moduli of the first- and second-type rods differ more than 30 times. The model has been generalized and is applicable for describing the deformation of not only individual nanotubes, but also of their systems and groups of graphene layers of different structure (hexagonal and rhombohedral). The generalization particularly required that additional rods different from the rods of types a and b be introduces to the equivalent rod system, in order to describe graphene plane bending accompanied by energy accumulation due to the S-orbital electron density variation on the both sides of the graphene layer. Various deformation processes in nanostructured objects have been numerically analyzed within the model. Among them are nanotube deformation in the matrix of a composite, deformation of systems of different-shaped nanotubes and a packet of graphene planes. Figure 6 illustrates simulation results for stability loss in a single-wall nanotube under axial compression. As one can see, depending on the ratio of the tube diameter to its length different ways of stability loss are possible, like in the classical problem of stability loss in an elastic rod. For short nanotubes

Fig. 6. Compression of thin tubes

a

 

   

   

the way of stability loss is similar to that observed for an elastic shell (Fig. 7). Note that the anisotropy of elastic properties of carbon nanotubes results from the anisotropy of graphite elasticity. The molecular basis of a single-layer carbon nanotube is a graphene plane one carbon atom thick. The anisotropy of elastic properties of multilayer carbon nanotubes and graphite rods is analyzed in detail elsewhere [13].

Fig. 7. Compression of short tubes

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2.3. Simulation of strength properties of carbon whiskers under tension Continuum mechanics approaches allow theoretical analysis of strength properties of graphite-like materials that have layered structure and exhibit anisotropic mechanical behavior [14]. Fibers of such materials are important for practical applications owing to their specific mechanical and other performance characteristics. This particularly refers to the micro- and nanoscale graphite whiskers whose basic structure, like in graphite, is made up of solid graphene layers weakly bound by the Van der Waals forces. By varying the technology and regimes, we can produce different-shaped tubes (circular cylindrical, toroidal and conical) as well as rolled and ribbon-like, spiral and braided multilayer rods. The commonness of the graphite-like layered atomic structure makes them similar in respect of elastic and strength properties. At the same time, the mechanical characteristics profoundly change with varying structure orientation and geometry of nano- and microspecimens. By the example of a thin bar cut out of a graphite-like material, we have studied the dependence of its strength properties on the angle of structure orientation relative to the bar tension direction. The static strength of the tensile bar is shown to change due to the varying contribution of two competing stability loss mechanisms related with the separation or relative displacement of layers at orientation angle variation. At small angles T between the tension axis and crystallographic axis of the crystal (Fig. 8) the mechanism of graphite layer separation is the leading one. At large angles (T > Tc , in the considered conditions Tc is close to 30q) the leading role goes to the mechanism of graphene layer sliding. The estimates and comparison to experimental data performed in [14] show that strength values for

Fig. 8. Filamentary crystal of a graphite-like material

graphene plane separation V c and sliding W c differ by an order of magnitude (V c | 50 MPa, W c ~ 0.5 MPa). Through accounting for the interrelation of the static and kinetic strength using Zhurkov’s formula, we have also obtained dependences of lifetime on temperature and angle of structure orientation relative to the tension direction. 2.4. Simulation of the nanoparticle filler influence on effective properties of composites One of the promising lines of nanotechnology development concerns the application of various particles as fillers in the production of different composite materials. Particularly, we know from experiments that composites consisting of a polymer matrix (e.g., polypropylene or polystyrene) that contains only several percent of montmorillonite nanoparticles have much higher strain characteristics than the initial polymer material. However, the theoretical estimates of effective characteristics of such composites, which are obtained within the classical approaches of composite materials mechanics, predict lower effective modulus values than the experimentally observed ones. Possible causes of the above discrepancies have been investigated. Paper [15] considers effects related with the formation of an intermediate phase at the “matrix – filler” interface, which appears as shells around filler nanoparticles. The influence of the intermediate phase on the effective strain characteristics of a nanoparticle-filled composite is simulated. According to the given approach, first, a nanoparticle and the surrounding layer of the intermediate phase are considered and effective properties of the two-phase medium are calculated. Then, the effective strain characteristics of the composite consisting of the matrix and inclusions of a homogeneous material are calculated; the properties of the material are estimated at the previous stage when considering the particle and surrounding intermediate phase. The approach allows the solution of both a direct and inverse problem. In the direct problem the effective elastic and strength properties of the composite are calculated by the known mechanical and geometrical characteristics of nanoparticles, the matrix and intermediate phase. In the inverse problem the mechanical properties of the composite and matrix are used to calculate the effective elastic properties of the intermediate phase and/or nanoparticles. The efficiency of the proposed approach is illustrated by the example of solving the inverse problem for a typical nanocomposite (the matrix contains 0.5 % of filler nanoplates with Young’s modulus exceeding 200 times the matrix modulus; Poisson’s ratios of the matrix and inclusions are respectively 0.45 and 0.33; increase of the composite Young’s modulus as compared to the matrix amounts to 70 %). In the case of no intermediate layer, the theory predicts a 50% increase of Young’s modulus. For the obtained inverse problem solution possible combinations of thickness and Young’s modulus of the intermediate phase layer which induce additional growth of the composite modulus are calculated. Poisson’s

