Analysis of elastic properties of cubic crystals of simple substances using the diagram a – ν0

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

The graphical diagram A – n0 proposed earlier by the authors was used to analyze the elastic properties of cubic crystals of simple substances. The elastic properties of crystals both at room temperature and their temperature dependences are considered. As the temperature increases, a general trend is observed for most crystals of simple substance: the points (A, n0) characterizing the elastic properties of crystals shift in the direction of the limiting angle of the diagram (A = 1.5, n0 = 0.5), i.e. in the direction of the region of special extrema being characteristic of metastable crystals, for example, such as crystals with shape-memory effect. The use of the A – n0 diagram made it possible to graphically represent and explain the relationships between the basic values of the elastic moduli of cubic crystals: Young’s modulus E0, shear modulus G0, and volumetric modulus of elasticity B.

About the authors

A. I. Epishin

Institute of Structural Macrokinetics and Materials Science named after A. G. Merzhanov of RAS

Author for correspondence.
Email: a.epishin2021@gmail.com
Chergonolovka, Russia

D. S. Lisovenko

Institute for Problems in Mechanics named after A. Yu. Ishlinsky of RAS

Email: lisovenk@ipmnet.ru
Moscow, Russia

References

  1. Haussühl S. Kristallphysik, Physik-Verlag, Weinheim, 1983. 434 p.
  2. Blackman M. On anomalous vibrational spectra // Proc. Roy. Soc. A. 1938. V. 164. № 916. P. 62–79. https://doi.org/10.1098/rspa.1938.0005
  3. Ledbetter H. Blackman diagrams and elastic-constant systematics: in Handbook of Elastic Properties of Solids, Liquids, and Gases, Ed. by M. Levy, H. Bass, and R. Stern (Academic Press, San Diego, 2000), V. II. Р. 57–64. https://doi.org/10.1016/B978-012445760-7/50029-0
  4. Svetlov I.L., Epishin A.I., Krivko A.I. et al. Anisotropy of Poisson ratio of nickel base alloy single crystals // Dokl. Akad. Nauk SSSR. 1988. V. 302. № 6. P. 1372–1375.
  5. Evans K., Nkansah M., Hutchinson I., Rogers S.C. Molecular network design // Nature. 1991. V. 353. № 6340. P. 124–125. https://doi.org/10.1038/353124a0
  6. Hayes M., Shuvalov A. On the extreme values of Young’s modulus, the shear modulus, and Poisson’s ratio for cubic materials // J. Appl. Mech. 1998. V. 65. № 3. P. 786–787. https://doi.org/10.1115/1.2789130
  7. Lim T.-C. Auxetic Materials and Structures. Singapore: Springer, 2015. https://doi.org/10.1007/978-981-287-275-3
  8. Kolken H.M.A., Zadpoor A.A. Auxetic Mechanical Metamaterials // RSC Adv. 2017. V. 7. № 9. P. 5111–5129. https://doi.org/10.1039/C6RA27333E
  9. Ren X., Das R., Tran P. et al. Auxetic Metamaterials and Structures: A Review // Smart Mater. Struct. 2018. V. 27. № 2. P. 023001. https://doi.org/10.1088/1361-665X/aaa61c
  10. Wu W., Hu W., Qian G. et al. Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review // Mater. Des. 2019. V. 180. P. 107950. https://doi.org/10.1016/j.matdes.2019.107950
  11. Gorodtsov V.A., Lisovenko D.S. Auxetics among materials with cubic anisotropy // Mech. Solids. 2020. V.55. № 4. P.461–474. https://doi.org/10.3103/S0025654420040044
