On analytical estimates of the effective elastic properties of polycrystalline silicon
DOI:
https://doi.org/10.21638/spbu01.2022.305Abstract
Several analytical approaches can be utilized to estimate the elastic properties of polycrystalline silicon. In experimental studies, the notion of an macroscopically isotropic aggregate is introduced while the single crystals obey cubic symmetry. We here give a synopsis on analytical approaches to predict elastic properties and apply them to estimate effective parameters of polycrystalline silicon. Here, the predictions are based on the parameters associated with shear solely. The results are juxtaposed in terms of the approaches applied, while different measures are introduced for evaluation. In comparison with experimental findings, the geometric mean implies a reasonable estimation for the elastic properties of polycrystalline silicon.Keywords:
silicon, cubic single crystals, polycrystalline aggregate, averaging methods, elasticity
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References
Литература
1. Sharpe W.N. Mechanical Properties of MEMS Materials. In: M. Gad-el-Hak (ed.). The MEMS Handbook. CRC Press. Chapter 3, 1-33 (2002).
2. Hill R. On Constitutive Macro-Variables for Heterogeneous Solids at Finite Strain. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 326 (1565), 131-147 (1972). Available at: http://www.jstor.org/stable/78044 (accessed: July 1, 2022).
3. Torquato S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. In Ser.: Interdisciplinary Applied Mathematics, vol. 16. New York, Springer (2002). https://doi.org/10.1007/978-1-4757-6355-3
4. Adams B., Olson T. The mesostructure-properties linkage in polycrystals. Progress in Materials Science 43 (1), 1-87 (1998). https://doi.org/10.1016/S0079-6425(98)00002-4
5. Voigt W. Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Wiesbaden, Springer (1910). https://doi.org/10.1007/978-3-663-15884-4
6. Reuss A. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizit¨atsbedingung f¨ur Einkristalle. Zeitschrift f¨ur Angewandte Mathematik und Mechanik 9 (1), 49-58 (1929). https://doi.org/10.1002/zamm.19290090104
7. Beran M. Statistical Continuum Theories. Monographs in statistical physics and thermodynamics. Interscience Publishers (1968).
8. Hooke R. De Potentia Restitutiva, or of Spring Explaining the Power of Springing Bodies. London, John Martyn (1678). Available at: http://name.umdl.umich.edu/A44322.0001.001 (accessed: July 1, 2022).
9. Hooke R. A Description of Helioscopes and Some other Instruments. London, John Martyn (1676).
10. Halmos P.R. Finite-Dimensional Vector Spaces. New York, Springer (1958). https://doi.org/10.1007/978-1-4612-6387-6
11. Rychlewski J. Unconventional approach to linear elasticity. Archives of Mechanics 47 (5), 149-171 (1995).
12. Kowalczyk-Gajewska K., Ostrowska-Maciejewska J. Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material. Engineering Transactions 57 (3-4), 145-1183 (2009). Available at: http://et.ippt.gov.pl/index.php/et/article/view/172 (accessed: July 1, 2022).
13. Nordmann J., Aßmus M., Altenbach H. Visualising Elastic Anisotropy: Theoretical Background and Computational Implementation. Continuum Mechanics and Thermodynamics 30 (4), 689- 708 (2018). https://doi.org/10.1007/s00161-018-0635-9
14. Aleksandrov K.S., Aizenberg L.A. A method of calculating the physical constants of polycrystalline materials. Doklady Akademii Nauk SSSR 167 (5), 1028-1031 (1966). Available at: http://mi.mathnet.ru/eng/dan/v167/i5/p1028 (accessed: July 1, 2022). (In Russian)
15. Nemat-Nasser S., Hori M. Micromechanics: Overall Properties of Heterogeneous Solids. New York, Elsevier (1993).
16. Jensen J.L.W.V. Sur les fonctions convexes et les in´egalit´es entre les valeurs moyennes. Acta Mathematica 30, 175-193 (1906). https://doi.org/10.1007/BF02418571
17. Hill R. The Elastic Behaviour of a Crystalline Aggregate. Proceedings of the Physical Society. Section A 65 (5), 349-354 (1952). https://doi.org/10.1088/0370-1298/65/5/307
18. Hashin Z., Shtrikman S. A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids 10 (4), 343-352 (1962). https://doi.org/10.1016/0022-5096(62)90005-4
19. Aßmus M., Gl¨uge R., Altenbach H. Hashin - Shtrikman Bounds of Cubic Crystalline Aggregate Elasticity for poly-Si Solar Cells. Technische Mechanik 41 (1), 24-33 (2021). https://doi.org/10.24352/ub.ovgu-2021-004
20. Huang M. Perturbation approach to elastic constitutive relations of polycrystals. Journal of the Mechanics and Physics of Solids 52 (8), 1827-1853 (2004). https://doi.org/10.1016/j.jmps.2004.02.006
21. Huang M., Man C.-S. Explicit Bounds of Effective Stiffness Tensors for Textured Aggregates of Cubic Crystallites. Mathematics and Mechanics of Solids 13 (5), 408-430 (2008). https://doi.org/10.1177/1081286507078299
22. Fokin A.G. Solution of statistical problems in elasticity theory in the singular approximation. Journal of Applied Mechanics and Technical Physics 13, 85-89 (1972). https://doi.org/10.1007/BF00852360
23. Matthies S., Humbert M. On the Principle of a Geometric Mean of Even-Rank Symmetric Tensors for Textured Polycrystals. Journal of Applied Crystallography 28 (3), 254-266 (1995). https://doi.org/10.1107/S0021889894009623
24. Mason W.P. Physical Acoustics and the Properties of Solids. Princeton, New Jersey, D. Van Nostrand Co. (1958). Available at: https://hdl.handle.net/2027/mdp.39015006372356 (accessed: July 1, 2022).