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ratio of the intermediate phase is shown to slightly influence the final results. The formation of thin layers with intermediate properties around nanoparticles is found in filled elastomers, which are usually filled with commercial carbon nanoparticles. The observed increase of the ultimate strain and strength of such materials is simulated in papers [16, 17]. Such effects are suggested to explain using a model that, according to the Fukahori hypothesis, assumes the formation of two boundary layers around the nanoparticle. In one of the layers the material is in a glassy state, and in the other — in viscous. Under deformation of the viscous boundary layer there occur orientation effects and transition from a planar orientation state of elastomer chains to a uniaxial orientation state. As a result, the ultimate strain and strength of the nanoparticle-filled elastomer increases. The hypothesis is theoretically justified. In connection with the aforementioned discrepancy between the observed effective elastic moduli and those predicted in the classical theories, of interest is to study the opportunity of further increasing the effective modulus values of the nanoparticle-filled composite through taking into account the role of surface stresses at the “nanofiller particles – matrix” interfaces (see, e.g., [18, 19]). For media with micro- and nanostructure the nonlocality and presence of internal degrees of freedom are also significant. In papers [20, 21] these effects are considered as applied to composites with polymer matrix and filler in the form of nanosized carbon and silicate inclusions. The effective strain and strength characteristics are calculated using various methods that allow describing multiscale processes. The microstructural peculiarities of a broad spectrum of polymer-based nanocomposites are described. The nanocomposites are classified according to their microproperties. The microcracking mechanism in large molecular composite aggregates is revealed. 2.5. Simulation of defects and cracks in micro-, nanoobjects and in nanocrystalline materials Since micro- and nanoscale objects are used as fillers in composite materials, particular attention is paid to the generation and evolution mechanisms of defects and cracks in these objects. Particularly, the presence of structural defects in carbon nanotubes is considered as one of the reasons of a large scatter of their elastic moduli. The discrete-continuous approach put forward in [10] for describing the deformation of micro- and nanoscale objects allows simulating the generation of structural defects, particularly in carbon nanotubes and graphene planes. For example, it allows determining preferable growth directions of defects with regard to their mutual arrangement. For the micro- and nanoscale objects important issues are the transformation of a structural defect to a crack and possibility of simulating limit equilibrium conditions and

