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Method of Images Solution for an Edge Dislocation and a Circular Cavity in Crystalline Solids

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Abstract

Mechanics of defects in solids across a wide span of length scales is commonly formulated using the dislocations theory. This paper revisits the classical problem of interaction between an elastic edge dislocation and a circular cavity. A heuristic, yet, mechanistic approach is taken to obtain the stress solution to this problem. The approach uses complex variable theory of elasticity, along with method of images. For this purpose, a definition and formulation of elastic dipole singularities similar to dipole charges in electrostatics is developed. It is shown that an image dislocation with Burger’s vector of the same strength as the real dislocation but in opposite direction, as well as a set of four singularities including a dislocation dipole, a moment-dilatation dipole, and two centers of dilatation would establish a circular, traction-free boundary in an infinite elastic medium. Adding a Volterra dislocation to the finite-length edge dislocation from this study would recover the related problem of interaction between an infinite-length edge dislocation and circular cavity. The interesting analogy between the considered elastic problem and the electrostatic problem of interaction between a line electric charge and a cylindrical conductor is discussed.

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REFERENCES

  1. Jackson, J.D., Classical Electrodynamics, John Wiley and Sons, 2007.

  2. Barber, J.R., Elasticity, Dordrecht: Kluwer Academic Publishers, 2002.

  3. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, Clarendon Press, 1992.

  4. Munson, B.R., Okiishi, T.H., Huebsch, W.W., and Rothmayer, A.P., Fluid Mechanics, Singapore: Wiley, 2013.

  5. Volterra, V., Note on the Application of the Method of Images to Problems of Vibrations, Proc. Lond. Math. Soc., 1905, vol. 2, no. 1, pp. 327–331.

  6. Maxwell, J.C., A Treatise on Electricity and Magnetism. V. 1, Clarendon Press, 1873.

  7. Eshelby, J.D., The Force on an Elastic Singularity, Philos. Trans. Roy Soc. Lond. A. Math. Phys. Sci., 1951, vol. 244, no. 877, pp. 87–112.

  8. Head, A.K., The Interaction of Dislocations and Boundaries, Philos. Mag. J. Sci., 1953, vol. 44, no. 348, pp. 92–94.

  9. Head, A.K., Edge Dislocations in Inhomogeneous Media, Proc. Phys. Soc. B, 1953, vol. 66, no. 9, p. 793.

  10. Dundurs, J., Elastic Interaction of Dislocations with Inhomogeneities, in Mathematical Theory of Dislocations, 1969, pp. 70–115.

  11. Taylor, R.I., The Force on a Screw Dislocation Due to a Series of Layers of Alternating Shear Modulus, Semiconduct. Sci. Technol., 1989, vol. 4, no. 8, p. 612.

  12. Friedman, L.H. and Chrzan, D.C., Scaling Theory of the Hall–Petch Relation for Multilayers, Phys. Rev. Lett., 1998, vol. 81, no. 13, p. 2715.

  13. Chou, Y.T., Pande, C.S., and Masumura, R.A., The Role of Harmonic Functions in Dislocation–Boundary Interactions by the Method of Images, Mater. Sci. Eng. A, 2007, vol. 452, pp. 99–102.

  14. Ma, C.C. and Lu, H.T., Theoretical Analysis of Screw Dislocations and Image Forces in Anisotropic Multilayered Media, Phys. Rev. B, 2006, vol. 73, no. 14, p. 144102.

  15. Wen, J. and Wu, M.S., Analysis of a Line Defect in a Multilayered Smart Structure by the Image Method, Mech. Mater., 2007, vol. 39, no. 2, pp. 126–144.

  16. Zhou, K. and Wu, M.S., Elastic Fields Due to an Edge Dislocation in an Isotropic Film–Substrate by the Image Method, Acta Mech., 2010, vol. 211, no. 3–4, pp. 271–292.

  17. Ogbonna, N., Force on a Screw Dislocation in a Multiphase Laminated Structure, Math. Mech. Solids, 2014, vol. 19, no. 6, pp. 694–702.

  18. Ogbonna, N., On Screw Dislocation in a Multiphase Lamellar Inclusion, J. Niger. Math. Soc., 2015, vol. 34, no. 1, pp. 32–39.

  19. Chou, T.W., Elastic Behavior of Disclinations in Nonhomogenous Media, J. Appl. Phys., 1971, vol. 42, no. 12, pp. 4931–4935.

  20. Chou, T.W. and Pan, Y.C., Elastic Energies of Disclinations in Hexagonal Crystals, J. Appl. Phys., 1973, vol. 44, no. 1, pp. 63–65.

