Dislocations in gradient elasticity revisited

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

Dislocations in gradient elasticity revisited Markus Lazar and Gérard A Maugin Proc. R. Soc. A 2006 462, 3465-3480 doi: 10.1098/rspa.2006.1699

References

This article cites 17 articles

Email alerting service

Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here

http://rspa.royalsocietypublishing.org/content/462/2075/3465.full. html#ref-list-1

To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions

This journal is © 2006 The Royal Society

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

Proc. R. Soc. A (2006) 462, 3465–3480 doi:10.1098/rspa.2006.1699 Published online 6 June 2006

Dislocations in gradient elasticity revisited B Y M ARKUS L AZAR *

AND

G E´ RARD A. M AUGIN

Laboratoire de Mode´lisation en Me´canique, Universite´ Pierre et Marie Curie, 4 Place Jussieu, Case 162, F-75252 Paris Cedex 05, France In this paper, we consider dislocations in the framework of first as well as second gradient theory of elasticity. Using the Fourier transform, rigorous analytical solutions of the twodimensional bi-Helmholtz and Helmholtz equations are derived in closed form for the displacement, elastic distortion, plastic distortion and dislocation density of screw and edge dislocations. In our framework, it was not necessary to use boundary conditions to fix constants of the solutions. The discontinuous parts of the displacement and plastic distortion are expressed in terms of two-dimensional as well as one-dimensional Fouriertype integrals. All other fields can be written in terms of modified Bessel functions. Keywords: gradient elasticity; dislocations; Fourier transform

1. Introduction In the theory of classical elasticity, it is well known that different displacement fields of a dislocation give the same elastic distortions but different plastic distortions. Thus, the displacement of a dislocation cannot be a physical state quantity. The reason why is that the displacement field of a dislocation is a multi-valued function. For example, the multi-valued displacement of a screw dislocation may be given in terms of y arctan ; x

or

x K arctan : y

ð1:1Þ

In classical elasticity, the displacement is fixed up to a constant displacement. One may reduce such a multi-valued function to a single-valued one by a branch cut which is of mathematical convenience, only (e.g. DeWit 1973; Teodosiu 1982). The single-valued function is then discontinuous because of the jump at the branch cut. In general, the branch cut may be arbitrary. The surface of the branch cut can be identified with the cut plane of the Volterra process. Thus, this branch cut cannot be detected experimentally and the elastic strains and stresses do not depend on it (e.g. Kro¨ner 1993). This fact is nothing but the main message of the Volterra theorem. Due to the discontinuity, the total distortion consists of a plastic part which depends on the cut plane. * Author and address for correspondence: Emmy Noether Research Group, Department of Physics, Darmstadt University of Technology, Hochschulstr. 6, Darmstadt 64289, Germany ([email protected]). Received 2 September 2005 Accepted 21 February 2006

3465

q 2006 The Royal Society

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3466

M. Lazar and G. A. Maugin

Modified solutions for displacements of dislocations were given in the framework of gauge theory of dislocations by Edelen (1996) and Lazar (2002a,b, 2003a). In this theory, one does not have a unique governing equation for the displacement and plastic distortion fields because they are not physical state quantities. Only, the elastic distortion and the dislocation density tensors are state quantities, which are gauge invariant. To determine and to modify the displacement, one may use local translations as gauge transformation and a gauge fixing. Thus, in gauge theory of dislocations the displacement and the plastic distortion are not unique due to physical reasons. A slightly different theory of extended elasticity is the so-called gradient elasticity. Strain gradient theories were introduced by Mindlin (1964, 1965) and Mindlin & Eshel 1968). Gradient elasticity is a generalization of linear elasticity which includes higher-order terms to account for microstructural effects. In a strain gradient theory, the strain energy depends on the elastic strain and gradients of the elastic strain. Due to the gradient terms, such a theory contains additional coefficients with the dimension of a length, which are called gradient coefficients. Ru & Aifantis (1993) postulated a governing equation for the displacement vector in the framework of a special gradient elasticity. Later, Lazar & Maugin (2005) showed how and under which conditions one can derive such an equation in gradient elasticity from a variational principle. In such a gradient elasticity, which is called first gradient elasticity of Helmholtz type, the field quantities must satisfy the following inhomogeneous Helmholtz equations: ð1K32 DÞui Z ui0 ;

ð1:2Þ

ð1K32 DÞbPij Z b0;P ij ;

ð1:3Þ

ð1K32 DÞbij Z b0ij ;

ð1:4Þ

ð1K32 DÞaij Z a0ij ;

ð1:5Þ

where 3 is a positive gradient parameter. ui0 , bij0;P , b0ij and a0ij are the displacement vector, plastic distortion, elastic distortion and dislocation density tensor, respectively, calculated in the theory of classical elasticity. On the other hand, ui , bPij , bij and aij are the displacement vector, plastic distortion, elastic distortion and dislocation density tensor, respectively, calculated in the theory of gradient elasticity. They fulfil P bT ij h vj ui Z bij C bij ;

