elastic models of defects in 3d and 2d crystals · elastic models of defects in 3d and 2d crystals...

130 A.L. Kolesnikova, M.Yu. Gutkin and A.E. Romanov

© 2017 Advanced Study Center Co. Ltd.

Rev. Adv. Mater. Sci. 51 (2017) 130-148

Corresponding author: A.E. Romanov, e-mail: [email protected]

ELASTIC MODELS OF DEFECTS IN 3DAND 2D CRYSTALS

A.L. Kolesnikova1,2, M.Yu. Gutkin1,2,3 and A.E. Romanov1,4

1ITMO University, Kronverkskiy pr. 49, Saint Petersburg, 197101, Russia2Institute of Problems of Mechanical Engineering, Russian Academy of Sciences,

Bolshoj pr. 61, V.O., Saint Petersburg, 199178, Russia3Peter the Great St.Petersburg Polytechnic University, Polytekhnicheskaya 29, Saint Petersburg,

195251, Russia4Ioffe Institute, Russian Academy of Sciences, Polytekhnicheskaya 26, Saint Petersburg, 194021, Russia

Received: April 24, 2017

Abstract. We present the elastic models of defects in crystals exploring the framework ofcontinuum mechanics of solids. The key feature of the defects attributed to zero-, one-, two- orthree-dimensional ones is the dimension of the region of their eigenstrains. It is shown how fromthe elastic fields of defects of a lower dimension to build, by integration, the elastic fields ofdefects of a higher dimension. On the base of the elastic fields of infinitesimal dislocation loops,the fields of the circular dilatation line, the circular dilatation disk, and the cylindrical andhemispherical inclusions are derived in a step-by-step manner. In the final part of the paper, theelastic fields of defects in 2D crystals are considered.

1. INTRODUCTION

The theory of defects of a continuous elastic me-dium, on the one hand, is a mathematical descrip-tion of the elastic behavior of defects in solids, andon the other hand, it is a tool for modeling theelastoplastic behavior of materials experiencing lo-cal transformations due to mechanical or thermaltreatments.

The theory of defects owes its modern appear-ance to the efforts of many scientists, beginningwith Volterra [1], who first presented elastic modelsof dislocations and disclinations. The foundationsof the theory were laid down in the fundamental stud-ies by Eshelby [2-4], Kröner [3], Kroupa [5],Indenbom [6], De Wit [7], Hirth [8], Lothe [8],Teodosiu [9], Mura [10], and many others.

At different stages of the defect theory develop-ment, various aspects of it become important. Cur-rently, topical problems include consideration of

defects in finite solids and films that is directly re-lated to physical and mechanical problems ofnanoheterostructures, nanowires, nanoparticles,nanocomposites, and 2D crystals, such asgraphene.

Nowadays it is known that for the solution ofboundary-value problems it is effective to use thevirtual defect technique, e.g. see Refs. [11-13]. Thistechnique originates from the method of surface dis-locations [14,15]. The technique operates with elas-tic fields of distributed virtual sources of elastic fieldsused to construct the boundary integral equations.Solving the equations with respect to the distribu-tion functions of defects one finds the solution forthe boundary-value elastic problem of the interest.Often it is not enough to have Volterra dislocationsand disclinations as virtual defects, and it is neces-sary to introduce other types of virtual defects ofvarious configurations, for example, the circular ra-dial Somigliana dislocation [11,13,16,17]. In addi-

mailto:[email protected]

131Elastic models of defects in 3D and 2D crystals

tion, we believe that circular dilatation and shearlines will serve as convenient virtual defects for solv-ing axisymmetric boundary-value problems. To findthe solutions to more general problems, one canuse dislocation half-loops [12], rays and infinitesi-mal dislocation loops [5].

The purpose of this paper is to provide a briefreview of the latest achievements in elastic descrip-tion of defects in 3D and 2D crystals. In the frame-work of the formal classification, based on dimen-sion of region of defect’s eigenstrain, it will be dem-onstrated how from the elastic fields of defects of alower dimension to get, by integration, the elasticfields of defects of a higher dimension. As an exam-ple, a step-by-step path from an infinitesimal dislo-cation loop through a circular dilatation line and acircular dilatation disk to axisymmetric inclusionsin the form of a finite circular cylinder and a hemi-sphere will be presented. The final part of the articleshows the progress of recent years in the construc-tion of elastic models of defects in 2D crystals.

2. CLASSIFICATION OF INTERNALSOURCES OF ELASTIC FIELDS IN3D CONTINUUM

Elastic fields in solids can be caused either by ex-ternal forces applied to the boundaries of the body,or by internal sources presented in the body. Inter-nal sources (or defects) distort the elastic medium,which is a crystal, and thereby generate elastic fieldsin it. We do not consider eigestrain-free inhomoge-neities, i.e. regions with other elastic modules butwith no eigenstrains, and voids as intrinsic defects.These inhomogeneities and voids themselves arenot the sources of elastic fields, but they only dis-tort the existing elastic fields.

In our opinion, there are three basic approachesfor specifying the internal sources of elastic fields:(a) using the distribution of forces with or without amoment (see, for example, [18]), (b) usingeigenstrain (see, for example, [2-4,10]), and (c) ex-ploring harmonic stress-functions, which are thesolutions of harmonic differential equations with pre-scribed source terms [9,16,17,19]. These ap-proaches are used to build mathematical models ofreal physical defects in the crystal lattice, e.g. pointdefects, dislocations and inclusions. The basic prin-ciple in all the approaches is that the elastic fieldsof the defect under consideration satisfies the equi-librium equations for stresses.

The “force” approach (a) makes it possible tofind the elastic fields of the defect using the volumeforce distributions [9,18]. This approach can be ef-

fectively applied to calculate the fields of pointcenters of dilatation, which in the real crystal latticemimic vacancies or impurity atoms. In the case ofdislocations, the force approach also works, al-though in a not so obvious way [20]. Direct and clearmethods to defect modeling are the approachesexploring eigenstrain (b) and harmonic functions ordisplacements (c). In this paper, we work in the frame-work of the “eigenstrain” approach.

In micromechanics of elastic continuum, theconcept of the eigenstrain tensor, which determinesthe way of specifying and region of localization ofthe defect, can be used to uniquely define any kindof defect. Eigenstrain is a complete characteristicof the defect. The response of an elastic medium toa given eigenstrain unambiguously determines theelastic fields of the defect in the medium. The math-ematical apparatus for finding the elastic fields ofthe defect from a given eigenstrain is well devel-oped [2-4,5,10].

The process of specifying eigenstrain can bepresented in the way that was first proposed byEshelby for inclusions [2]. The Eshelby procedurecan be outlined as follows: a cut is made in an elas-tic body, the material is removed, and then it isplastically deformed. Forces are applied to this de-formed volume of material in order to insert it intothe original place exactly at the place of the cut,then, the surfaces of the cut are glued together andthe forces are removed.

