On a continuum theory of dislocation equilibrium

International Journal of Engineering Science 106 (2016) 10–28

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On a continuum theory of dislocation equilibrium Victor L. Berdichevsky∗ Mechanical Engineering, Wayne State University, Detroit MI 48202, USA

a r t i c l e

i n f o

Article history: Received 31 March 2016 Revised 3 May 2016 Accepted 9 May 2016

Keywords: Continuum theory of dislocations Dislocation equilibrium Plastic deformation of twisted beams

a b s t r a c t A continuum theory of dislocations is suggested which is capable of predicting the equilibrium distributions of a large number of screw dislocations in anisotropic beams with arbitrary cross-section. The theory leads to a boundary value problem with unknown boundary. The problem is solved for isotropic beams with circular cross-sections and thin rectangular cross-sections. The solution for circular cross-sections is compared with the results of numerical simulations by [Weinberger, C. R., 2011. The structure and energetics of, and the plasticity caused by, Eshelby dislocations, International Journal of Plasticity, 27, 1391– 1408]. An extension of the theory to equilibrium of edge dislocations is discussed. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Search for continuum theory adequately describing the behavior of dislocation networks remains a fundamental incomplete task of solid mechanics. Among most recent works concerned with this task note the papers by Ghoniem, Tong, and Sun (20 0 0); El-Azab (20 0 0); Groma, Csikor, and Zaiser (20 03); Zaiser and Hochrainer (20 06); Berdichevsky (20 06a, b; 2016); Limkumnerd and Giessen (2008); Mesarovic, Baskaran, and Panchenko (2010); Geers, Peerlings, Peletier, and Scardia (2013); Poh, Peerlings, Geers, and Swaddiwudhipong (2013a, b); Hochrainer, Sandfeld, Zaiser, and Gumbsch (2014); Zaiser (2015); Hochrainer (2015, 2016); Le (2016); Mohamed, Larson, Tischler, and El-Azab (2015). In this paper one aspect of the subject is considered which does not seem to have been addressed previously. Evolution of dislocation networks ends up with some equilibrium states. Dynamical continuum theory of dislocations must predict, in particular, these equilibrium states. Therefore, it is interesting to see what is the continuum theory which describes the known equilibrium states of a large number of dislocations. Such continuum theory must be a consequence of a general continuum theory of dislocations, if the general theory exists. Herein, a continuum theory of dislocations is constructed to describe equilibria of a large number of screw dislocations in beams. Analysis of screw dislocation equilibria is the most simple problem of dislocation statics, and prediction of screw dislocation equilibria is a convenient proving ground for any continuum theory of dislocation. For isotropic beams with circular cross-sections the equilibrium positions of screw dislocations were found numerically by Weinberger (2011). Some of the equilibrium positions are shown in Fig. 1. At first glance, the existence of equilibrium positions is striking, because straight screw dislocations are similar to 2D charges, while according to Earnshaw’s theorem of electrostatics, there are no stable equilibrium positions of charges. The remarkable observation by Eshelby (1953) was that the analogy of screw dislocations with free charges holds only for zero twist; it breaks down if the beam is allowed to twist. Twist acts on screw dislocations as a potential well, and screw dislocations in a twisted beam behave as charges in a continuously charged

∗

Tel.: +15175223229. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.ijengsci.2016.05.003 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

11

Fig. 1. Equilibrium positions of screw dislocations in a beam with circular cross-section Weinberger, 2011; (a) - torque-free beam (10 0, 20 0 and 50 0 dislocations), (b) - twisted beam (50 dislocations, torque compresses the dislocation cloud; in the Figure torque increases from left to right, the starting equilibrium position is torque-free).

Fig. 2. Location of plastic zone according to classical plasticity theory.

media. Screw dislocations of the same sign repel and leave the beam,1 if twist is zero. In a twisted beam, dislocations cannot run away and unavoidably find equilibrium positions. Fig. 1 seems contradicting to classical plasticity theory and to the observed behavior of real metals, where plastic region is formed near the boundary while no plastic deformation is developed in the center of the beam (Fig. 2). The “classical” behavior is caused by a dense dislocation network which traps the dislocations nucleated near the boundary. The picture shown in Fig. 1 pertains to defect-free crystals when dislocations can move freely inside the beam; such case may be relevant for nanowires. This is the case which will be considered here. If the number of dislocations is large, then the averaged dislocation density could be approximated by a continuous function, and there should be a way to find equilibrium distribution of dislocations within the framework of a continuum theory. Note that phenomenological continuum theories for dissipative twist of nanowires were considered by Gao, Huang, and Hutchinson (1999); Hiang, Gao, Nix, and Hutchinson (20 0 0); Kaluza and Le (2011); Bardella Panteghini (2015). Here we focus only on dislocation equilibria and dissipation is ignored. Besides, Peierls forces are assumed to be negligible, while slip is not constrained by certain slip planes, thus dislocations can move freely in any direction. We find energy of continuum theory from homogenization reasoning. The basic result is that in the limit of an infinite number of dislocations the energy of the dislocation ensemble is just the energy an elastic body with eigenstrains,



Ue (ε − ε¯ p )dV,

(1)

where Ue is elastic energy density, ε the total strain, and the eigenstrain ε¯ p the averaged plastic strain of the dislocation ensemble. To find the equilibrium dislocation cloud one has to minimize energy not only with respect to displacements, but also with respect to plastic strains. The problem is reminiscent of problems of theory of phase transitions, where some “eigenstrains” ε p take certain values in each phase. In contrast to theory of phase transitions, in dislocation equilibrium ε¯ p are unknown and should be found by energy minimization. It is essential that variations of plastic strains are not arbitrary and should be generated by dislocation shifts from equilibrium positions. This yields a boundary value problem with

1 This is the case for traction-free boundary. If the boundary points are clamped, then dislocations are repelled from the boundary. For a large number of dislocations mutual repulsion of dislocations exceeds the boundary repulsion and dislocations concentrate near the boundary (Berdichevsky, 2005).

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unknown boundary. As in theory of phase transitions, the unknown boundary is determined by continuity of the normal component of energy-momentum tensor. The problem is formulated in quite general setting of anisotropic beams with arbitrary cross-sections. It is solved for beams with circular and thin rectangular cross-sections. Numerical data provided by Weinberger are compared with the solution obtained. The extension of the theory to edge dislocations is also discussed. We begin with the case of isotropic beams with circular cross-section (Section 2), then discuss two interpretations of dislocation density, analytical and probabilistic (Section 3); formulation of continuum theory for circular cross-section is completed in Section 4; in this section it is given also the solution of the problem for the circular cross-section; an alternative approach for derivation of energy relation (1) is presented in Section 5; the alternative approach admits a straightforward extension to anisotropic beams with arbitrary cross-sections (Section 6), and to the case of edge dislocations and (Section 8); equilibrium clouds of screw dislocation in anisotropic beams with thin rectangular cross-sections are found in Section 7. 2. Screw dislocations in circular isotropic beam Twist of elastic beam. Consider a long isotropic elastic circular beam referred to Cartesian coordinates x, y, z,

x2 + y2  R2 ,

0  z  L,

R  L.

