engineering fracture mechanics - apmath.spbu.ru filea periodic set of edge dislocations in an...

Engineering Fracture Mechanics 186 (2017) 423–435

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Engineering Fracture Mechanics

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A periodic set of edge dislocations in an elastic semi-infinitesolid with a planar boundary incorporating surface effects

https://doi.org/10.1016/j.engfracmech.2017.11.0050013-7944/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (M.A. Grekov).

M.A. Grekov ⇑, T.S. Sergeeva, Y.G. Pronina, O.S. SedovaDepartment of Computational Methods in Continuum Mechanics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 July 2017Received in revised form 1 November 2017Accepted 3 November 2017Available online 4 November 2017

Keywords:Edge dislocationsPoint forcesGreen functionsSurface stressNanomechanics

The 2-D problem of interacting periodic set of edge dislocations and point forces with pla-nar traction-free surface of semi-infinite elastic solid at the nanoscale is considered.Complex variable based technique and Gurtin-Murdoch model of surface elasticity, whichleads to the hypersingular integral equation in surface stress, are used. The solution of thisequation and explicit formulas for stress field (Green functions) are obtained in terms ofFourier series. The detailed numerical investigation of stress field induced by the disloca-tions at the nanometer distance from the surface and the force acting on each dislocationin classical and non-classical (with surface stress) solutions is presented. It is shown thatformulas derived for the periodic set of dislocations can be applied to the analysis of theinteraction of a single dislocation with the surface as well. The fundamental solutionsobtained in the work can be used for applying the boundary integral equation method toan analysis of defects such as cracks and inhomogeneities, periodically distributed at thenanometer distance from the boundary.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The study of elastic fields induced by dislocations located in the vicinity of a traction-free surface or interface is one of thesignificant parts of dislocation theory [1,2]. A lot of analytical solutions concerning the interaction of dislocations with sur-face/interface have been obtained within the framework of the classical theory of elasticity. Head [3] seems to be the firstone who gave such a solution for the edge dislocation parallel to the interface between two semi-infinite isotropic mediawith different elastic properties. This solution included the free surface as a limiting case. The peculiar method to obtain clas-sical solutions for the edge dislocations, based on analytical expressions for straight twist disclinations in semi-infinitemedia, has been addressed in [4,5]. Boundary value problems on the edge dislocations inside and outside of the circular[6,7] and elliptical [8,9] inclusions, near the free planar surface of a semi-infinite medium [10] and a planar interface[11], near a thin surface layer [11] and inside it [12] have been solved. In works [10,11], the stress and displacement fieldsarising due to the presence of radular arrays of edge dislocations and point forces near a planar surface or interface have beenobtained in a closed analytical form. Apart from these works, an interaction of an array of dislocations with a surface/inter-face has been studied in [13–19]. Such solutions make it possible to examine the influence of dislocations, which are the realdefects of crystalline materials, on mechanical and physical properties of these materials. For example, the rate of the

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424 M.A. Grekov et al. / Engineering Fracture Mechanics 186 (2017) 423–435

mechanochemical corrosion depends directly on the stress state of a traction-free surface [20,21] that can be produced bydislocations accumulating near the surface.

The value of the analytical elastic solutions for dislocations (and for point forces as well) is that they are fundamental(Green functions) and so are used for the analysis of different fracture mechanics and inhomogeneity problems with theboundary element method [22] involving the construction of a boundary integral equation [11,23,24]. The application ofthe dislocation theory to fracture modeling was discussed in [25].

In most theoretical descriptions, dislocations are attracted to the boundaries of elastically softer inhomogeneities, includ-ing the traction-free boundaries of a solid. So, dislocations can arise in a subsurface layer of a nanometer thickness. In thiscase, the classical continuum mechanics needs to be modified to account for the surface energy effects that are intrinsic tothe nanomaterials having at least one dimension in the range of 1–100 nm (nano-sized beams, plates, wires and films[26–29,31–33]) and nanostructures with nano-sized inhomogeneities (inclusions, voids, cracks, etc. [34–42]). The conceptof surface energy and surface stress in solids was first introduced by Gibbs [43] on the basis of the thermodynamics of solidsurfaces. Long before Gibbs formulated the surface stress theory in solids, the idea of surface tension in liquids had been pre-sented by Laplace [44] and Young [45]. Gurtin and Murdoch [46,47] elaborated the mathematical framework incorporatingsurface stress into continuummechanics. Miller and Shenoy [27] compared the results obtained by the simplified Gurtin andMurdoch’s surface elasticity model with those obtained by means of the embedded atom method for nanobeams andnanoplates and found that the results were in excellent agreement. In Gurtin and Murdoch’s theory, the surface/interfaceis modeled as a layer of vanishing thickness, which is bonded to the bulk material without slipping and has materialproperties different from the bulk.

