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Invariant-based hyperelastic constitutive model for hexagonal two-dimensional materials

Serge R. Maalouf*

Senthil S. Vel†

  • *Contact author: serge.maalouf@lau.edu.lb
  • †Contact author: senthil.vel@maine.edu
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Phys. Rev. B 112, 174120 – Published 24 November, 2025

DOI: https://doi.org/10.1103/35fw-813c

Abstract

A general hyperelastic constitutive model for hexagonal two-dimensional (2D) materials is presented, wherein the strain energy density is expressed in terms of the invariants of the Green–St. Venant strain tensor. The functional dependence of the strain energy density on the strain tensor is elucidated through the introduction of new invariants that capture the contributions of dilatational, deviatoric, and strain-induced anisotropy effects. The strain energy density is formulated in terms of these new invariants and the corresponding second Piola-Kirchhoff stress tensor is obtained. A specific form of the constitutive model, characterized by seven elastic constants, is presented and a methodology for systematically evaluating the elastic constants is provided. A general expression for the tangent stiffness tensor is provided based on the new invariants. Additionally, the constitutive model is linearized for infinitesimal deformations and expressions for the Young's modulus, Poisson's ratio, and shear modulus are derived. Expressions for the layer modulus and speed of sound of hexagonal 2D materials are presented in terms of the linearized elastic properties. The nonlinear elastic behavior of graphene and molybdenum disulfide is investigated using the proposed approach. The constitutive model is validated using density functional theory data and the linearized elastic properties of graphene and molybdenum disulfide are compared with values reported in the literature. The proposed constitutive model accurately captures the nonlinear elastic response of graphene and molybdenum disulfide and can be adapted to study other 2D materials of hexagonal symmetry.

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References (97)