If we apply the Eshelby procedure to regions ofdifferent dimensions, we can obtain defects of dif-ferent dimensions, and introduce a classification ofdefects in an elastic medium, based on the dimen-sion n of the region

n of eigenstrain n

ij

* (or self-distortion n

ij

* ).In 3D elastic space, defects can be subdivided

into four classes.Zero-dimensional or point defects (n = 0). Af-

ter Kroupa [5], we define infinitesimal dislocationloops as the elementary zero-dimensional defects(Figs. 1a and 1b). The self-distortion of an infinitesi-mal dislocation loop can be written as follows [21]:

ij j ib s i j x y z0 *

0( ), , , , , R R (1)

where bj is Burgers vector of dislocation loop, placed

in point R0, s

i is an area of the loop. The 3D Dirac

delta-function (R-R0) is defined by the product of

the 1D delta-functions: (R-R0)= (x-x

0) (y-y

0) (z-

z0). It has the following dimensionality: [(R-R

0)]=

[(x-x0)][(y-y

0)][(z-z

0)]=X-1X-1X-1=X-3, where X is a

unit of length. The distortion given by Eq. (1) can bedefined for any orthogonal coordinate system, for


example, in cylindrical or spherical ones (i, j=r,,zor i, j=R,,), respectively.

Equation (1) can be written in the general form:

ij ijv i j x y z0 * *

0( ), , , , , R R (2)

where ij

* is self-distortion of a small volume v with-out taking into account its location in an elasticmedium (Fig. 1c).

Consider now the defects with a small cut sur-face s

z. The surfaces of the cut are subjected to

arbitrary displacements written in the form of a com-bination of arbitrary shifts and rotations [10,22,23]:

ij j jpq p q q zb e x x s0 *

0 0( ) ( ), R R (3)

where in curly brackets the shift of the upper part ofthe cut

zs with respect to the fixed bottom part of

the cut z

s is shown, bj is the dislocation Burgers

vector; ejpq

is the Levi-Civita tensor; p is the

disclination Frank pseudo-vector; xq0

is the axis ofrotation. The sign of the distortion given by Eq. (3)is important. By definition, the Burgers vector is thedisclosure vector for a Burgers circuit surrounding

the defect line and crossing the selected cut sur-face s

z. The sign of the Burgers vector is related to

the definition of the normal to the cut surface, thejump in the displacements at the cut, and, ulti-mately, the sign of self-distortion.

According to Eq. (3), we obtain infinitesimal dis-location-disclination loops with distortions writtenin the same way as for finite-dimensional disloca-tion-disclination loops [10], but stitched to the point(x

0,y

0,z

0).

Two kinds of infinitesimal dislocation loops, firstinvestigated by Kroupa [5], are of particular inter-est. Using the Kroupa infinitesimal dislocation loops,as it will be shown below, it is possible to constructa point center of dilatation and some known defectsof greater dimensions: dislocations, disclinations,and inclusions. If the vector is directed along thenormal to the cut surface s

z, e.g. b=±b

ze

z, the in-

finitesimal loop is prismatic one (Fig. 1a). If the vec-tor b lies in the plane of the cut, e.g. b=±b

xe

x, the

infinitesimal loop is the glide dislocation loop (Fig.1b).

Fig. 1. Zero-dimensional defects in an elastic medium: (a) an infinitesimal prismatic dislocation loop; (b) aninfinitesimal dislocation glide loop; (c) a general point defect. Self-distortions of defects

ij

0 * and eigensrains

ij

0 * are shown. Reprinted with permission from A.L. Kolesnikova, R.M. Soroka,A.E. Romanov //MaterialsPhysics andMechanics 17(1) (2013) 71, (c) 2013Advanced Study Center Co., Ltd.


For a dilatation center, the eigenstrain is givenby

ii ii i ib s i x y z0 * 0 *

0( ), , , , R R (4)

The dilatation center with eigenstrainxx

0 * =yy

0 * =zz

0 *is the simplest elastic model of an impurity atom ora vacancy, for which one can write [7] (Fig. 1d):

xx yy zzbs

v

0 * 0 * 0 *

0

0

( )

1( ).

3

R R

R R (5)

Here v is the change in volume at the point of de-fect localization. If l is the initial linear size of a smallvolume v=l 3, then changing the linear dimension byl leads to a change in the volume by v = (l+ l )3-l 3 3l 2l . Assuming that s=l 2 and b= l , we obtainbs=1/3(v) and the last equality in Eq. (5). Thus,the dilatation center can be represented by threemutually perpendicular infinitesimal prismatic dis-location loops (Fig. 2). Note, that three mutuallyperpendicular pairs of forces also model the dilata-tion center [18] (Fig. 1d).

An increase in the dimension of the region

0

1 (

1L), where defect is localized, leads to

a one-dimensional defect.One-dimensional defects (n=1). The distortion

of a one-dimensional defect is logically to define asfollows: cut a tube of small cross-section along theline L of the future defect, plastically deform thetube and insert it back into the cut, gluing the edgesof the cut (Fig. 2a). As a result, the self-distortion ofthe one-dimensional defect is [21]:

ij ijs L1 * * ( ), (6)

where ij

* is plastic distortion of L; s is multiplierhaving the dimension of an area; (L) is a two-di-mensional delta-function.

One-dimensional defect can also be definedthrough zero-dimensional defects continuously dis-tributed with prescribed linear density

L along the

line L (Fig. 2b):

ij ij L ij L

L L

L v L

i j x y z

1 * 0 * *

0 0 0 0d ( ) ( ) d ,

, , , ,

R R R

(7)

where L0 is the coordinate of a probe point defect

along the line L.In the case when

ij

* in Eq. (7) does not dependon the coordinate of the distributed point defects,i.e. their linear density is constant, we obtain a sim-ple formula:

ij ij Lv L1 * * ( ). (8)

For a straight-line defect placed along the z-axis,(L)=(x-x

0)(y-y

0) and [(L)]=[(x-x

0)][(y-y

0)]=

X-1X-1=X-2. Obviously, replacing vL=s in Eq. (8) we

arrive to Eq. (6). It follows from Eqs. (7) and (8) thatby integrating the elastic fields of a zero-dimensionaldefect along the line with prescribed density, wecan find the elastic fields of the particular one-di-mensional defect. Representation of a one-dimen-sional defect via Eq. (7) has an explicit relationshipwith physical prototypes. For example, the lineardistribution of distortions from Eq. (2) models achain of vacancies or impurity atoms.

With a further increase in the dimension of thedistortion-setting region

1

2 (

2S), we obtain

two-dimensional defects, and, in particular,Somigliana and Volterra dislocations.

Fig. 2. One-dimensional defect (a) and its representation with the help of point defects (b), reprinted withpermission from A.L. Kolesnikova, R.M. Soroka, A.E. Romanov // Materials Physics and Mechanics 17(1)(2013) 71, (c) 2013 Advanced Study Center Co., Ltd.


Two-dimensional defects (n=2). The regiondefining the eigenstrain (or self-distortion) of two-dimensional defects is the surface.

