Research article Topical Sections

Stress field of a near-surface basal screw dislocation in elastically anisotropic hexagonal crystals

  • Received: 22 September 2017 Accepted: 16 November 2017 Published: 21 November 2017
  • In this study, we derive and analyze the analytical expressions for stress components of the dislocation elastic field induced by a near-surface basal screw dislocation in a semi-infinite elastically anisotropic material with hexagonal crystal lattice. The variation of above stress components depending on “free surface–dislocation” distance (i.e., free surface effect) is studied by means of plotting the stress distribution maps for elastically anisotropic crystals of GaN and TiB2 that exhibit different degrees of elastic anisotropy. The dependence both of the image force on a screw dislocation and the force of interaction between two neighboring basal screw dislocations on the “free surface–dislocation” distance is analyzed as well. The influence of elastic anisotropy on the latter force is numerically analyzed for GaN and TiB2 and also for crystals of such highly elastically-anisotropic materials as Ti, Zn, Cd, and graphite.
    The comparatively stronger effect of the elastic anisotropy on dislocation-induced stress distribution quantified for TiB2 is attributed to the higher degree of elastic anisotropy of this compound in comparison to that of the GaN. For GaN and TiB2, the dislocation stress distribution maps are highly influenced by the free surface effect at “free surface–dislocation” distances roughly smaller than ≈15 and ≈50 nm, respectively. It is found that, for above indicated materials, the relative decrease of the force of interaction between near-surface screw dislocations due to free surface effect is in the order Ti > GaN > TiB2 > Zn > Cd > Graphite that results from increase of the specific shear anisotropy parameter in the reverse order Ti < GaN < TiB2 < Zn < Cd < Graphite. The results obtained in this study are also applicable to the case when a screw dislocation is situated in the “thin film–substrate” system at a (0001) basal interface between the film and substrate provided that the elastic constants of the film and substrate are the same or sufficiently close to each other.

    Citation: Valeri S. Harutyunyan, Ashot P. Aivazyan, Andrey N. Avagyan. Stress field of a near-surface basal screw dislocation in elastically anisotropic hexagonal crystals[J]. AIMS Materials Science, 2017, 4(6): 1202-1219. doi: 10.3934/matersci.2017.6.1202

    Related Papers:

    [1] Chentong Li, Jinyan Wang, Jinhu Xu, Yao Rong . The Global dynamics of a SIR model considering competitions among multiple strains in patchy environments. Mathematical Biosciences and Engineering, 2022, 19(5): 4690-4702. doi: 10.3934/mbe.2022218
    [2] Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva . An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences and Engineering, 2010, 7(1): 123-147. doi: 10.3934/mbe.2010.7.123
    [3] Matthew D. Johnston, Bruce Pell, David A. Rubel . A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity. Mathematical Biosciences and Engineering, 2023, 20(9): 16083-16113. doi: 10.3934/mbe.2023718
    [4] Ali Mai, Guowei Sun, Lin Wang . The impacts of dispersal on the competition outcome of multi-patch competition models. Mathematical Biosciences and Engineering, 2019, 16(4): 2697-2716. doi: 10.3934/mbe.2019134
    [5] Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin . Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences and Engineering, 2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068
    [6] Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis . A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme. Mathematical Biosciences and Engineering, 2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679
    [7] Nancy Azer, P. van den Driessche . Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences and Engineering, 2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283
    [8] Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135
    [9] Yanxia Dang, Zhipeng Qiu, Xuezhi Li . Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences and Engineering, 2017, 14(4): 901-931. doi: 10.3934/mbe.2017048
    [10] Azmy S. Ackleh, Shuhua Hu . Comparison between stochastic and deterministic selection-mutation models. Mathematical Biosciences and Engineering, 2007, 4(2): 133-157. doi: 10.3934/mbe.2007.4.133
  • In this study, we derive and analyze the analytical expressions for stress components of the dislocation elastic field induced by a near-surface basal screw dislocation in a semi-infinite elastically anisotropic material with hexagonal crystal lattice. The variation of above stress components depending on “free surface–dislocation” distance (i.e., free surface effect) is studied by means of plotting the stress distribution maps for elastically anisotropic crystals of GaN and TiB2 that exhibit different degrees of elastic anisotropy. The dependence both of the image force on a screw dislocation and the force of interaction between two neighboring basal screw dislocations on the “free surface–dislocation” distance is analyzed as well. The influence of elastic anisotropy on the latter force is numerically analyzed for GaN and TiB2 and also for crystals of such highly elastically-anisotropic materials as Ti, Zn, Cd, and graphite.
    The comparatively stronger effect of the elastic anisotropy on dislocation-induced stress distribution quantified for TiB2 is attributed to the higher degree of elastic anisotropy of this compound in comparison to that of the GaN. For GaN and TiB2, the dislocation stress distribution maps are highly influenced by the free surface effect at “free surface–dislocation” distances roughly smaller than ≈15 and ≈50 nm, respectively. It is found that, for above indicated materials, the relative decrease of the force of interaction between near-surface screw dislocations due to free surface effect is in the order Ti > GaN > TiB2 > Zn > Cd > Graphite that results from increase of the specific shear anisotropy parameter in the reverse order Ti < GaN < TiB2 < Zn < Cd < Graphite. The results obtained in this study are also applicable to the case when a screw dislocation is situated in the “thin film–substrate” system at a (0001) basal interface between the film and substrate provided that the elastic constants of the film and substrate are the same or sufficiently close to each other.