  21. Eshelby, J.D., Screw Dislocations in Thin Rods, J. Appl. Phys., 1953, vol. 24, no. 2, pp. 176–179.

  22. Ogbonna, N., On Elastic Interaction of a Screw Dislocation with a Coated Cylindrical Inclusion, J. Eng. Math., 2016, vol. 99, no. 1, pp. 203–212.

  23. Wang, X. and Pan, E., Screw Dislocations in Piezoelectric Nanowires, Mech. Res. Comm., 2010, vol. 37, no. 8, pp. 707–711.

  24. Dundurs, J. and Mura, T., Interaction between an Edge Dislocation and a Circular Inclusion, J. Mech. Phys. Solids, 1964, vol. 12, no. 3, pp. 177–189.

  25. Dundurs, J. and Hetényi, M., The Elastic Plane with a Circular Insert, Loaded by a Radial Force, J. Appl. Mech., 1961, vol. 28, no. 1, p. 103.

  26. Hetényi, M. and Dundurs, J., The Elastic Plane with a Circular Insert, Loaded by a Tangentially Directed Force, J. Appl. Mech., 1962, vol. 29, no. 2, p. 362.

  27. List, R.D., A Two-Dimensional Circular Inclusion Problem, Math. Proc. Cambridge Philos. Soc., 1969, vol. 65, no. 3, pp. 823–830.

  28. Povstenko, Yu.Z., Interaction between an Edge Dislocation and a Circular Boundary in the Presence of an Alien Surface Layer, Sov. Appl. Mech., 1975, vol. 11, no. 3, pp. 272–277.

  29. Fukuzaki, K. and Shioya, S., On the Interaction between an Edge Dislocation and Two Circular Inclusions in an Infinite Medium, Int. J. Eng. Sci., 1986, vol. 24, no. 12, pp. 1771–1787.

  30. Chen, D.H., Green’s Functions for a Point Force and Dislocation Outside an Elliptic Inclusion in Plane Elasticity, ZAMP, 1996, vol. 47, no. 6, pp. 894–905.

  31. Warren, W.E., The Edge Dislocation Inside an Elliptical Inclusion, Mech. Mater., 1983, vol. 2, no. 4, pp. 319–330.

  32. Fang, Q.H., Liu, Y.W., and Jiang, C.P., Edge Dislocation Interacting with an Interfacial Crack Along a Circular Inhomogeneity, Int. J. Solids Struct., 2003, vol. 40, no. 21, pp. 5781–5797.

  33. Wang, X., Interaction between an Edge Dislocation and a Circular Inclusion with an Inhomogeneously Imperfect Interface, Mech. Res. Commun., 2006, vol. 33, no. 1, pp. 17–25.

  34. Dai, D.N., An Edge Dislocation Inside a Semi-Infinite Plane Containing a Circular Hole, Int. J. Solids Struct., 2018, vol. 136, pp. 295–305.

  35. Malvern, L.E., Introduction to the Mechanics of a Continuous, Englewood Clifs, N.J.: Prentice-Hall, 1969.

  36. Muskhelishvili N.I. Some Basic Problems of the Mathematical Theory of Elasticity, Graningen: Noordhoff, 1954.

  37. Ardakani, S.M. and Ulm, F.J., Chemoelastic Fracture Mechanics Model for Cement Sheath Integrity, J. Eng. Mech., 2014, vol. 140, no. 4, p. 04013009.

  38. Lardner, R.W., Mathematical Theory of Dislocations and Fracture, Toronto: University of Toronto Press, 1974.

  39. Chen, Y.Z. and Lin, X.Y., Potentials In-Plane Elasticity by Distribution of Dislocation Doublet or Force Doublet Along a Curve, Int. J. Eng. Sci., 1998, vol. 36, no. 1, pp. 23–31.

  40. Denda, M. and Kosaka, I., Dislocation and Point-Force-Based Approach to the Special Green’s Function BEM for Elliptic Hole and Crack Problems in Two Dimensions, Int. J. Numer. Meth. Eng., 1997, vol. 40, no. 15, pp. 2857–2889.

  41. Green, A.E. and Zerna, W., Theoretical Elasticity, New York: Dover Publications, 2012.

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  1. Department of Energy and Mineral Engineering, Earth and Mineral Sciences Energy Institute, The Pennsylvania State University, 16802, University Park, PA, USA

    K. Nguyen & A. Mehrabian

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  1. K. Nguyen
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  2. A. Mehrabian
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Correspondence to K. Nguyen.

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Russian Text © The Author(s), 2020, published in Fizicheskaya Mezomekhanika, 2020, Vol. 23, No. 4, pp. 31–42.

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Nguyen, K., Mehrabian, A. Method of Images Solution for an Edge Dislocation and a Circular Cavity in Crystalline Solids. Phys Mesomech 24, 20–31 (2021). https://doi.org/10.1134/S1029959921010045

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  • DOI: https://doi.org/10.1134/S1029959921010045

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