References
1. Sharpe W.N. Mechanical Properties of MEMS Materials. In: M. Gad-el-Hak (ed.). The MEMS Handbook. CRC Press. Chapter 3, 1-33 (2002).
2. Hill R. On Constitutive Macro-Variables for Heterogeneous Solids at Finite Strain. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 326 (1565), 131-147 (1972). Available at: http://www.jstor.org/stable/78044 (accessed: July 1, 2022).
3. Torquato S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. In Ser.: Interdisciplinary Applied Mathematics, vol. 16. New York, Springer (2002). https://doi.org/10.1007/978-1-4757-6355-3
4. Adams B., Olson T. The mesostructure-properties linkage in polycrystals. Progress in Materials Science 43 (1), 1-87 (1998). https://doi.org/10.1016/S0079-6425(98)00002-4
5. Voigt W. Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Wiesbaden, Springer (1910). https://doi.org/10.1007/978-3-663-15884-4
6. Reuss A. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizit¨atsbedingung f¨ur Einkristalle. Zeitschrift f¨ur Angewandte Mathematik und Mechanik 9 (1), 49-58 (1929). https://doi.org/10.1002/zamm.19290090104
7. Beran M. Statistical Continuum Theories. Monographs in statistical physics and thermodynamics. Interscience Publishers (1968).
8. Hooke R. De Potentia Restitutiva, or of Spring Explaining the Power of Springing Bodies. London, John Martyn (1678). Available at: http://name.umdl.umich.edu/A44322.0001.001 (accessed: July 1, 2022).
9. Hooke R. A Description of Helioscopes and Some other Instruments. London, John Martyn (1676).
10. Halmos P.R. Finite-Dimensional Vector Spaces. New York, Springer (1958). https://doi.org/10.1007/978-1-4612-6387-6
11. Rychlewski J. Unconventional approach to linear elasticity. Archives of Mechanics 47 (5), 149-171 (1995).
12. Kowalczyk-Gajewska K., Ostrowska-Maciejewska J. Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material. Engineering Transactions 57 (3-4), 145-1183 (2009). Available at: http://et.ippt.gov.pl/index.php/et/article/view/172 (accessed: July 1, 2022).
13. Nordmann J., Aßmus M., Altenbach H. Visualising Elastic Anisotropy: Theoretical Background and Computational Implementation. Continuum Mechanics and Thermodynamics 30 (4), 689- 708 (2018). https://doi.org/10.1007/s00161-018-0635-9
14. Aleksandrov K.S., Aizenberg L.A. A method of calculating the physical constants of polycrystalline materials. Doklady Akademii Nauk SSSR 167 (5), 1028-1031 (1966). Available at: http://mi.mathnet.ru/eng/dan/v167/i5/p1028 (accessed: July 1, 2022). (In Russian)
15. Nemat-Nasser S., Hori M. Micromechanics: Overall Properties of Heterogeneous Solids. New York, Elsevier (1993).
16. Jensen J.L.W.V. Sur les fonctions convexes et les in´egalit´es entre les valeurs moyennes. Acta Mathematica 30, 175-193 (1906). https://doi.org/10.1007/BF02418571
17. Hill R. The Elastic Behaviour of a Crystalline Aggregate. Proceedings of the Physical Society. Section A 65 (5), 349-354 (1952). https://doi.org/10.1088/0370-1298/65/5/307
18. Hashin Z., Shtrikman S. A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids 10 (4), 343-352 (1962). https://doi.org/10.1016/0022-5096(62)90005-4
19. Aßmus M., Gl¨uge R., Altenbach H. Hashin - Shtrikman Bounds of Cubic Crystalline Aggregate Elasticity for poly-Si Solar Cells. Technische Mechanik 41 (1), 24-33 (2021). https://doi.org/10.24352/ub.ovgu-2021-004
20. Huang M. Perturbation approach to elastic constitutive relations of polycrystals. Journal of the Mechanics and Physics of Solids 52 (8), 1827-1853 (2004). https://doi.org/10.1016/j.jmps.2004.02.006
21. Huang M., Man C.-S. Explicit Bounds of Effective Stiffness Tensors for Textured Aggregates of Cubic Crystallites. Mathematics and Mechanics of Solids 13 (5), 408-430 (2008). https://doi.org/10.1177/1081286507078299
22. Fokin A.G. Solution of statistical problems in elasticity theory in the singular approximation. Journal of Applied Mechanics and Technical Physics 13, 85-89 (1972). https://doi.org/10.1007/BF00852360
23. Matthies S., Humbert M. On the Principle of a Geometric Mean of Even-Rank Symmetric Tensors for Textured Polycrystals. Journal of Applied Crystallography 28 (3), 254-266 (1995). https://doi.org/10.1107/S0021889894009623
24. Mason W.P. Physical Acoustics and the Properties of Solids. Princeton, New Jersey, D. Van Nostrand Co. (1958). Available at: https://hdl.handle.net/2027/mdp.39015006372356 (accessed: July 1, 2022).
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Published
2022-10-10
How to Cite
Aßmus, M., & Altenbach, H. (2022). On analytical estimates of the effective elastic properties of polycrystalline silicon. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(3), 440–451. https://doi.org/10.21638/spbu01.2022.305
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On the anniversary of N.F. Morozov
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
