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A Non-Classical Analytical Approach for Vibration Analysis of Isotropic and Fgm Plate Containing a Star Shaped Crack

Published online by Cambridge University Press:  30 April 2020

Ankur Gupta Open the ORCID record for Ankur Gupta [Opens in a new window] ,
Shashank Soni  and
N. K. Jain
Ankur Gupta*
Affiliation:
Department of Mechanical Engineering, National Institute of Technology, Raipur, Chhattisgarh492010, India.
Shashank Soni
Affiliation:
Department of Mechanical Engineering, National Institute of Technology, Raipur, Chhattisgarh492010, India.
N. K. Jain
Affiliation:
Department of Mechanical Engineering, National Institute of Technology, Raipur, Chhattisgarh492010, India.
*
*Corresponding author ([email protected])
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Abstract

A non-classical analytical model for vibration analysis of thin isotropic and FGM plate containing multiple part-through cracks (star shaped) of arbitrary orientation is proposed. A plate containing four concentric cracks of arbitrary orientation in the form of continuous line is considered for analysis. The proposed governing equation is derived based on classical plate theory and modified couple stress theory. Line spring model is modified to accommodate all the crack terms. The application of Berger’s formulation introduces nonlinearities in the governing equation and then the Galerkin’s method is applied for solving final governing equation. Results for fundamental frequencies for different values of crack length, crack orientation, gradient index and material length scale parameters are presented for two different boundary conditions. Furthermore, to study the phenomenon of bending hardening/softening in a cracked plate, the frequency response curves are plotted for the parameters stated above. Based on the outcomes of this study, it can be concluded that stiffness of the plate is severely affected by the presence of multiple cracks and the stiffness goes on decreasing with increase in number of cracks thereby affecting the fundamental frequency.

Keywords

Star CrackMicro-PlateMultiple CrackFrequency
Type
Research Article
Information
Journal of Mechanics , Volume 36 , Issue 4 , August 2020 , pp. 465 - 484
Copyright
Copyright © 2020 The Society of Theoretical and Applied Mechanics