Lateral surface is traction-free. The beam ends are twisted for angle ϕL , and displacements u, v along x, y−axis at the beam ends are

u=v=0

u = ϕL y,

at z = 0,

v = −ϕL x,

at z = L.

For such deformation in-plane strains ε xx , ε xy , ε yy and elongation ε zz are zero everywhere except edge zones with widths of the order R. The only non-zero strain components are shears ε xz and ε yz ,

2εxz =

∂w ∂u + , ∂x ∂z

2εyz =

∂ w ∂v + , ∂y ∂z

w being the z−component of displacements, the warping. The elastic beam energy is

L



0

x2 +y2 R2

μ

E=



2

∂w ∂u + ∂x ∂z

2



∂ w ∂v + + ∂y ∂z

2 

dxdydz,

(2)

Minimization of energy (2) yields that warping w is zero while the beam cross-sections rotate as rigid bodies for an angle ϕ linearly increasing along the cylinder

u = −ϕ y,

v = ϕ x, ϕ =  z,

(3)

 = ϕL /L is the twist. Due to Saint–Venant principle the precise form of the end conditions is not essential. From (2) and (3) we find the beam energy

E=

1 μ 2 π R4 L. 4

(4)

Screw dislocation in elastic beam. Allow now a screw dislocation to enter the beam. This means that warping w(x, y) has a jump b at a strip, which is parallel to z−axis. In order to find warping one has to minimize energy per unit length

E = min w L



x2 +y2 R2

μ 2



∂w − y ∂x

2



∂w + + x ∂y

2 

dxdy

(5)

over all functions w(x, y) which have a prescribed jump on the slip line  (Fig. 3)

[w] = b on . Symbol [ϕ ] denotes the difference of values of function ϕ on the two sides + and − of the slip line . If b = const, then energy is diverging, and a regularization is needed. Regularization. Various regularizations are possible. It is convenient to do regularization in the variational principle which is dual to (5). Assume that the jump b is a smooth function on  vanishing at (x0 , y0 ). Using the general recipe for construction of dual variational principles (see, e.g., Berdichevsky, 2009 Sections 5, 6 and 6.10), one presents the integrand as the maximum value of a quadratic function of dual variables σ x , σ y :

2  2  ∂w ∂w − y + + x 2 ∂x ∂y      1 2 ∂w ∂w 2 = max σx −  y + σy + x − σ + σy . σx ,σy ∂x ∂y 2μ x

μ



(6)

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

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Fig. 3. Notation for one screw dislocation in a beam.

Apparently, σ x and σ y have the meaning of stress components σ xz and σ yz , respectively. Plugging (6) in (5) and minimizing over w(x, y) one obtains the dual variational problem

⎡

E = max ⎣ σx ,σy L







x2 +y2 R2

 1 2 −σx  y + σy  x − σ + σy2 dxdy − 2μ x



⎤

(σx nx + σy ny )bds⎦.

(7)



Here s is the arc length on , and maximum is sought over all functions σ x , σ y such that

∂σx ∂σy + =0 ∂x ∂y

(8)

and

σx nx + σy ny = 0 at x2 + y2 = R2 ,

[σ x ]n x + [σ y ]n y = 0

on .

(9)

The unit normal vector (nx , ny ) is directed on  from the side − to the side + . The general solution of (8), (9) can be conveniently written in terms of stress function ψ (x, y):

σx =

∂ψ , ∂y

σy = −

∂ψ , ∂x

[ψ ] = 0 on ,

 ψ  x2 +y2 =R2 = 0.

(10)

Let us introduce plastic distortions,

βx = bnx δ (),

βy = bny δ (),

(11)

where δ () is δ −function of the slip line. Then the last integral in (7) can be written as



(σx βx + σy βy )dxdy.

x2 +y2 R2

In terms of stress function ψ (x, y), the dual variational problem takes the form

E = − min L ψ





1 2μ

x2 +y2 R2



∂ψ ∂x

2



∂ψ + ∂y

Integrating by parts and using (10) we finally get

E = − min L ψ



x2 +y2 R2



1 2μ



∂ψ ∂x

2



∂ψ + ∂y

2 

 ∂ψ ∂ψ ∂ψ ∂ψ + x+ y + βx − βy dxdy. ∂x ∂y ∂y ∂x

2 

 − 2ψ + αψ dxdy.

Here α is the dislocation density of screw dislocations,2

α= 2

(12)

∂βy ∂βx − . ∂x ∂y

These 2D notation relates to 3D one as follows: β x ↔β zx , β y ↔β zy , α ↔α zz .

(13)

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Minimum is sought over all ψ vanishing at the boundary. The minimizer is a solution of the boundary value problem

 ψ  x2 +y2 =R2 = 0.

ψ = μ(α − 2 ),

(14)

If b = const, then α (x, y) becomes δ −function

α (x, y ) = bδ (x − x0 )(y − y0 ).

(15)

Note that dislocation density does not depend on the slip line. Energy diverges for dislocation density (15). To have finite energy we use the following regularized dislocation density,

b

α (x, y ) =

π a2

χ (x − x0 , y − y0 ),

(16)

where χ (x, y) is the characteristic function of the circle of radius a,



x2 + y2  a2 x2 + y2 > a.

1 0

χ (x, y ) =

Parameter a can be determined by the condition that elastic energy of the beam with a screw dislocation be equal to the sum of elastic energy outside of dislocation core and energy of dislocation core. Plastic deformation. If α = 0, then the minimizer of the variational problem (13) is

 1 ψ = μ R2 − x2 − y2 , 2

and energy takes the value of energy of elastic twisted beam (4). If α = 0, then energy can be reduced by screw dislocations entering the beam. In physical terms, the beam is deformed plastically. The corresponding plastic distortion is given by (11). First, following (Eshelby, 1953) we consider the case of one dislocation.3 One screw dislocation. For arbitrary dislocation density α (r), r = {x, y}, the solution of the boundary value problem (14) is



ψ (r ) = −μ

G(r, r˜)(α (r˜) − 2 )d2 r˜

(17)

where G(r, r˜) is Green’s function of the circle





 R2 − zz˜∗  1 , G(r, r˜) = ln  2π  R(z − z˜) 

(18)

z = x + iy,4 z˜ = x˜ + iy˜, r˜ = {x˜, y˜}, z∗ is complex conjugate of z. Energy is a functional of dislocation density,

I (α ) =

1 2



ψ (r )(2 − α (r ) )d2 r =

μ

 

2

G(r, r˜)(2 − α (r ) )(2 − α (r˜) )d2 rd2 r˜.