The study of material properties and elastic fields of nanostructures and nanomaterials is important for the further devel-opment of nanotechnology, which is focused on the creation of a wide range of advanced structural materials and soliddevice systems for optoelectronics, biomedical engineering, communications, mechanical engineering, etc. Based on the sur-face/interface elasticity approach, various boundary value problems have been solved for nanomaterials and nanostructures(e.g. [36–41,48–53]). In the analysis of size effect, most of the works exhibit the influence of a relevant geometric parameter(the size of an inhomogeneity [36–41,49,50,52], the period of a traction at a planar surface [48], or the period of surfaceasperities [51,53]) on the corresponding elastic field of a nanostructure. Such a size effect at the nanoscale is a direct resultof taking into account surface/interface stress. In addition, the dependence of nanomaterial properties on at least one of thespecimen dimensions at the nanoscale is also the size effect related to the surface/interface stress [26–35].

The present work is focused on two principle problems. First, we extend a complex variable based technique, used pre-viously by Grekov [10] in constructing periodic Green functions for the homogeneous elastic half-plane at the macrolevel, tothe same problem of nanomaterials containing point forces or edge dislocations close to a planar boundary, under the gen-eralized Young-Laplace boundary condition with unknown surface stress. We use the general Kolosov and Muskhelishvili’sapproach [11,54] and Gurtin and Murdoch’s theory of surface elasticity [46,47] that leads to the hypersingular integral equa-tion, similar to the equations derived in some other problems of nanomechanics [48,49,51–53]. Though an efficient way ofnumerical solution of such complex equations has been developed in [24,55], we shall give the analytical solution for theintegral equation in terms of the Fourier series with coefficients determined by quadratures. In contrast to the solutionobtained in the work [56] for the single dislocation and point force, we present the explicit formulas for an elastic field. Thesefundamental periodic solutions (Green functions) can be used for applying the boundary integral equation method to ananalysis of defects such as cracks and inhomogeneities periodically distributed near a boundary of a half-plane with surfacestresses.

Second, the interaction of the dislocations with the boundary at the nanoscale. The investigation of a surface stress effecton the elastic field around the dislocations and at the traction-free surface is very important for the prediction about thesurface cracking and the beginning of fracture. So, we have calculated the stress field and forces acting on the edge disloca-tions with dependence on the dislocations’ position and their Burgers vector orientation, using derived formulas. The numer-ical results and their analysis that we present in the paper have been obtained for the surface material constants determinedfor aluminium by Miller and Shenoy [27].

2. Problem formulation

We consider a semi-infinite elastic medium with a planar surface free from external load. The surface has elastic prop-erties that differ from the same properties of the volume and, according to the theory of surface elasticity [46,47], is repre-sented as a very thin film that adheres to the bulk material without slipping. The plane strain conditions are assumed to besatisfied under remote loading r1

ij and in the presence of the periodic set of straight edge dislocations with the Burgers vec-tor b or internal forces P, and the surface stress s (Fig. 1).

So, we come to the 2-D boundary value problem for the elastic half-plane X ¼ z : Imz < 0; Rez 2 ð�1;þ1Þf g of the com-plex variable z ¼ x1 þ ix2 (i is the imaginary unit) with the rectilinear boundary C. The positions of both dislocations andforces are zk ¼ ak� ih ða > 0; h > 0; k ¼ 0; �1; �2; . . .Þ, i.e., they are placed at the distance h from the surface. An arbitrarydirection of vectors b and P is defined by the components bj and Pj ðj ¼ 1;2Þ, respectively, in the Cartesian coordinates x1; x2.

Generalized Young-Laplace boundary condition [34,35] allowing for surface stress can readily be derived in the case of aplane problem considering an equilibrium of a traction-free boundary section [53]:

Fig. 1. Periodic set of edge dislocations with the Burgers vector b or forces P and the surface stress s in the half plane.

