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Statics, vibration, and buckling of sandwich plates with metamaterial cores characterized by negative thermal expansion and negative Poisson’s ratio

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Abstract

This paper proposes a three-dimensional (3D) Maltese cross metamaterial with negative Poisson’s ratio (NPR) and negative thermal expansion (NTE) adopted as the core layers in sandwich plates, and aims to explore the relations between the mechanical responses of sandwich composites and the NPR or NTE of the metamaterial. First, the NPR and NTE of the metamaterial are derived analytically based on energy conservation. The effective elastic modulus and mass density of the 3D metamaterial are obtained and validated by the finite element method (FEM). Subsequently, the general governing equation of the 3D sandwich plate under thermal environments is established based on Hamilton’s principle with the consideration of the von Kármán nonlinearity. The differential quadrature (DQ) FEM (DQFEM) is utilized to obtain the numerical solutions. It is shown that NPR and NTE can enhance the global stiffness of sandwich structures. The geometric parameters of the Maltese cross metamaterial significantly affect the responses of the thermal stress, natural frequency, and critical buckling load.

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References

  1. DONG, Y. H., ZHANG, Y. F., and LI, Y. H. An analytical formulation for postbuckling and buckling vibration of micro-scale laminated composite beams considering hygrothermal effect. Composite Structures, 170, 11–25 (2017)

    Article  Google Scholar 

  2. BOUAZZA, M., BENSEDDIQ, N., and ZENKOUR, A. M. Thermal buckling analysis of laminated composite beams using hyperbolic refined shear deformation theory. Journal of Thermal Stresses, 42(3), 332–340 (2018)

    Article  MATH  Google Scholar 

  3. GARG, A. and CHALAK, H. D. Analysis of non-skew and skew laminated composite and sandwich plates under hygro-thermo-mechanical conditions including transverse stress variations. Journal of Sandwich Structures & Materials, 23(8), 3471–3494 (2020)

    Article  Google Scholar 

  4. GARG, A., BELARBI, M. O., CHALAK, H. D., and CHAKRABARTI, A. A review of the analysis of sandwich FGM structures. Composite Structures, 258, 113427 (2021)

    Article  Google Scholar 

  5. TANG, H. and DAI, H. L. Dynamic instability zone analysis of laminated piezoelectric cylindrical shell with delamination under hygrothermal effects. Applied Mathematical Modelling, 99, 27–40 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. ZHANG, C., EYVAZIAN, A., ALKHEDHER, M., ALWETAISHI, M., and AMEER-AHAMMAD, N. Modified couple stress theory application to analyze mechanical buckling behavior of three-layer rectangular microplates with honeycomb core and piezoelectric face sheets. Composite Structures, 292, 115582 (2022)

    Article  Google Scholar 

  7. AL-MUKAHAL, F. H. H. and SOBHY, M. Wave propagation and free vibration of FG graphene platelets sandwich curved beam with auxetic core resting on viscoelastic foundation via DQM. Archives of Civil and Mechanical Engineering, 22, 12 (2021)

    Article  Google Scholar 

  8. ALLAM, M. N. M., RADWAN, A. F., and SOBHY, M. Hygrothermal deformation of spinning FG graphene sandwich cylindrical shells having an auxetic core. Engineering Structures, 251, 113433 (2022)

    Article  Google Scholar 

  9. KARIMIASL, M. and ALIBEIGLOO, A. Nonlinear free and forced vibration analysis of sandwich cylindrical panel with auxetic core and GPLRC facing sheets in hygrothermal environment. Thin-Walled Structures, 175, 109164 (2022)

    Article  Google Scholar 

  10. LI, Y. and TANG, Y. Application of Galerkin iterative technique to nonlinear bending response of three-directional functionally graded slender beams subjected to hygro-thermal loads. Composite Structures, 290, 115481 (2022)

    Article  Google Scholar 

  11. ZAITOUN, M. W., CHIKH, A., TOUNSI, A., AL-OSTA, M. A., SHARIF, A., AL-DULAIJAN, S. U., and AL-ZAHRANI, M. M. Influence of the visco-Pasternak foundation parameters on the buckling behavior of a sandwich functional graded ceramic-metal plate in a hygrothermal environment. Thin-Walled Structures, 170, 108549 (2022)

    Article  Google Scholar 

  12. WANG, Y. and ZHANG, W. On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams. Composite Structures, 296, 115880 (2022)

    Article  Google Scholar 

  13. ZHAO, B., LONG, C., PENG, X., CHEN, J., LIU, T., ZHANG, Z., and LAI, A. Size effect and geometrically nonlinear effect on thermal post-buckling of micro-beams: a new theoretical analysis. Continuum Mechanics and Thermodynamics, 34, 519–532 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. ATTIA, M. A. and MOHAMED, S. A. Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory. Engineering with Computers, 38(1), 525–554 (2022)

    Article  Google Scholar 

  15. LAKES, R. Cellular solids with tunable positive or negative thermal expansion of unbounded magnitude. Applied Physics Letters, 90, 221905 (2007)

    Article  Google Scholar 

  16. LIM, T. C. Mechanics of Metamaterials with Negative Parameters, Springer, Singapore, 351–424 (2020)

    Book  Google Scholar 

  17. RAMINHOS, J. S., BORGES, J. P., and VELHINHO, A. Development of polymeric anepectic meshes: auxetic metamaterials with negative thermal expansion. Smart Materials and Structures, 28(4), 045010 (2019)

    Article  Google Scholar 

  18. CARDOSO, J. O., BORGES, J. P., and VELHINHO, A. Structural metamaterials with negative mechanical/thermomechanical indices: a review. Progress in Natural Science: Materials International, 31(6), 801–808 (2021)

    Article  Google Scholar 

  19. CHEN, J., XU, W., WEI, Z., WEI, K., and YANG, X. Stiffness characteristics for a series of lightweight mechanical metamaterials with programmable thermal expansion. International Journal of Mechanical Sciences, 202–203, 106527 (2021)

    Article  Google Scholar 

  20. WEI, K., XIAO, X., CHEN, J., WU, Y., LI, M., and WANG, Z. Additively manufactured bimaterial metamaterial to program a wide range of thermal expansion. Materials & Design, 198, 109343 (2021)

    Article  Google Scholar 

  21. XU, W., XIAO, X., CHEN, J., HAN, Z., and WEI, K. Program multi-directional thermal expansion in a series of bending dominated mechanical metamaterials. Thin-Walled Structures, 174, 109147 (2022)

    Article  Google Scholar 

  22. LIM, T. C. Adjustable positive and negative hygrothermal expansion metamaterial inspired by the Maltese cross. Royal Society Open Science, 8, 210593 (2021)

    Article  Google Scholar 

  23. LIM, T. C. An auxetic metamaterial with tunable positive to negative hygrothermal expansion by means of counter-rotating crosses. Physica Status Solidi B, 258(8), 2100137 (2021)

    Article  Google Scholar 

  24. LIM, T. C. Auxetic and negative thermal expansion structure based on interconnected array of rings and sliding rods. Physica Status Solidi B, 254(12), 1600775 (2017)

    Article  Google Scholar 

  25. SAXENA, K. K., DAS, R., and CALIUS, E. P. Three decades of auxetics research — materials with negative Poisson’s ratio: a review. Advanced Engineering Materials, 18(11), 1847–1870 (2016)

    Article  Google Scholar 

  26. EVANS, K. E. and ALDERSON, A. Auxetic materials: functional materials and structures from lateral thinking. Advanced Materials, 12(9), 617–628 (2000)

    Article  Google Scholar 

  27. PRAWOTO, Y. Seeing auxetic materials from the mechanics point of view: a structural review on the negative Poisson’s ratio. Computational Materials Science, 58, 140–153 (2012)

    Article  Google Scholar 

  28. KARIMIASL, M. and ALIBEIGLOO, A. Vibration characteristics of composite sandwich cylindrical panel with double-V auxetic core subjected to the aerohygrothermal environment. Waves in Random and Complex Media, 2, 1–24 (2021)

    Article  Google Scholar 

  29. SOBHY, M. Analytical buckling temperature prediction of FG piezoelectric sandwich plates with lightweight core. Materials Research Express, 8(9), 095704 (2021)

    Article  Google Scholar 

  30. SOBHY, M. and ALAKEL-ABAZID, M. Mechanical and thermal buckling of FG-GPLs sandwich plates with negative Poisson’s ratio honeycomb core on an elastic substrate. European Physical Journal Plus, 137, 93 (2022)

    Article  Google Scholar 

  31. JIANG, W., ZHOU, J., LIU, J., ZHANG, M., and HUANG, W. Free vibration behaviours of composite sandwich plates with reentrant honeycomb cores. Applied Mathematical Modelling, 116, 547–568 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  32. WANG, H., YU, H., WANG, X., ZHOU, H., LEI, H., CHEN, M., and GUO, X. Load-bearing sandwiched metastructure with zero thermal-induced warping and high resonant frequency: mechanical designs, theoretical predictions, and experimental demonstrations. Mechanics of Materials, 177, 104531 (2023)