ð1:6Þ

aij Z ejkl vk bil ZKejkl vk bPil ;

ð1:7Þ

where bT ij is called the total distortion. Corresponding equations are valid for the fields in classical elasticity. Gutkin & Aifantis (1996, 1997, 1999) found solutions of equations (1.2)–(1.4) for screw and edge dislocations. They used the onedimensional Fourier transform with respect to x together with special ‘physically motivated’ boundary conditions. Unfortunately, the solutions for the displacement and the plastic distortion do not satisfy equations (1.2) and (1.3) at yZ0 and the plastic distortion is still singular due to Dirac delta terms. But, of course, Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

Dislocations in gradient elasticity

3467

a solution should be a solution everywhere. Only, their elastic distortions fulfil (1.4) everywhere and agree with the results given by Lazar (2003b) in the gauge theory of dislocation and in the theory of elastoplasticity. This is one motivation to re-investigate that matter. In the second gradient elasticity, one can show that the field quantities fulfil the following inhomogeneous partial differential equations of fourth order (Lazar et al. 2006): ð1K32 D C g4 DDÞui Z ui0 ;

ð1:8Þ

ð1K32 D C g4 DDÞbPij Z b0;P ij ;

ð1:9Þ

ð1K32 D C g4 DDÞbij Z b0ij ;

ð1:10Þ

ð1K32 D C g4 DDÞaij Z a0ij ;

ð1:11Þ

which can be factorized into the inhomogeneous bi-Helmholtz equations ð1Kc12 DÞð1Kc22 DÞui Z ui0 ;

ð1:12Þ

ð1Kc12 DÞð1Kc22 DÞbPij Z b0;P ij ;

ð1:13Þ

ð1Kc12 DÞð1Kc22 DÞbij Z b0ij ;

ð1:14Þ

ð1Kc12 DÞð1Kc22 DÞaij Z a0ij ;

ð1:15Þ

with 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 3 g4 c12 Z @1 C 1K4 4 A; 2 3

0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 3 g4 c22 Z @1K 1K4 4 A; 2 3

ð1:16Þ

and 32 Z c12 C c22 ;

g4 Z c12 c22 ;

ð1:17Þ

where g is a second gradient parameter of higher order. Only rigorous solutions of equations (1.14) and (1.15) have been given quite recently for dislocations by Lazar et al. (2006). But no solutions of (1.12) and (1.13) are given in the literature. In this paper, we will give exact solutions for screw and edge dislocations which satisfy equations (1.2)–(1.5) and (1.12)–(1.15) everywhere. We are using the two-dimensional Fourier transform to solve these equations. We are able to give closed form solutions for the elastic distortion tensor and the dislocation density tensor and one-dimensional integral representations for the displacement vector and the plastic distortion tensor. In our framework, it is not necessary to use boundary condition to fix the gradient term of the solution. Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3468

M. Lazar and G. A. Maugin y

x

Figure 1. Branch cut for w 0 ZKarctanðx=yÞ.

2. Screw dislocation (a ) Classical solution We consider a screw dislocation whose Burgers vector bz and dislocation line coincide with the z-axis. We use a discontinuous displacement field uz0 Z bz =½2p
w 0 ðx; yÞ which has the symmetry w 0 ðx; yÞ ZKw 0 ðKx; yÞ:

ð2:1Þ

Then it is given by (e.g. Leibfried & Dietze 1949) y p x w 0 ðx; yÞ Z arctan K ZKarctan ; x 2 y

ð2:2Þ

where Karctanðx=yÞ has the range ðKp=2; 3p=2Þ. Here, w 0 is a single-valued function with a discontinuity represented by a branch cut (see figure 1). Thus, the branch cut is given for y!0, Karctan

0H ZGp: y

ð2:3Þ

In the degenerate case, when yZ0 8p > x ! 0; > >2; > > x < Karctan Z undefined; x Z 0; y > > p > > > x O 0: :K 2 ; So, we have at yZ0: uz0 ðx ! 0; 0ÞZ bz =4 and uz0 ðx O 0; 0ÞZKbz =4. Proc. R. Soc. A (2006)

ð2:4Þ

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3469

Dislocations in gradient elasticity

The total distortion is just the gradient of the displacement (2.2)  bz  y 0 b0;T Z v u ZK C 2pdðxÞH ðKyÞ ; x z zx 2p r 2 0 b0;T zy Z vy uz Z

bz x ; 2p r 2

ð2:5Þ ð2:6Þ

where d denotes the Dirac delta function and H ðyÞ is the Heaviside step function defined by ( 0; y! 0; H ðyÞ Z ð2:7Þ 1; yO 0: Because the second part of (2.5) is discontinuous and singular at the branch cut, it may be identified with the plastic distortion. The plastic distortion gives rise to a dislocation density of the single screw dislocation according to a0zz Z vy b0;P zx Z bz dðxÞdðyÞ:

ð2:8Þ

It means that the dislocation is concentrated at x Z yZ 0. (b ) Gradient solution The governing equations (1.12)–(1.15) will be solved by using the twodimensional Fourier transform. The two-dimensional Fourier transform (Sneddon 1951) is defined by ðN ðN ~ F ðf Þðk1 ; k 2 Þ Z f ðkÞ Z f ðx; yÞeKiðxk1Cyk 2 Þ dx dy; ð2:9Þ KN KN

and the inverse Fourier transform by ðN ðN 1 F K1 ðf~Þðx; yÞ Z f ðxÞ Z f~ðk1 ; k 2 Þeiðxk1Cyk 2 Þ dk1 dk 2 ; ð2:10Þ ð2pÞ2 KN KN pffiffiffiffiffiffi where iZ K1. Taking the representation of (2.2) as a two-dimensional Fourier integral (Mura 1982) ð ð 1 N N k1 ik$x 0 w ZK e dk1 dk 2 ; k 2 Z k 21 C k 22 ; ð2:11Þ 2 2p KN KN k 2 k and substituting it into equation (1.12), we get an algebraic equation for ~ 1 ; k 2 Þ. With the help of the inverse Fourier transform, we obtain the wðk expression for wðx; yÞ. The solution for the displacement uz Z b=½2p
wðx; yÞ reads ð ð 1 N N k1 wðx; yÞ ZK eik$x dk1 dk 2 2 2 2p KN KN k 2 k ð1 C c1 k 2 Þð1 C c22 k 2 Þ ! ðN ðN 2 2 1 1 k c c 1 1 2 Z w0 C K eik$x dk1 dk 2 : ð2:12Þ 2p c12 Kc22 KN KN k 2 k 2 C c12 k 2 C c12 1

Proc. R. Soc. A (2006)

2

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3470

M. Lazar and G. A. Maugin

The total distortion bT zx is calculated as bT zx Zvx uz ð ð bz N N ik12 ZK eik$x dk1 dk 2 ð2pÞ2 KN KN k 2 k 2 ð1Cc12 k 2 Þð1Cc22 k 2 Þ  ð ð  bz N N ik 2 i Z K eik$x dk1 dk 2 2 2 2 2 2 2 2 2 2 2 ð2pÞ KN KN k ð1Cc1 k Þð1Cc2 k Þ k 2 ð1Cc1 k Þð1Cc2 k Þ b Z z 2 ð2pÞ

ðN ðN KN KN

ik 2 c12 ik 2 c22 ik 2 K C 2 2 2 2 2 2 2 1 c1 Kc2 k C c2 c1 Kc2 k C c12 k 1

i K

k 2 ð1Cc12 k 2 Þð1Cc22 k 2 Þ

2



eik$x dk1 dk 2

Zbzx CbPzx :

ð2:13Þ

With the help of the Fourier integral representations (Wladimirow 1971) ð ð 1 N N 1 ik$x e dk1 dk 2 ZKln r; ð2:14Þ 2p KN KN k 2 ð ð 1 N N 1 eik$x dk1 dk 2 Z K0 ðr=cÞ; ð2:15Þ 2 2p KN KN k C c12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r Z x 2 C y 2 , Kn denotes the modified Bessel function of order n and introducing the function   bz 1 2 2 FZ ½c1 K0 ðr=c1 ÞKc2 K0 ðr=c2 Þ
; ð2:16Þ ln r C 2 2p c1 Kc22 it can be decomposed into the elastic distortion   bz y 1 1K 2 ½c1 rK1 ðr=c1 ÞKc2 rK1 ðr=c2 Þ
; bzx ZKvy F ZK 2p r 2 c1 Kc22 and the plastic distortion ðN ðN bz i P bzx ZK eik$x dk1 dk 2 : 2 2 2 ð2pÞ KN KN k 2 ð1 C c1 k Þð1 C c22 k 2 Þ

ð2:17Þ

ð2:18Þ

In the limits c1 / 0 and c2 / 0, equation (2.18) converts into the classical plastic distortion given in the Fourier integral representation ðN ðN bz i ik$x 0;P bzx ZK e dk1 dk 2 ZKbz dðxÞH ðKyÞ: ð2:19Þ 2 ð2pÞ KN KN k 2 For the sake of convergence of the k 2 -integral in equation (2.19), it is customary to give k 2 an infinitesimally negative imaginary part. In addition, the plastic distortion (2.18) is a rigorous solution of (1.13) written in the double Fourier Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

Dislocations in gradient elasticity

3471

integral form. The other non-vanishing component of the total distortion reads bT zy Z vy uz ðN ðN bz ik1 ZK eik$x dk1 dk 2 2 2 2 2 2 2 ð2pÞ KN KN k ð1 C c1 k Þð1 C c2 k Þ ! ðN ðN bz 1 c12 1 c22 1 eik$x dk1 dk 2 ZK ik1 2 K 2 C c1 Kc22 k 2 C c12 c12 Kc22 k 2 C c12 k ð2pÞ2 KN KN 1 2 Z bzy C bPzy ; ð2:20Þ which has the decomposition   b x 1 bzy Z vx F Z z 2 1K 2 ½c rK ðr=c ÞKc rK ðr=c Þ
; 1 1 1 2 1 2 2p r c1 Kc22

