ISSN 10637850, Technical Physics Letters, 2009, Vol. 35, No. 11, pp. 985989. Pleiades Publishing, Ltd., 2009.Original Russian Text A.L. Kolesnikova, A.E. Romanov, 2009, published in Pisma v Zhurnal Tekhnichesko Fiziki, 2009, Vol. 35, No. 21, pp. 2332.

985

The mechanical response of small (in particular,nanodimensional) crystalline particles and atomicclusters to an external load is determined to a considerable degree by the presence of structural defects suchas dislocations and disclinations [1]. Elastic distortions introduced by these defects into the crystal lattice depend on their screening by the free surface,which becomes a determining factor for small particles.

A simple physical model for investigations of thebehavior of defects in small particles is offered by anelastic spherical body (spheroid) containing an individual defect. However, even in this simplified formulation, a boundaryvalue problem of the elasticity theory is threedimensional (3D) [2] and, hence, difficultfor analytical consideration. To the present, solutionshave been obtained only for a very restricted number ofproblems concerning defects in an elastic spheroid. Inparticular, Willis et al. [3] solved an axisymmetricproblem for a prismatic loop, while Polonsky et al.[4, 5] determined the elastic fields and energies of astraight wedge disclination and a screw dislocation,the lines of which coincided with the spheroid diameter. These solutions were obtained using the method ofspherical harmonics, which had been developed previously for solving nonsingular problems in the elasticitytheory of bodies with spherical boundaries [2].

This Letter presents a new solution, which has beenobtained by an original method of virtual defects forthe elastic fields and energy of a twist disclination loop(TDL) in a spheroid.

A circular TDL represents a defect, which appearsdue to the mutual rotation of banks of a circular cutaround an axis that is perpendicular to the cut

plane [6]. An analysis of these defects is important forboth basic physics (e.g., in the description of backbonetwist in polymers [7]) and applied mechanics (e.g., incalculations of the strength of rod structures [8]).

Originally, the mechanical stress fields of TDLs inan infinite elastic medium were calculated by Owenand Mura [9] using a representation via completeelliptic integrals. Later, the elastic fields of TDLs havebeen expressed in terms of the LifshitzHankel integrals [10], and this representation will be employedbelow. It should be noted that the LifshitzHankelintegrals can be used to describe the elastic fields ofcircular dislocationdisclination loops of the generaltype [11, 12]. Earlier, the boundaryvalue problemswere solved concerning the elastic behavior of TDLsnear a planar interface and in a plate of finite thickness[1115]. More recently, the authors determined theelastic fields and energy of a TDL in a cylinder [16].

The problem is considered in a geometry depictedin Fig. 1a, where the jump of displacements [u] in thesection, total displacements uj, and mechanicalstresses ij of a TDL in the cylindrical coordinate system (r, , z) can be expressed as follows [11, 12]:

(1)

(2a, 2b)

(2c)

(3a)

u[ ] z z0= rH 1rc

,=

ur uz 0,= =

uc z z0( )sgn

2J 2 1; 0,( ),=

rG z z0( )sgn

2J 2 2; 1,( ),=

Twist Disclination Loop in an Elastic Spheroid A. L. Kolesnikovaa and A. E. Romanovb, c*

a Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, 199178 Russia b Ioffe PhysicalTechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia

c Aristotle University of Thessaloniki, GR 54124 Thessaloniki, Greece *email: aer@mail.ioffe.ru

Received May 26, 2009

AbstractA solution of the boundaryvalue problem in the isotropic theory of elasticity for a twist disclination loop (TDL) in a spherical body (spheroid) has been obtained for the first time using the method of virtualdefects. The virtual defects are represented by TDLs with elastic fields, which are expanded into series withrespect to Legendre polynomials. The elastic fields and energy of a TDL are determined depending on itsposition in the spheroid.

PACS numbers: 61.72.Lk, 61.46.Hk

DOI: 10.1134/S1063785009110042

986

TECHNICAL PHYSICS LETTERS Vol. 35 No. 11 2009

KOLESNIKOVA, ROMANOV

(3b)

(3c3f)

where is the magnitude of the Frank vector = ezfor a TDL with the defect line oriented along e(e and ez are the corresponding coordinate unit vectors), c is the loop radius, H(1 r/c) is the Heavisidefunction, G is the shear modulus, is the Poissonratio, and J(m, n; p) are the LifshitzHankel integrals.The latter integrals are defined as

(4)

where Jm(x) and Jn(x) are the Bessel functions and z0 isthe TDL coordinate (see Fig. 1a).

To solve boundaryvalue problems of the elasticitytheory in spherically symmetric systems, it is convenient to use a representation of TDL fields in thespherical coordinate system (R, , ) (Fig. 1a). Uponsubstituting the quantities = /c, r = Rsin, z =Rcos, c = R0cos0, and z0 = R0cos0, separating theexponent into two components, expanding one of theBessel functions with an exponential weight intoseries, and integrating the series by parts, we obtain theLifshitzHankel integrals in the form of series withrespect to Legendre polynomials. Transforming thefield components from the cylindrical to sphericalcoordinates, replacing the LifshitzHankel integralsby the expansions in Legendre polynomials, and usingrecurrent relations for these polynomials, we eventu

zG2

J 2 1; 1,( ),=

rr zz rz 0,= = = =

J m n; p,( ) Jm ( )Jn rc

e

z z0

c

p ,d

0

=

ally arrive at the following expressions for the TDLfields:

(5a, 5b)

(5c)

(6a)

uR u 0,= =

uR0

2=

1( ) 1k k 1+( ) k 2+( )

R0R

k 1+

Pk2 0cos( )

k 2=

0Pk1 cos( ), R0sin

2R, /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 600 /MonoImageMinResolutionPolicy /Warning /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description >>> setdistillerparams> setpagedevice