(19)

It is convenient to write (19) extracting explicitly quadratic and linear functionals of dislocation density,

I (α ) = +

μ



2

π R4 4

1 2

μ 2 − μ



 α ( r ) R2 − x2 − y2 d 2 r

G(r, r˜)α (r )α (r˜)d2 rd2 r˜.

Here we used that



G(r, r˜)d2 r˜ =

 1 2 R − x2 − y2 , 4

(20)



G(r, r˜)d2 rd2 r˜ =

π R4 8

.

(21)

Consider one dislocation located at r0 = {x0 , y0 }. Plugging (16) and (18) in (20) and keeping only the leading terms of energy as a → 0, we find5

I ( r0 ) =

π R4 4

μ 2 −

μ bR2 2



1−

|r0 |2 R2



+

    μb2 R 1 |r0 |2 ln 1 − 2 + ln + . 4π a 4 R

(22)

3

The consideration here is slightly different from that of Eshelby. Further the beam axis coordinate z will not appear in our consideration, and this notation for complex variable z should not cause confusion. 5 In the computation of the convolution of Green’s function (18) with dislocation density it is used that for a → 0 the leading term of Green’s function is 1/2π ln(|R2 − |r0 |2 |/R|z − z˜| ). This is true for all points of unit circle except a small layer near the boundary |r| = R of the thickness of order a. In this layer our dislocation model fails anyway. The emerging integral, 4



 ln

|r|1 |r˜|1

1

d2 r d2 r˜

|r − r˜| π π

,

is equal to 1/4, the corresponding calculation is given in Berdichevsky (2009), p. 439, where this integral is found in computation of kinetic energy of vortex filaments.

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

15

/

Fig. 4. Energy relief for one dislocation as a function of dimensionless radius r/R; dotted line: ¯ = 1, upper solid line: ¯ = 2, low solid line: zero twist

¯ = 0, dashed line corresponds to zero torque.

One can check that the last term of (22), 1/4 (μb2 /4π ), is elastic energy of the tube of radius a surrounding the dislocation line. If this term is dropped in (22), one gets energy of the elastic beam outside of the tube. Let a be the radius of the dislocation core. Then, in order to obtain the total energy of the system, one has add the energy of the dislocation core, which is of the form cμb2 /4π , c being some numerical constant. The sum of μb2 ln (R/a)/4π and cμb2 /4π can be written as μb2 ln (R/a1 )/4π , where a1 is the ”effective” core radius, a1 = a exp(−c ). For example, from atomistic calculations for molybdenum by Park et al. (2012), a = 2b, c = 1.7, a1 = 0.36b. Finally, energy of screw dislocation in a twisted beam is

I ( r0 ) =

π R4 4

μ − 2

μ bR2 2



1−

|r0 |2 R2



    μb2 R |r0 |2 + ln 1 − 2 + ln . 4π a1 R

(23)

Denote by  the part of energy associated with dislocation position and referred to μb2 /4π , and by ¯ the dimensionless parameter ¯ = 2π R2  /b. The dimensionless energy is

1 1 I (r ) = ¯ 2 + (r0 ) + A, 4 μb2 /4π 0

(r0 ) = ¯

|r0 |2 R2



+ ln 1 −

|r0 |2 R2



,

A = −¯ + ln

R . a1

(24)

Function (r0 ) is shown in Fig. 4 for different values of ¯ . The qualitative change of the dislocation behavior occurs for ¯ = 1. If ¯ < 1 (including the case of negative ¯ , i.e. the case when twist ϰ and Burgers vector b have opposite signs), then there is only one equilibrium position of dislocation at the center of the beam. This position is unstable. If ¯ > 1, then this position becomes stable, and a circle of equilibrium unstable positions arises. The difference of beam energies with dislocation and without dislocation is equal to A. In order for a dislocation to enter the beam, A must be negative. If twist is large enough, then A < 0, and a dislocation can appear. To determine the minimum value of twist required for nucleation of dislocation is determined from the equation A = 0. The minimum value of dimensionless twist ϰR needed for nucleation of dislocation is

 R R  R = ln /2π . a1

b

This function is shown in Fig. 5 for a1 = 0.36b. The required twist decreases with the increase of the beam radius. To enter the beam from the lateral surface, a straight screw dislocation must overcome the energy barrier of infinite magnitude. Apparently, this is impossible, and most likely the screw dislocations penetrate through the energy barrier by means of jogs. The case of beams with zero torque is slightly different. Assume that there is a screw dislocation in the beam, and the beam ends are traction free. Then twist will adjust to minimize energy (22). The minimizing twist ϰ depends on the dislocation position,



b = π R2



1−

|r0 |2 R2



.

(25)

The screw dislocation becomes a helix with curvature (25). Energy of dislocation in a torque-free beam is

   2   μb2 R |r0 |2 |r0 |2 I ( r0 ) = − 1− 2 + ln 1 − 2 + ln . 4π a1 R R

(26)

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V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

/ Fig. 5. Minimum dimensionless twist ϰR needed for nucleation of a dislocation as a function of dimensionless radius of cross-section R/b.

The dislocation energy relief,

 2   |r |2 |r |2 (r0 ) = 1 − 1 − 02 + ln 1 − 02 , R

R

shifted for a constant to have (0 ) = 0, is shown in Fig. 4 by dashed line. Many screw dislocations. Continuum theory. As is seen from Fig. 4 sufficiently large twist creates a potential well. Clearly, if a set of dislocations with the same sign as twist is placed in the well, then dislocations find some equilibrium positions due to mutual repelling and impossibility to escape from the well. Of course, there must be some maximum number of dislocations which a given twist can keep in equilibrium. If the number of dislocations is large, then some continuum theory should describe the equilibrium state. Consider what kind of continuum dislocation theory arises in this problem from homogenization reasoning. Let the beam contain N dislocations located at points ri = {xi , yi }. Then

α (r ) =

N  b χ (r − ri ). π a2 i=1

Plugging this function in (20) and keeping the leading terms as a → 0, we get

π R4

I ( ri ) =

4

1 2

μ 2 − μb R2

N  i=1



1−

|ri |2 R2



+

N

μb2 

2



G ri , r j +

i= j

    N μb2  R |ri |2 ln 1 − 2 + ln . 4π a1 R i=1

(27)

The first term in (27) is elastic beam energy in absence of dislocations, the second term is the energy of the potential well introduced by twist ϰ, the third term is the interaction energy of dislocations, and the last one

is the sum of  the dislocation self-energies, μb2 (ln (R/a ) + 1/4 )/4π , and dislocation-boundary interaction energies, μb2 ln 1 − |ri |2 /R2 /4π . Interaction energy with boundaries is contained also in the third term; this is manifested by the dependence of Green’s function on R. Denote by α¯ (r ) the averaged dislocation density; by definition, α¯ (r )d2 r/b is the number of dislocations in a small circle of area d2 r with the center at the point r. If the total number of dislocations is N, then



α¯ (r )d2 r = Nb.