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propagation of such cracks within the continuous approximation. Note in this respect papers [22, 23] that simulate crack tip zone formation in a lattice model. A generalized Thomson-like atomistic model of a crack is put forward. An interface crack is assumed to divide four semi-infinite atomic chains. The interatomic interaction in each chain is described by springs with flexural rigidity, while the interaction of atoms of neighboring chains is modeled by bilinear bonding elements (springs) working in tension. The loading is performed by means of concentrated tensile forces applied to the extreme atoms of the outer atomic chains. The deformation and rupture of bonding elements close to the crack tip are analytically and numerically simulated. The cohesive zone size depending on crack length and bonding parameters is estimated. In a general case, the model assumes the formation of three zones: zone 1 with no atomic bonds (the crack zone and its length are characterized by the number m of ruptured bonds), zone 2 where interatomic interaction is described by the descending branch of the assumed force law (cohesive zone), and zone 3 where interatomic interaction occurs according to the ascending branch of the force law. The number of atoms in the cohesive zone s characterizes its length. The dependence of the cohesive zone length on crack length and two parameters of the bonding law (the first characterizes lattice rigidity, and the second — bond rupture work) for the bilinear law is estimated. It is shown that the cohesive zone size can be compared to the crack length for short cracks (the number of ruptured bonds in the crack m = 5–10), while for long cracks (m = 15–20) the cohesive zone size is much smaller ( s m ~ 0.1–0.2). The large cracks are nanoscopic. It can be thus expected that the continuous approach of linear fracture mechanics can be applied to simulate the equilibrium and growth of nanocracks. Note that the Thomson model contains only two atomic chains and implies no stress relaxation in the zone of unruptured bonds ahead the crack tip. As a result, the cohesive zone size remains almost unchanged at crack length variation. The nucleation and growth of defects and cracks in nanostructured materials have their own peculiarities. Certain defects can, on the one hand, play a negative role and induce fracture, while on the other can favor the evolution of deformation and enhance the ultimate strain characteristics of the material. A systematic study of defect generation (including nanocrack nucleation) and opportunities of suppressing this process is performed in papers [24–32], which propose the theory of new plastic deformation mechanisms in nanocrystalline materials with regard to their structural singularities. The presence of such structural elements in the nanocrystalline material as triple grain boundary junctions causes the formation of pores and cracks in these zones and subsequent intensive lattice sliding in the regions between na-

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nocracks and pores, which is in turn manifested in “pit structure” formation. The given processes are theoretically described. On the other hand, nanocrack generation in nanocrystalline materials is affected by diffusion. This effect arises under superplastic deformation. It is shown theoretically that intensive diffusion suppresses nanocrack generation in superplastically deformed nanocrystalline materials in certain domains of their structural parameters and deformation conditions. The deformation of nanocrystalline materials can occur through twinning, the generation of deformation twins is attributed to grain boundaries. A model describing parameters of the given process has been developed. In plastically deformed nanocomposites nanoparticles can induce dislocation emission that favors the development of plastic deformation. Theoretical dependences of nanostructural and dislocation emission parameters are obtained. The regularities of crack nucleation and generation in nanocomposite coatings are studied. Nanoparticles are shown to impede crack growth or change their propagation direction in a certain range of structural and phase parameters. A theoretical model of the rotational mode of plastic deformation, which relates to local grain boundary migration, has been developed for superhard nanocomposite coatings on the basis of nitrides and carbides. 2.6. Simulation of the influence of forces manifested in the nanoscale range on contact interaction characteristics Progress in engineering is more and more associated with the development of miniature systems, among which are micro- and nanoelectromechanical systems and atomic force microscopy systems. The design and analysis of such systems under operation make it necessary to simulate the interaction of contacting bodies with surfaces having nanoscopic roughness. In this connection papers [33, 34] simulate contact interaction mechanisms of deformed bodies with regard to adhesive forces of different origin induced by molecular adhesion of surfaces or capillary pressure in surface liquid films. Particularly, the contact interaction of asperities of elastic and visco-elastic bodies with regard to adhesion is studied. In the case of molecular adhesion, the Maugis–Dugdale model was used to describe adhesive surface interaction and, in the case of capillary adhesion, the Laplace formula was applied for which the strain-induced surface shape distortion was taken into account. An approach has been put forward which allows, with common notions, solving problems on the interaction of elastic bodies with regard to adhesion of different origin as well as studying adhesion at the interaction of bodies with regular surface relief. In the ap-

