Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T07:04:10.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributed Dislocation Method for Determining Elastic Fields of 2D and 3D Volume Misfit Particles in Infinite Space and Extension of the Method for Particles in Half Space

Published online by Cambridge University Press:  01 December 2014

J. D. Lerma ,
T. Khraishi ,
S. Kataria  and
Y.-L. Shen
J. D. Lerma
Affiliation:
Department of Mechanical Engineering, University of New Mexico New Mexico, USA
T. Khraishi*
Affiliation:
Department of Mechanical Engineering, University of New Mexico New Mexico, USA
S. Kataria
Affiliation:
Department of Mechanical Engineering, University of New Mexico New Mexico, USA
Y.-L. Shen
Affiliation:
Department of Mechanical Engineering, University of New Mexico New Mexico, USA
*
* Corresponding author (khraishi@unm.edu)
Get access

Abstract

A multitude of researchers have utilized a variety of techniques to formulate the stresses and deformations caused by volume misfit inclusions in infinite host media. Few of such techniques can also be extended to derive solutions for inclusions in a half space. In this manuscript we present a novel computational method for determining the elastic fields of two and three-dimensional inclusions of arbitrary shape in an infinite host matrix. The misfit strain is treated by a distribution of prismatic dislocation loops. A systematic numerical assessment illustrates that the discretization can yield excellent agreement with existing analytical solutions for certain particle geometries. This method is then further developed to solve for two-dimensional problems in a half space.

Keywords

EigenstrainDistributed dislocationsHalf spaceElastic field
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 3 , June 2015 , pp. 249 - 260
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Teodosiu, C., Elastic Models of Crystal Defects, Springer-Verlag, Berlin, Germany (1982).Google Scholar
2.Bert, N. A., Kolesnikova, A. L., Romanov, A. E. and Chaldyshev, V. V., “Elastic Behavior of a Spherical Inclusion with a Given Uniaxial Dilatation,” Physics of the Solid State, 44, pp. 2139–2148 (2002).CrossRefGoogle Scholar
3.Eshelby, J. D., “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems,” Proceedings of the Royal Society of London Series A, 221, pp. 376–396 (1957).Google Scholar
4.Eshelby, J. D., “Elastic Field Outside an Ellipsoidal Inclusion,” Proceedings of the Royal Society of London Series A, 252, pp. 561–569 (1959).Google Scholar
5.Pearson, G. S. and Faux, D. A., “Analytical Solution for Strain in Pyramidal Quantum Dots,” Journal of Applied Physics, 88, pp. 730–736 (2000).Google Scholar
6.Andreev, A. D., Downes, J. R., Faux, D. A. and O'Reilly, E. P., “Strain Distribution in Quantum Dots of Arbitrary Shape,” Journal of Applied Physics, 86, pp. 297–305 (1999).Google Scholar
7.Sheinerman, A. G. and Ovid'ko, I. A., “Elastic Fields of Inclusions in Nanocomposite Solids,” Reviews on Advanced Materials Science, 9, pp. 17–33 (2005).Google Scholar
8.Melan, E., “Der Spannungszustand Der Durch Eine Einzelkraft Im Innern Beanspruchten Halbscheibe,” Journal of Applied Mathematics and Mechanics, 12, pp. 343–346 (1932).Google Scholar
9.Mindlin, R. D., “Force at a Point in the Interior of a Semi-Infinite Solid,” Journal of Applied Physics, 7, pp. 195–202 (1936).Google Scholar
10.Mindlin, R. D. and Cheng, D. H., “Nuclei of Strain in the Semi-Infinite Solid,” Journal of Applied Physics, 21, pp. 926–930 (1950a).Google Scholar
11.Mindlin, R. D. and Cheng, D. H., “Thermoelastic Stress in the Semi-Infinite Solid,” Journal of Applied Physics, 21, pp. 931–933 (1950b).Google Scholar
12.Aderogba, K., “On Eigenstress in a Semi-Infinite Solid,” Mathematical Proceedings of the Cambridge Philosophical Society, 80, pp. 555–562 (1976).CrossRefGoogle Scholar
13.Chiu, Y. P., “On the Stress Field and Surface Deformation in a Half Space with a Cuboidal Zone in Which Initial Strains Are Uniform,” Journal of Applied Mechanics, 45, pp. 302–306 (1978).CrossRefGoogle Scholar
14.Hu, S. M., “Stress from a Parallelepipedic Thermal Inclusion in a Semispace,” Journal of Applied Physics, 66, pp. 2741–2743 (1989).CrossRefGoogle Scholar
15.Seo, K. and Mura, T., “The Elastic Field in a Half Space Due to Ellipsoidal Inclusions with Uniform Dilatational Eigenstrains,” Journal of Applied Mechanics, 46, pp. 568–572 (1979).Google Scholar
16.Masmura, R. A. and Chou, Y. T., “Antiplane Eigenstrain Problem of an Elliptic Inclusion in an An-isotropic Half Space,” Journal of Applied Mechanics, 49, pp. 52–54 (1982).Google Scholar
17.Hasegawa, H., Lee, V. G. and Mura, T., “Hollow Circular Cylindrical Inclusion at the Surface of a Half-Space,” Journal of Applied Mechanics, 60, pp. 33–40 (1993).Google Scholar
18.Wu, L. Z. and Du, S. Y., “The Elastic Field in a Half-Space with a Circular Cylindrical Inclusion,” Journal of Applied Mechanics, 63, pp. 925–932 (1996).Google Scholar
19.Ru, C. Q., “Analytic Solution for Eshelby's Problem of an Inclusion of Arbitrary Shape in a Plane or Halp-Plane,” Journal of Applied Mechanics, 66, pp. 315–322 (1999).Google Scholar
20.Glas, F., “Coherent Stress Relaxation in a Half Space: Modulated Layers, Inclusions, Steps, and a General Solution,” Journal of Applied Physics, 70, pp. 3556–3571 (1991).Google Scholar
21.Glas, F., “Elastic Relaxation of Truncated Pyramidal Quantum Dots and Quantum Wires in a Half Space: An Analytical Calculation,” Journal of Applied Physics, 90, pp. 3232–3241 (2001).CrossRefGoogle Scholar
22.Glas, F., “Analytical Calculation of the Strain Field of Single and Periodic Misfitting Polygonal Wires in a Half-Space,” Philosophical Magazine A, 82, pp. 2591–2608 (2002).Google Scholar
23.Glas, F., “Elastic Relaxation of Isolated and Interacting Truncated Pyramidal Quantum Dots and Quantum Wires in a Half Space,” Applied Surface Science, 188, pp. 9–18 (2002b).CrossRefGoogle Scholar
24.Glas, F., “Elastic Relaxation of a Truncated Circular Cylinder with Uniform Dilatational Eigenstrain in a Half Space,” Physical Status Solidi, 237, pp. 599–610 (2003).CrossRefGoogle Scholar
25.Romanov, A. E., Beltz, G. E., Fischer, W. T., Petroff, P. M. and Speck, J. S., “Elastic Fields of Quantum Dots in Subsurface Layers,” Journal of Applied Physics, 89, pp. 4523–4531 (2001).CrossRefGoogle Scholar
26.Pan, E. and Yang, B., “Elastostatic Fields in an Ani-sotropic Substrate Due to a Buried Quantum Dot,” Journal of Applied Physics, 90, pp. 6190–6196 (2001).CrossRefGoogle Scholar
27.Hirth, J. P. and Lothe, J., Theory of Dislocations, Krieger Publishing Company, Florida, U.S., pp. 86–88 (1982).Google Scholar
28.Hull, D. and Bacon, D. J., Introduction to Dislocations, Butterworth-Heinemann, Oxford U.K., , pp. 77–78 (1984).Google Scholar
29.Lerma, J., Khraishi, T., Shen, Y. L. and Wirth, B. D., “The Elastic Fields of Misfit Cylindrical Particles: A Dislocation-Based Numerical Approach,” Mechanics Research Communications, 30, pp. 325–334 (2003).Google Scholar
30.Devincre, B., “Three-Dimensional Stress-Field Expressions for Straight Dislocation Segments,” Solid State Communications, 93, pp. 875–878 (1995).Google Scholar
31.Bacon, D. and Groves, P. T., “The Dislocation in a Semi-Infinite Isotropic Medium,” Fundamental Aspects of Dislocation Theory, 1, pp. 35–45 (1970).Google Scholar
32.Jing, P., “Modeling Irradiation Damage in Crystalline Solids,” M.S. Thesis, University of New Mexico, New Mexico, U.S. (2003).Google Scholar
33.Weertman, J., Dislocation Based Fracture Mechanics, World Scientific, Singapore (1996).Google Scholar
34.Mura, T., Micromechanics of Defects in Solids, the Hague, Martinus Nijhoff Pub., the Netherlands (1982).Google Scholar