(28)

|r|R

For a large number of dislocations N and a smooth function α¯ (r ), the sums in (27) can be viewed as approximations of integrals. If the sums are replaced by integrals,

I (α¯ ) =

π R4

+

4

μ 2

μ 

2

   1 |r|2 2 − μ R 1 − 2 α¯ (r )d2 r+ 2 R   μb G(r, r˜)α¯ (r )α¯ (r˜)d2 rd2 r˜ + 4π

|r|R |r˜|R

|r|R

  ln 1 −

|r|2 R2

 + ln

R a1

 α¯ (r )d2 r,

(29)

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

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we arrive at a continuum theory of dislocations: equilibrium positions of dislocations should be stationary points of energy functional (29). However, this almost self-evident transition from (27) to (29) requires a more careful analysis. It is considered in the next section. 3. Two views on continuum description Replacement of the sum in (27) by integrals is possible only in the limit N → ∞. We have to specify the behavior of the parameters involved, b, ϰ and R, in this limit. We set ϰ and R to be independent on N. We assume also that dislocation density α¯ (τ ) is independent on N. Then due to (28), Burgers vector must tend to zero in such a way that the product γ = bN is finite. Parameter γ has the meaning of a “plastic warping”. It is convenient to refer energy to μγ 2 = μb2 N2 . We have

1

μγ

π

I ( ri ) = 2

4



 R2 γ

2

−

  N N  1  R2 1  1 

|ri |2 1− 2 + G r i , r j + , 2γ N N R 2N 2 i= j i=1

(30)

 being the sum of dimensionless self-energy and dislocation-boundary interaction energy,

=

N 

1 4π N

 

ln 1 −

i=1

|ri |2 R2





R 1 + ln + . a 4

Dimensionless parameter ϰR has the meaning of elastic shear; within the framework of linear elasticity it should not be taken larger than 10−1 . Dimensionless parameter γ /R could be of the order of ϰR. Therefore, the coefficient in (30), ϰR2 /γ is some finite dimensionless number. Twist-dislocation interaction energy, dislocation-boundary interaction energy and self-energy have the form N 1 (ri ), N

(31)

i=1

where (r) is a smooth function. We have to estimate the error in replacing the sum (31) by the integral



1 b

(r ) α¯ (τ )d2 r.

(32)

V

The error depends on two possible meanings of function α¯ (τ ), which we call analytical and probabilistic. In analytical interpretation, region V is divided in a large number n of boxes (squares) Bα , α = 1, ..., n. Boxes Bα are assumed to have the same area |B|. Let rα be the center of  Bα . It is assumed that Bα contains exactly b−1 α¯ (rα )|B| points ri , and α¯ (r ) is slowly changing on distances of order  = |B|. In probabilistic interpretation, points ri are randomly distributed over V. For different realizations, the number of points in each box Bα varies. The variation of the number of points in a box is huge: it ranges from 0 to N. The value of dislocation density at rα is equal to the average number of points in the box Bα × b/|B|, where averaging is made over all realizations. The dependence of α¯ on rα is smooth in the sense that the interpolation of α¯ (rα ) changes slowly on distances of order  . √ Two interpretations yield different errors. Assume that |B| ∼ |V|/N, thus  ∼ 1/ N. In analytical interpretation, the error order is smaller than 1/N. In probabilistic interpretation, the error order is larger than 1/N. If dislocations are placed in the circle randomly and independently with probability density f (r ) = α¯ (rα )/b, then according to the central limit theorem√of probability theory the sum (31) converges to its mathematical expectation (32) as N → ∞. The error is of the order 1/ N for most realizations. There are realizations, for which the error is of order unity, however probability to encounter such realizations decays as N increases. The double sum also converges to its mathematical expectation, N   1 

G r , r → G(r, r˜) f (r ) f (r˜)d2 rd2 r˜ i j N2

(33)

i= j

√ It is shown in Appendix that the error in approximation of the double sum by its mathematical expectation is also 1/ N. Thus, the error introduced by the transition from discrete dislocation energy to energy of a continuum (29) is larger than the last term in (29). Further we employ the probabilistic interpretation of dislocation density. Thus, there is no reason to keep the last term in (29), and it is dropped. The three remaining terms in (29) might be of the same order, because ϰR2 and γ are some finite numbers. Finally, in continuum theory energy of screw dislocations in circular beams is

I (α¯ ) =

π R4 4

1 2

μ 2 − μ R2





1−

|r|2 R2

  μ α¯ (r )d2 r + 2



|r|R, |r˜|R

G(r, r˜)α¯ (r )α¯ (r˜)d2 rd2 r˜.

(34)

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V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

Note also another form of energy. Due to the first Eq. (21) the twist-dislocation interaction term can be written as

μ R2





|r|2

1−

2

R2

  α¯ (r )d2 r = 2μ G(r, r˜)α¯ (r )d2 rd2 r˜.

Using also the second Eq. (21) obtain

I (α¯ ) =

μ



2

G(r, r˜)(α¯ (r ) − 2 )(α¯ (r˜) − 2 )d2 rd2 r˜.

(35)

Comparing (35) with (19), we see that homogenization yields the replacement of the true dislocation density α (r) by the averaged dislocation density α¯ (r ). This became possible due to smallness of dislocation-boundary interaction energy and self-energy. 4. Continuum theory of screw dislocations Local form of energy. Formula (35) presents energy as a non-local functional of α¯ (r ). Technicalities become simpler, if we put energy in a local form, using the formula for Dirichlet functional: for any function g(r)6

μ 2





G(r, r˜)g(r )g(r˜)d2 rd2 r˜ = max



ψ ∈(10 ) |r|R

|r|R |r˜|R





1 g(r )ψ (r ) − 2μ

∂ψ ∂x

2



∂ψ + ∂y

2 

d2 r.

Therefore, energy can be written as a functional of α¯ (r ) and ψ (r):

I (ψ , α¯ ) =





V

1 (2 − α¯ (r ) )ψ (r ) − 2μ



∂ψ ∂x

2



∂ψ + ∂y

2  d2 r.

(36)

The maximizer of energy functional (36) over ψ (r) is a solution of the boundary value problem

ψ = μ(α¯ (r ) − 2 ),

ψ ||r|=R = 0.

(37)

Admissible functions ψ (r) must be continuous, otherwise energy is infinite. Vanishing of energy variation over ψ (r) yields that the normal derivatives of the maximizer ψ are also continuous on any line inside the circle. Varying of dislocation density. Formula (36) is one of the two basic points in construction of continuum theory for dislocation equilibria. The second point concerns the way one can vary dislocation density α¯ (r ). The problem is that variations of α¯ (r ) are not arbitrary; they are generated by shifts of dislocations from their equilibrium positions. To describe dislocation shifts within the framework of a continuum theory, we introduce a two-dimensional continuum in such a way that dislocation positions coincide with positions of corresponding particles of the continuum. The ratio α¯ (r )/b has the meaning of the number of dislocations (particles) per unit Lagrangian volume. This number does not change when continuum moves. Therefore, variation of α¯ /b is similar to variation of mass density ρ in continuum mechanics. We have for Eulerian variation ∂ α¯ of α¯ (variation at a fixed space point) (see Berdichevsky (2009), p. 125)

∂ α¯ = −

∂ αδ ¯ xi ∂ xi

(38)

where δ xi are infinitesimal displacements of dislocation continuum. Formula (38) expresses, in particular, the fact that dislocations are not nucleated; if α¯ = 0 in some region, then ∂ α¯ = 0 in this region. This feature would not hold, if we allow arbitrary variations of α¯ (r ). Let dislocations occupy some region D, which is a subregion of V. In Dα¯ (r ) is not equal to zero. Variation of energy functional due to dislocation shifts is



∂ αψ ¯ (r )d2 r +



αψ ¯ (r )δ xi ni ds.

(39)

∂D

D

Here ni are the components of unit normal vector at the boundary ∂ D of region D. Plugging (38) in (39) and equating (39) to zero, we see that after integration by parts the boundary terms cancel out, and the only condition of stationarity is

α¯

∂ ψ ( r ) = 0. ∂ xi

This equation is satisfied identically outside of D, while in D

ψ (r ) = const. 6

Notation ψ ∈ (10) means that ψ obeys to the constraint (10).

(40)

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

19

On ∂ D also, as on any other line inside V,



[ψ ] = 0 ,

∂ψ n = 0 on ∂ D. ∂ xi i

(41)

It follows from (40) and (41) that

∂ψ n = 0 on ∂ D. ∂ ni i

(42)

Eqs. (37), (40) and (41) form a boundary value problem with unknown boundary ∂ D. If ∂ D were known, then to find ψ in V − D it would be enough to solve Eq. (37) with the boundary conditions

∂ψ n = 0 on ∂ D. ∂ xi i

ψ = 0 on ∂ D,

(43)

For a given ∂ D, solution of the boundary value problem (37), (43) does not satisfy the condition which follows from (40) and (41)

ψ = const

on

∂ D.

(44)

This additional equation serves to determine the unknown boundary ∂ D. Solution of the problem. The equilibrium distribution of dislocations is found from (37) and (40)

α¯ (r ) = 2 .

(45)

This is in compliance with visual inspection of Fig. 1. Denote by ξ the dimensionless radius, ξ = r/R, 0 ≤ ξ ≤ 1. Consider axisymmetric equilibrium dislocation distributions. The region D occupied by dislocations is a circle of some radius ξ 0 . The radius ξ 0 is determined by the given number of dislocations N , or a given plastic shift γ = Nb.



γ=

α¯ (r )d2 r = 2π R2 ξ02 .

(46)

|r|ξ0 R

Equilibrium is possible only for ξ 0 < 1, thus γ , ϰ and R are linked by the condition

γ  2π R2  .

(47)

This condition determines the maximum number of dislocations that can be held in equilibrium by twist ϰ in the cylinder of radius R

2π R2  . b

Nmax =

For given γ , ϰ and R obeying to (47) the radius of the circular region occupied by dislocations is



ξ0 =

γ

2π R2 

.

(48)

There are no dislocations for ξ > ξ 0 , therefore, as follows from (37), (41) and (40) for ξ > ξ 0 , ψ is determined by the boundary value problem

ψ = 2μ ,

ψ |ξ =1 = 0,

 ∂ψ  = 0. ∂ξ ξ =ξ

(49)

0

This determines ψ uniquely:

ψ =−

 μR2 1 − ξ 2 2

+ c ln ξ ,

c = −μR2 ξ02

for

ξ  ξ0 .

(50)

For ξ ≤ ξ 0 , ψ is a constant, which is found from (50) by setting ξ = ξ0 ,

ψ = −μR2

1

2





1 − ξ02 + ξ02 ln ξ0 .

(51)

Energy of the system can be obtained by integration by parts in the quadratic functional in (36). Using (37) we obtain

E 1 = L 2



(2 − α¯ )ψ d2 r.

|r|R

Plugging here (50) we find

 E 1 = π μ 2 R2 1 − 4ξ02 + 3ξ04 − 4ξ04 ln ξ0 . L 4

(52)

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V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

*

Fig. 6. Dimensionless master curve for torque-twist dependence.

The torque T is obtained by differentiation of energy over ϰ. From (36)

T =

∂ E/L =− ∂



2ψ (r )d2 r =

π μ R2

2

1 − ξ02

2

.

(53)

V

Unloaded beam. For zero torque and non-zero ϰ, as follows from (53)

ξ0 = 1 , i.e. the entire beam cross-section is occupied by dislocations. The corresponding twist is found from (46):

γ

=

2π R2

.

(54)

The stress function (51) is zero, and the beam is stress free. Energy (52) is zero as well. Torque-twist relation. The dependence torque on twist becomes nonlinear, if the beam contains dislocations. To write down this dependence in dimensionless form we introduce dimensionless plastic shift γ ∗ = γ /R = Nb/R, dimensionless twist  ∗ =  R and dimensionless torque T ∗ = T /(π μRγ ∗ ). For a given γ ∗ , i.e. a given number of dislocations, twist must be large enough to keep dislocations in equilibrium. From (47)

 ∗  γ ∗ /2π . For such twists, from (53)

T∗ =

1 ∗ 2 γ∗



1−

2

1 2π  ∗ /γ ∗

.

(55)

At the minimum possible twist  ∗ = γ ∗ /2π , torque is zero. To have a master curve, i.e. the dependence which is not affected the parameters of the problem, we introduce the twist variable X = 2π  ∗ /γ ∗ . Then

T∗ =



1 1 X 1− 4π X

2

.

This curve is shown in Fig. 6. Torque grows quadratically near X = 1. The torque-twist dependence becomes approximately linear for large twists. Comparison with numerical results. Weinberger plotted the dependence on N of normalized plastic twist per unit length

β¯ =

N 



1−

i=1

|ri |2 R2



.

In continuum theory, β¯ is given by formula

β¯ =





1−

 |r|2 α¯ (r ) R2

b

d2 r.

