Research Papers

Formation of a Prismatic Dislocation Loop in the Interface of a Circular Cylindrical Inclusion Embedded in a Thin Slab

[+] Author and Article Information
Jérôme Colin

Institut P’,
Université de Poitiers,
SP2MI-Téléport 2,
Futuroscope F86962, Chasseneuil Cedex, France
e-mail: jerome.colin@univ-poitiers.fr

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 31, 2015; final manuscript received October 19, 2015; published online November 13, 2015. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 83(2), 021006 (Nov 13, 2015) (7 pages) Paper No: JAM-15-1458; doi: 10.1115/1.4031895 History: Received August 31, 2015; Revised October 19, 2015

The introduction of a prismatic dislocation loop in the interface of an axisymmetric precipitate embedded in a thin slab of infinite lateral extension has been theoretically investigated. The critical misfit strain resulting from the lattice mismatch between the inclusion and the slab has been characterized for the loop formation versus the thickness of the slab and the radius of the inclusion. The case where the precipitate is embedded in a semi-infinite matrix is also discussed and a stability diagram of the structure is displayed with respect to the loop introduction versus the geometric and misfit parameters.

Copyright © 2016 by ASME
Topics: Slabs , Dislocations
Your Session has timed out. Please sign back in to continue.


Mura, T. , 1987, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht.
Nix, W. D. , 1989, “ Mechanical Properties of Thin Films,” Metall. Trans. A, 20(11), pp. 2217–2245. [CrossRef]
Shchukin, V. A. , and Bimberg, D. , 1999, “ Spontaneous Ordering of Nanostructures on Crystal Surfaces,” Rev. Mod. Phys., 71(4), pp. 1125–1171. [CrossRef]
Freund, L. B. , and Suresh, S. , 2003, Thin Film Materials: Stress, Defect Formation and Surface Evolution, Cambridge University Press, Cambridge, UK.
Seo, K. , and Mura, T. , 1979, “ The Elastic Field in a Half Space Due to Ellipsoidal Inclusions With Uniform Dilatational Eigenstrains,” ASME J. Appl. Mech., 46(3), pp. 568–572. [CrossRef]
Youngdahl, C. K. , and Sternberg, E. , 1966, “ Three-Dimensional Stress Concentration Around a Cylindrical Hole in a Semi-Infinite Elastic Body,” ASME J. Appl. Mech., 33(4), pp. 855–865. [CrossRef]
Hasegawa, H. , Lee, V.-G. , and Mura, T. , 1993, “ Hollow Circular Cylindrical Inclusion at the Surface of a Half-Space,” ASME J. Appl. Mech., 60(1), pp. 33–40. [CrossRef]
Wu, L. Z. , and Du, S. Y. , 1996, “ The Elastic Field in a Half-Space With a Circular Inclusion,” ASME J. Appl. Mech., 63(4), pp. 925–932. [CrossRef]
Kroupa, F. , 1960, “ Circular Edge Dislocation Loop,” Czech. J. Phys. B, 10(4), pp. 284–293. [CrossRef]
Chou, Y. T. , 1963, “ The Energy of Circular Dislocation Loops in Thin Plates,” Acta Metall., 11(8), pp. 829–834. [CrossRef]
Bastecká, J. , 1964, “ Interaction of Dislocation Loop With Free Surface,” Czech. J. Phys. B, 14(6), pp. 430–442. [CrossRef]
Salamon, N. J. , 1981, “ The Circular Glide Dislocation Loop Lying in an Interface,” J. Mech. Phys., 29(1), pp. 1–11. [CrossRef]
Matthews, J. W. , and Blakeslee, A. E. , 1974, “ Defects in Epitaxial Multilayers: I. Misfit Dislocations,” J. Cryst. Growth, 27, pp. 118–125.
Xin, X. J. , Daehn, G. S. , and Wagoner, R. H. , 1997, “ Equilibrium Configuration of Coaxial Prismatic Dislocation Loops and Related Size-Dependent Plasticity,” Acta Mater., 45(5), pp. 1821–1836. [CrossRef]
Giannattasio, A. , Senkader, S. , Falster, R. J. , and Wilshaw, P. R. , 2002, “ Generation of Dislocation Glides Loops in Czochralski Silicon,” J. Phys.: Condens. Matter, 14(48), pp. 12981–12987. [CrossRef]
Kolesnikova, A. L. , and Romanov, A. E. , 2004, “ Misfit Dislocation Loops and Critical Parameters of Quantum Dots and Wires,” Philos. Mag. Lett., 84(8), pp. 501–506. [CrossRef]
Kolesnikova, A. L. , and Romanov, A. E. , 2004, “ Misfit Dislocation Loop Nucleation at a Quantum Dot,” Tech. Phys. Lett., 30(2), pp. 126–128. [CrossRef]
Kolesnikova, A. L. , and Romanov, A. E. , 2004, “ Generation of Dislocation Loops in Strained Quantum Dots Embedded in a Heterolayer,” Phys. Solid. State, 46(9), pp. 1644–1648. [CrossRef]