ð2:21Þ

and bPzy Z 0. Like in classical elasticity, the screw dislocation has only one nonvanishing component of the plastic distortion tensor. The non-vanishing component of the plastic distortion gives rise to the following dislocation density ðN ðN bz 1 P azz Z vy bzx Z eik$x dk1 dk 2 2 2 2 2 2 ð2pÞ KN KN ð1 C c1 k Þð1 C c2 k Þ ! ðN ðN bz 1 1 1 eik$x dk1 dk 2 Z K ð2pÞ2 c12 Kc22 KN KN k 2 C c12 k 2 C c12 1

Z

2

bz 1 ½K ðr=c1 ÞKK0 ðr=c2 Þ
: 2 2p c1 Kc22 0

ð2:22Þ

In the limits c1 / 0 and c2 / 0, (2.22) goes to the classical expression (2.8). Additionally, (2.22) is a rigorous solution of (1.15). Equations (2.17) and (2.21) are proper solutions of the inhomogeneous bi-Helmholtz equation (1.14). We note that equations (2.17), (2.21) and (2.22) agree with the expressions recently given by Lazar et al. (2006) by means of the stress function method. In order to simplify the expressions for the displacement and the plastic distortion, we integrate out the variable k 2 in equations (2.12) and (2.18). So the double Fourier-type integral can be reduced to a single Fourier integral. Because of the symmetry of the single Fourier integrals the displacement may be written as Fourier-sine integral and the plastic distortion as Fourier-cosine integral. In addition, we re-label k1 Z s for convenience. In this way, the displacement is given as w Z w0 C

K

c12 c12 Kc22

c22 c12 Kc22

Proc. R. Soc. A (2006)

ðN 0

ðN 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s sinðsxÞ Kjyj s2Cð1=c12 Þ C 2H ðKyÞ
ds ½sgnðyÞe s2 C c12 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s sinðsxÞ Kjyj s2Cð1=c22 Þ C 2H ðKyÞ
ds; ½sgnðyÞe s2 C c12 2

ð2:23Þ

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3472

M. Lazar and G. A. Maugin

where sgnðyÞ denotes the signum function defined by ( K1 y! 0; sgnðyÞ Z 1 yO 0;

ð2:24Þ

or in terms of the Heaviside function sgnðyÞ Z 2H ðyÞK1:

ð2:25Þ

The plastic distortion reads ðN pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bz c12 cosðsxÞ P Kjyj s2Cð1=c12 Þ ½sgnðyÞe bzx ZK C 2H ðKyÞ
ds 2p c12 Kc22 0 1 C c12 s2 ðN pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bz c22 cosðsxÞ Kjyj s2Cð1=c22 Þ C C 2H ðKyÞ
ds: ð2:26Þ ½sgnðyÞe 2p c12 Kc22 0 1 C c22 s2 It is non-singular unlike the expressions for the plastic distortion calculated by Gutkin & Aifantis (1996). Of course, the cosine representation (2.26) of the plastic distortion gives the correct form of the dislocation density (2.22). The Fourier-cosine integral representation for the dislocation density has the form 8 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ðN < Kjyj s2Cð1=c12 Þ Kjyj s2Cð1=c22 Þ = b 1 e e qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K qffiffiffiffiffiffiffiffiffiffiffiffiffiffi azz Z z 2 cosðsxÞ ds: ð2:27Þ 2 > 2p c1 Kc2 0 ; : s2 C c12 s2 C c12 > 1

2

Using the Fourier-cosine integral pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eKjyj s Cð1=c Þ cosðsxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds Z K0 ð x 2 C y2 =cÞ; 0 s2 C c12

ð2:28Þ

the closed-form expression (2.22) is recovered in a consistent way. For further applications, the single Fourier-type integrals (2.23) and (2.26) may be evaluated numerically, e.g. by fast Fourier transform algorithms. When y/ 0, the integrals (2.23) and (2.26) may be evaluated in an explicit form. Using the integral relations ðN sinðsxÞ p ds Z sgnðxÞ; ð2:29Þ s 2 0 ðN s sinðsxÞ p ds Z sgnðxÞeKjxj=c ; ð2:30Þ 1 2 2 0 s C c2 the displacement has the form   p c12 c22 Kjxj=c1 Kjxj=c2 : wðx; 0Þ ZK sgnðxÞ 1K 2 e C 2 e 2 c1 Kc22 c1 Kc22

ð2:31Þ

The gradient terms which appear in (2.31) lead to a smoothing of the displacement profile unlike the jump occurring in the classical solution Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3473

Dislocations in gradient elasticity (a) 1.0

uz (x, 0)

0.5 0

– 0.5 –1.0 (b) 1.0

uz (x, 0)

0.5 0

– 0.5 –1.0 –6

–4

–2

0 x /c1

2

4

6

Figure 2. Displacement field uz in gradient elasticity of bi-Helmholtz type (solid line) and Helmholtz type (small dashed line) are given in units of bz =4: (a) c2 Z c1 =2, (b) c2 Z c1 . The dashed curves represent the classical component uz0 .