Somigliana dislocations and Volterra dislocations(i.e. translational dislocations and disclinations) aretwo-dimensional defects (Fig. 3à). For dislocation,the region of specifying its eigenstrain coincides withthe surface of the cut, whose edges are subjectedto relative displacements. Volterra dislocations donot have the discontinuity of the elastic strain/stressfields on the surface of the cut, but they have asingularity on the defect line bounding the surfaceof the cut. Therefore, they are traditionally calledlinear defects. The self-distortions for Somiglianaand Volterra dislocations have the form (middle andright hand part in the formula, respectively) [22,23]:

ij j i j jpq p q q iu S b e x x S2

0[ ] ( ) ( ) ( ), (9)

where [uj] is the jump of displacements specified

on the surface of the cut S with the normal ni,

Fig. 3. Two-dimensional defect: (a) the Somigliana dislocation (in the special case, the Volterra disloca-tion), (b) the two-dimensional defect of the general form, (c) the two-dimensional defect composed of pointdefects, reprinted with permission from A.L. Kolesnikova, R.M. Soroka, A.E. Romanov // Materials Physicsand Mechanics 17(1) (2013) 71, (c) 2013 Advanced Study Center Co., Ltd.

i(S) is the delta-function on the surface S:

i iSS S

0( ) ( ) d . R R For a plane surface S with

a normal nz, and a coordinate along the z-axis, we

obtain (S)=z(S)e

z=H(S)(z-z

0)e

z, where

x y SH S

x y S

1, ,( )

0, ,

is the Heaviside step-func-

tion.By analogy with Eqs. (2) and (6), for the two-

dimensional defect of general form (Fig. 3b) the fol-lowing expression can be written:

ij ijS2 * * ( ), l (10)

where l is factor that has the dimension of length.One can obtain a two-dimensional defect by dis-

tributing point defects on the surface S (Fig. 3c):

ij ij s

S

ij s

S

S

v S

2 * 0 *

0

*

0 0 0 0

d

( ) ( ) ( )d ,

R R R R (11)


where s is the surface density of distortions

ij

0 * , Sis the surface, over which the distortions are distrib-uted. With constant

ij

* (R0) and

s(R

0), and taking

into account the relation sv=l , from Eq. (11) one

gets Eq. (10).With the help of distributed infinitesimal prismatic

dislocation loops with self-distortions zz

0 * =bzs

z(R-

R0), a finite prismatic dislocation loop located in the

plane with the normal nz and occupying the region

S, is generated:

ij ij s z z s

S S

z z s

S

z z z

SdS b dS

b s x x y y z z dx dy

b H S z z b S

2 0

0 0 0

0 0 0 0 0

0

( )

( ) ( ) ( )

( ) ( ) ( ).

R R

Here it is assumed that s =1/s

z.

The final step in the chain 0

1

2

3

leads to volume defects.Volume defects or inclusions (n=3). For inclu-

sions, the eigenstrain ij

3 * or self-distortion ij

3 * iswritten with the help of a volume delta function

(3) = (V) =

V

V

1,

0,

R

R [7] in the same way as it

has been done in the previous cases (Fig. 4a):

ij ij ij ijV V i j x y z3 * * 3 * *( ), ( ), , , , . (12)

The components ij

* have the meaning of the rela-tive elongation in the direction j of the elementaryarea with the normal n

j (at i=j) or the shift in the

Fig. 4. The inclusion of the general form (a) and that composed of point defects (b), reprinted with permis-sion from A.L. Kolesnikova, R.M. Soroka, A.E. Romanov // Materials Physics and Mechanics 17(1) (2013)71, (c) 2013 Advanced Study Center Co., Ltd.

direction j of the elementary area with the normal nj

(at ij). The eigenstrain ij

* is a symmetrized distor-tion

ij

* .In a real crystal this means that, for example,

as a result of a phase transition in the region 3,

the crystal lattice has changed its constants a1,a

2,a

3

and, possibly, experienced some shear.Note that the components of

ij

* and ij

* givenby Eq. (12) need not necessarily be constant. Aclassical example of the inclusion of a not purelydilatational type is a particle with the intrinsic defor-mation of Marks-Ioffe [24], embedded in the matrix.

An inclusion can be composed of defects of lowerdimension. For example, continuously distributedpoint defects form the inclusion characterized byEq. (12), see Fig. 4b:

ij ij V

V

ij V ij

V

V

dV

dV V

3 0

0

0 0 0 0( ) ( ) ( ) ( ),

R R R R (13)

where v=1/v.

It is not difficult to see that in a 3D continuum,the defects of a dimension lower than 3 have self-distortions with some arbitrary dimensional factors.For the point defect, this is v in Eq. (2), for the lineardefect – s in Eq. (6), and for the 2D-defect – l in Eq.(10). For definiteness, we can consider such a fac-tor as the defect size before plastic deformation.So, v is the volume of the point, s is the cross-sectional area of the line, and l is the thickness ofthe 2D-defect.


3. ELASTIC FIELDS OF DEFECTS IN 3D MEDIUM

By using the self-distortion n

ij

* (or the eigenstrain n

ij

* ) of the defect, the Green function Gmk

and elasticmoduli C

jlmn of the elastic medium, the total displacements of the defect, as well as its elastic strains and

stresses can be found (see [7,10]).For example, the total displacements of defect are given by the following equation [7]:

t n

i j mn mn ij

V

u C G dV*

,, l l

R R R R (14)

where the derivative Gij,l(R-R’) is taken over the unprimed variable x

1, V is the volume of a solid, and the

summation is made over repeated indices. Note that Gij/x

l = -G

ij /x

l and, therefore, Eq. (14) can also be

written with the plus sign before the integral, bearing in mind the differentiation over the primed coordinatex

l. Taking into account the relations n

mn

* = n

nm

* =1/2 ( n

mn

* + n

nm

* )= n

nm

*

( ) and C

jlmn=C

jlnm, the substitution

n

mn

* (R’ ) n

mn

* (R’) in Eq. (14) can be done. Here, the upper left index n indicates the dimension of thedefect.

For an isotropic 3D elastic medium, the Green function and term Cjlmn

Gij,k

(|R-R’|) are expressed in theforms [10]:

i i j j

ij ij

x x x xG

G 2

( )( )1(3 4 ) ,

16 (1 )

R RR R R R

(15a)

j mn ij

mi n n ni m m mn i i n n m m i i

C G

x x x x x x x x x x x x

,

3 5

1

8 (1 )

( ) ( ) ( ) ( )( )( )(1 2 ) 3 .

l lR R

R R R R

(15b)

Here x x y y z z2 2 2( ) ( ) ( ) R R ; xn, x

m, x

i are x, y,, and z;

km is the Kronecker delta, G is the

shear modulus, and is the Poisson ratio.Equations (14) and (15) are convenient for calculating the displacements of point defects. For defects of

higher dimension, the formulas for displacements can be simplified by applying to them the direct andinverse Fourier transformations [22]. As a result, the universal working formula relating the self-distortion ofthe defect with its displacement field, takes the form [22]:

t n

i j mn ij mn x y zu i C G i d d dˆ ˆ( ) exp( ) ,

l lR ξ R (16)

where ij

G and n

ij

* are Fourier transforms of the Green function of the elastic medium and the self-distortion(eigenstrain) of the defect, respectively; R=

xx+

yy+

zz.