M.A. Grekov et al. / Engineering Fracture Mechanics 186 (2017) 423–435 425

r22ðx1Þ � ir12ðx1Þ ¼ �idsdx1

; ð1Þ

where rij ði; j ¼ 1;2Þ are the components of the stress tensor in coordinates x1; x2.At infinity, the stresses rij and the rotation angle x are defined as

limx2!�1

rij ¼ r1ij ; lim

x2!�1x ¼ 0: ð2Þ

Note that the values r1ij depend on a type of point disturbances considered in the paper and, as it will be shown here-

inafter, can not be arbitrarily assigned.The constitutive equations of surface linear elasticity for the first Piola-Kirchhoff stress tensor [35,38,46,47] and Hooke’s

law for the bulk material in the case of a plane strain [49,53] are thus

s ¼ rs11 ¼ ðks þ 2lsÞes11; rs

33 ¼ kses11; z 2 C; ð3Þ

r22 ¼ ðkþ 2lÞe22 þ ke11; r11 ¼ ðkþ 2lÞe11 þ ke22;

r12 ¼ 2le12; r33 ¼ kkþ l

r11 þ r22ð Þ; z 2 X:ð4Þ

In Eqs. (3) and (4), e22; e11; e12 are the strains of the bulk material; es11 is the surface strain; ks;ls are the surface elasticconstants similar to the Lame constants k;l.

The additional equation which enables us to find the surface stress s and solve the boundary value problem is the insep-arability condition of the surface and bulk, expressed in terms of hoop strains:

es11ðfÞ ¼ e11ðfÞ: ð5Þ
Note that the equilibrium Eq. (1) and constitutive equations of surface elasticity (3) can be derived from generalized
Young-Laplace law and original equations of Gurtin and Murdoch, respectevely, by equating surface tension (residual surfacestress) to zero and taking into account that the conditions of plane strain are satisfied and the surface is flat. For this purpose,one can use corresponding equations presented in [38] for the plane problems without any simplifications. Based on the factthat the surface is flat in the problem, we should assume that the surface tension equals zero. Otherwise, the presence of thesurface tension in this surface under unstrained conditions means that an external force, equal to this tension, is applied tothe surface at infinity. But it is natural to assume that no one external force acts under unstrained conditions. It’s quiteanother matter when a surface of a solid is curved and unbounded, as in the case of wires, rods, plain bores, etc. It has beenshown that the surface tension could become significant for the curved and rotated surface/interface [30,38].

3. Solution of the boundary value problem

3.1. Complex potentials

According to Muskhelishvili [54] and the superposition technique [10,57,58], the stresses and displacements in the elastichalf-plane X under the periodic system of forces or edge dislocations are related to the complex potentials U1;!1 and U0;W0

by the equality


Gðz;gÞ ¼ gU1ðzÞ þU1ðzÞ � !1ðzÞ þU1ðzÞ � z� zð ÞU01ðzÞ

� �e�2ia þ G0ðz;gÞ; ð6Þ

where z 2 X;Gðz;gÞ ¼ rnn þ irnt if g ¼ 1 and GðzÞ ¼ �2ldu=dz if g ¼ �,;rnn; rnt are the stress tensor components in thelocal Cartesian coordinates n; t with the angle a between the t and x1 axes, u ¼ u1 þ iu2.

The function U1 is holomorphic in the half-plane X and !1 — in the half-plane where Imz > 0. These functions determinestress and displacement fields arising due to the presence of the free surface and surface stress, and will be found hereafter.

The function G0 in Eq. (6) is defined as

G0ðz;gÞ ¼ gU0ðzÞ þU0ðzÞ þ zU00ðzÞ þW0ðzÞ

� �e�2ia; ð7Þ

where

U0ðzÞ ¼ �Hctgpðz� z0Þ

a;

W0ðzÞ ¼ ðvH þ HÞctg pðz� z0Þa

� pHa

ðzþ 2ihÞcosec2 pðz� z0Þa

ð8Þ

and

v ¼ �1; H ¼ ilðb1 þ ib2Það,þ 1Þ for the dislocations;

v ¼ ,; H ¼ P1 þ iP2

2að,þ 1Þ for the forces;

, ¼ ðkþ 3lÞ=ðkþ lÞ.Functions U0; W0 are the Goursat–Kolosov complex potentials corresponding to the infinite plane under periodically dis-

tributed edge dislocations or point forces. Following Eq. (8)

limx2!�1

U0ðzÞ ¼ �iH;

limx2!�1

W0ðzÞ ¼ limx2!�1

zU00ðzÞ þW0ðzÞ

� �¼ �i vH þ H

� �:

ð9Þ

Pass to the limit in Eq. (6) when x2 ! �0. Then, by setting a ¼ 0; g ¼ 1, and taking into account Eq. (1), we get the fol-lowing boundary equation for the function N holomorphic outside the boundary C:

Nþðx1Þ � N�ðx1Þ ¼ G0ðx1;1Þ þ is0ðx1Þ; x1 2 ð�1;þ1Þ: ð10Þ
Here N�ðx1Þ ¼ lim
x2!�0NðzÞ and

