Elastic properties of the f.c.c. hard sphere crystal with thick nanolayer-nanochannel inclusions formed by hard dumbbells
Narojczyk Jakub W. 1*, Tretiakov Konstantin V. 1,2, Wojciechowski Krzysztof W. 1
1 Institute of Molecular Physics
Polish Academy of Sciences
ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland
∗E-mail: narojczyk@ifmpan.poznan.pl2 President Stanisław Wojciechowski University of Kalisz
Polytechnic Faculty
ul. Nowy Świat 4, 62-800 Kalisz, Poland
Received:
Received: 14 October 2025; accepted: 8 December 2025
DOI: 10.12921/cmst.2025.0000019
Abstract:
The article investigates the possibility of modifying the elastic properties of materials by altering their structure at the atomic level. Such approach could prove useful in tailoring the properties of materials to particular applications. Advances in the nano-level techniques may prove this approach to be a cost-effective and fast to implement method of achieving this goal. The subject of this work are studies of elastic properties of f.c.c. crystals with inclusions containing di-atomic molecules, forming aperiodic phase within these inclusions. Recent studies showed that such phase may enhance auxetic properties of the crystal. It has been discovered that Poisson’s ratio in the direction of [110] has been decreased from −0.054 (pure f.c.c. crystal) down to −0.235 (inclusions with aperiodic phase of dumbbells). Here, different sized of the inclusions are investigated, in order to extend that study. It has been found that increasing the size of the inclusions will completely eliminate the auxetic properties when the diameters of the inclusion spheres increase. However, the decreasing atomic diameters show that auxetic properties can be switched between the in-plane and out-of-plane directions (with respect to inclusion nanolayers), by rotating the shape of the nanochannel around its longitudinal axis.
Key words:
auxetics, degenerate phase, elastic properties, hard dumbbells, hard spheres, nanoinclusions, Poisson’s ratio
References:
[1] L. D. Landau and E. M. Lifshitz. Theory of Elasticity. Pergamon Press, London, UK, 1986.
[2] R. S. Lakes. Foam structures with a negative Poisson’s ratio. Science, 235: pp. 1038–1040, 1987.
[3] K. W. Wojciechowski. Constant thermodynamic tension Monte Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers. Mol. Phys., 61: pp. 1247–1258, 1987.
[4] K. W. Wojciechowski. Two-dimensional isotropic model with a negative Poisson ratio. Phys. Lett. A, 137: pp. 60–64, 1989.
[5] K. E. Evans. Auxetic polymers: a new range of materials. Endeavour, 15: pp. 170–174, 1991.
[6] K. E. Evans and A. Alderson. Auxetic materials: Functional materials and structures from lateral thinking! Adv. Mater., 12: pp. 617–628, 2000.
[7] Y. Prawoto. Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio. Comput. Mater. Sci., 58: pp. 140–153, 2012.
[8] T.-C. Lim. Mechanics of Metamaterials with Negative Parameters. Springer: Singapore, 2020.
[9] J. B. Choi and R. S. Lakes. Design of a fastener based on negative poissons ratio foam. Cellular Polymers, 10(3): pp. 205–212, 1991.
[10] M. Kapnisi, C. Mansfield, C. Marijon, A. G. Guex, F. Perbellini, I. Bardi, E. J. Humphrey, J. L. Puetzer, D. Mawad, D. C. Koutsogeorgis, D. J. Stuckey, C. M. Terracciano, S. E. Harding, and M. M. Stevens. Auxetic cardiac patches with tunable mechanical and conductive properties toward treating myocardial infarction. Adv. Funct. Mater., 28:1800618, 2018.
[11] A. Kasal, T. Kuskun, and J. Smardzewski. Experimental and numerical study on withdrawal strength of different types of auxetic dowels for furniture joints. Materials, 13(19):4252, 2020.
[12] D. Tahir, M. Zhang, and H. Hu. Auxetic materials for personal protection: A review. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200324, 2022.
[13] R. F. Almgren. An isotropic three-dimensional structure with Poisson’s ratio =-1. J. Elast., 15: pp. 427–430, 1985.
[14] S. Bao, X. Ren, Y. J. Qi, H. R. Li, D. Han, W. Li, C. Luo, and Z. Z. Song. Quasi-static mechanical properties of a modified auxetic re-entrant honeycomb metamaterial. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200270, 2022.