proach adhesive pressure is represented as a piecewise constant function. This makes possible to consider arbitrary types of the adhesive interaction potential, including capillary adhesion cases, and to account for the presence of additional load that reflects the influence of surface geometry of contacting bodies (the influence of neighboring asperities at adhesion of rough bodies and bodies with regular surface relief). The obtained solution has provided the basis for analyzing the dependence of contact characteristics (contact pressure, contact surface element size, adhesion forces, etc.) on surface energy of contacting bodies, thickness and properties of surface liquid films. It is found that in the closed cycle of surface approach and withdrawal in the both considered cases (molecular and capillary adhesion) work is done and energy is lost. The dependence of energy dissipation on the bulk and surface properties of the contacting bodies is studied. The influence of surface roughness, thickness and properties of coatings on contact interaction characteristics with regard to adhesion is analyzed. A combined influence of adhesion and imperfect elasticity on the resistance of surfaces to relative displacement is investigated under sliding friction. 3. Development of experimental methods for studying mechanical characteristics of nanoscale objects One of the key problems in nanomechanics is the determination of mechanical and physical characteristics of nanoobjects. The difficulties arising in this case are well illustrated in a variety of publications devoted to the determination of elastic characteristics of carbon nanotubes (see, e.g., review [35]). The application of methods appropriate for microscale objects to the experimental investigation of nanoscale objects often runs into serious obstacles. The generalization of the mentioned methods for testing nanoscale objects is a nontrivial task. For example, a very efficient method of determining elastic moduli used in the mechanics of macroobjects is based on eigenfrequency measurement. The direct application of this method for nanosized objects is difficult. Papers [36, 37] suggest a method of eigenfrequency determination for certain nanostructures (nanotubes and nanocrystals). The method is based on the measurement of eigenfrequencies of a “large system”, which consists of a highly oriented array (network) of identical nanotubes or nanocrystals grown up on a microscopic substrate and positioned perpendicularly to the substrate, and of a substrate without nanoobjects. According to [36, 37], the eigenfrequency spectrum of the “large system” can be divided into two parts. One part of the system spectrum corresponds to eigenfrequencies of one nanoobject. At oscillation with these frequencies the substrate remains almost undisturbed.

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The other part of the system spectrum is a spectrum of eigenfrequencies close to those of the substrate without nanoobjects. At the given frequencies the nanoobject oscillation amplitude turns to be much lower than the substrate oscillation amplitude. In [38] two modified experimental procedures of eigenfrequency determination are proposed on the basis of papers [36, 37]. They imply the following steps: – To measure a few first eigenfrequencies of the system “nanotube or nanocrystal network – substrate”. To measure eigenfrequencies of the same substrate without nanoobjects. To compare the two obtained parameters. The frequencies in the system spectrum which are close to those of the substrate without nanoobjects are left aside. The frequencies in the system spectrum which do not correspond to any frequencies in the substrate spectrum are the nanoobject frequencies. – To measure the resonance frequencies of the system through recording the electromagnetic radiation of nanoobjects (this can be done because many nanoobjects are piezoelectrics). To simultaneously measure the substrate oscillation amplitude (this can be done because the substrate is a macroobject). The resonance frequencies at which the substrate oscillation amplitude is zero are the nanoobject eigenfrequencies. In [38] the above modifications were applied to the case of nanotubes fixed on the lateral side to a substrate. This allowed estimating the flexural rigidity of the nanofilm out of which the nanotubes were made. The calculation results (multilayer gallium arsenide films on a sapphire substrate were examined) are illustrated in Figs. 9, 10. The possibility of finding first eigenfrequencies out of the “large system” spectrum is shown in [36–38] through analyzing equations of oscillation of composite shells and plates as well as through finite-element calculation by means of 3D equations of the theory of elasticity and electric elasticity. Notice that to study natural oscillations of ordered arrays of nanoobjects is interesting not only in connection with the development of methods for the determination of their mechanical properties, it also has an independent significance because such arrays, particularly of semiconductor nanocrystals and nanotubes, show much promise for nanophotonics and nanoelectronics devices. The methodical difficulties of measuring strain characteristics of nanoobjects appear even greater in attempts to measure the strength of these objects. Currently, the performance of strength tests on carbon nanotubes is more in the nature of art rather than craft. This is, e.g., demonstrated in pioneer works [39, 40] on tensile strength measurement for single- and multilayer carbon nanotubes. In these experiments the loading devices are atomic force microscope components onto which a nanotube is “welded”, and the nanotube tension is examined in a high-resolution scanning electron microscope (the obtained tensile strength