Aifantis, K. E. , Kolesnikova, A. L. , and Romanov, A. E. , 2007, “ Nucleation of Misfit Dislocations and Plastic Deformation in Core/Shell Nanowires,” Philos. Mag. Lett., 87(30), pp. 4731–4757. [CrossRef]
Chaldyshev, V. V. , Bert, N. A. , Kolesnikova, A. L. , and Romanov, A. E. , 2009, “ Stress Relaxation Scenario for Buried Quantum Dots,” Phys. Rev. B, 79(23), p. 233304. [CrossRef]
Gutkin, M. Yu. , Kuzmin, K. V. , and Sheinerman, A. G. , 2011, “ Misfit Stresses and Relaxation Mechanisms in a Nanowire Containing a Coaxial Cylindrical Inclusion of Finite Length,” Phys. Stat. Sol. B, 248(7), pp. 1651–1657. [CrossRef]
Ovid'ko, I. A. , and Sheinerman, A. G. , 2004, “ Misfit Dislocation Loops in Composite Nanowires,” Philos. Mag., 84(20), pp. 2103–2118. [CrossRef]
Liang, Y. , Nix, W. D. , Griffin, P. B. , and Plummer, J. D. , 2005, “ Critical Thickness Enhancement of Epitaxial SiGe Films Grown on Small Structures,” J. Appl. Phys., 97(4), p. 043519. [CrossRef]
Chu, H. J. , Zhou, C. Z. , Wang, J. , and Beyerlein, I. J. , 2013, “ An Analytical Model for the Critical Shell Thickness in Core–Shell Nanowires Based on Crystallographic Slip,” J. Mech. Phys. Solids, 61(11), pp. 2147–2160. [CrossRef]
Kolesnikova, A. L. , Gutkin, M. Yu. , Krasnitckii, S. A. , and Romanov, A. E. , 2013, “ Circular Dislocation Loops in Elastic Bodies With Spherical Free Surfaces,” Int. J. Solids Struct., 50(10), pp. 1839–1857. [CrossRef]
Walfer, W. G. , and Drugan, W. J. , 1988, “ Elastic Interaction Energy Between a Prismatic Dislocation Loop and a Spherical Cavity,” Philos. Mag. A, 57(6), pp. 923–937. [CrossRef]
Ahn, D. C. , Sofronis, P. , and Minich, R. , 2006, “ On the Micromechanics of Void Growth by Prismatic-Dislocation Loop Emission,” J. Mech. Phys. Solids, 54(4), pp. 735–755. [CrossRef]
Willis, J. R. , Bullough, B. , and Stoneham, A. M. , 1983, “ The Effect of Dislocation Loop on the Lattice Parameter Determined by X-Ray Diffraction,” Philos. Mag. A, 48(1), pp. 95–107. [CrossRef]
Bondarenko, V. P. , and Litoshneko, N. V. , 1997, “ Stress–Strain State of a Spherical Layer With Circular Dislocation Loop,” Intern. Appl. Mech., 33(7), pp. 525–531. [CrossRef]
Gutkin, M. Yu. , Kolesnikova, A. L. , Krasnitckii, S. A. , and Romanov, A. E. , 2014, “ Misfit Dislocation Loops in Composite Core–Shell Nanoparticles,” Phys. Solid State, 56(4), pp. 723–730. [CrossRef]
Gutkin, M. Yu. , Kolesnikova, A. L. , Krasnitckii, S. A. , Romanov, A. E. , and Shalkovskii, A. G. , 2014, “ Misfit Dislocation Loops in Hollow Core–Shell Nanoparticles,” Scr. Mater., 83, pp. 1–4. [CrossRef]
Lubarda, V. A. , 2011, “ Emission of Dislocations From Nanovoids Under Combined Loading,” Int. J. Plast., 27, pp. 181–200. [CrossRef]
Gutkin, M. Yu. , Kolesnikova, A. L. , Krasnitckii, S. A. , Dorogin, L. M. , Serebryakova, V. S. , Vikarchuk, A. A. , and Romanov, A. E. , 2015, “ Stress Relaxation in Icosahedral Small Particles Via Generation of Circular Prismatic Dislocation Loops,” Scr. Metall., 105, pp. 10–13. [CrossRef]
Chang, T. , Guo, W. , and Dong, H. R. , 2001, “ Three-Dimensional Effects for Through-Thickness Cylindrical Inclusions in an Elastic Plate,” J. Strain Anal. Eng., 36(3), pp. 277–286. [CrossRef]
Timoshenko, S. , and Goodier, J. N. , 1951, Theory of Elasticity, 2nd ed., McGraw-Hill, New York.
Sneddon, J. N. , 1951, Fourier Transforms, McGraw Hill, New York.
Hirth, J. P. , and Lothe, J. , 1982, Theory of Dislocations, 2nd ed., Wiley, New York.
Kolesnikova, A. I. , and Romanov, A. E. , 2004, “ Virtual Circular Dislocation–Disclination Loop Technique in Boundary Value Problems in the Theory of Defects,” ASME J. Appl. Mech., 71(3), pp. 409–417. [CrossRef]
Dundurs, J. , and Salamon, N. J. , 1972, “ Circular Prismatic Dislocation Loop in a Two-Phase Material,” Phys. State Solids, 50(1), pp. 125–133. [CrossRef]
Salamon, N. J. , and Comninou, M. , 1979, “ The Circular Prismatic Dislocation Loop in an Interface,” Philos. Mag. A, 39(5), pp. 685–691. [CrossRef]
Ovid'ko, I. A. , and Sheinerman, A. G. , 2004, “ Misfit Dislocation Loops in Cylindrical Quantum Dots,” J. Phys.: Condens. Matter, 16(41), pp. 7225–7232. [CrossRef]
Wolfram Research, 2014, Mathematica, Version 10.1, Wolfram Research, Champaign, IL.