    [1] Hirth JP, Lothe J (1982) Theory of Dislocations, New York: John Wiley & Sons.
    [2] Morkoc H (2008) Handbook of Nitride Semiconductors and Devices, Berlin: Wiley-VCH.
    [3] Telling RH, Heggie MI (2003) Stacking fault and dislocation glide on basal plane of graphite. Phil Mag Lett 83: 411–421.
    [4] Jagannadham K, Marcinkowski MJ (1978) Comparison of the image and surface dislocation models. Phys Status Solidi A 50: 293–302. doi: 10.1002/pssa.2210500135
    [5] Cheng X, Shen Y, Zhang L, et al. (2012) Surface effect on the screw dislocation mobility over the Peierls barrier. Phil Mag Lett 92: 270–277. doi: 10.1080/09500839.2012.669053
    [6] Gars B, Markenscoff X (2012) The Peierls stress for coupled dislocation partials near a free surface. Philos Mag 92: 1390–1421. doi: 10.1080/14786435.2011.645900
    [7] Lee CL, Li S (2007) A half-space Peierls–Nabarro model and the mobility of screw dislocations in a thin film. Acta Mater 55: 2149–2157. doi: 10.1016/j.actamat.2006.11.015
    [8] Liu L, Meng Z, Xu G, et al. (2017) Surface effects on the properties of screw dislocation in nanofilms. Adv Mater Sci Eng 2017.
    [9] Eshelby JD, Read WT, Shockley W (1953) Anisotropic elasticity with applications to dislocations theory. Acta Metall 1: 251–259. doi: 10.1016/0001-6160(53)90099-6
    [10] Spence GB (1962) Theory of extended dislocations in symmetry directions in anisotropic infinite crystals and thin plates. J Appl Phys 33: 729–733. doi: 10.1063/1.1702496
    [11] Chou YT (1962) Interaction of parallel dislocations in a hexagonal crystal. J Appl Phys 33: 2747–2751. doi: 10.1063/1.1702541
    [12] Chou YT (1963) Characteristics of dislocation stress fields due to elastic anisotropy. J Appl Phys 34: 429–433. doi: 10.1063/1.1702625
    [13] Holec D (2008) Multi-Scale Modeling of III-Nitrides: from Dislocations to the Electronic Structure [PhD thesis]. University of Cambridge.
    [14] Chu HJ, Pan E, Wang J, et al. (2011) Three-dimensional elastic displacements induced by a dislocation of polygonal shape in anisotropic elastic crystals. Int J Solids Struct 48: 1164–1170. doi: 10.1016/j.ijsolstr.2010.12.015
    [15] Chu HJ, Wang J, Beyerlein IJ, et al. (2013) Dislocation models of interfacial shearing induced by an approaching glide dislocation. Int J Plasticity 41: 1–13. doi: 10.1016/j.ijplas.2012.08.005
    [16] Barnett DM, Lothe J (1974) An image force theorem for dislocations in anisotropic bicrystals. J Phys F Metal Phys 4: 1618–1635. doi: 10.1088/0305-4608/4/10/010
    [17] Wang J, Hoagland RG, Hirth JP, et al. (2008) Atomistic modeling of the interaction of glide dislocations with "weak" interfaces. Acta Mater 56: 5685–5693. doi: 10.1016/j.actamat.2008.07.041
    [18] Wang L, Liu Z, Zhuang Z (2016) Developing micro-scale crystal plasticity model based on phase field theory for modeling dislocations in heteroepitaxial structures. Int J Plasticity 81: 267–283.
    [19] Chou YT (1966) On dislocation–boundary interaction in an anisotropic aggregate. Phys Status Solidi B 15: 123–127. doi: 10.1002/pssb.19660150110
    [20] Chu H, Pan E (2014) Elastic fields due to dislocation arrays in anisotropic biomaterials. Int J Solids Struct 51: 1954–1961. doi: 10.1016/j.ijsolstr.2014.02.001
    [21] Shahsavari R, Chen L (2015) Screw dislocations in complex, low symmetry oxides: Core structures, energetics, and impact on crystal growth. ACS Appl Mater Interfaces 7: 2223–2234. doi: 10.1021/am5091808
    [22] Ruterana P, Albrecht M, Neugebauer J (2003) Nitride Semiconductors: Handbook on Materials and Devices, Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA.
    [23] Munro RG (2000) Material properties of titanium diboride. J Res Natl Inst Stan 105: 709–720.
    [24] Cheng TS, Davies A, Summerfield A, et al. (2016) High temperature MBE of graphene on sapphire and hexagonal boron nitride flakes on sapphire. J Vac Sci Technol B 34: 02L101.
    [25] Chung DH, Buessem WR (1968) The Elastic Anisotropy of Crystals, In: Vahldiek FW, Mersol SA, Anisotropy in Single-Crystal Refractory Compounds, New York: Plenum, 217–245.
    [26] Lethbridge ZAD, Walton RI, Marmier ASH, et al. (2010) Elastic anisotropy and extreme Poisson's ratios in single crystals. Acta Mater 58: 6444–6451.
    [27] Kube CM (2016) Elastic anisotropy of crystals. AIP Adv 6: 095209. doi: 10.1063/1.4962996
    [28] Specht P, Harutyunyan VS, Ho J, et al. (2004) Anisotropy of the elastic properties of wurtzite InN epitaxial films. Defect Diff Forum 226–228: 79–90.
    [29] Vurgaftman I, Meyer JR (2003) Band parameters for nitrogen-containing semiconductors. J Appl Phys 94: 3675–3696. doi: 10.1063/1.1600519
    [30] Wang HY, Xue FY, Zhao NH, et al. (2011) First-principles calculation of elastic properties of TiB2 and ZrB2. Adv Mater Res 150–151: 40–43.
    [31] Polian A, Grimsditch M, Grzegory I (1996) Elastic constants of gallium nitride. J Appl Phys 79: 3343–3344. doi: 10.1063/1.361236
    [32] Spoor PS, Maynard JD, Pan MJ, et al. (1997) Elastic constants and crystal anisotropy of titanium diboride. Appl Phys Lett 70: 1959–1961. doi: 10.1063/1.118791
    [33] Peselnick L, Meister R (1965) Variational method of determining effective moduli of polycrystals: (A) hexagonal symmetry, (B) trigonal symmetry. J Appl Phys 36: 2879–2884. doi: 10.1063/1.1714598
    [34] Watt JP, Peselnick L (1980) Clarification of the Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. J Appl Phys 51: 1525–1531. doi: 10.1063/1.327804
    [35] Simmons G, Wang H (1971) Single crystal elastic constants and calculated aggregate properties: a Handbook, Cambridge, Massachusetts: The MIT Press.
    [36] Cousins CSG, Heggie MI (2003) Elasticity of carbon allotropes. III. Hexagonal graphite: Review of data, previous calculations, and a fit to a modified anharmonic Keating model. Phys Rev B 67: 024109.
  • This article has been cited by:

    1. Yixiang Wu, Necibe Tuncer, Maia Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, 2017, 22, 1553-524X, 1167, 10.3934/dcdsb.2017057
    2. Junping Shi, Yixiang Wu, Xingfu Zou, Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction–Diffusion Chemostat Model, 2020, 32, 1040-7294, 1085, 10.1007/s10884-019-09763-0
    3. Yixiang Wu, Xingfu Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, 2016, 261, 00220396, 4424, 10.1016/j.jde.2016.06.028
    4. Lin Zhao, Zhi-Cheng Wang, Shigui Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, 2020, 51, 14681218, 102966, 10.1016/j.nonrwa.2019.102966
    5. Jing Ge, Ling Lin, Lai Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, 2017, 22, 1553-524X, 2763, 10.3934/dcdsb.2017134
    6. Yuan Lou, Rachidi B. Salako, Control Strategies for a Multi-strain Epidemic Model, 2022, 84, 0092-8240, 10.1007/s11538-021-00957-6
    7. Jinsheng Guo, Shuang-Ming Wang, Threshold dynamics of a time-periodic two-strain SIRS epidemic model with distributed delay, 2022, 7, 2473-6988, 6331, 10.3934/math.2022352
    8. Rachidi B. Salako, Impact of population size and movement on the persistence of a two-strain infectious disease, 2023, 86, 0303-6812, 10.1007/s00285-022-01842-z
    9. Yuan Lou, Rachidi B. Salako, Mathematical analysis of the dynamics of some reaction-diffusion models for infectious diseases, 2023, 370, 00220396, 424, 10.1016/j.jde.2023.06.018
    10. Jonas T. Doumatè, Tahir B. Issa, Rachidi B. Salako, Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023213
    11. Azmy S. Ackleh, Nicolas Saintier, Aijun Zhang, A multiple-strain pathogen model with diffusion on the space of Radon measures, 2025, 140, 10075704, 108402, 10.1016/j.cnsns.2024.108402
    12. Jamal Adetola, Keoni G. Castellano, Rachidi B. Salako, Dynamics of classical solutions of a multi-strain diffusive epidemic model with mass-action transmission mechanism, 2025, 90, 0303-6812, 10.1007/s00285-024-02167-9
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6028) PDF downloads(969) Cited by(0)

Article outline

Figures and Tables

Figures(7)  /  Tables(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog