Applied Mathematics and Computation 226 (2014) 194–197

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

A brief note on the self-balanced singular elastic field causedby an edge dislocation

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.10.059

E-mail address: swc@shmtu.edu.cn

Shi WeichenDepartment of Mechanical Engineering, Shanghai Maritime University, Haigang Avenue 1550, Shanghai 201306, PR China

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

Keywords:Edge dislocationConcentrated forceInteractionSW-integralPhysical quantity

Four physical quantities describing the self-balanced singular elastic field caused by anedge dislocation before or after loading are obtained by using SW-integral. These quantitiesare directly related to mechanisms of dislocation plasticity and are compared with that of aconcentrated force. An edge dislocation and a concentrated force interacting with the firststress invariant and the maximum shear stress at infinity, respectively, are also presented.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Traditional conservation integrals, such as J-integral, L-integral and M-integral, play an important role in finding out thephysical quantities of material points with singularities and the interaction among singularities in elastic field [1–5]. Forexample, apart from the crack extension force, Cherepanov [6] gives the known Peach–Koehler formula by calculating J-inte-gral around an edge dislocation as follows

Ji ¼ eij3r1jk bk; ði; j; k ¼ 1;2Þ; ð1Þ

where ðb1; b2Þ ¼ bðcos c; sin cÞ is the Burgers vector of an edge dislocation, r1jk are the stresses at infinity and eij3 is thepermutation symbol. The formula (1) means a configuration force acting on the dislocation. However, when lettingr1jk ¼ 0 at infinity, we know that Ji ¼ 0 in formula (1). That is, it is difficult for us to know whether there exists a self-balancedsingular elastic field caused by an edge dislocation by using J-integral without loading.

It is well known that mechanisms of dislocation plasticity are directly related to the self-balanced singular elastic fieldcaused by edge dislocations [7]. As mentioned in [8], dislocation motion on the primary slip system is severely hinderedby the presence of dislocations on the secondary slip systems, which is the reason for the high stage-II hardening rate inmaterials caused by the possibility of secondary slips. Therefore, it is significant to recognize the characteristics of disloca-tions before or after loading.

Recently, it is found that for any analytic function /ðzÞ, there exists SW-integral in the sense of Noether’s theorem [9–12]

IC

fðzÞ½/0ðzÞ�2dz ¼ 0: ð2Þ

Here, fðzÞ represents any conformal transformations, so that there are countless conserved quantities and path-indepen-dent integrals. Clearly, by adjusting the conformal transformation fðzÞ, a finite physical quantity can always be obtainedwhen calculating SW-integral (2) around a material point with any order singularity.

W. Shi / Applied Mathematics and Computation 226 (2014) 194–197 195

In this note, an edge dislocation with or without the loading at infinity is considered by using SW-integral (2) and somenew physical quantities are obtained. For comparison, a concentrated force acting at the origin is also calculated. The pur-pose is to show some light on edge dislocations with the measurement before or after loading.

2. An edge dislocation

Under the consideration of a plane stress or plane strain problem, suppose that an elastic body contains an edge disloca-tion. The complex Kolosov–Muskhelishvili potentials /ðzÞ and wðzÞ are given by [6,13]

/ðzÞ ¼ Gðb1 þ ib2Þpiðjþ 1Þ ln zþ A1z; ð3aÞ

wðzÞ ¼ �Gðb1 � ib2Þpiðjþ 1Þ ln zþ ðB1 þ iB2Þz; ð3bÞ

A1 ¼ ðr111 þ r122Þ=4;B1 ¼ ðr122 � r111Þ=2;B2 ¼ r112; ð3cÞ

where G is the shear modulus, ðb1 þ ib2Þ ¼ bðcos cþ i sin cÞ is a linear dislocation, j ¼ 3� 4m for plane strain,j ¼ ð3� mÞ=ð1þ mÞ for plane stress and m the Poisson’s ratio. Please note that the rigid body translation and rotation havebeen neglected in (3a) and (3b), which do not affect elastic deformation.

Applying the SW-integral (2) to the complex potentials (3a) and (3b) in an area surrounded by the dotted lines shown inFig 1(a), one has

SW ð/DÞ ¼I

C1

fðzÞ Gðb1 þ ib2Þpiðjþ 1Þz

� �2

þ 2A1Gðb1 þ ib2Þpiðjþ 1Þz þ A2

1

( )dz; ð4aÞ

SW ðwDÞ ¼I

C1

fðzÞ Gðb1 � ib2Þpiðjþ 1Þz

� �2

� 2ðB1 þ iB2ÞGðb1 � ib2Þpiðjþ 1Þz þ ðB1 þ iB2Þ2

( )dz; ð4bÞ

where the integral paths C0 and C1 are in an anticlockwise direction.In order to obtain some physical quantities, the following basic integral will be used

I

zndz ¼2pi; n ¼ �10; n–� 1

�ð5Þ

By letting fðzÞ ¼ z, it is derived from (4a) and (4b) that

SW ð/DÞz ¼ 2ðGbÞ2

pðjþ 1Þ2ðsin 2c� i cos 2cÞ; ð6aÞ

SW ðwDÞz ¼ � 2ðGbÞ2

pðjþ 1Þ2½sin 2cþ i cos 2c�: ð6bÞ

Fig. 1. (a) An edge dislocation b at the origin; (b) a concentrated force P acting at the origin.

196 W. Shi / Applied Mathematics and Computation 226 (2014) 194–197

These are four physical quantities from the self-balanced singular elastic field caused by an edge dislocation before orafter loading. By letting fðzÞ ¼ 1, the integration (4a) and (4b) result in

SW ð/DÞ1 ¼ Gb

jþ 1ðr111 þ r122Þðcos cþ i sin cÞ; ð7aÞ

SW ðwDÞ1 ¼ 2Gb

jþ 1ðr111 � r122Þ cos c� 2r112 sin c� i 2r112 cos cþ ðr111 � r122Þ sin c

� �� �: ð7bÞ

Expression (7a) represents the interaction between an edge dislocation and the first stress invariant at infinity. Expression(7b) means the interaction between an edge dislocation and the maximum shear stress smax ¼ ½ðr11 � r22Þ2 þ 4r2

12�12=2 at

infinity. This is a circumstantial evidence for that a dislocation motion on the primary slip system is severely hindered bythe presence of dislocations on the secondary slip systems, since the maximum shear stress smax may come from a slipsystem.

3. A concentrated force

Consider a thin elastic plate, and a concentrated force ðP1; P2Þ ¼ Pðcos c; sin cÞ is applied to the plate, calculated per unitthickness of the plate. The complex Kolosov–Muskhelishvili potentials /ðzÞ and wðzÞ are given by [6,13]

/ðzÞ ¼ � P1 þ iP2

2pðjþ 1Þ ln zþ A1z; ; ð8aÞ

wðzÞ ¼ jðP1 � iP2Þ2pðjþ 1Þ ln zþ ðB1 þ iB2Þz: ð8bÞ

Similarly to the above, applying the SW-integral (2) to the complex potentials (8a) and (8b) in an area surrounded by thedotted lines shown in Fig 1(b), it can be derived that

SW ð/PÞ ¼I

C1

fðzÞ P1 þ iP2

2pðjþ 1Þz

� �2

� 2A1P1 þ iP2

2pðjþ 1Þzþ A21

( )dz; ; ð9aÞ

SW ðwPÞ ¼I

C1

fðzÞ jðP1 � iP2Þ2pðjþ 1Þz

� �2

þ 2ðB1 þ iB2ÞjðP1 � iP2Þ2pðjþ 1Þzþ ðB1 þ iB2Þ2

( )dz: ð9bÞ

Letting fðzÞ ¼ z, we know from (9a) and (9b) that

SW ð/PÞz ¼ � P2

2pðjþ 1Þ2ðsin 2c� i cos 2cÞ; ð10aÞ

SW ðwPÞz ¼ j2P2

2pðjþ 1Þ2ðsin 2cþ i cos 2cÞ: ð10bÞ

By letting fðzÞ ¼ 1, the integration (9a) and (9b) give

SW ð/PÞ1 ¼ P

2ðjþ 1Þ r111 þ r122

ðsin c� i cos cÞ; ð11aÞ

SW ðwPÞ1 ¼ � jP

jþ 12r112 cos cþ ðr111 � r122Þ sin cþ i ðr111 � r122Þ cos c� 2r112 sin c

� �� �: ð11bÞ

Expression (11a) represents the interaction between a concentrated force and the first stress invariant at infinity, andexpression (11b) means the interaction between a concentrated force and the maximum shear stresssmax ¼ ½ðr11 � r22Þ2 þ 4r2

12�12=2 at infinity.

W. Shi / Applied Mathematics and Computation 226 (2014) 194–197 197

4. Discussion

Since the path independence of SW-integral (2) means a kind of physical and mathematical invariance, the real and imag-inary parts from expressions (6a)-(7b) and (10a)-(11b) are new physical quantities based on Noether’s theorem [9]. Espe-cially, the physical quantities (6a), (6b), (10a) and (10b) do not include the interference by the loading at infinity.Although the physical basis of self-balanced singular elastic field caused by an edge dislocation is different from that of aconcentrated force, some resemblances of the physical quantities (6a), (6b), (10a) and (10b) can be known as follows

SW ð/DÞz

SW ð/PÞz

¼ �4GbP

� �2

;SW ðwDÞ

z

SW ðwPÞz

¼ �4GbjP

� �2

: ð12Þ

Since ðGbÞ=P is a dimensionless value, these two expressions with inspecting (6a), (6b), (10a) and (10b) means that theexistence of an edge dislocation before loading can be checked by using the elastic field of a concentrated force to replace thefield caused by an edge dislocation. Comparisons of the interactions are given by (7a), (7b), (11a) and (11b), respectively, asfollows

SW ð/DÞ1

SW ð/PÞ1

¼ 2iGbP;

SW ðwDÞ1

SW ðwPÞ1

¼ 2iGbjP

: ð13Þ

Acknowledgements

This work is supported by Innovation Program of Shanghai Municipal Education Commission under Grant No. 2000SG31and No. 07ZZ98.

References

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