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

Number of dislocations (N)

21

Number of dislocations (N)

Fig. 7. Dimensionless plastic twist per unit length β¯ as a function of the number of dislocations for a torque-free beam. Asterisks and dots are numerical results (Weinberger, 2011). Asterisks represent the minimum energy configurations and open circles represent other metastable configurations. The right plot includes all of the data, the left plot shows a close up of the data for 50 of fewer dislocations. Solid lines on both plots correspond to continuum theory.

For torque-free beam, due to (45),

β¯ =

π  R2 b

.

(56)

Plugging here (54), we find the dependence of β¯ on N

1 2

β¯ = N. Comparison of this result of continuum theory with numerical simulations is shown in Fig. 7. It is seen that coincidence is getting better for large N. In comparison of the results for energy note that Weinberger used energy multiplied by N2 . He observed that the scaled energy changes linearly with N. In terms of continuum theory, these variations correspond to corrections of order 1/N. Since we dropped in continuum theory such corrections, it is not surprising that we obtain energy that is equal to zero and does not depend on N. This can be considered as another confirmation of continuum theory. 5. An alternative formula for energy In this section we present an alternative formula for energy which allows us to easily extend the theory to anisotropic beams with arbitrary cross-sections and to edge dislocations. Let us show that energy of a beam with screw dislocations (36) can be written in the form similar to (5)7 ,



E = min w L

μ 2

V

( p)



∂w ( p) −  y − 2ε¯xz ∂x

2



∂w ( p) + +  x − 2ε¯yz ∂y

2 

dxdy

(57)

( p)

where ε¯xz and ε¯yz are averaged plastic strains of the dislocation ensemble. Since for screw straight dislocations the compo( p)

( p)

nents of plastic distortion β xz and β yz are zero, plastic strains ε¯xz and ε¯yz , which are symmetric parts of plastic distortions, coincide with 12 βxz and 12 βyz , respectively. Accordingly,

 ( p)  ( p) ∂ ε¯yz ∂ β¯ y ∂ β¯ x ∂ ε¯xz α¯ = − =2 − . ∂x ∂y ∂x ∂y

(58)

To derive (57), we plug (58) in (36) and integrate by parts to obtain

E = max L ψ

 V



1 ∂ψ ( p) ∂ψ ( p) ∂ψ ∂ψ 2 ε¯ − 2 ε¯ −  x − y − ∂ x yz ∂ y xz ∂x ∂ y 2μ



∂ψ ∂x

2



∂ψ + ∂y

2  d2 r.

(59)

7 This formula is a special case of the general statement: if dislocation density tensor is split into the sum of averaged dislocation density tensor and fluctuations then, under some assumptions, energy of dislocation ensemble is also split into the sum of energy of averaged dislocation density and energy of fluctuations. Moreover, the former is given by formula (1) (Berdichevsky, 2006b, 2009).

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V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

The variational problem (59) can be put in another equivalent form

E = max σx ,σy L





( p)

( p)

−2σy ε¯yz − 2σx ε¯xz −  yσx −  xσy −

V



 1 2 σ + σy2 d2 r. 2μ x

Here maximum is sought over all σ x , σ y satisfying equilibrium equation

∂σx ∂σy + = 0 in V ∂x ∂y and boundary condition

σx nx + σy ny = 0 on ∂ V. The constraints can be taken care of by including into the functional the term with Lagrange’s multiplier w

  ∂w ∂w 2 + σy d r, σx ∂x ∂y



Then

E = max min σx ,σy w L

 V

     1 2 ∂w ∂w ( p) ( p) −  y − 2ε¯xz + σy +  x − 2ε¯yz − σx σx + σy2 d2 r. ∂x ∂y 2μ

Now we change the order of minimization and maximization. Maximum over σ x , σ y yields the integrand

μ



2

∂w ( p) −  y − 2ε¯xz ∂x

2



∂w ( p) + +  x − 2ε¯yz ∂y

2 

and we arrive at (57). ( p) ( p) Shift of dislocation positions yield variations of ε¯xz , ε¯yz . From (58)



∂ α¯ = 2

 ∂ ( p) ∂ ( p) ∂ ε¯yz − ∂ ε¯xz . ∂x ∂y

(60) ( p)

( p)

Comparing (60) and (38) we can introduce ∂ ε¯xz and ∂ ε¯yz as ( p)

∂ βzx = 2∂ ε¯xz = αδ ¯ y,

( p)

∂ βzy = 2∂ ε¯yz = −αδ ¯ x.

(61)

This is consistent with shifting of dislocations along the slip planes. Plastic deformations are varied only in the region where − ( p) − ( p) − ( p) − ( p) α¯ = 0; outside of this region ∂ εxz = ∂ εyz = 0, while εxz and εyz are not necessarily zero. We will obtain the closed system of equations following from variation of functional (57) in the next section for general √ case of anisotropic beams with arbitrary cross-section. 6. Screw dislocations in elastic anisotropic beams with arbitrary cross-sections Energy functional of elastic beams with arbitrary anisotropy and arbitrary cross-sections was obtained in Berdichevsky, 1981. If elastic properties are invariant with respect to inversion of the beam axis, as we will assume, then energy functional splits into the sum of twist energy and extension-bending energy. Here we consider twist energy,

I=

1 2







Gαβ (w,α +  eσ α xσ ) w,β +  eσ β xσ d2 x.

(62)

S

In (62) Greek indices run through values 1, 2 and correspond to projections on Cartesian cross-sectional coordinates xα , Gαβ are elastic moduli, ϰ the twist, eαβ the components of 2D Levi-Civita symbol (e12 = −e21 = 1, e11 = e22 = 0 ), comma in indices denotes partial derivatives, summation over repeated indices is always implied. The expression of Gαβ in terms of components of 3D tensor of elastic moduli can be found in (Berdichevsky, 1981; 2009, Chapter 15); for isotropic beams Gαβ = μδ αβ . Repeating reasoning of Sections 2 and 5, for a beam with screw dislocations we obtain energy functional

I=

1 2





Gαβ w,α +  eσ α xσ − 2ε¯α 3 ( p)





w,β +  eσ β xσ − 2ε¯β 3 d2 x ( p)

(63)

S ( p)

where ε¯α 3 are the components of averaged plastic strain tensor. For isotropic beams (63 ) transforms into energy functional of Section 5. Plastic strains are linked to averaged dislocation density of screw dislocations α¯ by the relation

α¯ = 2eμν ∂μ ε¯ν 3 , ( p)

∂μ ≡ ∂ /∂ xμ .