NðzÞ ¼ !1ðzÞ; Imz > 0;U1ðzÞ; Imz < 0:

�ð11Þ

Following Muskhelishvili [54], the solution of the Riemann-Hilbert boundary problem (10) can be presented as

NðzÞ ¼ 12pi

Z þ1

�1

G0ðx;1Þx� z

dt þ TðzÞ þ C; ð12Þ

where C is the constant which should be defined, and

TðzÞ ¼ 12pi

Z þ1

�1

is0ðtÞt � z

dt: ð13Þ

Based on the properties of Cauchy type integrals [11], one can express the function N, and as a consequence, functions U1

and !1 in terms of functions U0;W0 and so far unknown function TðzÞ and constant C:

NðzÞ ¼ TðzÞ þ C þ U0ðzÞ þ iHðv�1Þ2 ; Imz > 0;

�U0ðzÞ � zU00ðzÞ �W0ðzÞ þ iHðv�1Þ

2 ; Imz < 0;

(ð14Þ

3.2. Stress field relations

Taking into account Eq. (11) and substituting expressions (14), (7) and (8) in Eq. (6) under g ¼ 1; a ¼ 0 and a ¼ p=2, wearrive at the following formulas for the stresses:


r22 � ir12 ¼ 2Re U0ðzÞ �U0ðzÞ½ � þ zU00ðzÞ þW0ðzÞ � zU0

0ðzÞ �W0ðzÞ�ðz� zÞ U0

0ðzÞ � zU000ðzÞ �W0

0ðzÞ� �þ TðzÞ � TðzÞ þ ðz� zÞT 0ðzÞ;

r11 þ r22 ¼ 4Re U0ðzÞ �U0ðzÞ � zU00ðzÞ �W0ðzÞ

h iþ

4Re TðzÞ þ C½ � þ 2ð1� vÞImH

ð15Þ

Considering Eq. (6) at infinity (x2 ! �1) for a ¼ 0; a ¼ p=2;g ¼ 1 and g ¼ �,, and taking into account Eqs. (2), (7), (9),(11) and (14), one can obtain that

r122 � ir1

12 ¼ �2iðvþ 1ÞH; 4C ¼ r111 þ r1

22 þ 2iðvþ 3ÞH: ð16Þ
It is easy to see from Eq. (16) that the stresses r1
22; r112 equal zero in the case of dislocations (v ¼ �1) and relate to the

forces projections in the case of forces (v ¼ ,). In both cases, the stress r111 can be arbitrary. For example, the condition

of the displacements periodicity yields C ¼ iðv� 1ÞH=2 [10]. Then, as follows from Eq. (16), r111 ¼ 8ImH. In our numerical

calculations for dislocations, we assume that r111 ¼ 0, i.e C ¼ iH.

4. Solution for the surface stress

4.1. Integral equation

To find the surface stress and then the function T, pass to the limit in Eq. (15) when x2 ! �0. We come to the followingrelations for the stresses at the boundary C:

r22ðx1Þ � ir12ðx1Þ ¼ T�ðx1Þ � Tþðx1Þ ¼ �is0ðx1Þ;r11ðx1Þ ¼ 4Iðx1Þ � 4Re x1U

00ðx1Þ þW0ðx1Þ

� �þ 2Re 2C � ið1� vÞH½ �: ð17Þ

In Eq. (17), we denote T�ðx1Þ ¼ limx2!�0

TðzÞ defined by the Sokhotski-Plemelj formulas [54] as

T�ðx1Þ ¼ �12is0ðx1Þ þ Iðx1Þ; Iðx1Þ ¼ 1

2pi

Z þ1

�1

is0ðtÞt � x1

dt; ð18Þ

where the singular integral Iðx1Þ is understood in the sense of Cauchy principle value integral.After substituting Eq. (17) into Eq. (4) and taking into account the inseparability condition of the surface and bulk (5), Eq.

(3) can be transformed to the following integral equation:

sðx1Þ �Mð,þ 1ÞIðx1Þ ¼ Mð,þ 1ÞQðx1Þ; ð19Þ
where M ¼ ksþ2ls
2l and

Qðx1Þ ¼ 12Re 2C � ið1� vÞH½ � � Re x1U

00ðx1Þ þW0ðx1Þ

� �: ð20Þ

Differentiation of Eq. (19) leads to the hypersingular integral equation in the function s0

s0ðx1Þ �Mð,þ 1Þ2p

Z þ1

�1

s0ðtÞt � x1ð Þ2

dt ¼ Mð,þ 1ÞQ 0ðx1Þ; ð21Þ

where the hypersingular integral is understood in the sense of a finite part (Hadamard) integral (see [24,55]). This integral isthe result of the formal differentiation of the integral I in Eq. (18), that is valid when the function s0 has the first derivative ofHolder class [24].

According to the expressions (8) and (20), the continuous and periodic function Q 0 in the right hand size of Eq. (21) can beexpressed in terms of elementary functions as follows:

Q 0ðx1Þ ¼ paRe H þ vH þ 4Hf0ctgðn� f0Þ

� �cosec2ðn� f0Þ

; ð22Þ

where n ¼ px1=a; f0 ¼ �iph=a.

4.2. Solution of the integral equation

In order to evaluate analytically the integral Eq. (21), expand the function Q 0ðx1Þ into the following Fourier series

Q 0ðx1Þ ¼X1k¼1

Ak cos kkx1 þ Bk sin kkx1ð Þ; kk ¼ 2pk=a; ð23Þ

with coefficients


Ak ¼ 2a

Z a=2

�a=2Q 0ðtÞ cosðkktÞdt; Bk ¼ 2

a

Z a=2

�a=2Q 0ðtÞ sinðkktÞdt: ð24Þ

In the representation (23), we allowed for thatR a=2�a=2 Q

0ðtÞdt ¼ 0.

Inserting Eq. (23) into Eq. (21), one can obtain the function s0 in the form:

s0ðx1Þ ¼X1k¼1

ak cos kkx1 þ bk sin kkx1ð Þ; ð25Þ

where

ak ¼ 2Mð,þ 1Þ2þMð,þ 1Þkk Ak; bk ¼ 2Mð,þ 1Þ

2þMð,þ 1Þkk Bk: ð26Þ

After substituting the expression (25) into the formula (13), and using the properties of Cauchy type integrals, the func-tion T can be evaluated analytically as follows:

TðzÞ ¼ 12

X1k¼1

ðbk � iakÞ expð�ikkzÞ; TðzÞ ¼ TðzÞ; Imz < 0: ð27Þ

Eq. (27) and expressions (11) and (14) of functions UðzÞ; !ðzÞ, in terms of known complex potentials U0ðzÞ; W0ðzÞ, and thefunction TðzÞ provide a way of evaluating the periodic Green functions from Eq. (6), i.e., the displacements and stresses aris-ing due to the presence of edge dislocations or internal point forces.

If we ignore an existence of the surface stress, then TðzÞ � 0 and one can easy derive from Eq. (6) the same explicit closed-form formulas for the displacements and stresses that have been obtained in [10]. Incorporating surface effect leads to someextra summands in these formulas, depending on the function T as is seen, for example, in Eq. (15).

5. The dislocations stress field analysis

Since the function TðzÞ has been found, one can numerically study the stress field generated in a half-plane by the edgedislocations or internal point forces, using formulas (8) and (15). The investigation of the stress field produced by the inter-action of dislocations with a traction-free boundary is the most important function for applications as dislocations are thereal defects of crystalline materials. So, we present below a detailed analysis of such an interaction, assuming v ¼ �1 inEqs. (8), (15) and (2) and r1

ij ¼ 0 ði; j ¼ 1;2Þ in Eq. (2). For the numerical examination, we have chosen data related to thesurface elastic properties of Al[111]: c0 ¼ 1 N/m, ks ¼ 6:851 N/m, ls ¼ �0:376 N/m that was determined by Miller and She-noy [27] with the embedded atommethod. These surface parameters have been used in a number of investigations of surfacestress effects. The bulk elastic constants for aluminium are k ¼ 58:17 GPa and l ¼ 26:13 GPa.

To compute the sought quantities, we truncate the series (23) for the function Q 0ðx1Þ. The minimum length N of the trun-cated series

Q 0Nðx1Þ ¼

XNk¼1

Ak cos kkx1 þ Bk sin kkx1ð Þ ð28Þ

is defined by the following inequality:

jJ � JNjjJj < e; ð29Þ

where

J ¼ Qða=2� eaÞ � Qð0Þ; JN ¼ QNða=2� eaÞ � QNð0Þ: ð30Þ
Eq. (29) is the integral criterion. It means that the relative deference between the integrals of the function Q 0 and its
approximate expression Q 0N over the interval ½0; a=2� ea� should not exceed the prescribed value e. The integration interval

is just less than the interval ½0; a=2� because the integral J over the last one equals zero.All numerical results given below have been obtained for e ¼ 0:001. Besides e, the value of the parameter N depends on

the relation h=a and orientation of the Burgers vector b. So, if h=a ¼ 0:01, then N ¼ 261 for b ¼ ðb1;0Þ and N ¼ 337 forb ¼ ð0; b2Þ and, if h=a ¼ 0:02, then N ¼ 119 and N ¼ 159, respectively. At the same time, if h=a ¼ 1, then N ¼ 1 and N ¼ 3respectively, and are enough for the realization of the required accuracy e ¼ 0:001. If h=a P 1:4, then N ¼ 1 for any Burgersvector orientation.

