Analysis of natural frequency for bioinspired functional gradient plates

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

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)

    Article  Google 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)

    Article  Google Scholar 

  3. Barthelat, F.: Biomimetics for next generation materials. Philos. Trans. 365, 2907–2919 (2007)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  5. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Eng. 45, 601–620 (1999)

    Article  Google 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)

    Article  Google Scholar 

  7. Bruet, B.J., Song, J., Boyce, M.C., Ortiz, C.: Materials design principles of ancient fish armour. Nat. Mater. 7, 748 (2008)

    Article  Google 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)

    Article  Google Scholar 

  9. D’Ottavio, M.: A sublaminate generalized unified formulation for the analysis of composite structures. Compos. Struct. 142, 187–199 (2016)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  23. Stahl, B., Keer, L.: Vibration and stability of cracked rectangular plates. Int. J. Solids Struct. 8, 69–91 (1972)

    Article  Google Scholar 

  24. Tang, Z., Kotov, N.A., Magonov, S., Ozturk, B.: Nanostructured artificial nacre. Nat. Mater. 2, 413 (2003)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

Download references

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).

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Deju Zhu or Tinh Quoc Bui.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, C., Liu, P., Zhu, D. et al. Analysis of natural frequency for bioinspired functional gradient plates. Int J Mech Mater Des 16, 367–386 (2020). https://doi.org/10.1007/s10999-019-09466-w

Download citation

Keywords

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