(64)

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

23

To obtain equations for a equilibrium cloud of screw dislocations, we use the approach of Section 5: in addition to minimization of energy functional over w, we allow variations of plastic strains,

( p)  ∂ ε¯ν 3 = −α¯ eμν δ xμ ,

(65)

where δ xμ are infinitesimal displacements of the cloud. Eqs. (65) and (64) are consistent with the way the dislocation density is allowed to vary

∂ α¯ = −∂μ (αδ ¯ xμ ). ( p) ¯α 3

Variation of ε



(66)

yields the equations



( p) Gαβ w,β +  eσ β xσ − 2ε¯β 3 α¯ eαμ = 0.

This is equivalent to vanishing elastic strains and stresses inside the dislocation cloud

w,β +  eσ β xσ − 2ε¯β 3 = 0 ( p)

in D.

(67)

Varying w yields the boundary value problem



( p)  ∂α Gαβ w,β +  eσ β xσ − 2ε¯β 3 = 0 in V



Gαβ w,β +  eσ β xσ − 2ε¯β 3 nα = 0



( p)



on



Gαβ w,β +  eσ β xσ − 2ε¯β 3 nα = 0 ( p)

(68)

∂V

on

(69)

∂V

(70)

These equations must be complemented by the condition that there are no dislocations outside of D:

α¯ = 2eμν ∂μ ε¯ν 3 = 0 in V − D. ( p)

(71)

Varying of the dislocation cloud boundary yields Weierstrass condition, or, in physical terms, continuity of normal components of energy-momentum tensor Pαβ (see, e.g., Berdichevsky, 2009, p. 121 and sect. 7.4):





Pαβ nβ = 0 on

∂ D.

(72)

For functional (63) energy-momentum tensor is γ

Pαβ = Gα w,γ +  eγ σ xσ − 2ε¯γ 3 −

( p)



w,β +  eβσ xσ − 2ε¯βσ ( p)



(73)

1 γ δ

( p) 

( p)  G w,γ +  eγ σ xσ − 2ε¯γ 3 w,δ +  eγ σ xσ − 2ε¯δσ δαβ . 2

Condition (72) serves to find the boundary of the dislocation cloud. Let us analyze the equations obtained. Eq. (68) is satisfied identically in D. Therefore, it should be considered only in the region V − D with the boundary condition (69) and the boundary condition on ∂ D which follows from (70) and (67)





Gαβ w,β +  eσ β xσ − 2ε¯β 3 nα = 0 ( p)

on

∂ D.

(74)

According to (71) plastic strains are compatible in V − D, i.e. there exist plastic warping w(p) such that ( p)

2ε¯α 3 = ∂α w( p) .

(75)

Then (68), (69), (74) become a boundary value problem for elastic warping w(e ) = w − w( p) :

   ∂α Gαβ w,(βe) +  eσ β xσ = 0 in V − D 



Gαβ w,(βe ) +  eσ β xσ nα = 0

on

(76)

∂ (V − D ).

Two Eqs. (72) yield only one condition on ∂ D. Indeed, according to (74) and (67) the first term in (73) vanishes. Therefore, (72) is reduced to the equation





Gαβ w,α +  eασ xσ − 2ε¯α 3 ( p)



w,β +  eβσ xσ − 2ε¯β 3 ( p)



= 0 on

∂ D.

(77)

It is convenient to write it in terms of stress

( p)  σ α = Gαβ w,β +  eβσ xσ − 2ε¯β 3 .

We have





α β = 0 on G−1 αβ σ σ

∂ D.

(78)

24

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

α α β α α G−1 αβ denote the components of the universe tensor to Gαβ . Since σ nα = 0 on ∂ D and σα = 0 in D, σ = σ tβ t , t being tangent vector on ∂ D. Therefore, from (78) α β G−1 αβ t t



σ β tβ

2

= 0.

α β Tensor Gαβ is assumed to be definite-positive, and G−1 αβ t t > 0. Thus, (72) brings one condition

σ α tα = 0 on ∂ D.

(79)

Inside the dislocation cloud dislocation density is constant: from (67) and (64)

α¯ = 2 .

(80)

The boundary value problem can be written in terms of stress function ψ introduced by the equation

σ α = eαβ ψ,β .

(81)

The first Eq. (76) is satisfied identically. From (81) σμ w,β +  eσ β xσ − 2ε¯β 3 = G−1 βσ e ψ,μ . ( p)

(82)

The condition of solvability of (82) with respect to w(e) is obtained by applying to (82) operator eβλ ∂ λ . We have



σ μ βλ G−1 βσ e e ψ,μ



,λ

= α¯ − 2 .

(83)

At the boundaries

dψ =0 ds

on ∂ V and

∂ D.

(84)

s being arc length on ∂ D. Besides, from (79)

∂ψ = 0 on ∂ D. ∂n

(85)

Eqs. (83)–(85) comprise another form of the boundary value problem with unknown boundary. A complete solution for arbitrary anisotropy and arbitrary cross-section can be obtained for torque-free beam. In this case D = V, according to (67 ), everywhere in the beam elastic strains and elastic stresses are zero; for a given number of dislocations N twist is determined by (80),

1 2

 = α¯ =

Nb , 2A

A being cross-sectional area of the beam. The general solution of (64) for plastic strains is

1 4

ε¯ν 3 = α¯ eσ ν xσ + ∂ν w( p) . ( p)

Plastic warping w(p) is arbitrary and depends on the history of deformation. Total warping w coincides with w(p) . For non-zero torque a two-parametric analytical solution can be obtained from the solution of Section 2 by scaling the coordinates. Then elastic properties become anisotropic, while circular cross-section transforms into elliptical one. However, equilibrium of dislocations in a generic anisotropic beam with circular cross-section or isotropic beam with elliptic crosssection most likely admits only numerical study. 7. Screw dislocations in twisted beams with thin rectangular cross-sections The problem formulated in the previous section can be solved analytically for beams with thin rectangular cross-section in the leading approximation with respect small to parameter h/, h and  being the width and length of the cross-section. Let x and y be coordinates along the cross-section axes, |x| ≤ /2, |y| ≤ h/2. Suppose that the principle axes of the shear modulus tensor Gαβ coincide with (x, y )−axes. Then the dual functional for the functional (63) takes the form

I∗ =





(2 − α¯ )ψ

1 − 2Gx



∂ψ ∂y

2

1 − 2Gy



∂ψ ∂x

2 

dxdy.

(86)

The stress-function ψ changes fast in y−direction and slow in x− direction. Therefore, in the leading approximation one has to keep in (86) the derivatives ∂ ψ /∂ y only (this can be justified by the formal procedure of the variational-asymptotic method, see Berdichevsky (2009), Sect. 5.11). We obtain the variational problem

E = min max α¯ L ψ

h/2

−h/2



(2 − α¯ )ψ

1 − 2Gx



∂ψ ∂y

2 

dy.