For an isotropic medium, the terms of Eq. (16) are known [22]:

j mn j mn m jn n jm

GC G

2,

1 2

l l l l (17a)

ij i j

ijG

G

2

3/2 4

2(1 )1ˆ ,(2 ) 2(1 )

(17b)

where 2= 2x+2

y+2

z.

Thus, the way to find the elastic fields of an arbitrary defect is simple: one should take its eigenstrainand apply a well known formula, e. g., Eq. (14) or Eq. (16), to it.

Sometimes, however, it is easier to act in a different manner, by integrating the field of one defect oflower dimension to get the field of a defect of higher dimension. For example, based on of the field of thepoint dilatation center given by Eq. (14), we find the fields of other dilatation sources in the form of segment,straight line, circular line, circular disk, and axisymmetric inclusions [21,25]. In doing so, first, we find the


fields of an infinitesimal prismatic dislocation loop(IPDL) and compare them with the fields representedby Kroupa [5].

3.1. Infinitesimal prismatic dislocationloop (IPDL)

Consider an IPDL placed at the beginning of theCartesian coordinate system and characterized bythe following eigenstrain:

IDPL IDPL

zz zzbs0 * 0 * ( ). R (18)

The field of total displacements of this loopu

iIPDL(R), calculated by using Eqs. (14), (15), and

(18), reads:

IPDL

x

bs x zu

2

3 2(1 2 ) 3 ,

8 (1 ) | | | |

R R

(19a)

IPDL

y

bs y zu

2

3 2(1 2 ) 3 ,

8 (1 ) | | | |

R R

(19b)

IPDL

z

bs z zu

2

3 2(1 2 ) 3 ,

8 (1 ) | | | |

R R

(19c)

where x y z2 2 2 R .The displacements allow us to calculate the elas-

tic strain field ij = 1/2(u

j /x

i + u

i /x

j) - n

ij

* , andthen, through the Hooke law,

ij = 2G(

ij + (/(1-

2)ij), =

xx+

yy+

zz, i,j= x,y,z, the stress field:

IPDL

xx

Gbs

x z x z

3

2 2 2 2

2 4

3

4 (1 ) | |

4 1(1 2 ) 5 ,

3 | | | |

R

R R

(20a)

IPDL

yy

Gbs

y z y z

3

2 2 2 2

2 4

3

4 (1 ) | |

4 1(1 2 ) 5 ,

3 | | | |

R

R R

(20b)

IPDL

zz

Gbs

z z

3

2 4

2 4

3

4 (1 ) | |

12 5 ,

3 | | | |

R

R R

(20c)

IPDL

xy

Gbs xy z2

5 2

31 2 5 ,

4 (1 ) | | | |

R R

(20d)

IPDL

xz

Gbs xz z2

5 2

31 5 ,

4 (1 ) | | | |

R R

(20e)

IPDL

yz

Gbs yz z2

5 2

31 5 .

4 (1 ) | | | |

R R

(20f)

The calculated fields (19) and (20) are in accord-ance with the fields given in [5]. We note that inorder to find the elastic strains, we differentiatedthe total displacements (19), ignoring the singular-ity point (0,0,0).

3.2. Center of dilatation (P)

Combining the fields of three mutually perpendicu-lar IPDLs, given by Eqs. (19) and (20), we find thefields of the point dilatation source (Fig. 2) as:

P

x

bs xu

3

(1 ),

4 (1 ) | |

R (21a)

P

y

bs yu

3

(1 ),

4 (1 ) | |

R(21b)

P

z

bs zu

3

(1 );

4 (1 ) | |

R(21c)

P

xx

G b zs x y5

2 2 2(1 ) ( 2 ),

2 (1 | |)

R(22a)

P

yy

G b x y zs 2

5

2 2(1 ) ( ),

2 (1

2

| |)

R(22b)

P

xx

G b x ys z2

5

2 2(1 ) ( 2 ),

2 (1 |) |

R(22c)

P

xy

sG b xy5

3 (1 ),

2 ( | |1 )

R(22d)

P

xy

sG b xz5

3 (1 ),

2 ( | |1 )

R(22e)

P

xy

sG b yz5

3 (1 ),

2 ( | |1 )

R(22f)

P P P

xx yy zz0. (22g)


3.3. The dilatation segment (L), the dilatation ray (Li), and the dilatationalstraight line (Line)

Integrating the fields of the dilatation center along the straight line within the given limits, we obtain the fieldsof the dilatational segment. For example, the stress field of the segment oriented along the z-axis is:

z

L P

ij ij

z

x y z z z2

1

0 0( , , ) d , (23)

where is the linear density of the distribution of point defects. The stress components (23) have ananalytic representation via elementary functions.

From the stress field (23) of a segment, with limiting transition z2 we find the stress field of the

dilatation ray emerging from the coordinate origin (0,0,0). For z1=0 it reads:

Li

xx

G b x y z x x y z y y

y

s z

x x y

2 2 4 2 2 2 2 2 2

2 2 2 3 2 2 2

(1 ) (2 ( ) ( ))

|,

2 (1 ) ( ) (| )

R

(24a)

Li

yy

G b x z y y x zs x x

x y x y

y z2 2 4 2 2 2 2 2 2

2 2 2 3 2 2 2

(1 ) (2 ( ) ( ))

|,

2 (1 ) ( ) (| )

R

(24b)

Li

zz

G bs z3

(1 ),

2 (1 ) | |

R(24c)

Li

xy

G b xy xyz x y z

x y

s

y x

2 2 2

2 2 2 2 2 2 3

(1 ) 2 (3

|

3 2 ),

2 (1 ) ( ) |( )

R

(24d)

Li

xz

G bs x3

(1 ),

2 (1 ) | |

R(24e)

Li

yz

G bs y3

(1 ).

2 (1 ) | |

R(24f)

In addition, from the fields (24) of the segment, in the limiting transitions z1- and z

2, one can find

the stress field of an infinite dilatational straight line directed along the z-axis:

Line

xx

G bs x y

x y

2 2

2 2 2

(1 ) ( ),

(1 ) ( )

(25a)

Line

yy

G bs y x

x y

2 2

2 2 2

(1 ) ( ),

(1 ) ( )

(25b)

Line

xy

G bs xy

x y2 2 2

(1 ) 2,

(1 ) ( )

(25c)

Line Line Line

zz xz yz0. (25d,e,f)

In the cylindrical coordinate system (r,,z), the stress tensor (25) can be written as

Line Line

rr

G bs

r 2

(1 ) 1,

(1 )

(26a,b)


Line Line Line Line

zz r rz z0.

(26c-f)

In Eqs. (26 a,b), we can introduce a cutoff ra-dius around the singular point r=0 and rewrite thecoefficient at 1/r2 as

V

V V

V

G bs G

G G2 2

2

(1 ) (1 )

(1 ) 3 (1 )

(1 ) (1 ),

3(1 ) 3(1 )

thus finding the stress field caused by a cylindricalinclusion with a small core. This inclusion simu-lates an actual inclusion of cylindrical shape in thecase when the inclusion radius is small as compar-ing with other size parameters of the problem. Sucha model is adopted, for example, when consideringthe elastic behavior of the Abrikosov flux line in TypeII superconductors [26]. The Abrikosov flux line is acylindrical region of the normal phase, surroundedby a superconducting phase. A similar model wasused to model the phason imperfections inquasiperiodic grain boundaries in crystalline solids[27].

Fig. 5. From a point defect to an axisymmetric inclusion. (a) The point defect (red); (b) the circular line;(c) the circular disk, and (d) the axisymmetric inclusion.

3.4. Circular dilatation line (CL)

Egenstrain tensor of the dilatation center [see Eq.(4)] in the cylindrical coordinate system (r,,z) hasthe form:

rr zzs

s

b x x y y z

b r r zr

0 * 0 * 0 *

0 0

0 0

0

( ) ( ) ( )

1( ) ( ) ( ),

(27)

where (x0,y

0,0) and (r

0,

0,0) are the dilatation center

coordinates in the Cartesian and cylindrical coordi-nate systems, respectively (Fig. 5a).

Let us distribute the point centers of the dilata-tion along a circle of radius r

0 with a constant linear

density (Fig. 5b). As a result, we obtain a circulardilatation line (CL). In the cylindrical coordinatesystem (r,,z), the eigenstrain tensor of CL is writ-ten as:

rr zzs

s

b r rr

z r b r r z

2

1 * 1 * 1 *

0

0 0

0 0 0 0

1( )

( ) ( ) d ( ) ( ).

(28)


The displacements of the circular dilatation line canbe calculated by integrating the displacements (21)of the dilatation center with the coordinates (r

0,

0,0)

along the circle, the element of which is dl=r0d

0,

from 0 to 2 with the weight :

C L P

i iu u r i r z

2

0 0

0

d , , , .

(29)

The integration gives

C L

r

r kbsu

r r k

k r r z k rr k k

0

5/2 3 /2 2

0

2 2 2 2 2

0 0

(1 )

2 (1 ) 8 ( ) (1 )

( ) ( ) 4( )(1 ) ( ) ,

E K (30a)

C Lu 0,

(30b)

C L

z

s r z kbu k

k rr

3

0

2 3/2

0

(1 )( ),

2 (1 ) 4(1 )( )

E (30c)

where

tk

k t

k k t t

2

2 20

2

2 2

0

d( ) and

1 sin

( ) 1 sin d

K

E

are the complete elliptic integrals of the first andsecond kind [28], respectively, and

r rk

z r r

0

2 2

0

2.

( )

Displacements (30) can be rewritten in terms ofthe Lipschitz-Hankel integrals

z r pr

m n rJ m n p J J e 0

0

| | /

0

( , ; ) ( ) ( ) d

[29], where Jn() are the Bessel functions of the first

kind [28], as follows:

C L

r

bsu J

r0

(1 ) 1(0,1;1),

2(1 )

(31a)

C Lu 0,

(31b)

z

sb zu J

r

C L

0

(1 ) sgn( )(0,0;1),

2(1 )

(31c)

where sgn(z)={1, z>0; 0, z=0; -1, z<0}.In Fig. 6, the field of total displacements for a

circular dilatation line is shown.The displacements (31) give a possibility to find

the elastic strains ij and stresses

ij in the cylindri-

Fig. 6. Maps of displacements of a circular dilatation line. The maps are built in the plane x=0. The dis-placement values are given in units of ((1+)bs)/(2(1-)r

0) and the coordinates are normalized to the circle

radius r0. Here is the density of the dilatation centers that form the circular line; b and s are the values of

the small Burgers vector and the area of infinitesimal prismatic loops forming the dilatation center, respec-tively; is the Poisson ratio.


cal coordinate system by using the following equa-tions taking into account eigenstrain n

ij

* [18]:

nr

rr rr

u

r

* ,

(32a)

nru u

r r

* ,

(32b)

nz

zz zz

u

z

* ,

(32c)

z r

rz

u u

r z

1,

2

(32d)

r

r

u uu

r r r

1 1,

2

(32e)

z

z

uu

r z

1 1,

2

(32f)

and the Hooke law ij = 2G(

ij+(/(1-2))

ij), i,j =

r,,z [30], where =rr+

jj+

zz.

Here, for CL n

ij

* =ij

1 * , see Eq. (28).The resulting stresses of CL satisfy the equi-

librium equations.

Fig. 7. Maps of displacements of a circular dilatation disk located in the xy-plane. The maps are built in theplane x=0. Displacements are given in units of ((1+)b)/(2(1-)) and the coordinates are normalized to thedisk radius a. Here b is the value of the small Burgers vector of infinitesimal prismatic loops forming thedilatation centers which in turn form the dilatation disk; is the Poisson ratio.

3.5. Circular dilatation disk (CD)

The elastic fields of a circular dilatational disk (CD)can be calculated by integrating the elastic fields ofa CL along the radius r

0 from 0 to a with a constant

density (Fig. 5c).The displacements can be written as:

a

C D C L

i iu u r i r z

0

0

d , , , (33)

that gives

r

bu JCΔD (1 )

(1,1;0),2(1 )

(34a)

C Du 0,

(34b)

CΔD

z

bu z J

(1 )sgn( ) (1,0;0).

2(1 )

(34c)

In Fig. 7, the maps of total displacements for acircular dilatation disk are shown.

The elastic strains ij and stresses

ij of a CDD

can be calculated by Eqs. (32) and the Hooke lawtaking into account n

ij

* =ij

2 * ,:

rr zz

rb H z

a

2 * 2 * 2 * (1 ) ( ),

(35)


where H(1-r/a) is Heaviside step function.

3.6. Axisymmetric dilatational inclusions

With continuously distribution of the dilatational disks along the z-axis, we can construct axisymmetricinclusions (Fig. 5d) and calculate their elastic fields [25]. For example, eigenstrain and the displacementfield of a cylindrical dilatational inclusion of finite height can be written as:

ii

3 * * 1, inside inclusion ;0, outside inclusion

(36)

inside inclusion (r<c, -h/2<z<h/2):

in

r

c ru J J

c

*

(1) (2)(1 )- (1,1;-1)- (1,1;-1) ,

2(1- )

(37a)

inu 0, (37b)

in

z

cu z J J

*

(1) (2)(1 )sgn( ) (1,0;-1)- (1,0;-1) ;

2(1- )

(37c)

outside inclusion (z>h/2, or z<-h/2, or –h/2zh/2 and r>c):

h h

out

rc h h

r

J J z zc

uJ J z r c

(3) (2)* 2 2

(1) (2)

2 2

(1,1;-1)- (1,1;-1) , or(1 )

,2(1- ) - (1,1;-1)- (1,1;-1) , and

(38a)

outu 0,

(38b)

h h

out

zh h

J J z zc

u zJ J z r c

(3) (2)* 2 2

(1) (2)

