Mechanical stability conditions for 3D and 2D crystalsunder arbitrary load

Downloads

Authors

  • M. Maździarz Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland

Abstract

The paper gathers and unifies mechanical stability conditions for all symmetry classes of 3D and 2D materials under arbitrary load. The methodology is based on the spectral decomposition of the fourth-order stiffness tensors mapped to secondorder tensors using orthonormal (Mandel) notation, and the verification of the positivity of the so-called Kelvin moduli. An explicit set of stability conditions for 3D and 2D crystals of higher symmetry is also included, as well as a Mathematica notebook that allows mechanical stability analysis for crystals, stress-free and stressed, of arbitrary symmetry under arbitrary loads.

Keywords:

mechanical stability, Born’s stability, 2D materials, Kelvin moduli, orthonormal notation

Supplementary material

  • Download: Mechanical-stability.nb
    Mathematica notebook that allows mechanical stability analysis for crystals, stress-free and stressed, of arbitrary symmetry under arbitrary loads

References


  1. M. Born, On the stability of crystal lattices. I, Mathematical Proceedings of the Cambridge Philosophical Society, 36, 2, 160–172, 1940, https://doi.org/10.1017/S0305004100017138.

  2. F. Mouhat, F.-X. Coudert, Necessary and sufficient elastic stability conditions in various crystal systems, Physical Review B, 90, 224104, 2014, https://doi.org/10.1103/PhysRevB.90.224104.

  3. J.W. Morris Jr., C.R. Krenn, The internal stability of an elastic solid, Philosophical Magazine A, 80, 12, 2827–2840, 2000, https://doi.org/10.1080/01418610008223897.

  4. J.D. Clayton, Towards a nonlinear elastic representation of finite compression and instability of boron carbide ceramic, Philosophical Magazine, 92, 23, 2860–2893, 2012, https://doi.org/10.1080/14786435.2012.682171.

  5. C.S. Jog, K.D. Patil, Conditions for the onset of elastic and material instabilities in hyperelastic materials, Archive of Applied Mechanics, 83, 5, 661–684, 2013, https://doi.org/10.1007/s00419-012-0711-8.

  6. J.D. Clayton, K.M. Bliss, Analysis of intrinsic stability criteria for isotropic third-order Green elastic and compressible neo-Hookean solids, Mechanics of Materials, 68, 104–119, 2014, https://doi.org/10.1016/j.mechmat.2013.08.007.

  7. R.S. Elliott, N. Triantafyllidis, J.A. Shaw, Stability of crystalline solids – I: Continuum and atomic lattice considerations, Journal of the Mechanics and Physics of Solids, 54, 1, 161–192, 2006, https://doi.org/10.1016/j.jmps.2005.07.009.

  8. R.A. Cowley, Acoustic phonon instabilities and structural phase transitions, Physical Review B, 13, 11, 4877–4885, 1976. https://doi.org/10.1103/PhysRevB.13.4877.

  9. G.V. Sin’ko, N.A. Smirnov, Ab initio calculations of elastic constants and thermodynamic properties of bcc, fcc, and hcp Al crystals under pressure, Journal of Physics: Condensed Matter, 14, 29, 6989–7005, 2002, https://doi.org/10.1088/0953-8984/14/29/301.

  10. M. Wen, C.-Y. Wang, Lattice stability and the effect of Co and Re on the ideal strength of Ni: First-principles study of uniaxial tensile deformation, Chinese Physics B, 26, 9, 093106, 2017, https://doi.org/10.1088/1674-1056/26/9/093106.

  11. J. Nye, P. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford Science Publications, Clarendon Press, 1985.

  12. T. Ting, Anisotropic Elasticity: Theory and Applications, Applied Mathematics and Engineering Science Texts Series, John Wiley & Sons, 1992.

  13. J.D. Clayton, Nonlinear Mechanics of Crystals, Solid Mechanics and its Applications, Springer, Dordrecht, 2011, https://doi.org/10.1007/978-94-007-0350-6.

  14. S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Computational Physics, 117, 1, 1–19, 1995, https://doi.org/10.1006/jcph.1995.1039.

  15. S. Singh, L. Lang, V. Dovale-Farelo, U. Herath, P. Tavazohi, F.-X. Coudert, A.H. Romero, MechElastic: A Python library for analysis of mechanical and elastic properties of bulk and 2D materials, Computer Physics Communications, 267, 108068, 2021, https://doi.org/10.1016/j.cpc.2021.108068.

  16. Z.-L. Liu, C. Ekuma, W.-Q. Li, J.-Q. Yang, X.-J. Li, ElasTool: An automated toolkit for elastic constants calculation, Computer Physics Communications, 270, 108180, 2022, https://doi.org/10.1016/j.cpc.2021.108180.

  17. M. Carroll, Must elastic materials be hyperelastic?, Mathematics and Mechanics of Solids, 14, 4, 369–376, 2009, https://doi.org/10.1177/1081286508099385.

  18. R. Hetnarski, J. Ignaczak, The Mathematical Theory of Elasticity, 2nd ed., Taylor & Francis, 2010.

  19. A. Blinowski, J. Ostrowska-Maciejewska, J. Rychlewski, Two-dimensional Hooke’s tensors–isotropic decomposition, effective symmetry criteria, Archives of Mechanics, 48, 2, 325–345, 1996, https://am.ippt.pan.pl/am/article/view/v48p325.

  20. Q.C. He, Q.S. Zheng, On the symmetries of 2D elastic and hyperelastic tensors, Journal of Elasticity, 43, 3, 203–225, 1996, https://doi.org/10.1007/BF00042501.