  12. Shitikova M.V. Fractional operator viscoelastic models in dynamic problems of mechanics of solids: A Review // Mech. Solids. 2022. V. 57. № 1. P. 1–33. https://doi.org/10.3103/S0025654422010022 Ivanova S.Yu., Osipenko K.Yu., Banichuk N.V., Lisovenko D.S. Experimental study of the properties of metamaterials based on PLA plastic when perforated by a rigid striker // Mech. Solids. 2024. V. 59. № 4. P. 207–215.
  13. https://doi.org/10.1134/S0025654424604695
  14. Ting T.C.T., Barnett D.M. Negative Poisson’s ratios in anisotropic linear elastic media // J. Appl. Mech. 2005. V. 72. № 6. P. 929–931. http://dx.doi.org/10.1115/1.2042483
  15. Paszkiewicz T., Wolski S. Anisotropic properties of mechanical characteristics and auxeticity of cubic crystalline media // Phys. Status Solidi B. 2007. V. 244. № 3. P. 966–977. https://doi.org/10.1002/pssb.200572715
  16. Branka A.C., Heyes D.M., Wojciechowski K.W. Auxeticity of cubic materials // Phys. Status Solidi B. 2009. V. 246. № 9. P. 2063–2071. https://doi.org/10.1002/pssb.200982037
  17. Branka A.C., Heyes D.M., Wojciechowski K.W. Auxeticity of cubic materials under pressure// Phys. Status Solidi B. 2011. V. 248. № 1. P. 96–104. https://doi.org/10.1002/pssb.201083981
  18. Goldstein R.V., Gorodtsov V.A., Lisovenko D.S. Cubic auxetics // Doklady Physics. 2011. V. 56. № 7. P. 399-402. https://doi.org/10.1134/S1028335811120019
  19. Goldstein R.V., Gorodtsov V.A., Lisovenko D.S. Classification of cubic auxetics. Phys. Status Solidi B. 2013. V. 250. № 10. P. 2038–2043. https://doi.org/10.1002/pssb.201384233
  20. Goldstein R.V., Gorodtsov V.A., Lisovenko D.S., Volkov M.A. Negative Poisson’s ratio for cubic crystals and nano/microtubes // Phys. Mesomech. 2014. V. 17. № 2. P. 97–115. https://doi.org/10.1134/S1029959914020027
  21. Krivko A.I., Epishin A.I., Svetlov I.L., Samoilov A.I. Elastic properties of single crystals of nickel alloys // Problems of strength. 1988. № 2. P. 68–75. https://viam.ru/sites/default/files/scipub/1986/1986-199542.pdf
  22. Epishin A.I., Lisovenko D.S. Extreme values of Poisson’s ratio of cubic crystals. Tech. Phys. 2016. V. 61. № 10. P. 1516–1524. https://doi.org/10.1134/S1063784216100121
  23. Epishin A.I., Lisovenko D.S. Influence of the crystal structure and type of interatomic bond on the elastic properties of monatomic and diatomic cubic crystals // Mech. Solids. 2022. V. 57. № 6. P. 1344–1358. https://doi.org/10.3103/S0025654422060206
  24. Second and Higher Order Elastic Constants / Ed. by D.F. Nelson. Springer, 1992. https://doi.org/10.1007/b44185
  25. Norris A.N. Poisson’s ratio in cubic materials// Proc. Roy. Soc. A. 2006. V. 462. № 2075. P. 3385–3405. https://doi.org/10.1098/rspa.2006.1726
  26. Lisovenko D.S., Epishin A.I. Anisotropy of residual stress energy in two-component plate crystal structures // Mech. Solids. 2023. V.58. № 6. P. 2043–2057. https://doi.org/10.3103/S0025654423601179
  27. Schärer U., Jung A., Wachter P. Brillouin spectroscopy with surface acoustic waves on intermediate valent, doped SmS // Physica B. 1998. V. 244. P. 148. https://doi.org/10.1016/S0921-4526(97)00478-X
  28. Nash H.C., Smith C.S. Single-crystal elastic constants of lithium // J. Phys. Chem. Solids. 1959. V. 9. № 2. P. 113–118. https://doi.org/10.1016/0022-3697(59)90201-X