2 2

(1,0;-1)- (1,0;-1) , or(1 )

sgn( ) .2(1- ) (1,0;-1)- (1,0;-1) , and

(38c)

Here the Lipschitz-Hankel integrals have the following forms:

hp

m n

zrJ m n p J J d

c c1(1) 2

1

0

- | |( , ; )= ( ) ( )e , ,

(39a)

hp

m n

zrJ m n p J J d

c c2(2) 2

2

0

- | |( , ; )= ( ) ( )e , ,

(39b)

hp

m n

zrJ m n p J J d

c c3(3) 2

3

0

- | |( , ; )= ( ) ( )e , ,

(39c)

Note that 1>0 for –h/2zh/2,

2>0 for any z, and

3>0 for z>h/2 or z<-h/2; c and h are the radius and the

height of the cylinder, respectively.Fig. 8 shows the field of total displacements of the cylindrical dilatational inclusion in the central longi-

tudinal section x = 0. The field of elastic strains is determined from the field of total displacements given byEqs. (37)-(38) and relations (32), when

n

ii ii

* 3 * * 1, inside inclusion .0, outside inclusion


The dilatation of cylindrical inclusion can alsobe found from Eqs. (32) with taking into accountEqs. (37)-(38):

in in in

in in in in r r z

rr zz

u u u

r r z

*

*

* *

3

(1 ) 2(1-2 )3 ,

(1- ) (1- )

(40a)

out out out out

rr zzout out out

r r zu u u

r r z0.

(40b)

It follows from Eqs. (40 a,b) that the dilatation isconstant inside the inclusion and zero outside it.Analytical calculations show that hemispherical dila-tational inclusion has the same dilatation [25].

The stress field can be derived by using the Hookelaw for isotropic media, see Ref. [25] for details. Itis easy to show that the stress fields of theaxisymmetric inclusions under consideration sat-isfy the system of equilibrium equations [25].

The strain energy of dilatational inclusions isdetermined by formula [10]:

incl

in

ii ii incl

V

GE dV E V

*2

* 2 (1+ )1,

2 (1- )

(41)

Fig. 8. Maps of the displacements of a cylindrical dilatational inclusion: (a) radial displacement ur, (b) axial

displacement uz. The inclusion axis coincides with the z-axis. The displacement maps are built in the

central longitudinal section x = 0 of the inclusion. Displacement values are given in units of ((1+)*c)/(1-),where * is the eigenstrain of the inclusion, c is the inclusion radius, is the Poisson ratio.

where Vincl

is the inclusion volume.Our direct analytical calculations demonstrated

the independence of the strain energy on the shapeof the inclusion with dilatational eigenstrain, whichalso confirms the correctness of the solutions found.

4. DEFECTS IN 2D CRYSTALS

Two-dimensional crystals are films whose thicknesscan be neglected in the problem under considera-tion. Among them, one can distinguish crystallinecarbon films – graphenes [31,32], and carbon poly-hedral shells – fullerenes [33,34]. From the point ofview of continuum mechanics, such objects are 2Delastic media. A special feature of a 2D medium isits ability to have the nonzero curvature. In this way,the film reacts to external influences or to the ap-pearance of defects in it [35-37].

We can apply the classification of defects in a2D elastic medium based on the dimension of theregion, where the self-distortions (eigenstrains) ofthe defects are specified, in the same way as it isdone for defects in a 3D medium, see Part 2. Ac-cording to the dimensional principle, defects in a2D medium are subdivided into 3 types (Fig. 9).

Below, we denote the self-distortion of a defectas m n

ij

| * , where m is the dimension of the mediumand n is the dimension of the defect.


Fig. 9. Defects in 2D crystals and their elastic models. (a) Zero-dimensional defects are vacancy, self-interstitial, substitutional impurity, and interstitial impurity. (b) One-dimensional defects are dislocation,disclination, and misorientation boundary. (c) Two-dimensional inclusion. The main characteristic of a de-fect is its self-distortion n

ij

2| * , where n is the defect dimension and i, j = x, y, adapted from [38].

4.1. Classification of defects in 2Dcontinuum

Zero-dimensional (point) defects (n=0). First, letus define an infinitesimal dislocation loop which isan elementary zero-dimensional defect in both the2D and 3D media [38]. In the film lying in the xy-plane, the self-distortion of the infinitesimal disloca-tion loop becomes:

ij j ib l i j x y2|0 *

0( ), , , r r (42)

where bj is the dislocation Burgers vector; r

0 is the

coordinate of the loop on the xy-plane, li is the seg-

ment, which is an analog of the area si for an infini-

tesimal dislocation loop in the 3D medium, see Eq.

(1). The two-dimensional Dirac delta function is givenby the relation (r-r

0) = (r-r

0)(y-y

0).

Two mutually perpendicular infinitesimal dislo-cation loops form a dilatation center with the follow-ing distortion:

iib s i j x y2|0 *

0 0

1( ) ( ), , , ,

2 l r r r r (43)

Such a center is an elastic model of an impurityatom or a vacancy (Fig. 9a). If l the initial lineardimension of the loop, a change in the linear dimen-sion by an amount will lead to a change in the areaby an amount. Assuming that s=(l+l)2-l2 2lland b=l, we obtain the last equality in Eq. (43).


With a formal approach, the self-distortion of apoint defect in a film can be written:

ij ijs i j x y2|0 * *

0( ), , , , r r (44)

where s is a formal dimensional factor.One-dimensional defects (n=1). The self-dis-

tortion of a one-dimensional defect located, for ex-ample, on a segment [y

1,y

2] having a coordinate x

0,

reads:

ij ijx x H y y y2|1 * *

0 1 2( ) ( ), l (45)

where ij

* is the plastic distortion of the line seg-ment; l is a linear factor, H(y

1yy

2) is the Heaviside

function.In the film, the Somigliana dislocations are one-

dimensional defects, and the Volterra dislocationsand disclinations are degenerate one-dimensionaldefects (Fig. 9b).

One can rewrite the distortion of the dislocationshown in Fig. 9b, to the form typical for Volterradislocations [38]:

xx x

yx x

b x x H y yb x x H x x

2|1 *

0 02|1 *

0 0

( ) ( )or ( ) ( ).

(46)

A linear defect can be represented through pointdefects continuously distributed with some lineardensity along a line.

Two-dimensional defects (n=2). The area ofdetermining the self-distortion of two-dimensionaldefects is a part of the surface. In a film, two-dimen-sional defects are similar to inclusions in a 3D me-dium (Fig. 9c).

For a two-dimensional defect of general form,the following expression is true:

inclij ij

incl

SS

2|2 * * 1,.

0,

rr (47)

A two-dimensional defect can also be repre-sented through distribution of point defects over thesurface S

incl.

It is worth noting that in a 2D continuum, de-fects of dimension lower than 2 are described bydistortions which contain some arbitrary dimensionalfactors as is the case with defects of dimensionlower than 3 in a 3D medium.

4.2. Elastic fields of defects in 2Dplanar media

In the case of 2D elastic planar medium, the self-distortion n

ij

2| * (or eigenstrain n

ij

2| * ) allows to find,

by using the Green function 2Gij and elastic mod-

ules Cjlkm

, the total displacements of defects [10]:

t n

i jlkm km ij l

S

u C G S2 2| * 2

,

'

( ) ( ) (| |)d ', R R r - r (48)

where the left upper index 2 indicates the dimen-sion of the medium.

For the isotropic elastic medium, the Green func-tion 2G

ij and the elastic modules C

jlkm are [10]:

ij

i i j j

ij

GG

x x x xr

r

2

0

2

1(| |)

8 (1 )

( )( )(3 4 ) ln ,

r r

(49a)

jlkm jl km lk jm lm jk

GC G

2( ),

1 2

(49b)

where km

r x x y y2 2 2( ) ( ) , is the Kroneckerdelta, G is the shear modulus, and is the Poissonratio.

Since in our case the elastic space is a film withfree surfaces, which lies in the xy-plane, the stresstensor does not contain the components

zj. This

means that we deal with the plane-stress state andhence in both the Green function and Hooke law,the Young modulus E and the Poisson ratio shouldbe replaced by ratios E(1+2)/(1+)2 and /(1+),respectively [10]. Since the shear modulus G=E/[2(1+)], the replacement procedure does not af-fect it. In the transition from the 3D case to the 2Dcase, the unit of measurement of the Young andshear moduli change from N/m2 to N/m.

Using Eqs. (48) and (49), one can define thefields of an infinitesimal dislocation prismatic loopand a dilatation center.

4.2.1. Infinitesimal prismaticdislocation loop (IPDL) anddilatation center in film

The field of total displacements 2uiIPDL(x,y) of an IPDL,

placed in the origin of Cartesian coordinates (0, 0),is calculated from Eqs. (48) and (49) with replace-ment of the Poisson ratio by the expression /(1+) and results in

IPDL x

x

b xu y x

r

2 2 2

4( 1) (3 ) ,

4

l (50a)

IPDL x

y

b xu y x

r

2 2 2

4( 1) (3 ) ,

4

l (50b)


where r2=x2+y2 and bx is the value of the IPDL Burg-

ers vector.The displacements (50) make it possible to find

the elastic strains, and then, according to the Hookelaw, the elastic stresses

IPDL x

xx

G bx x y y

r

2 4 2 2 4

6

(1 )3 6 ,

2

l (51a)

IPDL x

yy

G bx x y y

r

2 4 2 2 4

6

(1 )6 ,

2

l(51b)

IPDL x

xy

G bx y

r

2 2 2

6

(1 )3 ,

2

l(51c)

Let us consider now the field of a dilatation centercharacterized by the self-distortion

P P

xx yys x y2|0 * 2|0 * * ( ) ( ). (52)

The displacement and stress fields of the dilatationcenter are:

P

x

s xu

r

*

2

2

(1 ),

2

(53a)

P

y

s yu

r

*

2

2

(1 ),

2

(53b)

P

xx

G s x y

r

* 2 2

2

4

(1 ) ( ),

(54a)

P

yy

G s x y

r

* 2 2

2

4

(1 ) ( ),

(54b)

P

zz

G s xy

r

*

2

4

2 (1 ),

(54c)

P P

xx yy

2 2 0. (54d)

The functional parts of the stress field (54) coincidewith those of the stress field caused by a dilatationline in the 3D medium, see Eqs. (25).

After replacement s=bxl in Eqs. (53) and (54),

it becomes clear that the dilatation center in the 2Dmedium is the sum of two mutually perpendicularIPDLs, see Eqs. (50) and (51), just as the dilatationcenter in the 3D medium is the sum of three mutu-ally perpendicular IPDLs.

It is also worth to note, that the fields of a biaxialdilatation line with self-distortion in the 3D medium

L L

xx yys x y3|1 * 3|1 * * ( ) ( ) precisely coincide with

fields (53) and (54) when replacing by /(1+).It is clear that defects causing a plane strain in a3D medium have their counterparts of smaller di-mension in the film. In this case, the fields of thesedefects in the film transform to those of their coun-terparts in the 3D crystal, when the elastic moduliare replaced in the appropriate way, and the condi-tion

zz=0 is taken into account.

4.2.3. Dislocation and wedgedisclination in film

Straight-line edge dislocations and wedgedisclinations in 3D media generate plane strainstates, see, for example, [7,39]. Therefore, the maindifference between the stress fields of these defectsin a film and in an infinite 3D medium is the ab-sence of the

zz component. The other stress com-

ponents of dislocations and disclinations coincidewith the accuracy to the factors associated with .

The solutions of the boundary-value problems foredge dislocations and wedge disclinations perpen-dicular to the free surfaces of a plate having a finitethickness, are presented in [11,40]. In particular, itwas shown in [41] that when the plate thicknesstends to zero, the elastic field generated by the edgedislocation becomes as that in the plane stressstate.

Notice that the disclination dipole is a charac-teristic element of grain boundaries in graphene (Fig.9b), see, for example, [36,37]. Using the rows ofdisclination dipoles with their elastic fields, it waspossible to calculate the strain energy of grainboundaries in graphene without exploiting the com-puter simulation [41].

4.2.4. Dilatation inclusion in the film

An example of a dilatation inclusion in a 2D me-dium is shown in Fig. 9c. On the base of the stressesof a biaxial cylindrical inclusion in a 3D mediumhaving eigenstrain

inclxx yy

incl

*3 * 3 * , ,

0,

RR

the stresses of circular dilatational inclusion in 2Dcrystal with eigenstrain

inclxx yy

incl

SS

*2|2 * 2|2 * , ,

0,

rr


can be written, replacing by /(1+) and neglect-ing

zz [21]:

incl

xx

x yr a

rG a

r aa

2 2

42 * 2

2

( ),

(1 ) ,2

,

(55a)

incl

yy

x yr a

rG a

r aa

2 2

42 * 2

2

( ),

(1 ) ,2

,

(55b)

incl

zz

xyr aG a r

r a

2 * 24

2,(1 ) ,

0,

(55c)

Here a is a radius of the inclusionIf we introduce the radius of the core for the bi-

axial dilatation center a and determine the coeffi-cient s in Eq. (54) as a2, then the stresses of dila-tation center are coincide with the stresses gener-ated by the circular biaxial inclusion in the surround-ing matrix r>a (55).

5. SUMMARY AND CONCLUSIONS

Classification of defects of an elastic continuum,presented in the paper, is based on the dimensionof the region, where the self-distortion or eigenstrainof the defect is localized. It allows to determine theplace of structural defects – inclusions, dislocationsand disclinations – in the hierarchy of defects of theelastic continuum. For example, if the scale param-eters of the elastic problem are much larger thanthe size of the defect, one can operate with point orone-dimensional defects.