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References

REFERENCES

Rice, J., Levy, N., “The part-through surface crack in an elastic plate,”. Journal of Applied Mechanics, 39(1), pp.185–194 (1972).CrossRefGoogle Scholar
King, R.B., “Elastic-plastic analysis of surface flaws using a simplified line-spring model,” Engineering Fracture Mechanics, 18, pp.217–231 (1983). doi: 10.1016/0013-7944(83)90108-X.CrossRefGoogle Scholar
Zhao-jing zeng, dai, Shu-ho, “Stress intensity factors for an inclined surface crack under biaxial,” Engineering Fracture Mechanics, 47, pp.281–289 (1994).Google Scholar
Bose, T., Mohanty, A.R., “Vibration analysis of a rectangular thin isotropic plate with a part-through surface crack of arbitrary orientation and position,” Journal of Sound and Vibration, 332, pp.7123–7141 (2013).CrossRefGoogle Scholar
Yu, S.W., Jin, Z.H., “On the equivalent relation of the line spring: A suggested modification,” Engineering Fracture Mechanics, 26, pp.75–82 (1987). doi: 10.1016/0013-7944(87)90081-6Google Scholar
Stahl, B., Keer, L.M., “Vibration and stability of cracked rectangular plates,” International Journal of Solids and Structures,” 8, pp.69–91 (1972). doi: 10.1016/0020-7683(72)90052-2.CrossRefGoogle Scholar
Solecki, R., “Bending vibration of a simply supported rectangular plate with a crack parallel to one edge,” Engineering Fracture Mechanics, 18, pp. 1111–1118 (1983). doi: 10.1016/0013-7944(83)90004-8.CrossRefGoogle Scholar
Liew, K.M., Hung, K.C., Lim, M.K., “A solution method for analysis of cracked plates under vibration,” Engineering Fracture Mechanics, 48, pp. 393–404 (1994). doi: 10.1016/0013-7944(94)90130-9.CrossRefGoogle Scholar
Khadem, S.E., Rezaee, M., “Introduction of Modified Comparison Functions for Vibration Analysis of a Rectangular Cracked Plate,” Journal of Sound and Vibration, 236, pp. 245–258 (2000). doi: 10.1006/jsvi.2000.2986.CrossRefGoogle Scholar
Maruyama, K., Ichinomiya, O., “Experimental study of free vibration of clamped rectangular plates with strength narrow slits,” JSME International Journal, 32(2), pp.187–193 (1989). doi: 10.1299/jsmec1988.32.187Google Scholar
Huang, C.S., Leissa, A.W., Li, R.S., “Accurate vibration analysis of thick, cracked rectangular plates,” Journal of Sound and Vibration, 330, pp. 2079–2093 (2011). doi: 10.1016/j.jsv.2010.11.007.CrossRefGoogle Scholar
Wu, D., Law, S.S., “Anisotropic damage model for an inclined crack in thick plate and sensitivity study for its detection,” International Journal of Solids and Structures, 41, pp.4321–4336 (2004). doi: 10.1016/j.ijsolstr.2004.03.001.CrossRefGoogle Scholar
Jha, D.K., Kant, T., Singh, R.K., “A critical review of recent research on functionally graded plates,” Composite Structures, 96, pp.833–849 (2013). doi: 10.1016/j.compstruct.2012.09.001.CrossRefGoogle Scholar
Israr, A., Cartmell, M.P., Krawczuk, M., Ostachowicz, W.M., Manoach, E., Trendafilova, I., Shishkina, E.V., Palacz, M., “On Approximate Analytical Solutions for Vibrations in Cracked Plates,”. Applied Mechanics and Materials, 5–6, pp.315–322 (2006). doi: 10.4028/www.scientific.net/AMM.5-6.315.CrossRefGoogle Scholar
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,” Journal of Applied Mechanics, 76, pp.1–9 (2009). doi: 10.1115/1.2998755.CrossRefGoogle Scholar
Ismail, R., Cartmell, M.P., “An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation, “Journal of Sound and Vibation, 331, pp.2929–2948 (2012). doi: 10.1016/j.jsv.2012.02.011.CrossRefGoogle Scholar
Jeyaraj, P., Padmanabhan, C., Ganesan, N., “Vibration and Acoustic Response of an Isotropic Plate in a Thermal Environment,” Journal of Vibration and Acoustics, 130, pp. 051005 (1-6) (2008). doi: 10.1115/1.2948387.CrossRefGoogle Scholar
Jeyaraj, P., Ganesan, N., Padmanabhan, C., “Vibration and acoustic response of a composite plate with inherent material damping in a thermal environment,” Journal of Sound and Vibration, 320, pp.322–338 (2009). doi: 10.1016/j.jsv.2008.08.013.CrossRefGoogle Scholar
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,” European Journal of Mechanics - A/Solids, 44, pp.136–147 (2014). doi: 10.1016/j.euromechsol.2013.10.003.CrossRefGoogle Scholar
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,” International Journal of Mechanical Sciences, 100, pp.269–282 (2015). doi: 10.1016/j.ijmecsci.2015.07.004.CrossRefGoogle Scholar
Gupta, A., Jain, N.K., Salhotra, R., Joshi, P.V., “Effect of crack location on vibration analysis of partially cracked isotropic and FGM micro-plate with non-uniform thickness : An analytical approach,” International Journal of Mechanical Sciences. 145, pp.410–429 (2018). doi: 10.1016/j.ijmecsci.2018.07.015.CrossRefGoogle Scholar
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,” International Journal of Mechanical Sciences. 105, pp.378–397 (2015). doi: 10.1016/j.ijmecsci.2015.11.020.CrossRefGoogle Scholar
Joshi, P.V., Jain, N.K., Ramtekkar, G.D., “Analytical modeling and vibration analysis of internally cracked rectangular plates,” Journal of Sound and Vibration, 333, pp.5851–5864 (2014). doi: 10.1016/j.jsv.2014.06.028.CrossRefGoogle Scholar
Joshi, P.V., Jain, N.K., Ramtekkar, G.D., “Analytical modelling for vibration analysis of partially cracked orthotropic rectangular plates,” European Journal of Mechanics - A/Solids, 50, pp.100–111 (2015). doi: http://dx.doi.org/10.1016/j.euromechsol.2014.11.007.CrossRefGoogle Scholar
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 Structures, 91, pp.38–49 (2015). doi: http://dx.doi.org/10.1016/j.tws.2015.02.004.CrossRefGoogle Scholar
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 Structures, 109, pp.143–158 (2016). doi: 10.1016/j.tws.2016.09.020.CrossRefGoogle Scholar
Soni, S., Jain, N.K., Joshi, P.V., “Vibration analysis of partially cracked plate submerged in fluid,” Jornal of Sound and Vibration, 412, pp.28–57 (2018).CrossRefGoogle Scholar
Soni, S., Jain, N.K., Joshi, P.V., “Vibration and deflection analysis of thin cracked and submerged orthotropic plate under thermal environment using strain gradient theory,” Nonlinear Dynamics, 96, pp.1575–1604 (2019). doi: 10.1007/s11071-019-04872-3.CrossRefGoogle Scholar
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 Dynamics, 90(1), pp. 137–170 (2017). doi: 10.1007/s11071-017-3652-5.CrossRefGoogle Scholar
Chakraverty, S., Pradhan, K.K., “Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions,” Aerospace Science and Technology, 36, pp.132–156 (2014). doi: 10.1016/j.ast.2014.04.005.CrossRefGoogle Scholar
Batra, R.C., Jin, J., “Natural frequencies of a functionally graded anisotropic rectangular plate,” Journal of Sound and Vibration, 282, pp.509–516 (2005). doi: 10.1016/j.jsv.2004.03.068.CrossRefGoogle Scholar
Vel, S., Batra, R.C., “Exact solution for thermoelastic deformations of functionally graded thick rectangular plates,” AIAA Journal, 40, pp.1421–1433 (2002). doi: 10.2514/3.15212.CrossRefGoogle Scholar
Ganapathi, M., Prakash, T., Sundararajan, N., “Influence of Functionally Graded Material on Buckling of Skew Plates under Mechanical Loads,” Journal of Engineering Mechanics, 132, pp.902–905 (2006). doi: 10.1061/(ASCE)0733-9399(2006) 132:8(902).CrossRefGoogle Scholar
Yin, S., Hale, J.S., Yu, T., Bui, T.Q., Bordas, S.P.A., “Isogeometric locking-free plate element: A simple first order shear deformation theory for functionally graded plates,” Composite Structures, 118, pp.121–138 (2014). doi: 10.1016/j.compstruct.2014.07.028.CrossRefGoogle Scholar
Natarajan, S., Baiz, P.M., Ganapathi, M., Kerfriden, P., Bordas, S, “Linear free flexural vibration of cracked functionally graded plates in thermal environment,” Computers and Structures, 89, pp.1535–1546 (2011). doi: 10.1016/j.compstruc.2011.04.002.CrossRefGoogle Scholar
Mindlin, R.D., Eshel, N.N., “On first strain-gradient theories in linear elasticity,” International Journal of Solids and Structures, 4, pp.109–124 (1968). doi: 10.1016/0020-7683(68)90036-X.CrossRefGoogle Scholar
Yin, L., Qian, Q., Wang, L., Xia, W., “Vibration analysis of microscale plates based on modified couple stress theory,” Acta Mechanica Solida Sinica, pp.386–393 (2010). doi: 10.1016/S0894-9166(10)60040-7.CrossRefGoogle Scholar
Jung, W., Park, W., Han, S., “Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modi fi ed couple stress theory,” International Journal of Mechanical Sciences, 87, pp.150–62 (2014). doi: 10.1016/j.ijmecsci.2014.05.025.CrossRefGoogle Scholar
Papargyri-Beskou, S., Beskos, D.E., “Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates,” Archive of Applied Mechanics, 78, pp.625–635 (2008). doi: 10.1007/s00419-007-0166-5.CrossRefGoogle Scholar
Mousavi, S.M., Paavola, J., “Analysis of plate in second strain gradient elasticity,” Archive of Applied Mechanics, 84(8), pp.1135–1143 (2014). doi: 10.1007/s00419-014-0871-9.CrossRefGoogle Scholar
Yang, S., Chen, W., “On hypotheses of composite laminated plates based on new modified couple stress theory,” Composite Structures, 133, pp.46–53 (2015). doi: 10.1016/j.compstruct.2015.07.050.CrossRefGoogle Scholar
Tsiatas, G.C., “A new Kirchhoff plate model based on a modified couple stress theory,” International Journal of Solids and Structures, 46, pp.2757–2764 (2009). doi: 10.1016/j.ijsolstr.2009.03.004.CrossRefGoogle Scholar
Gao, X.L., Zhang, G.Y., “A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects,” Continuum Mechanics and Thermodynamics, 28, pp.195–213 (2016). doi: 10.1007/s00161-015-0413-x.CrossRefGoogle Scholar
Szilard, R., “Theories and Applications of Plate Analysis. Hoboken”, NJ, USA: John Wiley & Sons, Inc. (2004). doi: 10.1002/9780470172872Google Scholar
Soni, S., Jain, N.K., Joshi, P.V., Gupta, A., “Effect of Fluid-Structure Interaction on Vibration and Deflection Analysis of Generally Orthotropic Submerged Micro-plate with Crack Under Thermal Environment: An Analytical Approach,” Journal of Vibration Engineering and Technologies, Article in Press (2019). doi.org/10.1007/s42417-019-00135-y.CrossRefGoogle Scholar