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Dislocations within elliptical holes
PII:
Acta mater. Vol. 47, No. 1, pp. 1±4, 1999 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00351-6 1359-6454/99 $19.00 + 0.00
DISLOCATIONS WITHIN ELLIPTICAL HOLES J. P. HIRTH School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 991642920, U.S.A. (Received 15 September 1998; accepted 1 October 1998) AbstractÐElastic ®elds are derived for edge and screw dislocations within holes of elliptical cross-section. The results are applicable to interactions with defects near the tips of blunted cracks and to hollow dislocation cores. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
in determining the equilibrium shape of the hole by free energy minimization [9].
Long ago, Frank [1] showed that dislocation cores could be hollow and derived the equilibrium size for a screw dislocation in a hole of circular crosssection. There were subsequently a number of observations of hollow cores for screw dislocations, with a current resurgence of interest in the area [2± 8]. For an edge dislocation, the cross-sectional shape should more nearly approximate an ellipse than a circle [9]. Indeed, Chisholm and Smith [10] have observed elliptically shaped amorphous cores for edge dislocations in a YBCO high Tc superconductor, a con®guration that has a ®eld analogous to that of a hollow core. Hence, the elastic ®eld of a dislocation in an elliptical hole is of interest. Both edge and screw ®elds are relevant for mixed dislocation cases or for cases where there is anisotropy in the surface energy [9]. The solution is also applicable to internal cracks containing edge or screw dislocations. Weertman [11] and Lin and Hirth [12] showed that the ®eld of a crack contained within a body is modi®ed when a dislocation is emitted at a crack tip. The antidislocation lies within the crack. In this case, the ®eld of the crack and the contained defect was derived for a planar crack [11, 12]. There is current interest in the atomistic simulation of cracks [13±16] and in such simulations the planar crack model is inadequate. Blunting at the crack tip, even in the elastic case because of nonlinear eects, creates a tip con®guration more closely approaching an ellipse. Also, for high temperature crack growth by diusive processes, as analyzed by Chuang et al. [17] for example, the crack tip con®guration is curved. The elastic ®elds, presented as follows, provide information for core±core dislocation interactions. In addition, for the hollow core case, the ®eld at the surface of the hole gives the strain energy contribution to the chemical potential there, of interest
2. EDGE DISLOCATION IN A HOLE WITH ELLIPTICAL CROSS-SECTION
The ®eld of an edge dislocation with a hollow core of elliptical cross-section can be derived in terms of complex potential functions f(z) and c(z) following the procedures of Muskhelishvili [18]. An ellipse with major and minor axes a and d, respectively, Fig. 1, is de®ned by the parameters RE a d =2 and d a ÿ d = a d . For mode II and III crack problems, d would be positive, while for mode I crack problems and the typical edge dislocation core, d would be negative. The complex coordinate z x iy is mapped onto a circle by z=RE z d=z, where z r exp ij. With Z m=4pRE 1 ÿ b2 ÿ ib1 , where m is the shear modulus, n is Poisson's ratio, and bi is the Burgers vector, the stress functions for the edge dislocation in an in®nite continuous medium, a plane strain case, are j z)f z Z ln z2 d=z c z)c z Z ln z2 d=z j 0 z Z z2 ÿ d=z z2 d:
1
Here the overbar indicates complex conjugate. Following the procedures of Ref. [19] and taking residues, we ®nd that the added stress functions needed to account for the presence of the hole are j0 ÿ Z ln
1 dz2 z
c0 ÿ Z ln
1 dz2 1 dz2 1 ÿ dz2 ÿZ 2 ÿZ 2 : 2 z z d z ÿ d
The total stress function is given by the sum of equations (1) and (2). The corresponding stresses 1
2
HIRTH: DISLOCATIONS WITHIN ELLIPTICAL HOLES
Fig. 1. Coordinates for elliptical hole.
for the plane strain case are
sjj
srr sjj F B F B, sjj ÿ srr 2isrj 2
2B 0 2 0 f$ BF B $ 0 BC Bg r $ B 0
0
0
3
0
where F B j B=$ B, C B c B=$ B, and $ 0 B B2 ÿ d=B2 . For the determination of the chemical potential for a hollow elliptical core, we only need the strain energy density at the surface of the hole where srr srj 0 and r 1. Thus, the relevant stresses are [20] sjj 4 Re j 0 z=o 0 z, szz sjj :
mb sin W 1 2d pRE 1 ÿ mb sin W 1 2d d cos 2W: pRH 1 ÿ
The corresponding strain energy density at the surface of the hole is [21] wel
1 1 sjj szz 2 ÿ sjj szz 4m 1 4m s2jj 1 ÿ : 4m
mb 1 ÿ d 1 d2 sin j : pRE 1 ÿ 1 d2 2 ÿ 4d2 cos2 2j
4
wel
sjj
1 d2 sin2 W 1 ÿ d2 cos2 W : 1 d4 sin2 W 1 ÿ d4 cos2 W
syz isxz j 0 z, uz Im 6
In the limit of a perturbed circular cross-section with dÿ ÿ40, ÿ this expression reduces to the form pertinent to Ref. [9]
9
Again we refer to the coordinates of Fig. 1. Muskhelishvili [22] presents the form for the antiplane strain stress function for the example of torsional loading as a special case of a Dirichlet problem. For dislocations and line forces, the simpler form [23] is
mb sin W 1 d2 pRH 1 ÿ
mb2 : ÿ d2
4p2 1
3. SCREW DISLOCATION IN A HOLE WITH ELLIPTICAL CROSS-SECTION
5
In terms of the real space coordinates in Fig. 1, since cos y 1 d RE =RH cos j, sin y 1 ÿ d RE =RH sin j, and R2H =R2E 1 d2 2d cos 2j 1 ÿ d2 2 = 1 d2 ÿ 2d cos 2W, equation (5) becomes
8
When y 0, wel 0. When y p=2, RH d and
For the present case of interest where b1 b, b2 0, and r 1, the stress sjj becomes sjj
7
j z : m
10
The transformation to the coordinates of Fig. 1 is accomplished by sjz isrz and
eij o 0 B syz isxz jo 0 Bj
11
HIRTH: DISLOCATIONS WITHIN ELLIPTICAL HOLES
j 0 z
0
j B : o 0 B
12
These relations combine to give sjz isrz
eij j 0 B: jo 0 Bj
13
Guided by the result for the edge dislocation, we ®nd that the appropriate stress function for the screw dislocation is j g ln
B2 d 1 dB2 ÿ g ln B B
14
with g mb=4pR. As for the edge dislocation case, this result reduces to the proper continuous in®nite body result j 2g ln z in the limit r=dÿ ÿ41. ÿ On the surface of the hole where r 1, the substitution of equation (14) into equation (13) gives the result srz 0 and sjz
mb 1 ÿ d2 2pR 1 d2 2 ÿ 4d2 cos2 2j mb 1 d2 sin2 W 1 ÿ d2 cos2 W3=2 : 2pRH 1 d4 sin2 W 1 ÿ d4 cos2 W 15
The strain energy density at the surface of the hole is w
s2jz : 2m
16
The solutions satisfy the free surface boundary conditions, give the proper Burgers vector as de®ned by a Burgers circuit surrounding the hole, and reduce to the in®nite body Volterra solution at positions remote from the hole.
such a manner [26]. As in the Peierls case [27], the ®elds can also be represented by a dislocation, diminished in magnitude of the Burgers vector, at the center of the ellipse and two image dislocations. In the present case, the images reside at the focal points of the ellipse. When d > 0, the focal points are at y 2c with c a2 ÿ d2 1=2 while for d<0, the focal points are at x 2c. For the hollow core case, where a minimization procedure is used to determine the equilibrium shape, the elliptical shape considered here is an approximation for most edge dislocation cases. In the absence of surface energy anisotropy, the zero strain energy result for equation (8) at y 0 prevents chemical potential equilibration for an elliptical shape [9]. However, the best ®t of an ellipse at several points [9] is an improvement relative to a circular cross-section and provides a starting point for atomistic calculations. As discussed by Muskhelishvili [28], a series of ellipses can be superposed to give a result for a hole of any shape, so an exact result can be found for a minimum energy shape other than a simple ellipse in an extension of the present methods. In summary, ®elds are derived for both edge and screw dislocations within an elliptical hole. The results are given in the isotropic elastic approximation. However, the physical interpretation of the solution indicates the corresponding form for the anisotropic elastic case. AcknowledgementsÐThe author is pleased to acknowledge the support of this research by the Defense Research Projects Agency through the Oce of Naval Research under Grant N00014-97-1-6007 as well as helpful discussions with N. Sridhar, D. J. Srolovitz and J. W. Cahn.
REFERENCES
4. DISCUSSION AND SUMMARY
The above results would apply directly to a dislocation within an elliptical crack or an elliptical core. In the crack case, the results can be coupled to the result of Vitek [24] for a dislocation emitted from a crack. For a ¯uid ®lled core, or an amorphous core in the case that shear stresses can relax in the core, the above solution gives a part of the complete ®eld. The other part of the ®eld in this case is associated with a possible isostatic stress in the core, with attendant shear stresses in the surrounding medium. The latter stresses are given by the Eshelby solution to the elliptical inclusion problem [25]. The total ®eld, according to the principle of superposition, is the sum of the two sub®elds. The physical basis for the above results is that the dislocation within the hole becomes smeared into a continuous distribution of in®nitesimal dislocations along the axis of the ellipse. An analogy is Eshelby's treatment of the Peierls dislocation in
3
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Frank, F. C., Acta crystallogr., 1951, 4, 497. Verma, A. R., Nature, 1951, 167, 939. Bhide, V. G., Physica, 1958, 24, 817. Giocondi, J., Rohrer, G. S., Skowronski, M., Balakrishna, V., Augustine, G., Hobgood, H. M. and Hopkins, R. H., J. Cryst. Growth, 1997, 181, 151. Qian, W., Rohrer, G. S., Skowronski, M., Doverspace, K., Rowland, L. B. and Gaskill, D. K., Appl. Phys. Lett., 1995, 66, 1252; 1995, 67, 2284. Ning, X., Chien, F. R., Pirouz, P., Yang, J. W. and Asif Khan, M., J. Mater. Res., 1996, 11, 580. Stein, R. A., Physica B, 1993, 185, 211. Pirouz, P., Phil. Mag., 1998, 78, 727. Srolovitz, D. J., Sridhar, N., Hirth, J. P. and Cahn, J. W., Scripta mater., 1998, 39, 379. Chisholm, M. F. and Smith, D. A., Phil. Mag., 1989, 59, 181. Weertman, J., Int. J. Fract., 1984, 185, 211. Lin, I. H. and Hirth, J. P., Phil. Mag. A, 1984, 50, L43. Cheetham, A. K., Suter, U. W., Winner, E. and Yip, S., J. Computer-Aid. Mater. Sci., 1996, 3, 1. Sun, Y., Beltz, G. E. and Rice, J. R., Mater. Sci. Engng, 1993, A170, 67. Hoagland, R. G., Phil. Mag. A, 1997, 76, 543.
4
HIRTH: DISLOCATIONS WITHIN ELLIPTICAL HOLES
16. Zhou, S. J., Lomdahl, P. S., Thomson, R. and Holian, B. L., Phys. Rev. Lett., 1996, 76, 2318. 17. Chuang, T. J., Kilgore, K. I., Rice, J. R. and Sills, L. B., Acta metall., 1979, 27, 265. 18. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 82, p. 347. 19. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 82, p. 351. 20. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 50, p. 193. 21. Hirth, J. P. and Lothe, J., Theory of Dislocations. Krieger, Melbourne, Florida, 1992, p. 42.
22. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 134, p. 602. 23. Rice, J. R., in Fracture, Vol. II, ed. H. Liebowitz. Academic Press, New York, 1968, p. 191. 24. Vitek, V., J. Mech. Phys. Solids, 1975, 24, 67. 25. Eshelby, J. D., in Progress in Solid Mechanics, Vol. II, ed. I. N. Sneddon and R. Hill. North-Holland, Amsterdam, 1961, p. 87. 26. Eshelby, J. D., Phil. Mag., 1949, 40, 903. 27. Hirth, J. P. and Lothe, J., Theory of Dislocations. Krieger, Melbourne, Florida, 1992, p. 42. 28. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 89, p. 385.
Acta mater. Vol. 47, No. 1, pp. 1±4, 1999 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00351-6 1359-6454/99 $19.00 + 0.00
DISLOCATIONS WITHIN ELLIPTICAL HOLES J. P. HIRTH School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 991642920, U.S.A. (Received 15 September 1998; accepted 1 October 1998) AbstractÐElastic ®elds are derived for edge and screw dislocations within holes of elliptical cross-section. The results are applicable to interactions with defects near the tips of blunted cracks and to hollow dislocation cores. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
in determining the equilibrium shape of the hole by free energy minimization [9].
Long ago, Frank [1] showed that dislocation cores could be hollow and derived the equilibrium size for a screw dislocation in a hole of circular crosssection. There were subsequently a number of observations of hollow cores for screw dislocations, with a current resurgence of interest in the area [2± 8]. For an edge dislocation, the cross-sectional shape should more nearly approximate an ellipse than a circle [9]. Indeed, Chisholm and Smith [10] have observed elliptically shaped amorphous cores for edge dislocations in a YBCO high Tc superconductor, a con®guration that has a ®eld analogous to that of a hollow core. Hence, the elastic ®eld of a dislocation in an elliptical hole is of interest. Both edge and screw ®elds are relevant for mixed dislocation cases or for cases where there is anisotropy in the surface energy [9]. The solution is also applicable to internal cracks containing edge or screw dislocations. Weertman [11] and Lin and Hirth [12] showed that the ®eld of a crack contained within a body is modi®ed when a dislocation is emitted at a crack tip. The antidislocation lies within the crack. In this case, the ®eld of the crack and the contained defect was derived for a planar crack [11, 12]. There is current interest in the atomistic simulation of cracks [13±16] and in such simulations the planar crack model is inadequate. Blunting at the crack tip, even in the elastic case because of nonlinear eects, creates a tip con®guration more closely approaching an ellipse. Also, for high temperature crack growth by diusive processes, as analyzed by Chuang et al. [17] for example, the crack tip con®guration is curved. The elastic ®elds, presented as follows, provide information for core±core dislocation interactions. In addition, for the hollow core case, the ®eld at the surface of the hole gives the strain energy contribution to the chemical potential there, of interest
2. EDGE DISLOCATION IN A HOLE WITH ELLIPTICAL CROSS-SECTION
The ®eld of an edge dislocation with a hollow core of elliptical cross-section can be derived in terms of complex potential functions f(z) and c(z) following the procedures of Muskhelishvili [18]. An ellipse with major and minor axes a and d, respectively, Fig. 1, is de®ned by the parameters RE a d =2 and d a ÿ d = a d . For mode II and III crack problems, d would be positive, while for mode I crack problems and the typical edge dislocation core, d would be negative. The complex coordinate z x iy is mapped onto a circle by z=RE z d=z, where z r exp ij. With Z m=4pRE 1 ÿ b2 ÿ ib1 , where m is the shear modulus, n is Poisson's ratio, and bi is the Burgers vector, the stress functions for the edge dislocation in an in®nite continuous medium, a plane strain case, are j z)f z Z ln z2 d=z c z)c z Z ln z2 d=z j 0 z Z z2 ÿ d=z z2 d:
1
Here the overbar indicates complex conjugate. Following the procedures of Ref. [19] and taking residues, we ®nd that the added stress functions needed to account for the presence of the hole are j0 ÿ Z ln
1 dz2 z
c0 ÿ Z ln
1 dz2 1 dz2 1 ÿ dz2 ÿZ 2 ÿZ 2 : 2 z z d z ÿ d
The total stress function is given by the sum of equations (1) and (2). The corresponding stresses 1
2
HIRTH: DISLOCATIONS WITHIN ELLIPTICAL HOLES
Fig. 1. Coordinates for elliptical hole.
for the plane strain case are
sjj
srr sjj F B F B, sjj ÿ srr 2isrj 2
2B 0 2 0 f$ BF B $ 0 BC Bg r $ B 0
0
0
3
0
where F B j B=$ B, C B c B=$ B, and $ 0 B B2 ÿ d=B2 . For the determination of the chemical potential for a hollow elliptical core, we only need the strain energy density at the surface of the hole where srr srj 0 and r 1. Thus, the relevant stresses are [20] sjj 4 Re j 0 z=o 0 z, szz sjj :
mb sin W 1 2d pRE 1 ÿ mb sin W 1 2d d cos 2W: pRH 1 ÿ
The corresponding strain energy density at the surface of the hole is [21] wel
1 1 sjj szz 2 ÿ sjj szz 4m 1 4m s2jj 1 ÿ : 4m
mb 1 ÿ d 1 d2 sin j : pRE 1 ÿ 1 d2 2 ÿ 4d2 cos2 2j
4
wel
sjj
1 d2 sin2 W 1 ÿ d2 cos2 W : 1 d4 sin2 W 1 ÿ d4 cos2 W
syz isxz j 0 z, uz Im 6
In the limit of a perturbed circular cross-section with dÿ ÿ40, ÿ this expression reduces to the form pertinent to Ref. [9]
9
Again we refer to the coordinates of Fig. 1. Muskhelishvili [22] presents the form for the antiplane strain stress function for the example of torsional loading as a special case of a Dirichlet problem. For dislocations and line forces, the simpler form [23] is
mb sin W 1 d2 pRH 1 ÿ
mb2 : ÿ d2
4p2 1
3. SCREW DISLOCATION IN A HOLE WITH ELLIPTICAL CROSS-SECTION
5
In terms of the real space coordinates in Fig. 1, since cos y 1 d RE =RH cos j, sin y 1 ÿ d RE =RH sin j, and R2H =R2E 1 d2 2d cos 2j 1 ÿ d2 2 = 1 d2 ÿ 2d cos 2W, equation (5) becomes
8
When y 0, wel 0. When y p=2, RH d and
For the present case of interest where b1 b, b2 0, and r 1, the stress sjj becomes sjj
7
j z : m
10
The transformation to the coordinates of Fig. 1 is accomplished by sjz isrz and
eij o 0 B syz isxz jo 0 Bj
11
HIRTH: DISLOCATIONS WITHIN ELLIPTICAL HOLES
j 0 z
0
j B : o 0 B
12
These relations combine to give sjz isrz
eij j 0 B: jo 0 Bj
13
Guided by the result for the edge dislocation, we ®nd that the appropriate stress function for the screw dislocation is j g ln
B2 d 1 dB2 ÿ g ln B B
14
with g mb=4pR. As for the edge dislocation case, this result reduces to the proper continuous in®nite body result j 2g ln z in the limit r=dÿ ÿ41. ÿ On the surface of the hole where r 1, the substitution of equation (14) into equation (13) gives the result srz 0 and sjz
mb 1 ÿ d2 2pR 1 d2 2 ÿ 4d2 cos2 2j mb 1 d2 sin2 W 1 ÿ d2 cos2 W3=2 : 2pRH 1 d4 sin2 W 1 ÿ d4 cos2 W 15
The strain energy density at the surface of the hole is w
s2jz : 2m
16
The solutions satisfy the free surface boundary conditions, give the proper Burgers vector as de®ned by a Burgers circuit surrounding the hole, and reduce to the in®nite body Volterra solution at positions remote from the hole.
such a manner [26]. As in the Peierls case [27], the ®elds can also be represented by a dislocation, diminished in magnitude of the Burgers vector, at the center of the ellipse and two image dislocations. In the present case, the images reside at the focal points of the ellipse. When d > 0, the focal points are at y 2c with c a2 ÿ d2 1=2 while for d<0, the focal points are at x 2c. For the hollow core case, where a minimization procedure is used to determine the equilibrium shape, the elliptical shape considered here is an approximation for most edge dislocation cases. In the absence of surface energy anisotropy, the zero strain energy result for equation (8) at y 0 prevents chemical potential equilibration for an elliptical shape [9]. However, the best ®t of an ellipse at several points [9] is an improvement relative to a circular cross-section and provides a starting point for atomistic calculations. As discussed by Muskhelishvili [28], a series of ellipses can be superposed to give a result for a hole of any shape, so an exact result can be found for a minimum energy shape other than a simple ellipse in an extension of the present methods. In summary, ®elds are derived for both edge and screw dislocations within an elliptical hole. The results are given in the isotropic elastic approximation. However, the physical interpretation of the solution indicates the corresponding form for the anisotropic elastic case. AcknowledgementsÐThe author is pleased to acknowledge the support of this research by the Defense Research Projects Agency through the Oce of Naval Research under Grant N00014-97-1-6007 as well as helpful discussions with N. Sridhar, D. J. Srolovitz and J. W. Cahn.
REFERENCES
4. DISCUSSION AND SUMMARY
The above results would apply directly to a dislocation within an elliptical crack or an elliptical core. In the crack case, the results can be coupled to the result of Vitek [24] for a dislocation emitted from a crack. For a ¯uid ®lled core, or an amorphous core in the case that shear stresses can relax in the core, the above solution gives a part of the complete ®eld. The other part of the ®eld in this case is associated with a possible isostatic stress in the core, with attendant shear stresses in the surrounding medium. The latter stresses are given by the Eshelby solution to the elliptical inclusion problem [25]. The total ®eld, according to the principle of superposition, is the sum of the two sub®elds. The physical basis for the above results is that the dislocation within the hole becomes smeared into a continuous distribution of in®nitesimal dislocations along the axis of the ellipse. An analogy is Eshelby's treatment of the Peierls dislocation in
3
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Frank, F. C., Acta crystallogr., 1951, 4, 497. Verma, A. R., Nature, 1951, 167, 939. Bhide, V. G., Physica, 1958, 24, 817. Giocondi, J., Rohrer, G. S., Skowronski, M., Balakrishna, V., Augustine, G., Hobgood, H. M. and Hopkins, R. H., J. Cryst. Growth, 1997, 181, 151. Qian, W., Rohrer, G. S., Skowronski, M., Doverspace, K., Rowland, L. B. and Gaskill, D. K., Appl. Phys. Lett., 1995, 66, 1252; 1995, 67, 2284. Ning, X., Chien, F. R., Pirouz, P., Yang, J. W. and Asif Khan, M., J. Mater. Res., 1996, 11, 580. Stein, R. A., Physica B, 1993, 185, 211. Pirouz, P., Phil. Mag., 1998, 78, 727. Srolovitz, D. J., Sridhar, N., Hirth, J. P. and Cahn, J. W., Scripta mater., 1998, 39, 379. Chisholm, M. F. and Smith, D. A., Phil. Mag., 1989, 59, 181. Weertman, J., Int. J. Fract., 1984, 185, 211. Lin, I. H. and Hirth, J. P., Phil. Mag. A, 1984, 50, L43. Cheetham, A. K., Suter, U. W., Winner, E. and Yip, S., J. Computer-Aid. Mater. Sci., 1996, 3, 1. Sun, Y., Beltz, G. E. and Rice, J. R., Mater. Sci. Engng, 1993, A170, 67. Hoagland, R. G., Phil. Mag. A, 1997, 76, 543.
4
HIRTH: DISLOCATIONS WITHIN ELLIPTICAL HOLES
16. Zhou, S. J., Lomdahl, P. S., Thomson, R. and Holian, B. L., Phys. Rev. Lett., 1996, 76, 2318. 17. Chuang, T. J., Kilgore, K. I., Rice, J. R. and Sills, L. B., Acta metall., 1979, 27, 265. 18. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 82, p. 347. 19. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 82, p. 351. 20. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 50, p. 193. 21. Hirth, J. P. and Lothe, J., Theory of Dislocations. Krieger, Melbourne, Florida, 1992, p. 42.
22. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 134, p. 602. 23. Rice, J. R., in Fracture, Vol. II, ed. H. Liebowitz. Academic Press, New York, 1968, p. 191. 24. Vitek, V., J. Mech. Phys. Solids, 1975, 24, 67. 25. Eshelby, J. D., in Progress in Solid Mechanics, Vol. II, ed. I. N. Sneddon and R. Hill. North-Holland, Amsterdam, 1961, p. 87. 26. Eshelby, J. D., Phil. Mag., 1949, 40, 903. 27. Hirth, J. P. and Lothe, J., Theory of Dislocations. Krieger, Melbourne, Florida, 1992, p. 42. 28. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordho, Leyden, 1975, Section 89, p. 385.