[15] H. Sun, H. Zhou, J. Zhang, L. Wang, and T. Wang. Multiple crashworthiness evaluation of double-v npr filled square tube under axial impact. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200388, 2022.
[16] E. A. Korznikova, K. Zhou, L. Kh. Galiakhmetova, E. G. Soboleva, A. A. Kudreyko, and S. V. Dmitriev. Partial auxeticity of laterally compressed carbon nanotube bundles. Phys. Status Solidi – Rapid Research Letters, 16(1):2100189, 2022.
[17] N. Novak, M. Nowak, M. Vesenjak, and Z. Ren. Structural optimization of the novel 3d graded axisymmetric chiral auxetic structure. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200409, 2022.
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[19] P.-S. Farrugia, R. Gatt, and J. N. Grima. Creating a three-dimensional auxetic system using torsion springs. Phys. Status Solidi B-Basic Solid State Phys., 261(12):2400212, 2024.
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[21] O. Duncan, F. Clegg, A. Essa, A. M. T. Bell, L. Foster, T. Allen, and A. Alderson. Effects of heat exposure and volumetric compression on Poisson’s ratios, Young’s moduli, and polymeric composition during thermo-mechanical conversion of auxetic open cell polyurethane foam. Phys. Status Solidi B-Basic Solid State Phys., 256(1):1800393, 2019.
[22] S. Mohsenizadeh, Z. Ahmad, R. Alipour, R. A. Majid, and Y. Prawoto. Quasi tri-axial method for the fabrication of optimized polyurethane auxetic foams. Phys. Status Solidi B-Basic Solid State Phys., 256(10):1800587, 2019.
[23] O. Duncan, A. Alderson, and T. Allen. Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams. Smart Mater. Struct., 30(3):035014, 2021.
[24] Q. Zhang, X. Yu, Y. Xia, D. Zhang, R. S. Lakes, K. W. Wojciechowski, and F. Scarpa. The shear performance of uniaxially thermoformed auxetic polymer foams. Composites Part B, 286: 111791, 2024.
[25] A. Zulifqar, T. Hua, and H. Hu. Single- and double-layered bistretch auxetic woven fabrics made of nonauxetic yarns based on foldable geometries. Phys. Status Solidi B-Basic Solid State Phys., 257(10):1900156, 2020.
[26] S. Zhao, Y. Chang, Y. Yang, H. Kamrul, M. Zhang, and H. Hu. Deformation behaviors of auxetic warp knitted fabrics based on reentrant geometry. Phys. Status Solidi B-Basic Solid State Phys., 258(12):2000580, 2021.
[27] Y. Chang and H. Hu. 3d fabrics with negative Poisson’s ratio: A review. Applied Composite Materials, 29: pp. 95–108, 2022.
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[31] F. Portone, M. Amorini, M. Montanari, R. Pinalli, A. Pedrini, R. Verucchi, R. Brighenti, and E. Dalcanale. Molecular auxetic polymer of intrinsic microporosity via conformational switching of a cavitand crosslinker. Advanced Functional Materials, 33(51):2307605, 2023.
[32] K. Bertoldi, V. Vitelli, J. Christensen, and M. van Hecke. Flexible mechanical metamaterials. Nature Reviews Materials, 2:17066, 2017.
[33] Y. Y. Chen, T. T. Li, F. Scarpa, and L. F. Wang. Lattice metamaterials with mechanically tunable Poisson’s ratio for vibration control. Physical Review Applied, 7: 024012, 2017.
[34] K. K. Dudek, J. A. I. Martinez, G. Ulliac, and M. Kadic. Micro-scale auxetic hierarchical mechanical metamaterials for shape morphing. Advanced Materials, 34(14):2110115, 2022.
[35] K. K. Dudek, J. A. I. Martinez, G. Ulliac, L. Hirsinger, L. Wang, V. Laude, and M. Kadic. Micro-scale mechanical metamaterial with a controllable transition in the Poisson’s ratio and band gap formation. Advanced Materials, 35:2210993, 2023.
[36] R. Galea Mifsud, K. K. Dudek, P.-S. Farrugia, J. N. Grima, and R. Gatt. Egg-rack-like active magnetomechanical metamaterial. Phys. Status Solidi B-Basic Solid State Phys., Early view:2400528, 2025.
[37]J. N. Grima and K. E. Evans. Auxetic behavior from rotating squares. J. Mater. Sci. Lett., 19: pp. 1563–1565, 2000.
[38] Wm. G. Hoover and C. G. Hoover. Searching for auxetics with DYNA3D and ParaDyn. Phys. Status Solidi B-Basic Solid State Phys., 242(3): pp. 585–594, 2005.
[39] R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko. The elastic properties of hexagonal auxetics under pressure. Phys. Status Solidi B-Basic Solid State Phys., 253(7): pp. 1261–1269, 2016.
[40] R. V. Goldstein, V. A. Gorodtsov, D. S. Lisovenko, and M. A. Volkov. Auxeticity in nano/microtubes produced from orthorhombic crystals. Smart Mater. Struct., 25(5):05006, 2016.
[41] R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko. Longitudinal elastic tension of two-layered plates from isotropic auxetics-nonauxetics and cubic crystals. European Journal of Mechanics A-Solids, 63:pp. 122–127, 2017.
[42] K. K. Dudek, R. Gatt, L. Mizzi, M. R. Dudek, D. Attard, K. E. Evans, and J. N. Grima. On the dynamics and control of mechanical properties of hierarchical rotating rigid unit auxetics. Sci Rep, 7: 46529, 2017.
[43] V. A. Gorodtsov, M. A. Volkov, and D. S. Lisovenko. Out-of-plane tension of thin two-layered plates of cubic crystals. Phys. Status Solidi B-Basic Solid State Phys., 258(12):2100184, 2021.
[44] S. Czarnecki and T. Łukasiak. Auxetic properties of the stiffest elastic bodies as a result of topology optimization and microstructures recovery based on homogenization method. Phys. Status Solidi B-Basic Solid State Phys., 261(12):2300495, 2024.
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The article investigates the possibility of modifying the elastic properties of materials by altering their structure at the atomic level. Such approach could prove useful in tailoring the properties of materials to particular applications. Advances in the nano-level techniques may prove this approach to be a cost-effective and fast to implement method of achieving this goal. The subject of this work are studies of elastic properties of f.c.c. crystals with inclusions containing di-atomic molecules, forming aperiodic phase within these inclusions. Recent studies showed that such phase may enhance auxetic properties of the crystal. It has been discovered that Poisson’s ratio in the direction of [110] has been decreased from −0.054 (pure f.c.c. crystal) down to −0.235 (inclusions with aperiodic phase of dumbbells). Here, different sized of the inclusions are investigated, in order to extend that study. It has been found that increasing the size of the inclusions will completely eliminate the auxetic properties when the diameters of the inclusion spheres increase. However, the decreasing atomic diameters show that auxetic properties can be switched between the in-plane and out-of-plane directions (with respect to inclusion nanolayers), by rotating the shape of the nanochannel around its longitudinal axis.
Key words:
auxetics, degenerate phase, elastic properties, hard dumbbells, hard spheres, nanoinclusions, Poisson’s ratio
References:
[1] L. D. Landau and E. M. Lifshitz. Theory of Elasticity. Pergamon Press, London, UK, 1986.
[2] R. S. Lakes. Foam structures with a negative Poisson’s ratio. Science, 235: pp. 1038–1040, 1987.
[3] K. W. Wojciechowski. Constant thermodynamic tension Monte Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers. Mol. Phys., 61: pp. 1247–1258, 1987.
[4] K. W. Wojciechowski. Two-dimensional isotropic model with a negative Poisson ratio. Phys. Lett. A, 137: pp. 60–64, 1989.
[5] K. E. Evans. Auxetic polymers: a new range of materials. Endeavour, 15: pp. 170–174, 1991.
[6] K. E. Evans and A. Alderson. Auxetic materials: Functional materials and structures from lateral thinking! Adv. Mater., 12: pp. 617–628, 2000.
[7] Y. Prawoto. Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio. Comput. Mater. Sci., 58: pp. 140–153, 2012.
[8] T.-C. Lim. Mechanics of Metamaterials with Negative Parameters. Springer: Singapore, 2020.
[9] J. B. Choi and R. S. Lakes. Design of a fastener based on negative poissons ratio foam. Cellular Polymers, 10(3): pp. 205–212, 1991.
[10] M. Kapnisi, C. Mansfield, C. Marijon, A. G. Guex, F. Perbellini, I. Bardi, E. J. Humphrey, J. L. Puetzer, D. Mawad, D. C. Koutsogeorgis, D. J. Stuckey, C. M. Terracciano, S. E. Harding, and M. M. Stevens. Auxetic cardiac patches with tunable mechanical and conductive properties toward treating myocardial infarction. Adv. Funct. Mater., 28:1800618, 2018.
[11] A. Kasal, T. Kuskun, and J. Smardzewski. Experimental and numerical study on withdrawal strength of different types of auxetic dowels for furniture joints. Materials, 13(19):4252, 2020.
[12] D. Tahir, M. Zhang, and H. Hu. Auxetic materials for personal protection: A review. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200324, 2022.
[13] R. F. Almgren. An isotropic three-dimensional structure with Poisson’s ratio =-1. J. Elast., 15: pp. 427–430, 1985.
[14] S. Bao, X. Ren, Y. J. Qi, H. R. Li, D. Han, W. Li, C. Luo, and Z. Z. Song. Quasi-static mechanical properties of a modified auxetic re-entrant honeycomb metamaterial. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200270, 2022.
[15] H. Sun, H. Zhou, J. Zhang, L. Wang, and T. Wang. Multiple crashworthiness evaluation of double-v npr filled square tube under axial impact. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200388, 2022.
[16] E. A. Korznikova, K. Zhou, L. Kh. Galiakhmetova, E. G. Soboleva, A. A. Kudreyko, and S. V. Dmitriev. Partial auxeticity of laterally compressed carbon nanotube bundles. Phys. Status Solidi – Rapid Research Letters, 16(1):2100189, 2022.
[17] N. Novak, M. Nowak, M. Vesenjak, and Z. Ren. Structural optimization of the novel 3d graded axisymmetric chiral auxetic structure. Phys. Status Solidi B-Basic Solid State Phys., 259(12):2200409, 2022.
[18] T.-C. Lim. Auxetic and non-auxetic metamaterial model from interconnected rotating parallelograms and triangles. Phys. Status Solidi B-Basic Solid State Phys., 261(12):2300413, 2024.
[19] P.-S. Farrugia, R. Gatt, and J. N. Grima. Creating a three-dimensional auxetic system using torsion springs. Phys. Status Solidi B-Basic Solid State Phys., 261(12):2400212, 2024.
[20] T.-C. Lim. Auxetic metamaterial model made from rotating rectangles, hexagons, and triangles. Phys. Status Solidi B-Basic Solid State Phys., Early view:2400343, 2025.
[21] O. Duncan, F. Clegg, A. Essa, A. M. T. Bell, L. Foster, T. Allen, and A. Alderson. Effects of heat exposure and volumetric compression on Poisson’s ratios, Young’s moduli, and polymeric composition during thermo-mechanical conversion of auxetic open cell polyurethane foam. Phys. Status Solidi B-Basic Solid State Phys., 256(1):1800393, 2019.
[22] S. Mohsenizadeh, Z. Ahmad, R. Alipour, R. A. Majid, and Y. Prawoto. Quasi tri-axial method for the fabrication of optimized polyurethane auxetic foams. Phys. Status Solidi B-Basic Solid State Phys., 256(10):1800587, 2019.
[23] O. Duncan, A. Alderson, and T. Allen. Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams. Smart Mater. Struct., 30(3):035014, 2021.
[24] Q. Zhang, X. Yu, Y. Xia, D. Zhang, R. S. Lakes, K. W. Wojciechowski, and F. Scarpa. The shear performance of uniaxially thermoformed auxetic polymer foams. Composites Part B, 286: 111791, 2024.
[25] A. Zulifqar, T. Hua, and H. Hu. Single- and double-layered bistretch auxetic woven fabrics made of nonauxetic yarns based on foldable geometries. Phys. Status Solidi B-Basic Solid State Phys., 257(10):1900156, 2020.
[26] S. Zhao, Y. Chang, Y. Yang, H. Kamrul, M. Zhang, and H. Hu. Deformation behaviors of auxetic warp knitted fabrics based on reentrant geometry. Phys. Status Solidi B-Basic Solid State Phys., 258(12):2000580, 2021.
[27] Y. Chang and H. Hu. 3d fabrics with negative Poisson’s ratio: A review. Applied Composite Materials, 29: pp. 95–108, 2022.
[28] D. M. Kochmann and G. N. Venturini. Homogenized mechanical properties of auxetic composite materials in finite-strain elasticity. Smart Mater. Struct., 22:084004, 2013.
[29] I. Shufrin, E. Pasternak, and A. V. Dyskin. Effective properties of layered auxetic hybrids. Composite Structures, 209: pp. 391–400, 2019.
[30] N. Novak, M. Vesenjak, G. Kennedy, N. Thadhani, and Z. Ren. Response of chiral auxetic composite sandwich panel to fragment simulating projectile impact. Phys. Status Solidi B-Basic Solid State Phys., 257(10):1900099, 2020.
[31] F. Portone, M. Amorini, M. Montanari, R. Pinalli, A. Pedrini, R. Verucchi, R. Brighenti, and E. Dalcanale. Molecular auxetic polymer of intrinsic microporosity via conformational switching of a cavitand crosslinker. Advanced Functional Materials, 33(51):2307605, 2023.
[32] K. Bertoldi, V. Vitelli, J. Christensen, and M. van Hecke. Flexible mechanical metamaterials. Nature Reviews Materials, 2:17066, 2017.
[33] Y. Y. Chen, T. T. Li, F. Scarpa, and L. F. Wang. Lattice metamaterials with mechanically tunable Poisson’s ratio for vibration control. Physical Review Applied, 7: 024012, 2017.
[34] K. K. Dudek, J. A. I. Martinez, G. Ulliac, and M. Kadic. Micro-scale auxetic hierarchical mechanical metamaterials for shape morphing. Advanced Materials, 34(14):2110115, 2022.
[35] K. K. Dudek, J. A. I. Martinez, G. Ulliac, L. Hirsinger, L. Wang, V. Laude, and M. Kadic. Micro-scale mechanical metamaterial with a controllable transition in the Poisson’s ratio and band gap formation. Advanced Materials, 35:2210993, 2023.
[36] R. Galea Mifsud, K. K. Dudek, P.-S. Farrugia, J. N. Grima, and R. Gatt. Egg-rack-like active magnetomechanical metamaterial. Phys. Status Solidi B-Basic Solid State Phys., Early view:2400528, 2025.
[37]J. N. Grima and K. E. Evans. Auxetic behavior from rotating squares. J. Mater. Sci. Lett., 19: pp. 1563–1565, 2000.
[38] Wm. G. Hoover and C. G. Hoover. Searching for auxetics with DYNA3D and ParaDyn. Phys. Status Solidi B-Basic Solid State Phys., 242(3): pp. 585–594, 2005.
[39] R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko. The elastic properties of hexagonal auxetics under pressure. Phys. Status Solidi B-Basic Solid State Phys., 253(7): pp. 1261–1269, 2016.
[40] R. V. Goldstein, V. A. Gorodtsov, D. S. Lisovenko, and M. A. Volkov. Auxeticity in nano/microtubes produced from orthorhombic crystals. Smart Mater. Struct., 25(5):05006, 2016.
[41] R. V. Goldstein, V. A. Gorodtsov, and D. S. Lisovenko. Longitudinal elastic tension of two-layered plates from isotropic auxetics-nonauxetics and cubic crystals. European Journal of Mechanics A-Solids, 63:pp. 122–127, 2017.
[42] K. K. Dudek, R. Gatt, L. Mizzi, M. R. Dudek, D. Attard, K. E. Evans, and J. N. Grima. On the dynamics and control of mechanical properties of hierarchical rotating rigid unit auxetics. Sci Rep, 7: 46529, 2017.
[43] V. A. Gorodtsov, M. A. Volkov, and D. S. Lisovenko. Out-of-plane tension of thin two-layered plates of cubic crystals. Phys. Status Solidi B-Basic Solid State Phys., 258(12):2100184, 2021.
[44] S. Czarnecki and T. Łukasiak. Auxetic properties of the stiffest elastic bodies as a result of topology optimization and microstructures recovery based on homogenization method. Phys. Status Solidi B-Basic Solid State Phys., 261(12):2300495, 2024.
[45] R. H. Baughman, J. M. Shacklette, A. A. Zakhidov, and S. Stafstrom. Negative Poisson’s ratios as a common feature of cubic metals. Nature, 392: pp. 362–365, 1998.
[46] A. C. Brańka, D. M. Heyes, and K. W. Wojciechowski. Auxeticity of cubic materials under pressure. Phys. Status Solidi B-Basic Solid State Phys., 248: pp. 96–104, 2011.
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