(87)

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

25

/2

-

- /2 Fig. 8. Notation for beams with thin rectangular cross-sections.

Maximum over ψ is sought over ψ vanishing at y = ±h/2. Admissible dislocation densities obey the condition

h/2

α¯ dy =

−h/2

Nb . 

(88)

We assume that α¯ is zero outside of some segment [−c, c], c ≤ h/2 (Fig. 8). The variational problem (87) yields the system of equations:

1 Gx

α¯

∂ 2ψ = α¯ − 2 ∂ y2

∂ψ =0 ∂y

|y|  h/2,

ψ |±h/2 = 0,

 ∂ψ  =0 ∂ y ±h/2

|y|  h/2.

(89)

(90)

The solution of (89) and (90) is



α¯ =

2 0

|y|  h/2 |y| > c

⎧ ⎪  ⎨ψ0  2   h h ψ= 2 −y − 2 Gx  c − |y| ⎪ ⎩Gx  2

 ψ0 = Gx 

h −c 2

2

|y|  c |y|  c

(91)

2 .

The thickness of dislocation cloud 2c is determined by a given number of dislocations N and parameters of the problem from (88):

c=

Nb . 4 

(92)

Since c ≤ h/2, the maximum number of dislocations which can be kept in equilibrium by twist ϰ is

Nmax =

2 h . b

(93)

Derivative of energy with respect to ϰ is equal to torque T. Thus,

T =2 L

h/2

ψ dy.

(94)

−h/2

From

T 1 2 = + Gx  (h − 2c ) (h + c ). L 3

(95)

26

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

*

Fig. 9. Master curve for torque-twist dependence for beams with thin rectangular cross-sections.

Fig. 10. To discussion of equilibrium of edge dislocations.

For zero torque, c = h/2, i.e. dislocations are distributed homogeneously over the entire cross-section. The torque-twist dependence becomes universal, i.e. independent on parameters of geometry h, , shear modulus Gx , the number of dislocations N and Burgers vector b, if it is written in terms of dimensionless parameters

T∗ = We have

T∗ =

T Gx LNbh2



1 1 X 1− 6 X

and X =

2 h . Nb

2 



1+

1 . 2X

This dependence is shown in Fig. 9. Since c/h = 1/2X and c ≤ h/2, parameter X is larger than 1. The value X = 1 corresponds to zero torque. Near the point X = 1, torque grows quadratically, T ∗ = (X − 1 )2 /4. For large X, the torque-twist dependence is linear, T ∗ = X/6. 8. Generalizations The continuum theory can be easily extended to edge dislocations. There is, however, a new issue which we discuss for the case of traction-free two-dimensional body shown in Fig. 10. The body contains edge dislocations with slip planes y = const. The side x = 0 is clamped. To make the technicalities simple, the body is assumed to be isotropic with zero Lamé constant λ. Denote by ϕ the stress function and by β the only nonzero component of plastic distortion. Dislocation density α¯ is equal to ∂ β /∂ x. Then the energy functional is (see, e.g. Berdichevsky, 2009, Section 6.11)



V

  1 2 ϕ, xy β − ϕ, xx + 2ϕ,2xy + ϕ,2yy dxdy. 4μ

(96)

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28

27

Admissible functions ϕ are such that

ϕ = ϕ, y = 0 at y = 0, H;

ϕ = ϕ, x = 0 at x = .

Varying β we get

ϕ,x y = 0. Varying ϕ we obtain the equation

α¯ , y +

1 2

ϕ=0 2μ

(97)

and additional boundary condition at x = 0

ϕ, x x = 0,

β, y +

1

μ

ϕ, xyy +

1 ϕ, xxx = 0. 2μ

As easy to see, equilibrium distributions of edge dislocations are possible only for zero stresses, ϕ ≡ 0. In contrast to the case of screw dislocations, the dislocation density of edge dislocations is not determined uniquely: as follows from (97), it can be an arbitrary function of x . On microlevel the independence α¯ on y corresponds to formation of equilibrium states which slowly change in x−direction, like dislocation walls shown in Fig. 9(b). Distances between the walls can be arbitrary: local stresses exponentially decay away from the walls, and the wall interaction is a small effect which is not included in the leading approximation considered here. To find the dependence of α¯ on x one has to include in energy the small terms of order N −1 . From the perspective of a general theory of functionals depending on small parameters (see Berdichevsky, 2009, Section 5.11) the difference between equilibrium of screw and edge dislocations is as follows: For screw dislocations, the leading term of energy has isolated stationary points. Therefore, the next terms of energy yield only small corrections. For edge dislocations, the leading term of energy has non-isolated stationary points (arbitrary α¯ (x )), and the terms of energy of order 1/N are necessary to find a unique solution. Equilibrium of three-dimensional dislocation networks exhibit the similar feature: in a generic case, the stationary points of energy are not isolated, and incorporation of small terms is necessary. The construction of small terms will be discussed elsewhere. Appendix A. Fluctuations of interaction energy Consider the function of N variables, r1 , ..., rN , N 1  G ( ri , r j ). 2 2N

(98)

i= j

Let ri be distributed randomly and independently with probability f(r). We want to determine the order of deviation of the sum (98) from its mathematical expectation

M

 1  N−1 G ( r , r ) = G(r, r ) f (r ) f (r )drdr . i j 2N 2N 2

(99)

i= j

Let us present G(r, r ) in the form

G(r, r ) = G˜ (r, r ) + w(r ) + w(r ) + h

(100)

where functions G˜ (r, r ) and w(r) have zero mathematical expectations,



G˜ (r, r ) f (r )d2 r = 0,



G˜ (r, r ) f (r )d2 r = 0,



w ( r ) f ( r )d 2 r = 0,

(101)

and h is a constant. Conditions (100) and (101) determine w(r) and h uniquely. Indeed, integrating (100) with the weight f(r) we get



h=

G(r, r ) f (r ) f (r )d2 rd2 r ,

w (r ) =



G(r, r ) f (r )d2 r − h.

(102)

The constant h/2 coincide with mathematical expectation of the sum (98) in the limit N → ∞. If G(r, r ) has the meaning of Green’s function of Laplace operator with zero boundary conditions, then w(r) is determined by the boundary value problem

w(r ) = − f (r ) in V ,

w(r ) = −h

on ∂ V.

Eq. (100) can be considered as the definition of function G˜ (r, r ). Plugging the presentation (100) into (33) we obtain for interaction energy

(103)

28

V.L. Berdichevsky / International Journal of Engineering Science 106 (2016) 10–28 N 1  1 1 1 G ( r i , r j ) = h + θ1 + √ θ2 2 2 N 2N N i= j

θ1 =

N 1  G˜ (ri , r j ), 2N i= j

N 1 

θ2 = √

N

w(ri ).

i=1

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