In Figs. 2–9, classical and non-classical solutions for the dimensionless stress components rmij ¼ rijM=lbm � 102

ðm ¼ 1;2Þ are plotted for two different Burgers vector orientations b ¼ ðb1;0Þ and b ¼ ð0; b2Þ. Only the solutions are shownwithin a half period due to the symmetry and antisymmetry about the x2-axis. The normal stresses r11 and r22 are

Fig. 2. Normalized hoop stress at the surface for the Burgers vector orientation along the x1-axis (a) and x2-axis (b) when period a ¼ 3 nm.



symmetric for the first type of orientation and antisymmetric for the second, whereas the tangential stress r12 is antisym-metric for b ¼ ðb1;0Þ and symmetric for b ¼ ð0; b2Þ.

5.1. Hoop stress at the free surface

The distribution of the hoop stress rm11 at the surface is plotted in Figs. 2–4 for different values of the relation h=awhen the

period a is fixed (a ¼ 3;10 or 20 nm), and in Figs. 5, 6 for different values of awhen the distance of dislocations to the bound-ary is fixed (h ¼ 0:5 nm or h ¼ 1 nm). The dimensionless stress field along the x1-direction in the layer between the plane ofdislocations and the surface is presented in Figs. 7–9.

As follows from Figs. 2–4, the maximum relative difference between classical and non-classical solutions depends on therelation h=a. The effect of the surface stress increases when h=a decreases. One of the reasons for such a behavior is thedependence of the hoop stress r11 in both solutions on the distance of dislocations h to the free surface when the perioda is fixed. But, one can see from Fig. 3(a) (curves for h=a ¼ 0:2) and Fig. 4(a) (curves for h=a ¼ 0:1) that the stress distributionand surface effect depend on the period a as well. It is worth noting that the influence of the surface stress on the stress r11

becomes negligible when h=a ¼ 1, independently of the magnitude of the parameter a. So, one can say that the surface effectdisappears when h=a > 1.


Fig. 5. Normalized hoop stress at the surface for the Burgers vector orientation along the x1-axis (a) and x2-axis (b) when the distance of dislocations fromthe surface h ¼ 0:5 nm.


To investigate the influence of the period a on the hoop stress r11 at the surface in classical and non-classical solutions,two sets of numerical results are presented in Figs. 5, 6, when the magnitude of the distance h is invariable. As is seen, theincrease of the period a leads to the stabilization of the maximum difference of r11 in classical and non-classical solutions,and this difference takes place in the point x1 ¼ 0 under the dislocation core if b ¼ ðb1;0Þ, and in the vicinity of this point ifb ¼ ð0; b2Þ. The point of the extremal values of the hoop stress r11 in both solutions tends to the point x1 ¼ 0 when hdecreases (see Figs. 2–4) or a increases (see Figs. 5, 6) for both Burgers vector orientations. By moving away from the pointx1 ¼ 0, the hoop stress r11 at the surface exhibits more smooth changes, and the surface stress effect becomes negligible inthe vicinity of the point x1 ¼ 0:5a.

Fig. 6. Normalized hoop stress at the surface for the Burgers vector orientation along the x1-axis (a) and x2-axis (b) when the distance of dislocations fromthe surface h ¼ 1 nm.

Fig. 7. Normalized normal stress oi11 at the distance of jx2j ¼ const from the surface for the Burgers vector orientation along the x1-axis (a) and x2-axis (b).

Curves 1,2,3,4,5 correspond to x2=h ¼ �f0;0:25;0:5;0:75;1g.


5.2. Stress field in the bulk

In order to compare the classical and non-classical stress solutions in the points out of the surface, the distribution of thenormal r11; r22 and tangential r12 stresses in the layer�h 6 x2 6 0 along the x1-direction are plotted in Figs. 7–9 for both theBurgers vector orientations, a ¼ 10 nm and h ¼ 1 nm. One can conclude from the results shown in Figs. 7–9 that the stressr11 has the most influence of the surface stress and this influence decreases when the distance to the surface increases.Besides r11, the classical and non-classical solutions for the tangential stress r12 differ noticeably close to the surface andespecially at the surface where r12 ¼ 0 in the classical solution.

