On Nix's Theorem for Two Skew Dislocations in Anisotropic Elastic Half-Spaces and Bimaterials

Abstract

In the earlier papers, the net interaction force F between two skew dislocations with Burgers vectors b, b separated by a distance h in an infinite anisotropic elastic space, in a half space, or in a dissimilar bimaterial has been presented. In this paper, the authors employ Green's functions to obtain new explicit expressions of F. The new solutions provide new physical interpretations. The authors show that, for the infinite space and the half-space with a traction-free surface, F is independent of b₂, b₂ and that F = 0 if b or b is along the x2-axis. F is independent of the skew angle Oif the projections of b, b on the plane X2 = 0 lie along a pair of conjugate radii of an ellipse. For other half-spaces and bimaterials, F(on b) = 0 when b and b x v are along the x2-axis where v is the axial vector of a skew-symmetric matrix. Hence b is arbitrary if v = 0, which is the case for the half-space with a slippery surface and for certain bimaterials. F(on b) is independent of 0 when b is along the x2-axis and b is coplanar with v and the xi-axis. More physical interpretations are presented in the paper.

Keywords

dislocations interaction force anisotropic elasticity half-space bimaterials

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References

[1] Orlov, S. S. and Indenbom, V L. : The interaction of non-parallel dislocations in an anisotropic medium. Kristallo-grafiya, 14, 780-783 (1969).

[2] Nix, W D. : Private communications (1998).

[3] Barnett, D. M. : The net interaction force between two skew dislocations in an elastically anisotropic half-space.

[4] Bamett, D. M. , Ting, TC.T , and Kirchner, H.O.K. : The net interaction force between two skew dislocations in an anisotropic linear elastic bimetallic medium. Materials Science and Engineering A Proc. of the Symposium in honor of T Mori's 65th Birthday held in Seattle, WA, July 20-22, in press (2000).

[5] Stroh, A. N. : Dislocations and cracks in anisotropic elasticity. Phil. Mag., 3, 625-646 (1958).

[6] Ting, T.C.T. : Anisotrmpic Elasticity: Theory and Applications. Oxford University Press, New York, 1996.

[7] Barnett, D. M. and Lothe, J. : Synthesis of the sextic and the integral formalism for dislocations, Greens function and surface waves in anisotropic elastic solids. Phys. Norv., 7, 13-19 (1973).

[8] Chadwick, P. and Smith, G. D. : Foundations of the theory of surface waves in anisotropic elastic materials. Adv Appl. Mech., 17, 303-376 (1977).

[9] Ting, T.C.T. : New explicit expression of Barnett-Lothe tensors for anisotropic linear elastic materials. J. Elasticity, 47, 207-218 (1997).

10.

[10] Ting, TC.T. : Image singularities of Green's functions for anisotropic elastic half-spaces and bimaterials. Q. J. Mech. Appl. Math., 45, 119-139 (1992).

11.

[11] Ting, T.C.T. and Wang, M. Z. : Generalized Stroh formalism for anisotropic elasticity for general boundary conditions. Acta Mechanica Sinica, 8, 193-207 (1992).

12.

[12] Ingebrigtsen, K. A. and Tonning, A. : Elastic surface waves in crystal. Phys. Rev., 184, 942-951 (1969).

13.

[13] Ting, T.C.T : Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int. J. Solids Structures, 22, 965-983 (1986).

14.

[14] Qu, Jianmin and Bassani, J. L. : Cracks on bimaterial and bicrystal interfaces. J. Mech. Phys. Solids, 37, 417-433 (1989).

15.

[15] Ting, TC.T. : Some identities and the structure of Ni in the Stroh formalism of anisotropic elasticity. Q. Appl. Math., 46, 109-120 (1988).

16.

[16] Barnett, D. M. and Chadwick, P. : The existence of one-component surface waves and exceptional subsequent transonic states of types 2, 4, and El in anisotropic elastic media, in Modern Theory of Anisotmopic Elasticity and Applications, pp. 199-214, ed. J. J. Wu , TC.T. Ting , and D. M. Barnett , SIAM Proceedings Series, SIAM, Philadelphia, 1991.

17.

[17] Gibbs, J. W. : Elements of ector Analysis, Dover, New York, 1961.

18.

[18] Boulanger, Ph . and Hayes, M. : Bivectors and inhomogeneous plane waves in anisotropic elastic bodies. Modern Theory of Anisotmopic Elasticity and Applications, pp. 280-289, ed. J. J. Wu , T.C.T Ting , and D. M. Barnett , SIAM Proceedings Series, SIAM, Philadelphia, 1991.