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Fig. 9. Eigenmodes localized in nanotubes

values of single-layer nanotubes turned to be 13–52 GPa, while for the outer layer of multilayer nanotubes 11– 63 GPa). Shock-wave loading methods seem to be very promising for the investigation of nanomaterial resistance, particularly metals and alloys with nanoscale structure, to deformation and fracture. The methods are developed by the authors of [41–43]. For example, they have measured the spalling strength of copper in various structural states and established correlation between the spectrum of defects in the material and its resistance to deformation and fracture. It is shown that fracture resistance is defined by the ratio of extension rate, on the one hand, to the rate of discontinuity nucleation and growth, on the other. The temperature dependences of spalling strength observed in [41–43] can be described with the criterion of the incubation time of fracture [44–47]. The criterion is convenient for the analysis of fracture processes occurring at different scales and allows tracing the transition of fracture from one scale to another. The interrelation between spalling strength and strain rate is studied by means of molecular dynamics methods elsewhere [48].  

Fig. 10. Eigenmode corresponding to the first bending mode of the substrate

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The mechanical characteristics, micro- and nanostructural parameters of materials can also be measured using wave diagnostics methods. For example, papers [49, 50] illustrate that the information about structural parameters can be gathered with surface waves of new types inherent in a medium with internal rotation (the Cosserat continuum). A new type of the surface wave is found –– shear one –– which has a transverse component of displacement relative to the wave propagation direction and lies in the surface plane. The diagnostics with the use of this wave type becomes easier because it has pronounced dispersion properties. The effects of deformation of nanostructured objects which are probably associated with the influence of internal rotations are analyzed in paper [51]. For nanostructured materials general identification methods of material properties can be adopted with regard to the multilevel structure. Such methods are put forward in papers [52–54] that develop methods of defining extreme states of a deformed layered medium on the basis of distinguishing and compiling classes of heterogeneities having preferable layered orientation, which cause vibration localization in the layered structure and are able to induce its fracture. The development of experimental methods for the investigation of strength and crack growth resistance properties of nanoscale specimens is an acute problem in nanomechanics. Nanotechnology development requires certification of performance characteristics, including mechanical properties, of both the nanoscale objects themselves and composites and products on their basis. 4. Modeling of nanotechnology elements Models and methods of nanomechanics prove to be useful for modeling some elements and revealing optimal regimes of nanotechnologies. An example is found in papers [55, 56] that model the nanotube synthesis technology through rolling a strained double-layer film at buffer layer etching, which is suggested elsewhere [57, 58]. In [56] molecular dynamics methods and in [55] elasticity theory methods are applied. The elastic modulus and diameter of the nanotubes are calculated. The calculation and experimental values are found to agree. The results can be used to select the technological parameters necessary for the production of nanotubes with specified elastic and geometric characteristics. The theoretical grounds for the probable technology of nanothick film growing are developed in [59]. Thin films are proposed to grow by governing diffusion mass transfer along the substrate surface. In model construction account is taken of the competition between surface tension effects and effects arising due to lattice misfit. The possibilities of governing ultimate strain characteristics of systems with coatings or hardened surface layers

through direct variation of geometric parameters and elastic properties of the joined materials are demonstrated in [60]. Based on theoretical and experimental findings it is shown that the strength and plasticity of materials with hardened surface layers or hardening coating depend on the character of multiple cracking of the hardened layer (or coating). The surface hardening that gives rise to a system of transverse opening-mode cracks in the surface layer of the loaded material is always accompanied by material plasticity reduction. On the other hand, multiple cracking of the hardened surface layer (coating) in the form of a network of cracks oriented in the conjugate directions of maximum tangential stresses provides uniform loading of the deformed material (specimen) at the mesoscale. Such multiple cracking increases both the strength and plasticity of the material. The observed effect is shown to depend on the hardened surface layer (coating) thickness and profile (plane/serrated) of the “hardened layer (coating) – substrate” interface. Along with coating deposition technologies, the technologies of fine and superfine wire production have been developed. The technology of superfine (less than 1 Pm) wire production of hard-to-machine materials, e.g., titanium and its alloys [61, 62], was proposed. The research results obtained in the above papers laid at the basis of an invention patent and favorable decision has been already obtained. 5. Conclusions The given investigation findings show promise for applying approaches of solid mechanics and the mechanics of strength and fracture to model and optimize the functional characteristics of nanoobjects and nanomaterials as well as to model and optimize nanotechnologies. Emphasize that we primarily considered the issues directly concerning the nanoscopic response of materials to external mechanical action. The investigation results for scale interaction under deformation and fracture require further consideration. In summary, we would like to define some problems in the mechanics of nanomaterials and nanotechnology which acquire greater importance in connection with the growing worldwide attention to nanotechnology. – The simulation of the mechanical behavior of nanocomposites (and generally nanomaterials) has just begun. We need to develop models that account for the influence of the intermediate zone between filler particles and the matrix on nanocomposite mechanical properties as well as take into account the filler type and its volume content. Such models and corresponding constitutive relations allow predicting the mechanical properties of nanocomposites and developing materials with specified properties. – The development of methods for testing the mechanical properties of nanocomposites, filler nanoparticles, intermediate zones between nanoparticles and the matrix.

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– The development of a metrological basis for the mechanical testing of nanoscopic objects. – Experimental study and theoretical simulation of the mechanical behavior of nanoparticles and their conglomerates under mechanical loading, influence of physical fields and active media. – The development of the mechanics of multilayer nanoelements and coatings in connection with the prospects of their application in nanoelectronics, in machine building for the deposition of gradient heat- and corrosion-resistant coatings and production of tribotechnical junctions with increased wear resistance. – The development of nanotechnology mechanics in order to obtain quantitative dependences between mechanical properties and technological parameters, to find optimal technological regimes for the production of nanomaterials and products on their basis with specified performance characteristics. – The development of a mechanical basis for the technology of measuring mechanical and physical quantities using executive nano- and microelements in sensors and measuring systems. We are deeply thankful to the participants of the DPEMBCP RAS Fundamental Research Program “Structural Mechanics of Materials and Structural Components: Interaction of Nano-, Micro-, Meso- and Macroscales under Deformation and Fracture” (coordinated by Academician N.F. Morozov) for the assistance, support and provision of materials in preparing a report at the IX All-Russian Congress on Theoretical and Applied Mechanics, Nizhnii Novgorod, 22–28 August 2006. The report laid at the basis of the present paper. References [1] N.F. Morozov and A.M. Krivtsov, Anomalies of the mechanical characteristics of nanoscale objects, Dokl. Phys., 46, No. 11 (2001) 345. [2] A.M. Krivtsov and N.F. Morozov, On mechanical characteristics of nanocrystals, Phys. Solid State, 44, No. 12 (2002) 2260. [3] E.A. Ivanova, A.M. Krivtsov, and N.F. Morozov, Peculiarities of the bending-stiffness calculation for nanocrystals, Dokl. Phys., 47, No. 8 (2002) 620. [4] A.M. Krivtsov, Deformation and Fracture of Solids with Microstructure, Fizmatlit, Moscow, 2007 (in Russian). [5] C.T. Sun and Zhang Haitao, Size-dependent elastic moduli of platelike nanomaterials, J. Appl. Phys., 93, No. 2 (2002) 1212. [6] R.E. Miller and V.B. Shenoy, Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 11 (2000) 139. [7] L.G. Zhou and H. Huang, Are surfaces elasticity softer or stiffer?, Appl. Phys. Lett., 84, No. 11 (2004) 1940. [8] E.I. Golovneva, I.F. Golovnev, and V.M. Fomin, Peculiarities of application of continuum mechanics methods to the description of nanostructures, Phys. Mesomech., 8, Nos. 5–6 (2005) 41. [9] R.A. Johnson, Alloy models with the embedded-atom method, Phys. Rev. B, 39 (1989) 12554. [10] R.V. Goldstein and A.V. Chentsov, Discrete-continuous model of the nanotube, Mech. Solids, 40, No. 4 (2005) 45. [11] G.M. Odegard, T.S. Gates, L.M. Nicholson, and K.E. Wise, Equivalent-Continuum Modeling of Nano-Structured Materials, in Techni-

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