Anomalies in Mechanical Characteristics of Nanometer-Size ObjectsA. M. Krivtsov and AcademicianN. F. MorozovReceived July 18, 2001St. Petersburg State Technical University,ul. Politechnicheskaya 29, St. Petersburg, 195251 RussiaMECHANICS Anomalies in Mechanical Characteristics of Nanometer Size ObjectsA M Krivtsov and AcademicianN F
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826DOKLADY PHYSICSVol. 46No. 112001KRIVTSOV, MOROZOVstrains, forces acting within the crystal can be approxi-mately written out in the form(3)where C has a meaning of the rigidity of an interatomicbond, while the symbol ∆ implies the deviation of a cer-tain quantity from its value corresponding to the undis-torted crystal. We denote the crystal strain in the x andy directions as ε1and ε2, where(4)The substitution of relationships (2), (3), and (4) intoexpressions (1) and (1a) yields the elasticity relations(5)As is seen from relations (5), the crystal under consid-eration is anisotropic. We recall that the infinite crystalwith the HCP crystal lattice is isotropic and, hence, theanisotropy indicated is a manifestation of the scale fac-tor. Furthermore, we denoteF a( )C∆a, F b( )C∆b, CF' a0( ) 0,>===defε1∆aa0-------, ε2∆hh0-------, h03a02------------≡.==defdefσ1312-------CN*------- 9N 1–()ε13 N 1–()ε2+(),=σ234-------C ε13ε2+().=ν1ε2ε1----–=σ20=, E1σ1ε1-----=σ20=;defdefν2ε1ε2----σ10=, E2–σ2ε2-----σ10=.==defdefHere, ν1and E1are the Poisson’s ratio and Young mod-ulus for tension along the x axis; the quantities ν2andE2correspond to tension along the y axis. Using rela-tionships (5), we obtainwhere, ν∞= and E∞= are values of the Poisson’sratio and Young modulus, which correspond to the infi-nite crystal [6, 7]. We now analyze the formulasobtained. Under tension along atomic layers, the Youngmodulus E1substantially depends on the quantity N*,i.e., on a method for determining the thickness of thenanocrystal strip. If we assume that N*= N (N is themaximal value of N*) then, under tension along atomiclayers, the Poisson’s ratio and Young modulus are inde-pendent of a number of layers. Evidently, this is associ-ated with the fact that in the longitudinal direction, thecrystal under consideration is infinite. By contrast, theYoung modulus corresponding to the minimalvalue N*= N – 1, is not constant. It increases with adecrease in the number of atomic layers and for N = 2attains a value twice as large as E∞(see table). Thus, theambiguity in determining the Young modulus turns outto be rather substantial for small values of N. In the caseof tension in the direction perpendicular to atomic lay-ers, both the Poisson’s ratio and the Young modulusdepend on N, the former decreasing and the latterν1ν∞, E1=NN*-------E∞;=ν2N 1–N 1/9–------------------ν∞,=E2NN 1/9–------------------E∞,=13---2C3-------E1maxyxH= (N–1)hhbaABQH= NhQhhhhhhhhTwo-dimensional single-crystal strip.