  29. Slotwinski T., Trivisonno J. Temperature dependence of the elastic constants of single crystal lithium // J. Phys. Chem. Solids. 1969. V. 30. № 5. P. 1276–1278. https://doi.org/10.1016/0022-3697(69)90386-2
  30. Bolef D.I., Smith R.E., Miller J.G. Elastic properties of vanadium. I. Temperature dependence of the elastic constants and the thermal expansion // Phys. Rev. B. 1971. V. 3. № 12. P. 4100-4108. https://doi.org/10.1103/PhysRevB.3.4100
  31. Walker E. Anomalous temperature behaviour of the shear elastic constant C44 in vanadium // Solid State Communications. 1978. V. 28. № 7. P. 587–589. https://doi.org/10.1016/0038-1098(78)90495-7
  32. Talmor Y., Walker E., Steinemann S. Elastic constants of niobium up to the melting point // Solid State Communications. 1977. V. 23. № 9. P. 649–651. https://doi.org/10.1016/0038-1098(77)90541-5
  33. Featherston F.H., Neighbours J.R. Elastic Constants of tantalum, tungsten, and molybdenum // Phys. Rev. 1963. V. 130. № 4. P. 1324–1333. https://link.aps.org/doi/10.1103/PhysRev.130.1324
  34. Walker E., Bujard P. Anomalous temperature behaviour of the shear elastic constant C44 in tantalum // Solid State Communications. 1980. V. 34. № 8. P. 691–693. https://doi.org/10.1016/0038-1098(80)90957-6
  35. Alers G.A., Neighbours J.R., Sato H. Temperature dependent magnetic contributions to the high field elastic constants of nickel and an Fe-Ni alloy // J. Phys. Chem. Solids. 1960. V. 13. № 1–2. P. 40–55. https://doi.org/10.1016/0022-3697(60)90125-6
  36. Renaud Ph., Steinemann S.G. High temperature elastic constants of fcc Fe-Ni invar alloys // Physica B: Condensed Matter. 1990. V. 161. № 1–3. P. 75–78. https://doi.org/10.1016/0921-4526(89)90107-5
  37. Rayne J.A., Chandrasekhar B.S. Elastic Constants of Iron from 4.2 to 300°K // Phys. Rev. 1961. V. 122. № 6. P. 1714–1716. https://link.aps.org/doi/10.1103/PhysRev.122.1714
  38. Dever D.J. Temperature dependence of the elastic constants in α-iron single crystals: relationship to spin order and diffusion anomalies // J. Appl. Phys. 1972. V. 43. № 8. P. 3293–3301. https://doi.org/10.1063/1.1661710
  39. Neighbours J.R., Alers G.A. Elastic constants of silver and gold // Phys. Rev. 1958. V. 111. № 3. P. 707–712. https://link.aps.org/doi/10.1103/PhysRev.111.707
  40. Chang Y.A., Himmel L. Temperature dependence of the elastic constants of Cu, Ag, and Au above room temperature // J. Appl. Phys. 1966. V. 37. № 9. P. 3567–3572. https://doi.org/10.1063/1.1708903
  41. Collard S.M., McLellan R.B. High-temperature elastic constants of gold single-crystals // Acta Metall. Mater. 1991. V. 39. № 12. P. 3143–3151. https://doi.org/10.1016/0956-7151(91)90048-6
  42. MacFarlane R.E., Rayne J.A., Jones C.K. Anomalous temperature dependence of shear modulus c44 for platinum // Phys Lett. 1965. V .18. № 2. P. 91–92. https://doi.org/10.1016/0031-9163(65)90659-1
  43. Collard S.M., McLellan R.B. High-temperature elastic constants of platinum single crystals // Acta Metall. Mater. 1992. V. 40. № 4. P. 699–702. https://doi.org/10.1016/0956-7151(92)90011-3
  44. Rayne J. A. Elastic constants of Palladium from 4.2-300°K // Phys. Rev. 1960. V. 118. № 6. P. 1545–1549. https://link.aps.org/doi/10.1103/PhysRev.118.1545
  45. Walker E., Ortelli J., Peter M. Elastic constants of monocrystalline alloys of Pd-Rh and Pd-Ag between 4.2°K and 300°K // Phys. Lett. A. 1970. V. 31. № 5. P. 240–241. https://doi.org/10.1016/0375-9601(70)90949-7