Grahic Jump Location
Fig. 1

An axisymmetric precipitate of radius R is embedded in a thin slab of thickness h. The elastic coefficients of both phases are equal. A prismatic dislocation loop of Burgers vector (0, 0, b) is located in the interface between the inclusion and the slab at a distance d from the upper free-surface.

Grahic Jump Location
Fig. 2

Total energy variation ΔẼtot versus the loop distance from the upper free-surface d̃ for different values of the misfit strain parameter ε*ν. For ε*ν=0.001, ΔẼtot is positive for all d̃ values. A minimum in energy is observed when the loop is emitted in the interface at d = h/2 for ε*ν=0.006. ΔẼtot cancels at d = h/2 when the misfit parameter reaches a critical value ε*ν,c=0.01196. For ε*ν=0.015 > ε*ν,c, the formation of the interface dislocation loop should occur in the horizontal plane of symmetry of the structure.

Grahic Jump Location
Fig. 3

Critical misfit strain parameter ε*ν,c versus the slab thickness h̃, for different values of the inclusion radius R̃

Grahic Jump Location
Fig. 4

Total energy variation ΔẼtot versus the distance d̃ between the loop and the free-surface in the case of a semi-infinite matrix and for different values of the misfit parameter ε*ν. For ε*ν=0.002, ΔẼtot is positive. For ε*ν=0.008, ΔẼtot is negative for d̃ > 92 and the introduction of the loop is favorable.

Grahic Jump Location
Fig. 5

Critical distance from the free-surface d̃c versus the misfit strain parameter ε*ν, for different values of the inclusion radius R̃

Grahic Jump Location
Fig. 6

Critical misfit strain parameter ε*ν,c versus the distance of the loop from the free-surface d̃ for different values of the radius R̃

Grahic Jump Location
Fig. 7

Three-dimensional contourplot of ΔẼtot energy versus (R̃,d̃,ε*ν), the surface of energy corresponding to ΔẼtot=0. Region 1 is dislocation-free, the dislocation formation is favorable in region 2.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In