We have discussed how to create defects ofhigher dimension from defects of smaller dimension.On the base of the elastic fields of the infinitesimaldislocation loops, the fields of the circular dilatationline, the circular dilatation disk, and finite cylindri-cal and hemispherical inclusions are derived in astep-by-step manner.

We have also presented the elastic models ofdefects in flat 2D crystals. Using the self-distortion,the elastic fields of an infinitesimal dislocation loopand a dilatational center in the film are calculated. Itis shown that the defects creating a plane-strainstate in a 3D elastic medium have their counter-parts in a 2D medium.

If to consider the theory of defects wider, we canextend it beyond the framework of the elastic field

and apply it to the fields of different physical nature:electromagnetic and gravitational. Based on theknown field equations with a known Green functionand specifying a counterpart of the eigenstrain, forexample, the density of electric charge and the den-sity of mass, etc., it is possible to calculate thefields of defects of any dimension and, in particular,to take into account the time coordinate. The for-malized approach allows not only to calculate thefields of defects, but also makes it possible to findparallels in the behavior of defects of different na-ture.

ACKNOWLEDGEMENTS

Authors thank the Ministry of Education and Sci-ence of the Russian Federation for its support(Project No 3.3194.2017/4.6).

REFERENCES

[1] V. Volterra // Annales Scientifique de l’ÉcoleNormale Supérieure. Paris 24 (4) (1907) 401.

[2] J.D. Eshelby // Proceedings of the RoyalSociety of London A 241 (1957) 376.

[3] J.D. Eshelby, Continual Theory of Dislocations(Publisher of Foreign Literature, Moscow,1963), In Russian.

[4] Collected Works of J.D. Eshelby. TheMechanics of Defects and Inhomogeneties,ed. by Xanthippi Markenscoff and AnuragGupta (Springer, Notherlands, 2006).

[5] F. Kroupa, In: Theory of Crystal Defects.Proceedings of the Summer School held inHrazany in September 1964 (Academia -Publishing House of the CzechoslovakAcademy of Sciences, Prague, 1966), p. 276.

[6] V.L. Indenbom, In: Theory of Crystal Defects.Proceedings of the Summer School held inHrazany in September 1964 (Academia -Publishing House of the CzechoslovakAcademy of Sciences, Prague, 1966), p. 257.

[7] R. De Wit, Continual Theory of Disclinations(Mir, Moscow, 1977), In Russian.

[8] J.P. Hirth and J. Lothe, Theory of dislocations(John Wiley and Sons, New York/ Chichester/Brisbane/ Toronto/ Singapore, 1982).

[9] C. Teodosiu, Elastic Models of Crystal Defects(Springer-Verlag, 1982).

[10] T. Mura, Micromechanics of Defects in Solids(Martinus Nijhoff Publishers, Dordrecht/Boston/Lancaster, 1987).

[11] A.L. Kolesnikova and A.E. Romanov, Circulardislocation-disclination loops and their


application to boundary problem solution inthe theory of defects (Phys.-Tech. Inst.,Preprint No 1019, Leningrad, 1986),In Russian.

[12] A.Yu. Belov, In: Elastic Strain Fields andDislocation Mobility, ed. by V.L. Indenbomand J. Lothe (Elsevier, 1992).

[13] A.L. Kolesnikova and A.E. Romanov //Journal of Applied Mechanics 71 (3) (2004)409.

[14] N. Louat // Nature (London) 196 (4859) (1962)1081.

[15] M.J. Marcinkowski, Unified Theory of theMechanical Behavior of Matter (John Wileyand Sons, New York, 1979).

[16] A.M. Korsunsky // Journal of Elasticity 44(1996) 97.

[17] R.J.H. Paynter, D.A. Hills and A.M.Korsunsky // International Journal of Solidsand Structures 44 (2007) 6653.

[18] Yu.N. Rabotnov, Mechanics of the DeformedSolids (Nauka, Moscow, 1979), In Russian.

[19] R.J. Asaro and V.A. Lubarda, Mechanics ofSolids and Materials (Cambridge UniversityPress, New York, 2006).

[20] P.P. Groves and D.J. Bacon // Journal ofApplied Physics 40 (1969) 4207.

[21] A.L. Kolesnikova, R.M. Soroka and A.E.Romanov // Materials Physics and Mechanics17(1) (2013) 71.

[22] T. Mura, In: Advanced in Materials Research,ed. by H. Herman (Interscience Publ., NewYork, 1968), Vol. 3, p. 1.

[23] T. Mura // Archives of Mechanics 24 (3)(1972) 449.

[24] A. Howie and L.D. Marks // PhilosophicalMagazine A 49 (1984) 95.

[25] A.L. Kolesnikova, M.Yu. Gutkin and A.E.Romanov // International Journal of Solidsand Structures (2017) (submitted).

[26] A.M. Campbell and J. Evetts, CriticalCurrents in Superconductors (Taylor& Francis Ltd, United Kingdom, 1972).

[27] M.Yu. Gutkin and I.A. Ovid’ko // TheEuropean Physical Journal B 1(4) (1998) 429.

[28] E. Jahnke, F Emde and F. Losch, Tafelnhöherer Funktionen (Stuttgart, Teubner,1960).

[29] G. Eason, B. Noble and I.N. Sneddon //Philosophical Transactions of the RoyalSociety of London 247 (935) (1955) 529.

[30] A.I. Lur’e, Spatial Problems of Theory ofElasticity (State Publishing House ofScientific and Technical Literature, Moscow,1955), In Russian.

[31] K.S. Novoselov, D. Jiang, F. Schedin, T.J.Booth, V.V. Khotkevich, S.V. Morozov andA.K. Geim // Proceedings of the NationalAcademy of Sciences of the United States ofAmerica 102 (2005) 10451.

[32] M.I. Katsnelson, Graphene: Carbon in TwoDimensions (New York: Cambridge UniversityPress, 2012).

[33] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F.Curl and R.E. Smalley // Nature 318 (1985)162.

[34] J. Baggott, Perfect Symmetry: TheAccidental Discovery ofBuckminsterfullerene (Oxford UniversityPress, 1995).

[35] A.L. Kolesnikova and A.E. Romanov //Physics of the Solid State 40(6) (1998) 1075.

[36] B.I. Yakobson and F. Ding // ACS NANO5 (2011) 1569.

[37] J. Zhang and J. Zhao // Carbon 55 (2013)151.

[38] A.L. Kolesnikova, T.S. Orlova, I. Hussainovaand A.E. Romanov // Physics of the SolidState 56(12) (2014).

[39] A.E. Romanov and V.I. Vladimirov, In:Dislocations in Solids, ed. by F.R.N. Nabarro(North-Holland, Amsterdam, 1992), Vol. 9,p. 191.

[40] A.L. Kolesnikova and A.E. Romanov //Technical Physics Letters 3(6) (1987) 272.

[41] A.E. Romanov, A.L. Kolesnikova, T.S. Orlova,I. Hussainova, V.E. Bougrov and R.Z. Valiev //Carbon 81(1) (2015) 223.

elastic models of defects in 3d and 2d crystals · elastic models of defects in 3d and 2d crystals...

Documents