    Article  Google Scholar 

  33. TOROPOVA, M. M. and STEEVES, C. A. Adaptive bimaterial lattices to mitigate thermal expansion mismatch stresses in satellite structures. Acta Astronautica, 113, 132–141 (2015)

    Article  Google Scholar 

  34. LIU, K. J., LIU, H. T., and LI, J. Thermal expansion and bandgap properties of bi-material triangle re-entrant honeycomb with adjustable Poisson’s ratio. International Journal of Mechanical Sciences, 242, 108015 (2023)

    Article  Google Scholar 

  35. WEI, K., PENG, Y., QU, Z., ZHOU, H., PEI, Y., and FANG, D. Lightweight composite lattice cylindrical shells with novel character of tailorable thermal expansion. International Journal of Mechanical Sciences, 137, 77–85 (2018)

    Article  Google Scholar 

  36. YU, H., LIANG, B., ZHAO, Z., LIU, P., LEI, H., SONG, W., CHEN, M., and GUO, X. Metamaterials with a controllable thermal-mechanical stability: mechanical designs, theoretical predictions and experimental demonstrations. Composites Science and Technology, 207, 108694 (2021)

    Article  Google Scholar 

  37. YU, B., XU, Z., MU, R., WANG, A., and ZHAO, H. Design of large-scale space lattice structure with near-zero thermal expansion metamaterials. Aerospace, 10, 294 (2023)

    Article  Google Scholar 

  38. ALAKEL-ABAZID, M. 2D magnetic field effect on the thermal buckling of metal foam nanoplates reinforced with FG-GPLs lying on Pasternak foundation in humid environment. European Physical Journal Plus, 135, 910 (2020)

    Article  Google Scholar 

  39. KARIMIASL, M., EBRAHIMI, F., and MAHESH, V. Postbuckling analysis of piezoelectric multi-scale sandwich composite doubly curved porous shallow shells via homotopy perturbation method. Engineering with Computers, 37(1), 561–577 (2021)

    Article  Google Scholar 

  40. GUPTA, A., VERMA, S., and GHOSH, A. Static and dynamic NURBS-based isogeometric analysis of composite plates under hygrothermal environment. Composite Structures, 284, 115083 (2022)

    Article  Google Scholar 

  41. FEI, B. The Buckling Analysis Based on 3-D Elastic Solid Structure and Its Implementation with Boundary Face Method (in Chinese), M. Sc. dessertation, Hunan University, Changsha, 18–22 (2012)

    Google Scholar 

  42. XING, Y. and LIU, B. High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain. International Journal for Numerical Methods in Engineering, 80(13), 1718–1742 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. XING, Y., LIU, B. O., and LIU, G. A differential quadrature finite element method. International Journal of Applied Mechanics, 2(1), 207–227 (2010)

    Article  Google Scholar 

  44. CHANG, W. H. Calculation of the natural vibration frequency of rectangular thin plates with four edges supported. Shanxi Architechture, 38(5), 63–65 (2012)

    Google Scholar 

  45. BAO, S. Y. and CAO, J. R. Elastic buckling analysis of rectangular plates with arbitrary elastic boundary conditions. Chinese Journal of Ship Research, 15(6), 162–169 (2020)

    Google Scholar 

  46. BEDFORD, A. and LIECHTI, K. M. Mechanics of Materials, 2nd ed., Springer, Cham, Switzerland, 729–781 (2020)

    Google Scholar 

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Authors and Affiliations

  1. National Key Laboratory of Strength and Structural Integrity, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China

    Qiao Zhang & Yuxin Sun

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  1. Qiao Zhang
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  2. Yuxin Sun
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Correspondence to Yuxin Sun.

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Citation: ZHANG, Q. and SUN, Y. X. Statics, vibration, and buckling of sandwich plates with metamaterial cores characterized by negative thermal expansion and negative Poisson's ratio. Applied Mathematics and Mechanics (English Edition), 44(9), 1457–1486 (2023) https://doi.org/10.1007/s10483-023-3024-6

Project supported by the National Natural Science Foundation of China (No. 11872098)

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Zhang, Q., Sun, Y. Statics, vibration, and buckling of sandwich plates with metamaterial cores characterized by negative thermal expansion and negative Poisson’s ratio. Appl. Math. Mech.-Engl. Ed. 44, 1457–1486 (2023). https://doi.org/10.1007/s10483-023-3024-6

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  • DOI: https://doi.org/10.1007/s10483-023-3024-6

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