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Free vibration analysis for V-notched Mindlin plates with free or clamped radial edges

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Abstract

The free vibration analysis for the moderately thick V-notched plate based on Mindlin’s first-order shear deformation theory is examined here. The Ritz method is employed with two sets of admissible functions assumed for the generalized displacement components to evaluate the frequency and mode shape of the V-notched plate, where the complete algebraic trigonometric polynomials and the singularity characteristic angular functions are chosen to guarantee the accuracy and convergence. The singularity characteristic angular functions of the V-notch are derived from the singularity characteristic equations, which are established by introducing the series asymptotic expansions of the generalized displacement components into the equilibrium equations and corresponding boundary conditions. It is found that the addition of the characteristic angular functions substantially enhances the convergence and accuracy of the calculated result of the frequency compared with the classical Ritz method. The influences of the notch opening angle, relative plate thickness and relative notch depth on the frequency of the V-notched Mindlin plate under the free and clamped radial boundary conditions are, respectively, investigated. It is found that the frequency increases with the increase in the opening angle and relative thickness. The frequency increases with the relative depth of the V-notch with the clamped radial edges, while the conclusion is on the contrary for the V-notch with free radial edges.

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References

  1. Abualnour, M., Houari, M.S.A., Tounsi, A., Mahmoud, S.R.: A novel quasi-3d trigonometric plate theory for free vibration analysis of advanced composite plates. Compos. Struct. 184, 688–697 (2018)

    Article  Google Scholar 

  2. Chen, W.Q., Ding, H.J.: On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mech. 153(3), 207–216 (2002)

    Article  Google Scholar 

  3. Tahouneh, V., Yas, M.H.: 3-d free vibration analysis of thick functionally graded annular sector plates on Pasternak elastic foundation via 2-d differential quadrature method. Acta Mech. 223(9), 1879–1897 (2012)

    Article  MathSciNet  Google Scholar 

  4. Kumar, R., Partap, G.: Free vibration of microstretch thermoelastic plate with one relaxation time. Theor. Appl. Fract. Mech. 48(3), 238–257 (2007)

    Article  Google Scholar 

  5. 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). Theor. Appl. Fract. Mech. 72, 150–163 (2014)

    Article  Google Scholar 

  6. Liew, K.M., Liu, F.: Differential quadrature method for vibration analysis of shear deformable annular sector plates. J. Sound Vib. 230(2), 335–356 (2000)

    Article  Google Scholar 

  7. Hosseini-Hashemi, S., Taher, H.R.D., Akhavan, H.: Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations. Compos. Struct. 92(7), 1734–1743 (2010)

    Article  Google Scholar 

  8. Harik, I.E., Molaghasemi, H.R.: Analytical solution to free vibration of sector plates. J. Eng. Mech. 115(12), 2709–2722 (1989)

    Article  Google Scholar 

  9. Kim, C.S., Dickinson, S.M.: On the free, transverse vibration of annular and circular, thin, sectorial plates subject to certain complicating effects. J. Sound Vib. 134(3), 407–421 (1989)

    Article  Google Scholar 

  10. Thang, P.T., Lee, J.: Free vibration characteristics of sigmoid-functionally graded plates reinforced by longitudinal and transversal stiffeners. Ocean Eng. 148, 53–61 (2018)

    Article  Google Scholar 

  11. Thang, P.T.: Analytical solution for thermal buckling analysis of rectangular plates with functionally graded coatings. Aerosp. Sci. Technol. 55, 465–473 (2016)

    Article  Google Scholar 

  12. Thang, P., Nguyen-Thoi, T., Lee, J.: Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium. Compos. Struct. 143, 143–150 (2016)

    Article  Google Scholar 

  13. Leissa, A.W., Mcgee, O.G., Huang, C.S.: Vibrations of circular plates having V-notches or sharp radial cracks. J. Sound Vib. 161, 227–239 (1993)

    Article  Google Scholar 

  14. Mcgee, O.G., Leissa, A.W., Huang, C.S., Kim, J.W.: Vibrations of circular plates with clamped V-notches or rigidly constrained radial cracks. J. Sound Vib. 181(2), 185–201 (1995)

    Article  Google Scholar 

  15. Mcgee, O.G., Kim, J.W., Kim, Y.S.: Influence of boundary stress singularities on the vibration of clamped and simply-supported sectorial plates with arbitrary radial edge conditions. J. Sound Vib. 329(26), 5563–5583 (2010)

    Article  Google Scholar 

  16. Williams, M.L.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension. ASME J. App. Mech. 19, 526–528 (1952)

    Article  Google Scholar 

  17. Yang, Y., Huang, Y., Pan, W., Yao, S., Cheng, C.: Singularity characteristic analysis for anti-plane accelerated propagating v-notches. Eng. Fract. Mech. 219, 106620 (2019)

    Article  Google Scholar 

  18. Theocaris, P.S., Ioakimidis, N.I.: The V-notched elastic half-plane problem. Acta Mech. 32(1), 125–140 (1979)

    Article  MathSciNet  Google Scholar 

  19. Carpinteri, A., Cornetti, P., Pugno, N., Sapora, A.: On the most dangerous V-notch. Int. J. Solids Struct. 47(7–8), 887–893 (2010)

    Article  Google Scholar 

  20. Zhang, Y., Ren, H., Areias, P., Zhuang, X., Rabczuk, T.: Quasi-static and dynamic fracture modeling by the nonlocal operator method. Eng. Anal. Bound. Elem. 133, 120–137 (2021)

    Article  MathSciNet  Google Scholar 

  21. Torabi, A.R., Campagnolo, A., Berto, F.: Experimental and theoretical investigation of brittle fracture in key-hole notches under mixed mode I/II loading. Acta Mech. 226(7), 2313–2322 (2015)

    Article  Google Scholar 

  22. Amiri, F., Millán, D., Shen, Y., Rabczuk, T., Arroyo, M.: Phase-field modeling of fracture in linear thin shells. Theor. Appl. Fract. Mech. 69, 102–109 (2014)

    Article  Google Scholar 

  23. Yuan, F.G., Yang, S.: Asymptotic crack-tip fields in an anisotropic plate subjected to bending, twisting moments and transverse shear loads. Compos. Sci. Technol. 60(12–13), 2489–2502 (2000)

    Article  Google Scholar 

  24. Rabczuk, T., Areias, P.M.A., Belytschko, T.: A simplified mesh-free method for shear bands with cohesive surfaces. Int. J. Numer. Methods Eng. 69(5), 993–1021 (2007)

    Article  Google Scholar 

  25. Rabczuk, T., Belytschko, T.: A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput. Methods Appl. Mech. 196(29), 2777–2799 (2007)

    Article  MathSciNet  Google Scholar 

  26. Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61(13), 2316–2343 (2004)

    Article  Google Scholar 

  27. Rabczuk, T., Areias, P.M.A., Belytschko, T.: A meshfree thin shell method for non-linear dynamic fracture. Int. J. Numer. Methods Eng. 72(5), 524–548 (2007)

    Article  MathSciNet  Google Scholar 

  28. Rabczuk, T., Zi, G., Bordas, S., Nguyen-Xuan, H.: A simple and robust three-dimensional cracking-particle method without enrichment. Comput. Methods Appl. Mech. 199(37), 2437–2455 (2010)

    Article  Google Scholar 

  29. Huang, C.S.: Stress singularities at angular corners in first-order shear deformation plate theory. Int. J. Mech. Sci. 45(1), 1–20 (2003)

    Article  Google Scholar 

  30. Mindlin, R.D., Deresiewicz, H.: Thickness shear and flexural vibrations of a circular disk. J. Appl. Phys. 25(10), 1329–1332 (1954)

    Article  MathSciNet  Google Scholar 

  31. Niu, Z., Ge, D., Cheng, C., Ye, J., Recho, N.: Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials. Appl. Math. Model. 33(3), 1776–1792 (2009)

    Article  MathSciNet  Google Scholar 

  32. Mcgee, O.G., Kim, J.W.: Three-dimensional vibrations of cylindrical elastic solids with V-notches and sharp radial cracks. J. Sound Vib. 329(4), 457–484 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12172114) and the China Scholarship Council (No. 202006690032).

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Authors and Affiliations

  1. Department of Engineering Mechanics, Hefei University of Technology, Hefei, 230009, China

    Yongyu Yang, Changzheng Cheng, Zhongrong Niu & Zongjun Hu

  2. Institute of Structural Mechanics, Bauhaus-University Weimar, 99423, Weimar, Germany

    Yongyu Yang

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Correspondence to Changzheng Cheng.

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Yang, Y., Cheng, C., Niu, Z. et al. Free vibration analysis for V-notched Mindlin plates with free or clamped radial edges. Acta Mech 233, 2271–2285 (2022). https://doi.org/10.1007/s00707-022-03211-9

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  • DOI: https://doi.org/10.1007/s00707-022-03211-9

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