(see figure 2). Of course, the smoothing depends on the two gradient coefficients c1 and c2 . The asymptotic limits are uz ðx; 0Þ Z

bz 4

b as x /KN and uz ðx; 0Þ ZK z 4

as x /N:

ð2:32Þ

With this smoothing of the displacement profile the width of the dislocation (dislocation core radius) may be defined as a function of the gradient parameters c1 and c2 . By using the integral ðN 0

cosðsxÞ p eKjxj=c ds Z ; 2 c 1 C c2 s2

ð2:33Þ

the plastic distortion reads b p 1 ½c eKjxj=c1 Kc2 eKjxj=c2
: bPzx ðx; 0Þ ZK z 2 2p 2 c1 Kc22 1

ð2:34Þ

In contrast to the classical plastic distortion which is singular at xZ0 due to dðxÞ in equation (2.19), equation (2.34) is smooth there (see figure 3). For completeness, the expressions for the first gradient elasticity are listed. They are obtained from the second gradient results in the limit c2 / 0 and c1 Z 3. Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3474

M. Lazar and G. A. Maugin 1.0 –bzPx (x, 0)

0.8 0.6 0.4 0.2 0 1.0

–bzPx (x, 0)

0.8 0.6 0.4 0.2 0 –6

–4

–2

0 x/ c1

2

4

6

Figure 3. Plastic distortion in gradient elasticity of bi-Helmholtz type (solid line) and Helmholtz type (small dashed line) are given in units of bz =4: (a) c2 Z c1 =2, (b) c2 Z c1 .

Of course, they satisfy equations (1.2)–(1.5). They read ð ð 1 N N k1 1 0 w Zw C eik$x dk1 dk 2 2 2p KN KN k 2 k C 312 0

Zw C

ðN 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s sinðsxÞ Kjyj s2Cð1=32 Þ C 2H ðKyÞ
ds; ½sgnðyÞe s2 C 312

o b y n r bzx ZK z 2 1K K1 ðr=3Þ ; 3 2p r o bz x n r 1K ðr=3Þ ; K 1 3 2p r 2 ðN ðN b i bPzx ZK z 2 eik$x dk1 dk 2 2 2 k ð1 C 3 k Þ ð2pÞ KN KN 2 ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bz N cosðsxÞ Kjyj s2Cð1=32 Þ C 2H ðKyÞ
ds; ZK ½sgnðyÞe 2p 0 1 C 32 s2

bzy Z

azz Z Proc. R. Soc. A (2006)

bz 1 K ðr=3Þ: 2p 32 0

ð2:35Þ

ð2:36Þ ð2:37Þ

ð2:38Þ

ð2:39Þ

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

Dislocations in gradient elasticity

3475

Let us compare these results with the formulae given by Gutkin & Aifantis (1996). Equations (2.36) and (2.37) have the same form as their result. On the other hand, equations (2.35) and (2.38) look similar as their expression and, anyway, they have an important difference. Due to the choice Karctanðx=yÞ instead of arctanðy=xÞ for w 0 an additional term, 2H ðKyÞ appears in equations (2.35) and (2.38). This Heaviside term guarantees the satisfying of the inhomogeneous Helmholtz equations and it cancels the classical dðyÞ-terms which would appear due to the differentiation of the sgnðyÞ-function. In addition, it can be seen in figures 2 and 3 that the second order expressions are smoother than the first order results. The elastic strain, stress and the higher order stresses of a screw dislocation in the second gradient elasticity have been given by Lazar et al. (2006). 3. Edge dislocation (a ) Classical solution Consider now an edge dislocation whose Burgers vector is parallel to the x-axis and the dislocation line is along the z-direction. We are using the classical displacement with the branch cut at xZ0 and for y!0 given by Leibfried & Lu ¨cke (1949) (e.g. Seeger 1955)   bx xy 0 0 ux Z ; ð3:1Þ w ðx; yÞ C 2p 2ð1KnÞr 2   bx x2 0 uy ZK ð3:2Þ ð1K2nÞln r C 2 ; 4pð1KnÞ r where w 0 ðx; yÞ is given by equation (2.2) and n is the Poisson’s ratio. The displacement (3.1) was also given by Nabarro (1967). But his uy0 differs by a constant, because he used Ky 2 =r 2 instead of x 2 =r 2 . Of course, this constant value is not significant in elasticity theory. We note that only wðx; yÞ is discontinuous due to the jump. All other parts of the displacements (3.1) and (3.2) are continuous. In addition, the first part of (3.2) has a logarithmic singularity. The expression (3.2) was also given by Mura (1969) and DeWit (1973). The nonvanishing components of the elastic distortion are (DeWit 1973)   bx y 2x 2 0 bxx ZK ð3:3Þ ð1K2nÞ C 2 ; 4pð1KnÞ r 2 r   bx x 2y2 0 bxy Z ð3K2nÞK 2 ; ð3:4Þ 4pð1KnÞ r 2 r   bx x 2y 2 0 ð3:5Þ ð1K2nÞ C 2 ; byx ZK 4pð1KnÞ r 2 r   bx y 2x 2 0 ð3:6Þ ð1K2nÞK 2 ; byy ZK 4pð1KnÞ r 2 r Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3476

M. Lazar and G. A. Maugin

which are singular at rZ0. The plastic distortion reads b0;P xx ZKbx dðxÞH ðKyÞ;

ð3:7Þ

which is caused by the jump of wðx; yÞ. The dislocation density tensor has the following non-vanishing component (e.g. DeWit 1973) a0xz Z vy b0;P xx Z bx dðxÞdðyÞ:

ð3:8Þ

(b ) Gradient solution Using the same technique to solve the inhomogeneous bi-Helmholtz equations (1.12)–(1.15) for the edge dislocation as in §2 for the screw dislocation, we obtain rigorous solutions for all field quantities. Here, we just give the results in order to avoid the same technical details. The solution for the displacement reads  bx xy xy ux Z 2ð1KnÞwðx; yÞ C 2 K4ðc12 C c22 Þ 4 4pð1KnÞ r r  2 xy 2 ½c K ðr=c1 ÞKc22 K2 ðr=c2 Þ
; C 2 2 2 1 2 ð3:9Þ c1 Kc2 r    bx 1 ð1K2nÞ lnr C 2 uy ZK ½c12 K0 ðr=c1 ÞKc22 K0 ðr=c2 Þ
4pð1KnÞ c1 Kc22  x2 x 2 Ky2 1 x 2 Ky2 2 2 C 2 K2ðc12 Cc22 Þ C ½c K ðr=c ÞKc K ðr=c Þ
; ð3:10Þ 1 2 1 2 2 2 c12 Kc22 r 2 r r4 where wðx;yÞ is given by (2.12) and (2.23). The displacement (3.10) is plotted in figure 4. It is interesting to note that if we substitute Ky2 =r 2 instead of x 2 =r 2 in the second part of equation (3.10), the expression that depends on the gradient coefficients remains the same. The elastic distortion is calculated as  bx y 2x 2 4ðc12 Cc22 Þ 2 bxx ZK ð1K2nÞC C ðy K3x 2 Þ 4pð1KnÞ r 2 r2 r4 2ðy2 Knr 2 Þ ½c1 rK1 ðr=c1 ÞKc2 rK1 ðr=c2 Þ
r 2 ðc12 Kc22 Þ  2ðy 2 K3x 2 Þ 2 2 K 2 ½c1 K2 ðr=c1 ÞKc2 K2 ðr=c2 Þ
; ðc1 Kc22 Þr 2  bx x 2y2 4ðc12 Cc22 Þ 2 bxy Z ð3K2nÞK K ðx K3y2 Þ 2 2 4 4pð1KnÞ r r r K

2ðy2 Cð1KnÞr 2 Þ ½c1 rK1 ðr=c1 ÞKc2 rK1 ðr=c2 Þ
ðc12 Kc22 Þr 2  2ðx 2 K3y 2 Þ 2 2 C 2 ½c1 K2 ðr=c1 ÞKc2 K2 ðr=c2 Þ
; ðc1 Kc22 Þr 2

ð3:11Þ

K

Proc. R. Soc. A (2006)

ð3:12Þ

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3477

Dislocations in gradient elasticity (a)

0

uy (x, 0)

– 0.5 –1.0 –1.5 –2.0 (b)

0

uy (x, 0)

– 0.5 –1.0 –1.5 –2.0 –10

–5

0 x/c1

5

10

Figure 4. Displacement field uy in gradient elasticity of bi-Helmholtz type (solid line) and Helmholtz type (small dashed line) are given in units of bx =½4pð1KnÞ
and nZ 0:3. The dashed curves represent the classical component uy0 .

bx x byx ZK 4pð1KnÞ r 2 C

 2y 2 4ðc12 C c22 Þ 2 ð1K2nÞ C 2 C ðx K3y 2 Þ r r4

2ðy 2 Kð1KnÞr 2 Þ ½c1 rK1 ðr=c1 ÞKc2 rK1 ðr=c2 Þ
ðc12 Kc22 Þr 2

 2ðx 2 K3y 2 Þ 2 2 ½c1 K2 ðr=c1 ÞKc2 K2 ðr=c2 Þ
; K 2 ðc1 Kc22 Þr 2 bx y byy ZK 4pð1KnÞ r 2 K

 ð1K2nÞK

2x 2 4ðc12 C c22 Þ 2 K ðy K3x 2 Þ r2 r4

2ðx 2 Knr 2 Þ ½c1 rK1 ðr=c1 ÞKc2 rK1 ðr=c2 Þ
r 2 ðc12 Kc22 Þ

 2ðy2 K3x 2 Þ 2 2 ½c1 K2 ðr=c1 ÞKc2 K2 ðr=c2 Þ
: C 2 ðc1 Kc22 Þr 2 Proc. R. Soc. A (2006)

ð3:13Þ

ð3:14Þ

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3478

M. Lazar and G. A. Maugin

The plastic distortion is given by ðN ðN b i bPxx ZK x 2 eik$x dk1 dk 2 2 2 ð2pÞ KN KN k 2 ð1 C c1 k Þð1 C c22 k 2 Þ b c2 ZK x 2 1 2 2p c1 Kc2 b c2 C x 2 2 2 2p c1 Kc2

ðN 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðsxÞ Kjyj s2Cð1=c12 Þ C 2H ðKyÞ
ds ½sgnðyÞe 1 C c12 s2

ðN 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðsxÞ Kjyj s2Cð1=c22 Þ C 2H ðKyÞ
ds: ð3:15Þ ½sgnðyÞe 1 C c22 s2

It can be seen that it has the same form as the plastic distortion (2.18) and (2.26) for the screw dislocation. The dislocation density of a single edge dislocation reads axz Z vy bPxx Z

bx 1 ½K ðr=c1 ÞKK0 ðr=c2 Þ
; 2p c12 Kc22 0

ð3:16Þ

which has the same form as the expression for a screw dislocation (2.22). Finally, we give the expressions for the first gradient elasticity obtained from the second gradient results in the limit c2 / 0 and c1 / 3. They fulfil equations (1.2)–(1.5). The displacements have the following form:   bx xy 2xy 2 xy ux Z 2ð1KnÞwðx; yÞ C 2 K43 4 C 2 K2 ðr=3Þ ; ð3:17Þ 4pð1KnÞ r r r   2 2 bx x2 x 2 Ky2 2 x Ky C K2 ðr=3Þ ; uy ZK ð1K2nÞðln r C K0 ðr=3ÞÞ C 2 K23 4pð1KnÞ r r2 r4 ð3:18Þ where wðx; yÞ is given by (2.31). Equation (3.17) has a similar form as the expression given by Gutkin & Aifantis (1997). Only, the expression for the discontinuous function wðx; yÞ is different because we used w 0 ðx; yÞ. Equation (3.18) is in full agreement with the formula given by Gutkin & Aifantis (1997). The elastic distortion is obtained as  bx y 2x 2 432 2 2ðy 2 K3x 2 Þ 2 bxx ZK ð1K2nÞ C C ðy K3x ÞK K2 ðr=3Þ 4pð1KnÞ r 2 r2 r2 r4  2ðy2 Knr 2 Þ ð3:19Þ K K1 ðr=3Þ ; 3r 

2y2 432 2 2ðx 2 K3y 2 Þ K 4 ðx K3y2 Þ C K2 ðr=3Þ 2 r2 r r  2ðy 2 C ð1KnÞr 2 Þ ð3:20Þ K1 ðr=3Þ ; K 3r

bx x bxy Z 4pð1KnÞ r 2

Proc. R. Soc. A (2006)

ð3K2nÞK

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

Dislocations in gradient elasticity

3479



2y 2 432 2 2ðx 2 K3y2 Þ 2 C ðx K3y ÞK K2 ðr=3Þ r2 r2 r4  2ðy 2 Kð1KnÞr 2 Þ ð3:21Þ K1 ðr=3Þ ; C 3r

bx x byx ZK 4pð1KnÞ r 2

ð1K2nÞ C

 2x 2 432 2ðy2 K3x 2 Þ ð1K2nÞK 2 K 4 ðy2 K3x 2 Þ C K2 ðr=3Þ r2 r r  2ðx 2 Knr 2 Þ ð3:22Þ K K1 ðr=3Þ ; 3r

bx y byy ZK 4pð1KnÞ r 2

which agree with the formulae given by Lazar (2003a,b). The plastic distortion and the dislocation density read ðN ðN bx i P bxx ZK eik$x dk1 dk 2 2 ð2pÞ KN KN k 2 ð1 C 32 k 2 Þ ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bz N cosðsxÞ ð3:23Þ Kjyj s2Cð1=32 Þ ZK ½sgnðyÞe C 2H ðKyÞ
ds; 2 2 2p 0 1 C 3 s bx 1 K ðr=3Þ: ð3:24Þ 2p 32 0 The elastic strain, stress and the higher order stresses of an edge dislocation in the second gradient elasticity have been recently calculated by Lazar et al. (2006). axz Z

4. Conclusions In this paper, we used special theories of first and second strain gradient elasticity. The second strain gradient theory is a generalization of the first strain gradient elasticity with only one gradient parameter. This gradient theory of second order has two gradient coefficients, only. Using the Fourier transform technique, rigorous solutions for the displacement, plastic distortion, elastic distortion, dislocation density of screw and edge dislocations have been derived in the theory of gradient elasticity. The gradient solutions for the elastic distortion and the dislocation density are given in closed form in terms of modified Bessel functions and they agree with results obtained with a slightly different mathematical technique. The formulae for the displacement and plastic distortion have been obtained as double Fourier as well as single Fourier integrals. An advantage of our solutions for the displacement and plastic distortion is that they are solutions of the governing equations everywhere unlike the solutions obtained earlier by Gutkin & Aifantis (1996, 1997) which do not satisfy the corresponding inhomogeneous Helmholtz equations at yZ0. Our solutions for the plastic distortion of screw and edge dislocations are non-singular and they do not contain Dirac delta terms. In addition, we have found our solutions without using boundary conditions unlike Gutkin & Aifantis (1996, 1997). Proc. R. Soc. A (2006)

Downloaded from rspa.royalsocietypublishing.org on May 15, 2011

3480

M. Lazar and G. A. Maugin

M.L. acknowledges the support by the Laboratoire de Mode´lisation en Me´canique and the Universite´ Pierre et Marie Curie in Paris. G.A.M. benefits from a Max-Planck-Award for international cooperation (2002–2006).

References DeWit, R. 1973 Theory of disclinations IV. J. Res. Natl Bur. Stand. (US) 77A, 607–658. Edelen, D. G. B. 1996 A correct, globally defined solution of the screw dislocation problem in the gauge theory of defects. Int. J. Eng. Sci. 34, 81–86. (doi:10.1016/0020-7225(95)00081-X) Gutkin, M. Yu. & Aifantis, E. C. 1996 Screw dislocation in gradient elasticity. Scripta Mater. 35, 1353–1358. (doi:10.1016/1359-6462(96)00295-3) Gutkin, M. Yu. & Aifantis, E. C. 1997 Edge dislocation in gradient elasticity. Scripta Mater. 36, 129–135. (doi:10.1016/S1359-6462(96)00352-1) Gutkin, M. Yu. & Aifantis, E. C. 1999 Dislocations in gradient elasticity. Scripta Mater. 40, 559–566. (doi:10.1016/S1359-6462(98)00424-2) Kro¨ner, E. 1993 Theory of crystal defects and their impact on material behaviour. In Modelling of defects and fracture mechanics (ed. G. Herrmann), pp. 61–117. Wien, Austria: Springer. Lazar, M. 2002a An elastoplastic theory of dislocations as a physical field theory with torsion. J. Phys. A: Math. Gen. 35, 1983–2004. (doi:10.1088/0305-4470/35/8/313) Lazar, M. 2002b Screw dislocations in the field theory of elastoplasticity. Ann. Phys. (Leipzig) 11, 635–649. (doi:10.1002/1521-3889(200210)11:9!635::AID-ANDP635O3.0.CO;2-8) Lazar, M. 2003a A nonsingular solution of the edge dislocation in the gauge theory of dislocations. J. Phys. A: Math. Gen. 36, 1415–1437. (doi:10.1088/0305-4470/36/5/316) Lazar, M. 2003b Dislocations in the field theory of elastoplasticity. Comput. Mater. Sci. 28, 419–428. (doi:10.1016/j.commatsci.2003.08.003) Lazar, M. & Maugin, G. A. 2005 Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184. (doi:10.1016/ j.ijengsci.2005.01.006) Lazar, M., Maugin, G. A. & Aifantis, E. C. 2006 Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817. (doi:10.1016/j.ijsolstr.2005.07.005) Leibfried, G. & Dietze, H.-D. 1949 Zur Theorie der Schraubenversetzung. Z. Phys. 126, 790–808. (doi:10.1007/BF01368757) ¨ ber das Spannungsfeld einer Versetzung. Z. Phys. 126, 450–464. Leibfried, G. & Lu ¨cke, K. 1949 U (doi:10.1007/BF01669489) Mindlin, R. D. 1964 Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78. (doi:10.1007/BF00248490) Mindlin, R. D. 1965 Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438. (doi:10.1016/0020-7683(65)90006-5) Mindlin, R. D. & Eshel, N. N. 1968 On first strain gradients theory in linear elasticity. Int. J. Solids Struct. 4, 109–124. (doi:10.1016/0020-7683(68)90036-X) Mura, T. 1969 Methods of continuously distributed dislocations. In Mathematical theory of dislocations (ed. T. Mura). New York, NY: The American Society of Mechanical Engineers. Mura, T. 1982 Micromechanics of defects in solids. Dordrecht: Martinus Nijhoff. Nabarro, F. R. N. 1967 Theory of crystal dislocations. Oxford, UK: Oxford University Press. Ru, C. Q. & Aifantis, E. C. 1993 A simple approach to solve boundary-value problems in gradient elasticy. Acta Mech. 101, 59–68. (doi:10.1007/BF01175597) Seeger, A. 1955 Theorie der Gitterfehlstellen. In Handbuch der Physik VII/1 (ed. S. Flu ¨gge), pp. 383–665. Berlin: Springer. Sneddon, I. N. 1951 Fourier transform. New York, NY: McGraw-Hill. Teodosiu, C. 1982 Elastic models of crystal defects. Berlin: Springer. Wladimirow, W. S. 1971 Equations of mathematical physics. Berlin: VEB Deutscher Verlag der Wissenschaften. [In German.]

Proc. R. Soc. A (2006)