Advertisement

Vibration and deflection analysis of thin cracked and submerged orthotropic plate under thermal environment using strain gradient theory

  • Shashank Soni
  • N. K. Jain
  • P. V. JoshiEmail author
Original Paper
  • 29 Downloads

Abstract

Based on a non-classical plate theory, a nonlinear analytical model is proposed to analyze transverse vibration of thin partially cracked and submerged orthotropic plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory in conjunction with the strain gradient theory of elasticity. The effect of centrally located surface crack is deduced using appropriate crack compliance coefficients based on the simplified line spring model, whereas the effect of thermal environment is introduced using moments and in-plane forces. The influence of fluidic medium is incorporated in the governing equation in the form of fluid forces associated with its inertial effects. The equation has been solved by transforming the lateral deflection in terms of modal functions. The shift in primary resonance due to crack, length scale parameter and temperature has also been derived with central deflection. To demonstrate the accuracy of the present model, a few comparison studies are carried out with the published literature. The variation in fundamental frequency of the cracked plate is studied considering various parameters such as crack length, plate thickness, level of submergence, temperature and length scale parameter. It has been concluded that the frequency is affected by crack length, temperature and level of submergence. A comparison has also been made for the results obtained from the classical plate theory and Strain gradient theory. Furthermore, the variation in frequency response and peak amplitude of the cracked plate is studied using method of multiple scales to show the phenomenon of bending hardening or softening as affected by level of submergence, temperature, crack length and length scale parameter .

Keywords

Vibration Crack Temperature Fluid–structure interaction Virtual added mass 

Notes

Acknowledgements

This research work is not funded by any organization.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Lamb, H.: On the vibrations of an elastic plate in contact with water author. Proc. R. Soc. Lond. Ser. A 98, 205–216 (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kwak, M.K.: Hydroelastic vibration of rectangular plates. J. Appl. Mech. 63, 110 (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Kwak, M.K., Kim, K.C.: Axisymmetric vibration of circular plates in contact with fluid. J. Sound Vib. 146, 381–389 (1991)CrossRefGoogle Scholar
  4. 4.
    Amabili, M., Frosali, G., Kwak, M.K.: Free vibrations of annular plates coupled with fluids. J. Sound Vib. 191, 825–846 (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Haddara, M.R., Cao, S.: A study of the dynamic response of submerged rectangular flat plates. Mar. Struct. 9, 913–933 (1996)CrossRefGoogle Scholar
  6. 6.
    Kerboua, Y., Lakis, A.A., Thomas, M., Marcouiller, L.: Vibration analysis of rectangular plates coupled with fluid. Appl. Math. Model. 32, 2570–2586 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hosseini-Hashemi, S., Karimi, M., Rokni, H.: Natural frequencies of rectangular Mindlin plates coupled with stationary fluid. Appl. Math. Model. 36, 764–778 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu, T., Wang, K., Dong, Q.W., Liu, M.S.: Hydroelastic natural vibrations of perforated plates with cracks. Proc. Eng. 1, 129–133 (2009)CrossRefGoogle Scholar
  9. 9.
    Si, X.H., Lu, W.X., Chu, F.L.: Modal analysis of circular plates with radial side cracks and in contact with water on one side based on the Rayleigh–Ritz method. J. Sound Vib. 331, 231–251 (2012)CrossRefGoogle Scholar
  10. 10.
    Si, X., Lu, W., Chu, F.: Dynamic analysis of rectangular plates with a single side crack and in contact with water on one side based on the Rayleigh–Ritz method. J. Fluids Struct. 34, 90–104 (2012)CrossRefGoogle Scholar
  11. 11.
    Murphy, K.D., Ferreira, D.: Thermal buckling of rectangular plates. Int. J. Solids Struct. 38, 3979–3994 (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Yang, J., Shen, H.S.: Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. J. Sound Vib. 255, 579–602 (2002)CrossRefGoogle Scholar
  13. 13.
    Li, Q., Iu, V.P., Kou, K.P.: Three-dimensional vibration analysis of functionally graded material plates in thermal environment. J. Sound Vib. 324, 733–750 (2009)CrossRefGoogle Scholar
  14. 14.
    Kim, Y.-W.: Temperature dependent vibration analysis of functionally graded rectangular plates. J. Sound Vib. 284, 531–549 (2005)CrossRefGoogle Scholar
  15. 15.
    Viola, E., Tornabene, F., Fantuzzi, N.: Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape. Compos. Struct. 106, 815–834 (2013)CrossRefGoogle Scholar
  16. 16.
    Natarajan, S., Chakraborty, S., Ganapathi, M., Subramanian, M.: A parametric study on the buckling of functionally graded material plates with internal discontinuities using the partition of unity method. Eur. J. Mech. A/Solids 44, 136–147 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Papargyri-Beskou, S., Beskos, D.E.: Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch. Appl. Mech. 78, 625–635 (2007)CrossRefzbMATHGoogle Scholar
  18. 18.
    Movassagh, A.A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech./A Solids 40, 50–59 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46, 2757–2764 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Yin, L., Qian, Q., Wang, L., Xia, W.: Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech. Solida Sin. 23, 386–393 (2010)CrossRefGoogle Scholar
  21. 21.
    Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)CrossRefzbMATHGoogle Scholar
  22. 22.
    Gao, X.L., Zhang, G.Y.: A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects. Contin. Mech. Thermodyn. 28, 195–213 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gupta, A., Jain, N.K., Salhotra, R., Joshi, P.V.: Effect of microstructure on vibration characteristics of partially cracked rectangular plates based on a modified couple stress theory. Int. J. Mech. Sci. 100, 269–282 (2015)CrossRefGoogle Scholar
  24. 24.
    Gupta, A., Jain, N.K., Salhotra, R., Rawani, A.M., Joshi, P.V.: Effect of fibre orientation on non-linear vibration of partially cracked thin rectangular orthotropic micro plate: an analytical approach. Int. J. Mech. Sci. 105, 378–397 (2015)CrossRefGoogle Scholar
  25. 25.
    Rice, J., Levy, N.: The part-through surface crack in an elastic plate. J. Appl. Mech. 39, 185–194 (1972)CrossRefzbMATHGoogle Scholar
  26. 26.
    Israr, A., Cartmell, M.P., Manoach, E., Trendafilova, I., Ostachowicz, W., Krawczuk, M., Zak, A.: Analytical modelling and vibration analysis of cracked rectangular plates with different loading and boundary conditions. J. Appl. Mech. 76, 1–9 (2009)CrossRefGoogle Scholar
  27. 27.
    Ismail, R., Cartmell, M.P.: An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation. J. Sound Vib. 331, 2929–2948 (2012)CrossRefGoogle Scholar
  28. 28.
    Joshi, P.V., Jain, N.K., Ramtekkar, G.D.: Analytical modelling for vibration analysis of partially cracked orthotropic rectangular plates. Eur. J. Mech. A/Solids 50, 100–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Joshi, P.V., Jain, N.K., Ramtekkar, G.D., Virdi, G.S.: Vibration and buckling analysis of partially cracked thin orthotropic rectangular plates in thermal environment. Thin Walled Struct. 109, 143–158 (2016)CrossRefGoogle Scholar
  30. 30.
    Soni, S., Jain, N.K., Joshi, P.V.: Vibration analysis of partially cracked plate submerged in fluid. J. Sound Vib. 412, 28–57 (2018)CrossRefGoogle Scholar
  31. 31.
    Soni, S., Jain, N.K., Joshi, P.V.: Analytical modeling for nonlinear vibration analysis of partially cracked thin magneto-electro-elastic plate coupled with fluid. Nonlinear Dyn. 90, 137–170 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jones, R.M.: Buckling of Bars, Plates, and Shells. Bull Ridge Corporation, Blacksburg (2006)Google Scholar
  33. 33.
    Luo, Y., Karney, B.W.: Virtual testing for modal and damping ratio identification of submerged structures using the PolyMAX algorithm with two-way fluid–structure interactions. J. Fluids Struct. 54, 1–18 (2016)Google Scholar
  34. 34.
    Joshi, P.V., Jain, N.K., Ramtekkar, G.D.: Effect of thermal environment on free vibration of cracked rectangular plate: an analytical approach. Thin Walled Struct. 91, 38–49 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentNational Institute of TechnologyRaipurIndia
  2. 2.Department of Basic Sciences and EngineeringIndian Institute of Information TechnologyNagpurIndia

Personalised recommendations