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Analysis of natural frequency for bioinspired functional gradient plates

  • Chaohui Zhang
  • Peng Liu
  • Deju ZhuEmail author
  • Le Van Lich
  • Tinh Quoc BuiEmail author
Article
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Abstract

Biological materials-fish scales exhibit ultra-flexibility due to its functionally graded materials. Inspired by the hierarchical gradient structure of fish scales, a new flexible gradient model that can adequately describe the characteristics of the bioinspired hierarchical structures is proposed in this work. To assess the flexibility of the proposed gradient model, a combination of extended finite element method (XFEM) and stabilized discrete shear gap (DSG) is established to analyze the vibration of bioinspired gradient plates with/without cracks. The DSG technique is employed to eliminate the shear locking phenomenon, while the XFEM is used for a mesh-independent modelling of crack. The combined approach is applicable to both moderately thick and thin plates, and is insensitive to mesh distortion. Functionally gradient plates take two types: power law function (Type I) and bioinspired hierarchical mode (Type II). For Type I, the natural frequencies decrease by increasing the gradient factor, i.e., the exponent of the power law. When the gradient factor is larger than one, the improvement of the plate stiffness by material gradient is restricted. For Type II, the natural frequencies are mostly independent of the step smoothing factor, yet quite sensitive to the number of step layers, providing an additional degree of freedom in tailoring the material properties. In addition, the natural frequencies of the bioinspired gradient plate are lower than that of the homogeneous ceramic plate. By using Type II, the stiffness of the plate can be reduced effectively, making the plate prone to deformation, which coincides with the flexible scale design. Therefore, the present study provides an incisive method and instructive guideline for a new era of artificially designed flexible materials inspired by natural (or biological) materials and structures.

Keywords

Biological materials Flexible structure Bio-inspired hierarchical gradient plate XFEM DSG 

Notes

Funding

Funding was provided by National Defense Science and Technology Innovation District Program (Grant No. 19-H863-03-ZT-003-026-01), High-level Talent Gathering Project in Hunan Province (Grant No. 2018RS3057), Key Technology Research and Development Program of Shandong (Grant No. 2017GK2130), Natural Science Foundation of Hunan Province (Grant No. 2019JJ50063), China Postdocral Science Fundation (2018M632957) and Hunan Provincial Innovation Foundation for Postgraduate (Grant No. 541109080163).

References

  1. Ballarini, R., Kayacan, R., Ulm, F.J., Belytschko, T., Heuer, A.H.: Biological structures mitigate catastrophic fracture through various strategies. Int. J. Fract. 135, 187–197 (2005)CrossRefGoogle Scholar
  2. Banić, D., Bacciocchi, M., Tornabene, F., Ferreira, A.J.: Influence of Winkler–Pasternak foundation on the vibrational behavior of plates and shells reinforced by agglomerated carbon nanotubes. Appl. Sci. 7, 1228 (2017)CrossRefGoogle Scholar
  3. Barthelat, F.: Biomimetics for next generation materials. Philos. Trans. 365, 2907–2919 (2007)MathSciNetCrossRefGoogle Scholar
  4. Barthelat, F., Espinosa, H.D.: An experimental investigation of deformation and fracture of nacre-mother of pearl. Exp. Mech. 47, 311–324 (2007)CrossRefGoogle Scholar
  5. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Eng. 45, 601–620 (1999)CrossRefzbMATHGoogle Scholar
  6. Bletzinger, K.-U., Bischoff, M., Ramm, E.: A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput. Struct. 75, 321–334 (2000)CrossRefGoogle Scholar
  7. Bruet, B.J., Song, J., Boyce, M.C., Ortiz, C.: Materials design principles of ancient fish armour. Nat. Mater. 7, 748 (2008)CrossRefGoogle Scholar
  8. Chen, P.-Y., Schirer, J., Simpson, A., Nay, R., Lin, Y.-S., Yang, W., et al.: Predation versus protection: fish teeth and scales evaluated by nanoindentation. J. Mater. Res. 27, 100–112 (2012)CrossRefGoogle Scholar
  9. D’Ottavio, M.: A sublaminate generalized unified formulation for the analysis of composite structures. Compos. Struct. 142, 187–199 (2016)CrossRefGoogle Scholar
  10. Fantuzzi, N., Brischetto, S., Tornabene, F., Viola, E.: 2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels. Compos. Struct. 154, 573–590 (2016)CrossRefzbMATHGoogle Scholar
  11. Gao, H., Ji, B., Jager, I.L., Arzt, E., Fratzl, P.: Materials become insensitive to flaws at nanoscale: lessons from nature. Proc. Natl. Acad. Sci. U.S.A. 100, 5597–5600 (2003)CrossRefGoogle Scholar
  12. Liew, K., Hung, K., Lim, M.: A solution method for analysis of cracked plates under vibration. Eng. Fract. Mech. 48, 393–404 (1994)CrossRefGoogle Scholar
  13. Lin, Y.S., Wei, C.T., Olevsky, E.A., Meyers, M.A.: Mechanical properties and the laminate structure of Arapaima gigas scales. J. Mech. Behav. Biomed. Mater. 4, 1145–1156 (2011)CrossRefGoogle Scholar
  14. Liu, P., Bui, T.Q., Zhu, D., Yu, T.T., Wang, J.W., Yin, S.H., et al.: Buckling failure analysis of cracked functionally graded plates by a stabilized discrete shear gap extended 3-node triangular plate element. Compos. B Eng. 77, 179–193 (2015)CrossRefGoogle Scholar
  15. Liu, Z., Meyers, M.A., Zhang, Z., Ritchie, R.O.: Functional gradients and heterogeneities in biological materials: design principles, functions, and bioinspired applications. Prog. Mater Sci. 88, 467–498 (2017)CrossRefGoogle Scholar
  16. Marino, C.G.A., La, R.G., Zhang, D., Niu, L.N., Tay, F.R., Majd, H., et al.: On the mechanical behavior of scales from Cyprinus carpio. J. Mech. Behav. Biomed. Mater. 7, 17–29 (2012)CrossRefGoogle Scholar
  17. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46, 131–150 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Natarajan, S., Ferreira, A., Bordas, S., Carrera, E., Cinefra, M., Zenkour, A.: Analysis of functionally graded material plates using triangular elements with cell-based smoothed discrete shear gap method. Math. Probl. Eng. 2014, 1–13 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Neves, A., Ferreira, A., Carrera, E., Cinefra, M., Roque, C., Jorge, R., et al.: A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos. Struct. 94, 1814–1825 (2012)CrossRefGoogle Scholar
  20. Nguyen-Thoi, T., Rabczuk, T., Lam-Phat, T., Ho-Huu, V., Phung-Van, P.: Free vibration analysis of cracked Mindlin plate using an extended cell-based smoothed discrete shear gap method (XCS-DSG3). Theoret. Appl. Fract. Mech. 72, 150–163 (2014)CrossRefGoogle Scholar
  21. Silva, E.C.N., Walters, M.C., Paulino, G.H.: Modeling bamboo as a functionally graded material: lessons for the analysis of affordable materials. J. Mater. Sci. 41, 6991–7004 (2006)CrossRefGoogle Scholar
  22. Smith, B.L., Schäffer, T.E., Viani, M., Thompson, J.B., Frederick, N.A., Kindt, J., et al.: Molecular mechanistic origin of the toughness of natural adhesives, fibres and composites. Nature 399, 761–763 (1999)CrossRefGoogle Scholar
  23. Stahl, B., Keer, L.: Vibration and stability of cracked rectangular plates. Int. J. Solids Struct. 8, 69–91 (1972)CrossRefzbMATHGoogle Scholar
  24. Tang, Z., Kotov, N.A., Magonov, S., Ozturk, B.: Nanostructured artificial nacre. Nat. Mater. 2, 413 (2003)CrossRefGoogle Scholar
  25. Tornabene, F.: Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput. Methods Appl. Mech. Eng. 198, 2911–2935 (2009)CrossRefzbMATHGoogle Scholar
  26. Tornabene, F., Liverani, A., Caligiana, G.: FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations. Int. J. Mech. Sci. 53, 446–470 (2011)CrossRefGoogle Scholar
  27. Tornabene, F., Brischetto, S., Fantuzzi, N., Bacciocchi, M.: Boundary conditions in 2D numerical and 3D exact models for cylindrical bending analysis of functionally graded structures. Shock Vib. 2016, 1–17 (2016)CrossRefGoogle Scholar
  28. Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E., Reddy, J.N.: A numerical investigation on the natural frequencies of FGM sandwich shells with variable thickness by the local generalized differential quadrature method. Appl. Sci. 7, 131 (2017)CrossRefGoogle Scholar
  29. Wang, C.M., Lim, G.T., Reddy, J.N., Lee, K.H.: Relationships between bending solutions of Reissner and Mindlin plate theories. Eng. Struct. 23, 838–849 (2001)CrossRefGoogle Scholar
  30. Zghal, S., Frikha, A., Dammak, F.: Mechanical buckling analysis of functionally graded power-based and carbon nanotubes-reinforced composite plates and curved panels. Compos. B Eng. 150, 165–183 (2018)CrossRefGoogle Scholar
  31. Zhu, D., Ortega, C.F., Motamedi, R., Szewciw, L., Vernerey, F., Barthelat, F.: Structure and mechanical performance of a “modern” fish scale. Adv. Eng. Mater. 14, B185–B194 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Key Laboratory for Green and Advanced Civil Engineering Materials and Application Technology of Hunan Province, College of Civil EngineeringHunan UniversityChangshaPeople’s Republic of China
  2. 2.International Science Innovation Collaboration Base for Green and Advanced Civil Engineering Materials of Hunan ProvinceHunan UniversityChangshaPeople’s Republic of China
  3. 3.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaPeople’s Republic of China
  4. 4.School of Materials Science and TechnologyHanoi University of Science and TechnologyHanoiVietnam
  5. 5.Institute for Research and DevelopmentDuy Tan UniversityDa Nang CityVietnam
  6. 6.Department of Civil and Environmental EngineeringTokyo Institute of TechnologyTokyoJapan

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