  21. M.M. Mehrabadi, S.C. Cowin, Eigentensors of linear anisotropic elastic materials, The Quarterly Journal of Mechanics and Applied Mathematics, 43, 1, 15–41, 1990, https://doi.org/10.1093/qjmam/43.1.15.

  22. L. Morin, P. Gilormini, K. Derrien, Generalized Euclidean distances for elasticity tensors, Journal of Elasticity, 138, 2, 221–232, 2020, https://doi.org/10.1007/s10659-019-09741-z.

  23. J. Gao, Q.-J. Liu, B. Tang, Elastic stability criteria of seven crystal systems and their application under pressure: Taking carbon as an example, Journal of Applied Physics, 133, 13, 135901, 2023, https://doi.org/10.1063/5.0139232.

  24. R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, New York, 2013.

  25. J. Rychlewski, On Hooke’s law, Journal of Applied Mathematics and Mechanics, 48, 3, 303–314, 1984, https://doi.org/10.1016/0021-8928(84)90137-0.

  26. K. Kowalczyk-Gajewska, J. Ostrowska-Maciejewska, Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material, Engineering Transactions, 57, 3-4, 145–183, 2009, https://doi.org/10.24423/engtrans.172.2009.

  27. M. Moakher, A.N. Norris, The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry, Journal of Elasticity, 85, 3, 215–263, 2006, https://doi.org/10.1007/s10659-006-9082-0.

  28. W. Thomson, XXI. Elements of a mathematical theory of elasticity, Philosophical Transactions of the Royal Society of London, 146, 481–498, 1856, https://doi.org/10.1098/rstl.1856.0022.

  29. R. Ogden, Non-linear Elastic Deformations, Dover Civil and Mechanical Engineering, Dover Publications, 1997.

  30. J. Griesser, L. Frérot, J.A. Oldenstaedt, M.H. Müser, L. Pastewka, Analytic elastic coefficients in molecular calculations: Finite strain, nonaffine displacements, and many-body interatomic potentials, Physical Review Materials, 7, 073603, 2023, https://doi.org/10.1103/PhysRevMaterials.7.073603.

  31. C. Malgrange, C. Ricolleau, M. Schlenker, Symmetry and Physical Properties of Crystals, Springer, Netherlands, 2014, https://doi.org/10.1007/978-94-017-8993-6.

  32. R.E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure, Oxford University Press, 2004, https://doi.org/10.1093/oso/9780198520757.001.0001.

  33. M. Maździarz, M. Gajewski, Estimation of isotropic hyperelasticity constitutive models to approximate the atomistic simulation data for aluminium and tungsten monocrystals, Computer Modeling in Engineering & Sciences, 105, 2, 123–150, 2015, https://doi.org/10.3970/cmes.2015.105.123.

  34. P. Chadwick, M. Vianello, S.C. Cowin, A new proof that the number of linear elastic symmetries is eight, Journal of the Mechanics and Physics of Solids, 49, 11, 2471–2492, 2001, the Jean-Paul Boehler Memorial Volume, https://doi.org/10.1016/S0022-5096(01)00064-3.

  35. M. Maździarz, Comment on ‘The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals’, 2D Materials, 6, 4, 048001, 2019, https://doi.org/10.1088/2053-1583/ab2ef3.

  36. M. Maździarz, S. Nosewicz, Atomistic investigation of deformation and fracture of individual structural components of metal matrix composites, Engineering Fracture Mechanics, 298, 109953, 2024, https://doi.org/10.1016/j.engfracmech.2024.109953.

  37. S.H. Zhang, R.F. Zhang, AELAS: Automatic ELAStic property derivations via high-throughput first-principles computation, Computer Physics Communications, 220, 403–416, 2017, https://doi.org/10.1016/j.cpc.2017.07.020.

  38. S.C. Cowin, M.M. Mehrabadi, Anisotropic symmetries of linear elasticity, Applied Mechanics Reviews, 48, 5, 247–285, 1995, https://doi.org/10.1115/1.3005102.

  39. J. Wang, S. Yip, S.R. Phillpot, D. Wolf, Crystal instabilities at finite strain, Physical Review Letters, 71, 4182–4185, 1993, https://doi.org/10.1103/PhysRevLett.71.4182.

  40. J. Wang, J. Li, S. Yip, S. Phillpot, D. Wolf, Mechanical instabilities of homogeneous crystals, Physical Review B, 52, 12627–12635, 1995, https://doi.org/10.1103/PhysRevB.52.12627.

  41. M. Itskov, Tensor Algebra and Tensor Analysis for Engineers, Springer, Berlin, Heidelberg, 2007, https://doi.org/10.1007/978-3-319-16342-0.

  42. Wolfram Research, Inc., Mathematica, Version 13.1, Champaign, IL, 2022.

  43. G. Van Rossum, F.L. Drake, Python 3 Reference Manual, CreateSpace, Scotts Valley, CA, 2009.

  44. J. Bezanson, A. Edelman, S. Karpinski, V.B. Shah, Julia: A fresh approach to numerical computing, SIAM Review, 59, 1, 65–98, 2017, https://doi.org/10.1137/141000671.

  45. M. Maździarz, Mechanical stability.nb, June, 2025, https://figshare.com/articles/journal_contribution/Mechanical_stability_nb/29356658., https://doi.org/10.6084/m9.figshare.29356658.v1.

  46. G.P. Purja Pun, Y. Mishin, Development of an interatomic potential for the Ni-Al system, Philosophical Magazine, 89, 34–36, 3245–3267, 2009, https://doi.org/10.1080/14786430903258184.

  47. R.M. Hackett, Hyperelasticity Primer, Springer, Cham, 2018, https://doi.org/10.1007/978-3-319-73201-5.