Note that the surface effect becomes negligible at the distance h from the surface. So, we don’t give numerical results forthe region x2 < �h. It should be emphasized that at x2 ¼ �h, the stress r12 is infinite at the dislocation core if b ¼ ð0; b2Þ (seeFig. 8(a)), whereas for b ¼ ðb1;0Þ, the stresses r11; r22 are infinite at the dislocation core as well (see Figs. 7(b), 9(b)).

It is worth comparing the classical and non-classical stress fields arising due to the interaction of a single dislocation withthe free surface that was obtained in [56] for the same properties of the surface and bulk materials as in the present work,and our results. Only one position (h ¼ 0:15288 nm) of the dislocation has been considered in the work [56]. Since the stress

Fig. 8. Normalized tangential stress oi12 at the distance of jx2j ¼ const from the surface for the Burgers vector orientation along the x1-axis (a) and x2-axis

(b). Curves 1,2,3,4 correspond to x2=h ¼ �f0;0:1;0:5;1g.

Fig. 9. Normalized normal stress oi22 at the distance of jx2j ¼ const from the surface for the Burgers vector orientation along the x1-axis (a) and x2-axis (b).

Curves 1,2,3,4 correspond to x2=h ¼ �f0:25;0:5;0:75;1g.


field at the surface is absent in this work, we have compared maximum absolute values of the stress r11 at the linesx2 ¼ const into the layer �h < x2 < 0, with corresponding values from [56], allowing for the difference of designations.We have ascertained that the results practically coincide when a=h > 50 in the case b ¼ ðb1;0Þ and a=h > 10 for b ¼ ð0; b2Þ.

5.3. Force acting on dislocation

The force on dislocation is defined as the negative gradient of the interaction energy [2,59]. In the present problem, thisenergy for each dislocation is the elastic strain energy generated by the rest dislocations, surface stress and image disloca-tions corresponding to the set of all real dislocations. The force on dislocation is of primary importance in studying the dislocations–surface/interface interaction. This force can be resolved into a glide component along the dislocation Burgers vec-tor, and a climb component perpendicular to the Burgers vector. Exceeding the Peierls force, the glide component of the forcecan lead to the depletion of dislocations from regions near the surface of large crystals and to a completely dislocation–freenanocrystal. The climb component of the force is expected to play a role at high temperatures when the dislocation climbbecomes feasible [60].


It is clear that the interaction of periodically distributed edge dislocations in an infinite isotropic medium does not pro-duce the force on each dislocation because of symmetry. In addition, the movement of the dislocations parallel to the planarsurface does not change the interaction energy. So, the force on each dislocation is perpendicular to the surface.

The force on the edge dislocation in the point ð�h; akÞ is commonly determined by the Peach–Koehler formula [61], whichin the case of the 2-D problem can be written as follows:

Fig. 10.dislocat

F1 � iF2 ¼ b1r�12ð�h; akÞ þ b2r�

22ð�h; akÞ þ i b1r�11ð�h; akÞ þ b2r�

12ð�h; akÞ� �

; ð31Þ
where r�
ij ði; j ¼ 1;2Þ are the stress tensor components determined by the Eq. (6) for a ¼ 0; a ¼ p=2 and G0 ¼ 0.Following Weertman [62], replace the stress tensor in Eq. (31) by its deviator to allow for the inelastic change in the solid

volume accompanying the dislocation climb:

F1 � iF2 ¼ b1r�12ð�h; akÞ þ b2r�

22ð�h; akÞþi b1r�

11ð�h; akÞ þ b2r�12ð�h; akÞ

� ��3kþ 2l6ðkþ lÞ ðb1 þ ib2Þ r�

11ð�h; akÞ þ r�22ð�h; akÞ

� �:

ð32Þ

Taking into account that F1 ¼ 0 and denoting F ¼ F2, rewrite the expression for the force (32) in more suitable form:

F ¼ b3kþ 2l6ðkþ lÞ r�

11ð�h; akÞ þ r�22ð�h; akÞ

� �sinu� r�

11ð�h; akÞ cosu� r�12ð�h; akÞ sinu

� �: ð33Þ

Here, b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21 þ b2

2

qand u is the angle between Burgers vector b and the x1-axis. So, F is the climb force on the dislocation if

u ¼ 0, and F is the glide one if u ¼ p=2.The dependence of the normalized force acting on each dislocation F� ¼ FM=lb2 � 102 on the Burgers vector orientation

(angle u) is shown in Fig. 10 for two distances of the dislocations to the surface – h ¼ 0:5 nm (Fig. 10(a)) and h ¼ 1 nm(Fig. 10(b)), and different values of the period a. Since the function F is symmetric about the u-axis and p-periodic, we pre-sent this dependence only for u 2 ½0;p=2�.

As is seen from Fig. 10, the climb force at the angleu ¼ 0 is appreciably greater than the glide one atu ¼ p=2, both for theclassical and non-classical solution when h=a ¼ 0:1. But such a difference decreases when the ratio h=a decreases. The pres-ence of the surface stress decreases the force on dislocation, irrespective of the Burgers vector orientation. At the same time,the surface effect decreases with increasing h=a. It is interesting to note that unlike the climb force, the glide force in both thesolutions is scarcely affected by the period a and is closed to the corresponding value of the force acting on a single dislo-cation located near the planar surface.

It is worth comparing the image force acting on the single edge dislocation near the planar surface, which is defined in the

classical solution by the formula Fs ¼ lb2=½phð1þ ,Þ�, with corresponding our solution. It is evident from Fig. 10, that the

force in classical and non-classical solutions on each dislocation is negligibly influenced by the Burgers vector orientationwhen h=a ¼ 0:02. For this ratio, the difference between our classical solution for the climb force at the angle u ¼ 0 andthe force Fs for h ¼ 0:5 nm and h ¼ 1 nm does not exceed 2%, and for the glide force at u ¼ p=2 – 0.3%. The same comparisonfor the relation h=a ¼ 0:01 and the distance h ¼ 0:5 nm gives 0.5% and 0.2%, accordingly, that is in the range of the specifiedcalculation accuracy. So, follows from this comparison and the comparison with the work [56], the stress field in the vicinity

Dependence of the normalized image force F� acting on each dislocation on the angle u of the Burgers vector orientation when the distance ofions from the surface h ¼ 0:5 nm (a) and h ¼ 1 nm (b).


of the single dislocation located near a planar surface can be obtained by considering the periodic set of dislocations with theratio h=a 6 0:02, using both classical and non-classical approach.

6. Summary and conclusions

The technique developed in [10] to construct periodic Green functions for the homogeneous elastic half-plane at themacrolevel has been extended in this paper to solve the same problem at the nanoscale. Incorporating the surface stress,we have derived the hypersingular integral equation in the derivative of this stress to solve the 2-D problem of the interac-tion of periodically distributed edge dislocations or internal forces located at nanometer distance from the planar surfacewith this surface. We have obtained the exact solution of this equation in terms of the Fourier series that has allowed usto give the explicit formulas for the stress tensor components. These formulas are virtually Green functions that can beapplied to solve corresponding boundary value problems at the nanoscale for different defects like periodically distributed(or single) holes and cracks.

A numerical investigation of the elastic field arising in a subsurface at the nanoscale due to the presence of periodicallydistributed edge dislocations, and the force acting on each dislocation, has been performed on the basis of the classical andnon-classical (with surface stress) solution. We have demonstrated that the stress field, the force on dislocation and the sur-face effect depend on the position of the dislocation, the ratio of the distance between dislocations and the surface to theperiod and Burgers vector orientation.

We have shown that the hoop stress at the surface is most influenced by the presence of the surface stress. In addition,unlike the classical solution with the free surface, allowing fore the surface stress leads to the existence of tangential stressesat the surface. The closer the dislocations to the surface, the more significant the surface effect. The maximum relative dif-ference between the hoop stresses in classical and non-classical solutions depends not only on the position of the disloca-tions, but also on the period and the ratio of the period to the distance between dislocations and the surface. The effect of thesurface stress increases when this ratio decreases. The influence of surface stress on the stress field vanishes rapidly awayfrom the surface and is practically absent at the plane of the dislocations disposition.

For the given material properties, incorporating the surface stress decreases the force on dislocation calculated within theclassical elasticity, regardless of the dislocations position, Burgers vector orientation and the period. We have revealed thatin both the classical and non-classical solutions the glide force acting on each dislocation is scarcely affected by the periodand is close to the corresponding magnitude of the force acting on a single dislocation placed near a free planar surface.

It should be emphasized, that the explicit formulas for the stresses derived in this work, for the periodic edge dislocationsand point forces, can be used for the analysis of elastic fields arising due to the interaction of a single dislocation or force witha traction-free planar surface. The ability to analyze such an interaction and to precisely evaluate the corresponding elasticfields provides new opportunities in several applications. One of such applications is the determination of the strength of asubsurface layer and prediction of the fracture beginning that requires detailed information about the stress and strain fieldsresulting from the interactions of different perturbation sources.

Acknowledgements

This research was supported by the Russian Foundation for Basic Research under Grant 14-01-00260.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.en-gfracmech.2017.11.005.

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