  46. Weinmann C., Steinemann S. Lattice and electronic contributions to the elastic constants of palladium // Solid State Communications. 1974. V. 15. № 2. P. 281–285. https://doi.org/10.1016/0038-1098(74)90758-3
  47. Yoshihara M., McLellan R.B., Brotzen F.R. The high-temperature elastic properties of palladium single crystals // Acta Metall. 1987. V. 35. № 3. P. 775–780. https://doi.org/10.1016/0001-6160(87)90204-5
  48. Kamm G.N., Alers G.A. Low-temperature elastic moduli of aluminum // J. Appl. Phys. 1964. V. 35. № 2. P. 327–330. https://doi.org/10.1063/1.1713309
  49. Gerlich D., Fisher E.S. The high temperature elastic moduli of aluminum // J. Phys. Chem. Solids. 1969. V. 30. № 5. P. 1197–1205. https://doi.org/10.1016/0022-3697(69)90377-1
  50. Waldorf D.L., Alers G.A. Low-temperature elastic moduli of lead // J. Appl. Phys. 1962. V. 33. № 1. P. 3266–3269. https://doi.org/10.1063/1.1931149
  51. Vold C.L., Glicksman M.E., Kammer E.W., Cardinal L.C. The elastic constants for single-crystal lead and indium from room temperature to the melting point // J. Appl. Phys. 1977. V. 38. № 2. P. 157–160. https://doi.org/10.1016/0022-3697(77)90159-7
  52. Zouboulis E.S., Grimsditch M., Ramdas A.K., Rodriguez S. Temperature dependence of the elastic moduli of diamond: A Brillouin-scattering study // Phys. Rev. B. 1998. V. 57. № 5. P. 2889–2896. https://doi: 10.1103/PhysRevB.57.2889
  53. Epishin A.I., Lisovenko D.S. Comparison of isothermal and adiabatic elasticity characteristics of the single crystal nickel-based superalloy CMSX-4 in the temperature range between room temperature and 1300 C // Mech. Solids. 2023. V. 58. № 5. P. 1587–1598. https://doi.org/10.3103/S0025654423601301
  54. Daniels W.B. Pressure variation of the elastic constants of sodium // Phys. Rev. 1960. V. 119. № 4. P. 1246–1252. https://doi.org/10.1103/PhysRev.119.1246
  55. Diederich M.E., Trivisonno J. Temperature dependence of the elastic constants of sodium // J. Phys. Chem. Solids. 1966. V. 27. № 4. P. 637–642. https://doi.org/10.1016/0022-3697(66)90214-9
  56. Martinson R.H. Variation of the elastic constants of sodium with temperature and pressure // Phys. Rev. 1969. V. 178. № 3. P. 902–913. https://doi.org/10.1103/PhysRev.178.902
  57. Marquardt W.R., Trivisonno J. Low temperature elastic constants of potassium // J. Phys. Chem. Solids. 1965. V. 26. № 2. P. 273–278. https://doi.org/10.1016/0022-3697(65)90155-1
  58. Gutman E.J., Trivisonno J. Temperature dependence of the elastic constants of rubidium // J. Phys. Chem. Solids. 1967. V. 28. № 5. P. 805–809. https://doi.org/10.1016/0022-3697(67)90009-1
  59. Barrett C.S. X-ray study of the alkali metals at low temperatures // Acta Cryst. 1956. V. 9. № 8. P. 671–677. https://doi.org/10.1107/S0365110X56001790
  60. Ernst G., Artner C., Blaschko O., Krexner G. Low-temperature martensitic phase transition of bcc lithium // Phys. Rev. B. 1986. V. 33. № 9. P. 6465–6469. https://doi.org/10.1103/PhysRevB.33.6465
  61. Pichl W., Krystian M. Martensitic transformation and mechanical deformation of high-purity lithium // Mater. Sci. Eng. A. 1999. V. 273–275. P. 208–212. https://doi.org/10.1016/S0921-5093(99)00372-X

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences