Dislocation pileups: topological features of stresses and strains

Theoretical and Applied Fracture Mechanics 35 (2001) 237±242

www.elsevier.com/locate/tafmec

Dislocation pileups: topological features of stresses and strains S.D. Borisova, I.I. Naumov * Institute of Strength Physics and Materials Science, Siberian Branch of Russian Academy of Sciences, pr. Akademicheskii, 2/1, Tomsk 634021, Russian Federation

Abstract When averaged over half-spaces, the elastic stress and strain ®elds from dislocation pileups are shown to take a peculiar topological features of the form y=jyj, i.e., they undergo an abrupt change when passing through the pileup plane. The average ®elds appear to be equivalent to the Somigliana dislocations and provides a description of strain boundaries in crystals such as the torsion boundary which represents a surface where the average rotation ®eld experience a topological jump. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction The development of physical mesomechanics [1] has received considerable attention in recent years. The underlying concepts can be used to describe the plastic ¯ow and failure of solids including the hierarchy of scale and structural levels. For this approach, it is essential to consider the process of averaging, especially when higher scale levels are considered. Averaging of a ®eld entails smoothing, i.e., to have a more gradual ®eld variation. Such an intuitive anticipation, however, is not necessarily valid. Suppose an elastic ®eld is spatially ordered, then conversion to the average cannot lead to smoothing. On the contrary, it may cause sudden change or even give singular behavior. The dislocation pileups in crystals under plastic strain as a result of stress relaxation along phase boundaries and interfaces undeniably belong to ordered plastic ®eld sources. In what follows it will be shown that when averaged, the ®elds induced

*

Corresponding author. E-mail address: [email protected] (I.I. Naumov).

by the dislocation pileups take a peculiar topological feature of the form y=jyj which does not result from singularities of individual dislocations. For an individual dislocation, the displacement ®eld appears to be discontinuous. In the case of dislocation pileups, both the displacement ®elds and the average stress and strain ®elds entail discontinuity. The stress and strain ®elds are identical to those of Somigliana dislocations which represent the most general form. The mutual correspondence between dislocation pileups and Somigliana dislocations will be re®ned in this work. In the limit that y ! 1 with y being the distance from a pileup, the elastic ®elds of dislocation pileups are equivalent to the Somigliana dislocation ®elds [2±6]. The average ®elds of dislocation pileups (in contrast to the exact ones) are not asymptotically equivalent but equivalent to the Somigliana dislocation ®elds. Expression for elastic ®elds of homogeneous pileups of edge and screw dislocations will be given. Unlike those in [6±8], they are given as Fourier series rather than in terms of hyperbolic sines and cosines. The resulting ®elds are averaged while their topological features and the relationship with

0167-8442/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 1 6 7 - 8 4 4 2 ( 0 1 ) 0 0 0 4 7 - 7

238

S.D. Borisova, I.I. Naumov / Theoretical and Applied Fracture Mechanics 35 (2001) 237±242

the corresponding Somigliana dislocation ®elds are provided. The results are discussed with reference to the strain boundaries in crystals.

where D ˆ l=2p…1 m†, l is the shear modulus and m is Poisson's ratio. Using the identities 1 X

…1†k e

ka

cos kx ˆ

kˆ1

2. Parallel edge dislocations

1 X

Consider an in®nite set of identical parallel edge dislocations lying in the same slip plane y ˆ 0 at an equal distance h from each other, Fig. 1. The dislocations are parallel to the z-axis so that their Burgers vectors b are directed along the x-axis. The resulting y-component stress generated by all the dislocations at point x is 1 X rij ˆ r0ij …x nh; y†; …1†

kˆ1

nˆ 1

where r0ij …x; y† is the stress developed by an individual dislocation. Following the work in [9] and using Poisson's summation formula Z 1 1 X X f …x† e2p ikx dx; f …n† ˆ …2† nˆ1

kˆ 1

exact stress expressions for an elastic isotropic solid are obtained: "  1  X bD y 2pkjyj rxx ˆ 2p 1‡ 2 h jyj h kˆ1 #   2pkx ;  e 2pkjyj=h cos h ryy ˆ