  1. C. Tan, X. Cao, X.-J. Wu, Q. He, J. Yang, X. Zhang, J. Chen, W. Zhao, S. Han, G.-H. Nam, M. Sindoro, and H. Zhang, Recent advances in ultrathin two-dimensional nanomaterials, Chem. Rev. 117, 6225 (2017).
  2. V. Shanmugam, R. A. Mensah, K. Babu, S. Gawusu, A. Chanda, Y. Tu, R. E. Neisiany, M. Försth, G. Sas, and O. Das, A review of the synthesis, properties, and applications of 2D materials, Part. Part. Syst. Charact. 39, 2200031 (2022).
  3. S. Das, M. Kim, J. won Lee, and W. Choi, Synthesis, properties, and applications of 2-D materials: A comprehensive review, Crit. Rev. Solid State Mater. Sci. 39, 231 (2014).
  4. K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Two-dimensional atomic crystals, Proc. Natl. Acad. Sci. USA 102, 10451 (2005).
  5. H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, Phosphorene: An unexplored 2D semiconductor with a high hole mobility, ACS Nano 8, 4033 (2014).
  6. D. Akinwande, C. J. Brennan, J. S. Bunch, P. Egberts, J. R. Felts, H. Gao, R. Huang, J.-S. Kim, T. Li, Y. Li, K. M. Liechti, N. Lu, H. S. Park, E. J. Reed, P. Wang, B. I. Yakobson, T. Zhang, Y.-W. Zhang, Y. Zhou, and Y. Zhu, A review on mechanics and mechanical properties of 2D materials–Graphene and beyond, Extreme Mech. Lett. 13, 42 (2017).
  7. Q. Cao, X. Geng, H. Wang, P. Wang, A. Liu, Y. Lan, and Q. Peng, A review of current development of graphene mechanics, Crystals 8, 357 (2018).
  8. S. J. Kim, K. Choi, B. Lee, Y. Kim, and B. H. Hong, Materials for flexible, stretchable electronics: Graphene and 2D materials, Annu. Rev. Mater. Res. 45, 63 (2015).
  9. A. Chhetry, M. Sharifuzzaman, H. Yoon, S. Sharma, X. Xuan, and J. Y. Park, MoS2-decorated laser-induced graphene for a highly sensitive, hysteresis-free, and reliable piezoresistive strain sensor, ACS Appl. Mater. Interfaces 11, 22531 (2019).
  10. S. Weng, T. Peng, K. Gao, H. Zhu, Y. Yan, H. Yu, and D. He, Bidirectional large strain monitoring using a novel graphene film-based patch antenna sensor, Smart Mater. Struct. 33, 075015 (2024).
  11. N. Tang, C. Du, Q. Wang, and H. Xu, Strain engineering in bilayer WSe2 over a large strain range, Microelectron. Eng. 223, 111202 (2020).
  12. M. Pandey, C. Pandey, R. Ahuja, and R. Kumar, Straining techniques for strain engineering of 2D materials towards flexible straintronic applications, Nano Energy 109, 108278 (2023).
  13. S. R. Maalouf and S. S. Vel, Anisotropic elastic properties of triclinic 2D materials using density functional theory with application to rhenium disulfide, Comput. Condens. Matter 34, e00790 (2023).
  14. H. Sun, P. Agrawal, and C. V. Singh, A first-principles study of the relationship between modulus and ideal strength of single-layer, transition metal dichalcogenides, Mater. Adv. 2, 6631 (2021).
  15. C. Lee, X. Wei, J. W. Kysar, and J. Hone, Measurement of the elastic properties and intrinsic strength of monolayer graphene, Science 321, 385 (2008).
  16. X. Liu, T. H. Metcalf, J. T. Robinson, B. H. Houston, and F. Scarpa, Shear modulus of monolayer graphene prepared by chemical vapor deposition, Nano Lett. 12, 1013 (2012).
  17. R. C. Cooper, C. Lee, C. A. Marianetti, X. Wei, J. Hone, and J. W. Kysar, Nonlinear elastic behavior of two-dimensional molybdenum disulfide, Phys. Rev. B 87, 035423 (2013).
  18. F. Liu, P. Ming, and J. Li, Ab initio calculation of ideal strength and phonon instability of graphene under tension, Phys. Rev. B 76, 064120 (2007).
  19. T. Li, Ideal strength and phonon instability in single-layer MoS2, Phys. Rev. B 85, 235407 (2012).
  20. L. B. Drissi, K. Sadki, and M.-H. Kourra, Mechanical response of SiC sheet under strain, Mater. Chem. Phys. 201, 199 (2017).
  21. J.-W. Jiang and H. S. Park, Mechanical properties of single-layer black phosphorus, J. Phys. D: Appl. 47, 385304 (2014).
  22. E. Cadelano, P. L. Palla, S. Giordano, and L. Colombo, Nonlinear elasticity of monolayer graphene, Phys. Rev. Lett. 102, 235502 (2009).
  23. X. Wei, B. Fragneaud, C. A. Marianetti, and J. W. Kysar, Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description, Phys. Rev. B 80, 250407 (2009).
  24. F. D. Murnaghan, Finite deformations of an elastic solid, Am. J. Math. 59, 235 (1937).
  25. Y. Hiki, Higher order elastic constants of solids, Annu. Rev. Mater. Sci. 11, 51 (1981).
  26. F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71, 809 (1947).
  27. R. Thurston, in Physical Acoustics: Principles and Methods, Part A, edited by W. P. Mason (Academic Press, New York, 1964), Chap. 1, Vol. 1, pp. 36.
  28. W. S. Slaughter, The Linearized Theory of Elasticity, 1st ed. (Birkhäuser, Boston, 2002).
  29. E. B. Tadmor, R. E. Miller, and R. S. Elliott, Continuum Mechanics and Thermodynamics: From Fundamental Principles to Governing Equations (Cambridge University Press, Cambridge, 2012).
  30. S. R. Maalouf and S. S. Vel, Nonlinear elastic analysis of 2D materials of arbitrary symmetries with application to black phosphorus, Mech. Mater. 165, 104159 (2022).
  31. S. R. Maalouf and S. S. Vel, Nonlinear elastic behavior of 2D materials using molecular statics and comparisons with first principles calculations, Phys. E: Low Dimens. Syst. Nanostruct. 148, 115633 (2023).
  32. Q. Peng, High-order nonlinear mechanical properties of g-SiC, Mech. Mater. 148, 103473 (2020).
  33. Q. Peng, C. Liang, W. Ji, and S. De, A first-principles study of the mechanical properties of g-GeC, Mech. Mater. 64, 135 (2013).
  34. Q. Peng, W. Ji, and S. De, Mechanical properties of the hexagonal boron nitride monolayer: Ab initio study, Comput. Mater. Sci. 56, 11 (2012).
  35. M. Faghihnasiri, H. Jafari, A. Ramazani, M. Shabani, S. M. Estalaki, and R. G. Larson, Nonlinear elastic behavior and anisotropic electronic properties of two-dimensional borophene, J. Appl. Phys. 125, 145107 (2019).
  36. Q. Peng, W. Ji, and S. De, Mechanical properties of graphyne monolayers: A first-principles study, Phys. Chem. Chem. Phys. 14, 13385 (2012).
  37. M. Xue, W. He, Q. Gong, M. Yi, and W. Guo, Nonlinear elasticity and strain-tunable magnetocalorics of antiferromagnetic monolayer MnPS3, Extreme Mech. Lett. 57, 101900 (2022).
  38. L. R. G. Treloar, The elasticity of a network of long-chain molecules–II, Trans. Faraday Soc. 39, 241 (1943).
  39. M. Mooney, A theory of large elastic deformation, J. Appl. Phys. 11, 582 (1940).
  40. R. Ogden, Non-linear Elastic Deformations, 1st ed. (Ellis Horwood Limited, Toronto, 1984).
  41. M. Xu, J. T. Paci, J. Oswald, and T. Belytschko, A constitutive equation for graphene based on density functional theory, Intl. J. Solids Struct. 49, 2582 (2012).
  42. S. Kumar and D. M. Parks, On the hyperelastic softening and elastic instabilities in graphene, Proc. R. Soc. A: Math. Phys. Eng. Sci. 471, 20140567 (2015).
  43. H. Hencky, Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen, Z. Techn. Phys. 9, 215 (1928).
  44. H. Hencky, The law of elasticity for isotropic and quasi-isotropic substances by finite deformations, J. Rheology 2, 169 (1931).
  45. H. Hencky, The elastic behavior of vulcanized rubber, Rubber Chem. Technol. 6, 217 (1933).
  46. K. Heiduschke, The logarithmic strain space description, Intl. J. Solids Struct. 32, 1047 (1995).
  47. Q.-S. Zheng, Theory of representations for tensor functions–A unified invariant approach to constitutive equations, Appl. Mech. Rev. 47, 545 (1994).
  48. R. Ghaffari, F. Shirazian, M. Hu, and R. A. Sauer, A nonlinear hyperelasticity model for single layer blue phosphorus based on ab initio calculations, Proc. R. Soc. A: Math. Phys. Eng. Sci. 475, 20190149 (2019).
  49. R. Höller, V. Smejkal, F. Libisch, and C. Hellmich, Energy landscapes of graphene under general deformations: DFT-to-hyperelasticity upscaling, Intl. J. Eng. Sci. 154, 103342 (2020).
  50. R. C. Andrew, R. E. Mapasha, A. M. Ukpong, and N. Chetty, Mechanical properties of graphene and boronitrene, Phys. Rev. B 85, 125428 (2012).
  51. L. Xue, Y. Ren, J.-R. He, Y. Zhao, S.-L. Xu, Y. Hu, and C.-B. Hua, The mechanical and thermal parameters of two-dimensional hexagonal materials evaluated using elastic properties: Monolayer MoS2 as an example, Results Phys. 57, 107418 (2024).
  52. R. Atkin and N. Fox, An Introduction to the Theory of Elasticity, 1st ed. (Dover, Mineola, 1980).
  53. G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering (Wiley, Chichester, 2000).
  54. A. W. Knapp, Basic Algebra, 1st ed. (Birkhäuser, Boston, 2006).
  55. Y. Xiong, Z. Gong, and H. Jin, Strain tuning of the nonlinear anomalous Hall effect in MoS2 monolayer, J. Phys.: Condens. Matter 37, 195301 (2025).
  56. E. B. Tadmor and R. E. Miller, Modelling Materials: Continuum, Atomistic and Multiscale Techniques, 1st ed. (Cambridge University Press, Cambridge, UK, 2016).
  57. F. Gantmacher, The Theory of Matrices, 1st ed., Vol. 1 (AMS Chelsea Publishing, New York, 1959).
  58. N. J. Higham, Functions of Matrices: Theory and Computation, 1st ed. (SIAM, Philadelphia, 2008).
  59. Y. P. Varshni, Comparative study of potential energy functions for diatomic molecules, Rev. Mod. Phys. 29, 664 (1957).
  60. T.-C. Lim and R. Udyavara, Relations between Varshni and Morse potential energy parameters, Centr. Eur. J. Phys. 7, 193 (2009).
  61. R. C. Batra, Elements of Continuum Mechanics (American Institute of Aeronautics and Astronautics, Reston, VA, 2006).
  62. A. Politano, A. R. Marino, D. Campi, D. Farías, R. Miranda, and G. Chiarello, Elastic properties of a macroscopic graphene sample from phonon dispersion measurements, Carbon 50, 4903 (2012).
  63. E. Bolen, E. Deligoz, and H. Ozisik, Origin of low thermal conductivity in monolayer PbI2 Solid State Commun. 327, 114223 (2021).
  64. R. M. Hackett, Hyperelasticity Primer, 2nd ed. (Springer Cham, Cham, Switzerland, 2018).
  65. P. Helnwein, Some remarks on the compressed matrix representation of symmetric second-order and fourth-order tensors, Comput. Methods Appl. Mech. Eng. 190, 2753 (2001).
  66. W. Voigt, Lehrbuch Der Kristallphysik: Mit Ausschluss Der Kristalloptik (B.G. Teubner, Berlin, 1910).
  67. B. Mousavi, A. Sadeghirad, and V. Lotfi, A computational investigation of applicability of nonlinear fracture mechanics in nano-scale fracture of graphene, Mater. Today Commun. 31, 103687 (2022).
  68. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phy. Rev. B 136, B864 (1964).
  69. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  70. O. H. Nielsen and R. M. Martin, Quantum-mechanical theory of stress and force, Phys. Rev. B 32, 3780 (1985).
  71. P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. D. Jr., A. Ferretti, A. Floris, G. Fratesi, G. Fugallo et al., Advanced capabilities for materials modelling with QUANTUM ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
  72. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos et al., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  73. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  74. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  75. A. Dal Corso, Pseudopotentials periodic table: From H to Pu, Comput. Mater. Sci. 95, 337 (2014).
  76. K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. Dal Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. Di Marco, C. Draxl, M. Dułak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese et al., Reproducibility in density functional theory calculations of solids, Science 351, aad3000 (2016).
  77. G. Prandini, A. Marrazzo, I. E. Castelli, N. Mounet, and N. Marzari, Precision and efficiency in solid-state pseudopotential calculations, Npj Comput. Mater 4, 72 (2018).
  78. H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  79. R. P. Feynman, Forces in molecules, Phys. Rev. 56, 340 (1939).
  80. D. F. Shanno, Conditioning of quasi-Newton methods for function minimization, Math. Comp. 24, 647 (1970).
  81. D. Goldfarb, A family of variable-metric methods derived by variational means, Math. Comp. 24, 23 (1970).
  82. R. Fletcher, A new approach to variable metric algorithms, Comput. J. 13, 317 (1970).
  83. C. G. Broyden, The convergence of a class of double-rank minimization algorithms 1. General considerations, IMA J. Appl. Math. 6, 76 (1970).
  84. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, New York, 2006).
  85. C. T. Kelley, Iterative Methods for Optimization (Society for Industrial and Applied Mathematics, Philadelphia, 1999).
  86. S. Singh, C. Espejo, and A. H. Romero, Structural, electronic, vibrational, and elastic properties of graphene/MoS2 bilayer heterostructures, Phys. Rev. B 98, 155309 (2018).
  87. A. R. Hall, L. An, J. Liu, L. Vicci, M. R. Falvo, R. Superfine, and S. Washburn, Experimental measurement of single-wall carbon nanotube torsional properties, Phys. Rev. Lett. 96, 256102 (2006).
  88. X. Cong, Q.-Q. Li, X. Zhang, M.-L. Lin, J.-B. Wu, X.-L. Liu, P. Venezuela, and P.-H. Tan, Probing the acoustic phonon dispersion and sound velocity of graphene by Raman spectroscopy, Carbon 149, 19 (2019).
  89. T. Sohier, M. Calandra, C.-H. Park, N. Bonini, N. Marzari, and F. Mauri, Phonon-limited resistivity of graphene by first-principles calculations: Electron-phonon interactions, strain-induced gauge field, and Boltzmann equation, Phys. Rev. B 90, 125414 (2014).
  90. A. Kokalj, XCrySDen–a new program for displaying crystalline structures and electron densities, J. Mol. Graph. Model. 17, 176 (1999).
  91. M. Born, On the stability of crystal lattices. I, Math. Proc. Cambridge Philos. Soc. 36, 160 (1940).
  92. O. K. Le, V. Chihaia, M.-P. Pham-Ho, and D. N. Son, Electronic and optical properties of monolayer MoS2 under the influence of polyethyleneimine adsorption and pressure RSC Adv. 10, 4201 (2020).
  93. J. van Baren, G. Ye, J.-A. Yan, Z. Ye, P. Rezaie, P. Yu, Z. Liu, R. He, and C. H. Lui, Stacking-dependent interlayer phonons in 3R and 2H MoS2, 2D Mater. 6, 025022 (2019).
  94. D. Çakır, F. M. Peeters, and C. Sevik, Mechanical and thermal properties of h-MX2 (M = Cr, Mo, W; X = O, S, Se, Te) monolayers: A comparative study, Appl. Phys. Lett. 104, 203110 (2014).
  95. Q. Peng and S. De, Outstanding mechanical properties of monolayer MoS2 and its application in elastic energy storage, Phys. Chem. Chem. Phys. 15, 19427 (2013).
  96. S. Bertolazzi, J. Brivio, and A. Kis, Stretching and breaking of ultrathin MoS2, ACS Nano 5, 9703 (2011).
  97. X. Li, J. T. Mullen, Z. Jin, K. M. Borysenko, M. Buongiorno Nardelli, and K. W. Kim, Intrinsic electrical transport properties of monolayer silicene and MoS2 from first principles, Phys. Rev. B 87, 115418 (2013).

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Phys. Rev. B 112, 174120– Published 24 November, 2025
  • Received 17 May 2025
  • Revised 13 August 2025
  • Accepted 30 September 2025
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