1 bD 4p2 X y ke h h kˆ1

1  bD X 2p 1 rxy ˆ h kˆ1

 2pkjyj=h

 2pkjyj e h

cos

 2pkx ; h 

2pkjyj=h

sin

…3†

 2pkx ; h

Fig. 1. Homogeneous pileup of parallel edge dislocations.

k

…1† e

ka

sin kx ˆ

1 sh a 2 ch a  cos x

1 ; 2

…4†

1 sin x : 2 cos x  ch a

Eqs. (3) can be written in the conventional form [6±8]:  2p sh…2py=h† rxx ˆ Db h Bx  py ch…2py=h† cos…2px=h† 1 ; h B2x ryy ˆ

2p2 y Db 2 h



ch…2py=h† cos…2px=h† B2x

1

 ; …5†

 p sin…2px=h† rxy ˆ Db h Bx  2py sh…2py=h† sin…2px=h† ; h B2x where Bx ˆ ch…2py=h†

cos…2px=h†:

…6†

The series in Eqs. (3) are ordinary Fourier series. It is seen that the amplitudes of ®eld harmonic with k P 1 decrease exponentially as the distance y from the dislocation pileup plane increases. The higher the harmonic number, the faster the decrease. Even with the harmonic k ˆ 1, the ®eld amplitude decreases by a factor 104 at distances on the order of y  2h. Another feature is that y ˆ 0, the contribution to rxx and rxy from all the harmonics are comparable in magnitude even for k  1 or a spacing much shorter than h. In general, reference can be made to the oscillatory stresses along the xaxis. Those decrease exponentially decreasing along the y-axis are microstresses; they are significant only when y < 2h. Unlike ryy and rxy , the component rxx has a zero (k ˆ 0) harmonic y=jyj being responsible for stresses at large distances y > 2h (macrostresses). This harmonic possesses a peculiar topological feature. It experiences a jump

S.D. Borisova, I.I. Naumov / Theoretical and Applied Fracture Mechanics 35 (2001) 237±242

jbl=h…1 m†j when moving through a dislocation pileup y ˆ 0. The total value rxx , however, does not undergo any jump (except for transitions through the point where dislocations occur, i.e., through x ˆ nh, where n is an integer. The jump of the zero harmonic is compensated by the jump with an opposite sign formed by all the other harmonics with k P 1. The fact that there is no jump is clearly evident in Eqs. (5). The situation changes radically when passing to the average values. Consider averaging the stresses in Eqs. (3) for small volumes such that they lie entirely in the upper y > 0 or lower y < 0 half-space. If the linear size of such volumes in the x-direction is much bigger than h, then all the harmonics with k P 1 will make no contribution to the average stresses. The latter will depend only on the zero harmonic. The same results are obtained from averaging over the planes y ˆ const: 6ˆ 0 according to convention. This gives hrxx i ˆ hrxy i ˆ 0;

bl y ; h…1 m† jyj

hryy i ˆ 0;

hrzz i ˆ mhrxx i ˆ

mbl y h…1 m† jyj

…7†

whose characteristics are determined by the ratio b=h. Note that the Somigliana dislocation appears as a peculiar coherent phase boundary separating the ``phases'' with the same moduli but with different lattice parameters. In the x-direction, the lattice has a mis®t of Da=a ˆ b=h. It should be kept in mind that the jump under consideration is actually localized in a layer of the order of 10a with a being the lattice parameter) rather than in an in®nitely small layer. It is at distances P10a where this continuum description of lattice defects is valid. Note that the zero harmonics given by Eq. (7) correspond to the stresses generated by a continuous dislocation distribution with a density 1=h. Hence, the macrostresses of a ®nite homogeneous dislocation pileup can be estimated from Z 1 L=2 0 rij …x; y† ˆ r …x n; y† dn …9† h L=2 ij only if L > h while is h considerably larger than b. Direct calculation gives # L=2 " bD …x n†y x n ; rxx ˆ 2arctg h …x n†2 ‡ y 2 y L=2

while the corresponding average strains are b y ; hexx i ˆ 2h jyj m b m y heyy i ˆ hexx i ˆ ; 1 m 2h 1 m jyj hexy i ˆ 0; hezz i ˆ 0:

239

ryy ˆ …8†

Eqs. (7) and (8) describe the elastic state resulting from · cutting along the plane y ˆ 0, · uniformly compressing the upper half plane up to the exx ˆ b=2h and uniformly extending the lower half plane up to exx ˆ b=2h, · joining the halves along the cut, and · relaxing the stress ryy …hryy i ! 0† such that the zdirection dimension remains unchanged …hezz i ˆ 0†. These operations would reproduce a Somigliana dislocation [4±6]. Hence, the average ®elds of a homogeneous dislocation pileup are equivalent to the corresponding Somigliana dislocation ®elds

" # L=2 bD …x n†y ; h …x n†2 ‡ y 2 L=2

" bD rxy ˆ h …x

…10†

q# L=2 2 : ‡ ln …x n† ‡ y 2 2 2 n† ‡ y y2

L=2

For the sake of brevity, the limits were not substituted into the integrands. For a ®nite pileup, the component rxx has the same topological feature: when moving through a dislocation pileup, rxx undergoes a jump bl=h …1 m†. This jump is provided by the function arctg‰…x  L=2†=yŠ with each limit making an equal contribution. No jump occurs if the x-axis crosses beyond the dislocation wall. Jumps corresponding to varying limits with l…1 ‡ m†=h…1 m†, i.e., above and below the pileup plane. Finite pileups are also equivalent to the Somigliana dislocations [4±6].

240

S.D. Borisova, I.I. Naumov / Theoretical and Applied Fracture Mechanics 35 (2001) 237±242

Note that the stresses in Eqs. (10) satisfy the equilibrium condition orij =oxj ˆ 0 in the entire space, even at the edges of the dislocation wall (x ˆ L=2, L=2, y ˆ 0) where each of the individual derivatives oryy =oy , oryx =ox, orxx =ox, and orxy =oy becomes in®nite. In the latter case, the equilibrium condition is   orij lim ˆ 0: …11† x!L=2 oxj y!0

Eq. (7) can be expressed in polar coordinates centered at the right and left edges of the dislocation wall. This gives rxx ˆ

ryy ˆ rxy ˆ

2bD ‰ …sin 2u2 sin u1 † ‡ arctg…ctg u2 † h arctg…ctg u1 †Š; 2bD ‰sin 2u2 h bD ‰sin2 u2 h

sin 2u1 Š;

Consider now an in®nite set of parallel screw dislocations lying along the z-axis at equal distance h, Fig. 3. The exact equations for the stress components are rxz ˆ

bD1 p y h jyj " 1 X  1‡2 e

# 2pkx cos ; h kˆ1   1 bD1 X 2pkx 2pkjyj=h 2p ryz ˆ e sin ; h h kˆ1 

2pkjyj=h

…13†

and rotations xzx ˆ …12†

sin2 u1 ‡ ln…q2 =q1 †Š:

Here u1 and u2 are the angles, the left …x ˆ L=2† and right …x ˆ L=2† edges of the dislocation wall can be seen from a given point while q1 and q2 are the distances from these edges. Fig. 2. It is seen that in the limit u1 ! 0, u2 ! 0, and q2 =q1 ! 1, Eqs. (12) are transformed to Eqs. (7). Moreover, at the starting and end points of the dislocation accumulation, the stresses are ambiguous, i.e., their values are determined by the angle of approach to these points.

Fig. 2. Dislocation wall in polar coordinate system.

3. Parallel screw dislocations

b y 4h jyj "



1 X

2pkx  1‡2 e 2pkjyj=h cos h kˆ1   1 b X 2pkx e 2pkjyj=h sin ; xyz ˆ 2h kˆ1 h

# ;

…14†

where D1 ˆ l=2p. Now, the singularity corresponds to the zero harmonics of rxz and xxz . In passing through the pileup plane, the harmonics experience the jumps jbl=hj and jb=2hj. Much as in the case of edge dislocation pileups, the stresses in Eqs. (13) are averaged over small volumes lying in the upper y > 0 or lower y < 0 half-spaces (or over the planes y ˆ const: 6ˆ 0). The result is hrxz i ˆ

bl y ; 2h jyj

hryz i ˆ 0:

…15†

Fig. 3. Homogeneous pileup of screw dislocations lines are normal to xy-plane.

S.D. Borisova, I.I. Naumov / Theoretical and Applied Fracture Mechanics 35 (2001) 237±242

The corresponding associated average rotation components are given by hxzx i ˆ

b y ; 4h jyj

hxyz i ˆ 0:

…16†

The foregoing jumps refer to the average shear stresses and rotations when passing through the dislocation pileup plane. Note that the jump in hxzx i is the angle of disorientation for the two regions separated by the pileup plane, Fig. 4. From the standpoint of average ®elds, the pileup plane represents a macroscopic crystalline torsion boundary with the angle b=2h. The torsion boundary being a two-dimensional macroscopic defect can be treated as a Somigliana dislocation [4±6]. As in the case of edge dislocations, the derivation, the macro®elds for a ®nite homogeneous dislocation pileup with length L can be obtained: L=2 bD1 x n rxz ˆ arctg ; y L=2 h …17† q L=2 bD1 2 2 ln …x n† ‡ y ; rzy ˆ h L=2 L=2 b x n arctg xzx ˆ ; 4ph y L=2 q L=2 b ln …x n†2 ‡ y 2 : xyz ˆ 4ph L=2

…18†

It follows from Eqs. (17) and (18) that when passing over the plane x ˆ 0 with L=2 6 x 6 L=2, the components rxz and xxz experience the jumps jbl=hj and jb=2hj, respectively. These values are the

Fig. 4. Torsion boundary formation under rotation of one part of the crystal relative to another.

241

same as those for an in®nite crystal, i.e., they are equivalent to the corresponding components of the elastic ®eld of Somigliana dislocation [4±6]. Note that at L ! 1, Eqs. (17) and Eqs. (18) are transformed to Eqs. (15) and Eqs. (16) respectively. 4. Discussions The fact that singularity appears in the average ®elds of dislocation pileups may be unexpected because accurate initial values have singularities only at points where individual dislocations are located …y ˆ 0 at x ˆ nh with n being an integer). However, it should not be surprising that singularity occurs. It results from ``coherent'' behavior of aligned dislocations. A similar e€ect is also produced when the ®elds due to point defects are averaged. It has been shown in [10] for a uniform point defect distribution that the average elastic displacement along the rectangular contour of a macroscopic length d` parallel to an arbitrary z-axis has the same singularity: hui …z†i ˆ

1 z dd`Gik Ezk ; 2jzj

…19†

where d is the bulk defect density, Gik is the Green tensor components and the tensor Ezk is related to the dipole strength tensor Pik and elastic moduli kik`m refers to the relation Eik ˆ kik`m P`m . As mentioned earlier, the edge dislocation pileup yields in much the same macroscopic stresses as does the coherent phase boundary. This accounts for the generation of similar pileups such as coherent stress realization at the interface. Assume that in response to a martensitic transformation there appears a two-phase coherent mixture characterized by the lattice mis®t Da=a. At the macroscopic scale, the resulting stress state can be described by the stresses in Eq. (7). Therefore, for complete stress relaxation, it suces to introduce a homogeneous set of mis®t dislocations with the linear density 1=h ˆ Da=ab. Homogeneous edge dislocation pileups develop not only at phase boundaries or layer boundaries of heterogeneous structures produced by arti®cial means [11±13], but also under plastic strain. In a number of alloys and metals, especially those with

242

S.D. Borisova, I.I. Naumov / Theoretical and Applied Fracture Mechanics 35 (2001) 237±242

low stacking faults, plane (or quasi-plane) dislocation pileups of the same sign are an essential feature of the so-called localized shear bands [14,15]. Hence, stress and strain jumps must be observed in such bands; this fact must be taken into account when describing the development of plastic ¯ow. The average ®elds introduced above prove to be useful in describing torsion boundaries under strain. In a number of relatively simple cases, the torsion boundaries are formed by screw dislocation pileups belonging to two (or more ) slip systems, that is, by dislocation networks [8]. A characteristic feature of these boundaries is the absence of the long-range elastic ®eld. The generation of the screw dislocation network is obvious from the discussion to follow. In e€ect, although a dislocation pileup of one slip system can cause disorientation of di€erent regions in the crystal, its macroscopic stresses rxz are of long-range character. Refer to Eqs. (15) and (17). It is not dicult to speculate how this long-range ®eld can be eliminated without breaking the singularity in the rotation ®eld. Add the same pileup rotated through an angle p=2 around the y-axis to the existing one. This leads to a network of mutually orthogonal dislocation systems. The pileup just added is characterized by average values with required properties to yield. bl y ; hrxy i ˆ 0; 2h jyj b y ; hxyx i ˆ 0: hxzx i ˆ 4h jyj

hrxz i ˆ

…20†

In fact, combining Eqs. (15), (16), and (20), the overall shear stress would vanish, and the total torsion angle would be doubled be to become b=h: this angle is a topological jump of the average rotation ®eld.

5. Conclusions Conclusions when passing to an average description of dislocation pileups, singularities appear in the elastic ®elds. This is because averaging tends to wipe out the microscopic (fast oscillating)

components such that only the macroscopic components remain. They have topological singularities and re¯ect the properties of the pileups as an entity. In the context of these macroscopic ®elds, the dislocation pileups are indistinguishable from the ®elds of surface defects, the Somigliana dislocations. The results obtained in this work are of methodological importance for developing approaches and methods of physical mesomechanics. The embarkment of a higher structural level may not simply reduce to a trivial ®eld of smoothing. On the contrary, it may give rise to new ®eld singularities. The latter re¯ect the properties of a ®eld source ensemble of lower structural level as an entity. References [1] V.E. Panin (Ed.), Physical Mesomechanics and ComputerAided Design of Materials, in two volumes, Nauka, Novosibirsk, 1995 (in Russian). [2] E.H. Mann, An elastic theory of dislocations, Proc. R. Soc. Ser. A 199 (1949) 376±394. [3] J. Bogdano€, On the theory of dislocations, J. Appl. Phys. 21 (1950) 1258±1263. [4] J.D. Eshelby, The continuum theory of lattice defects, Solid State Phys. 3 (1956) 79±144. [5] V.I. Vladimirov, A.E. Romanov, Disclinations in Crystals, Nauka, Leningrad, 1986 (in Russian). [6] C. Teodosiu, Elastic Models of Crystals Defects, Springer, Berlin, 1982. [7] J. Firth, J. Rothe, Theory of Dislocations, McGraw-Hill, New York, 1968. [8] A. Prevoditelev, N.A. Tyapunina, G.M. Zinenkova, G.V. Bushueva, Physics of Crystals with Defects, Izd. MGU, Moscow, 1986 (in Russian). [9] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Nauka, Moscow, 1965, p. 204 (in Russian). [10] F. Andreev, Points defects and long range order, Pisma. Zh. Tekh. Fiz. 62 (1995) pp. 123±128. [11] Ya. Lyubov, Di€usion Processes in Nonhomogeneous Solid Media, Nauka, Moscow, 1981 (in Russian). [12] L. Roytburd, Principal concepts of martensitic theory, J. Phys. IV, Colloq. 8 (5) (1995) 21±30. [13] M.Yu. Gutkin, A.E. Romanov, Straight edge dislocations in thin two-phase, Plate Phys. Stat. Sol. (a) 129 (1992) 363± 377. [14] S.V. Harren, H.E. Deve, R.J. Asaro, Shear band formation in plane strain compession, Acta Metall. 36 (1988) 2435± 2480. [15] M.P. Kashchenko, L.A. Teplyakova, Physical nature of macro-shear bands in fcc crystals in plain strain deformation, Russ. Phys. J. 40 (3) (1997) 81±87.