### Vibration, Buckling and Deflection Analysis of Cracked Thin Magneto Electro Elastic Plate Under Thermal Environment

Document Type: Research Paper

Authors

1 National Institute of Technology, Raipur, Chhattisgarh, India

2 Department of Basic Sciences and Engineering, Indian Institute of Information Technology, Nagpur, India

10.22034/jsm.2019.665912

Abstract

The Magneto-Electro-Elastic (MEE) material exhibits pyroelectric and pyromagnetic effects under thermal environment. The effects of such pyroelectric and pyromagnetic behavior on vibration, buckling and deflection analysis of partially cracked thin MEE plate is presented and discussed in this paper. The aim of the study is to develop an analytical model for the vibration and geometrically linear thermal buckling analysis of cracked MEE plate based on the classical plate theory (CPT). The line spring model (LSM) is modified for the crack terms to accommodate the effect of electric and magnetic field rigidities, whereas the effect of thermal environment is accommodated in the form of thermal moment and in-plane forces. A classical relation for thermal buckling phenomenon of cracked MEE plate is also proposed. The governing equation for cracked MEE plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack and temperature rise. The results evaluated for natural frequencies as affected by crack length, plate aspect ratio and critical buckling temperature are presented for first four modes of vibration. The obtained results reveal that the fundamental frequency of the cracked plate decreases with increase in temperature and crack length. Furthermore the variation of the critical buckling temperature with plate aspect ratio and crack length is also established for different modes of vibration.

Keywords

[1] Chang T.P., 2013, On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid, Applied Mathematical Modelling 37: 2503-2515.
[2] Liu M., 2011, An exact deformation analysis for the magneto-electro-elastic fiber-reinforced thin plate, Applied Mathematical Modelling 35: 2443-2461.
[3] Liu M., Chang T., 2010, Closed form expression for the vibration problem of a transversely isotropic magneto-electro- elastic plate, Journal of Applied Mechanics 77: 1-8.
[4] Chen Z., Yu S., Meng L., Lin Y., 2006, Effective properties of layered magneto-electro-elastic composites, Composite Structures 57: 177-182.
[5] Li J.Y., 2000, Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials, International Journal of Engineering Science 38: 1993-2011.
[6] Xue C.-X., Pan E., 2013, On the longitudinal wave along a functionally graded magneto-electro-elastic rod, International Journal of Engineering Science 62: 48-55.
[7] Wu T.-L., Huang J.H., 2000, Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases, International Journal of Solids and Structures 37: 2981-3009.
[8] Pan E., 2001, Exact solution for simply supported and multilayered magneto-electro-elastic plates, Journal of Applied Mechanics 68: 608.
[9] Pan E., Heyliger P.R., 2002, Free vibrations of simply supported and multilayered magneto-electro-elastic plates, Journal of Sound and Vibration 252: 429-442.
[10] Ramirez F., Heyliger P.R., Pan E., 2006, Free vibration response of two-dimensional magneto-electro-elastic laminated plates, Journal of Sound and Vibration 292: 626-644.
[11] Ramirez F., Heyliger P.R., Pan E., 2006, Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates, Mechanics of Advanced Materials and Structures 13: 249-266.
[12] Simões Moita J.M., Mota Soares C.M., Mota Soares C.A., 2009, Analyses of magneto-electro-elastic plates using a higher order finite element model, Composite Structures 91: 421-426.
[13] Milazzo A., 2012, An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics, Composite Structures 94: 2078-2086.
[14] Milazzo A., 2014, Layer-wise and equivalent single layer models for smart multilayered plates, Composites Part B: Engineering 67: 62-75.
[15] Milazzo A., 2014, Refined equivalent single layer formulations and finite elements for smart laminates free vibrations, Composites Part B: Engineering 61: 238-253.
[16] Kattimani S.C., Ray M.C., 2015, Control of geometrically nonlinear vibrations of functionally graded magneto-electro-elastic plates, International Journal of Mechanical Sciences 99: 154-167.
[17] Li Y., Zhang J.,2014, Free vibration analysis of magnetoelectroelastic plate resting on a Pasternak foundation, Smart Materials and Structures 23: 25002.
[18] Razavi S., Shooshtari A., 2015, Nonlinear free vibration of magneto-electro-elastic rectangular plates, Composite Structures 119: 377-384.
[19] Shooshtari A., Razavi S., 2014, Nonlinear free and forced vibrations of anti-symmetric angle-ply hybrid laminated rectangular plates, Journal of Composite Materials 48: 1091-1111.
[20] Shooshtari A., Razavi S., 2015, Nonlinear vibration analysis of rectangular magneto-electro-elastic thin plates, International Journal of Engineering 28: 136-144.
[21] Phoenix S.S., Satsangi S.K., Singh B.N., 2009, Layer-wise modelling of magneto-electro-elastic plates, Journal of Sound and Vibration 324: 798-815.
[22] Guan Q., He S.R., 2006,Three-dimensional analysis of piezoelectric/piezomagnetic elastic media, Composite Structures 72: 419-428.
[23] Shooshtari A., Razavi S., 2016,Vibration analysis of a magnetoelectroelastic rectangular plate based on a higher-order shear deformation theory, Latin American Journal of Solids and Structures 13: 554-572.
[24] Rao S.S., Sunar M., 1993, Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures, AIAA Journal 31: 1280-1284.
[25] Dunn J.Y., Dunn M.L., 1998, Anisotropic coupled field inclusion and inhomogeneity problem, Philosophical Magazine A 77: 1341-1350.
[26] Li J.Y., 2003, Uniqueness and reciprocity theorems for linear thermo-electro-magneto-elasticity, Journal of Mechanics and Applied Mathematics 56: 35-43.
[27] Chen W.Q., Lee K.Y., Ding H.J., 2005, On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates, Journal of Sound and Vibration 279: 237-251.
[28] Hou P.F., Leung A.Y., Ding H.J., 2008, A point heat source on the surface of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material, International Journal of Engineering Science 46: 273-285.
[29] Hou P.F., Teng G.H., Chen H.R., 2009,Three-dimensional Green’s function for a point heat source in two-phase transversely isotropic magneto-electro-thermo-elastic material, Mechanics of Materials 41: 329-338.
[30] Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2014, Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory, Acta Mechanica Sinica/Lixue Xuebao 30: 516-525.
[31] Rice J., Levy N., 1972, The part-through surface crack in an elastic plate, Journal of Applied Mechanics 39(1): 185-194.
[32] Delale F., Erdogan F., 1981, Line-spring model for surface cracks in a reissner plate, International Journal of Engineering Science 19: 1331-1340.
[33] Zheng Z., Dai S., 1994, Stress intensity factors for an inclined surface crack under biaxial, Engineering Fracture Mechanics 47: 281-289.
[34] Khadem S.E., Rezaee M., 2000, Introduction of modified comparison functions for vibration analysis of a rectangular cracked plate, Journal of Sound and Vibration 236: 245-258.
[35] Israr A., Cartmell M.P., Manoach E., Trendafilova I., Ostachowicz W., Krawczuk M., Zak A., 2009, Analytical modelling and vibration analysis of cracked rectangular plates with different loading and boundary conditions, Journal of Applied Mechanics 76: 1-9.
[36] Ismail R., Cartmell M.P., 2012, An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation, Journal of Sound and Vibration 331: 2929-2948.
[37] Joshi P.V., Jain N.K., Ramtekkar G.D., 2014, Analytical modeling and vibration analysis of internally cracked rectangular plates, Journal of Sound and Vibration 333: 5851-5864.
[38] Joshi P. V., Jain N.K., Ramtekkar G.D., 2015, Analytical modelling for vibration analysis of partially cracked orthotropic rectangular plates, European Journal of Mechanics - A/Solids 50: 100-111.
[39] Gupta A., Jain N.K., Salhotra R., Joshi P.V., 2015, Effect of microstructure on vibration characteristics of partially cracked rectangular plates based on a modified couple stress theory, International Journal of Engineering Science 100: 269-282.
[40] Gupta A., Jain N.K., Salhotra R., Rawani A.M., Joshi P.V., 2015, Effect of fibre orientation on non-linear vibration of partially cracked thin rectangular orthotropic micro plate: An analytical approach, International Journal of Engineering Science 105: 378-397.
[41] Joshi P. V., Jain N.K., Ramtekkar G.D., 2015, Effect of thermal environment on free vibration of cracked rectangular plate: An analytical approach, Thin-Walled Structures 91: 38-49.
[42] Joshi P.V., Jain N.K., Ramtekkar G.D., Singh Virdi G., 2016, Vibration and buckling analysis of partially cracked thin orthotropic rectangular plates in thermal environment, Thin-Walled Structures 109: 143-158.
[43] Soni S., Jain N.K., Joshi P. V., 2017, Analytical modeling for nonlinear vibration analysis of partially cracked thin magneto-electro-elastic plate coupled with fluid, Nonlinear Dynamics 90: 137-170.
[44] Kondaiah P., Shankar K., Ganesan N., 2012, Studies on magneto-electro-elastic cantilever beam under thermal environment, Coupled Systems Mechanics 1: 205-217.
[45] Zhang C.L., 2013, Discussionâ€¯: Closed form expression for the vibration problem of a transversely isotropic, Journal of Applied Mechanics 80: 15501.
[46] Gao C., Noda N., 2004, Thermal-induced interfacial cracking of magnetoelectroelastic materials, International Journal of Engineering Science 42: 1347-1360.
[47] Berger H., 1954, A New Approach to the Analysis of Large Deflections of Plates, Dissertation (Ph.D.), California Institute of Technology.
[48] Szilard R., 2004, Theories and Applications of Plate Analysis, John Wiley & Sons, Inc., Hoboken, NJ, USA.
[49] Jones R.M., 2006, Buckling of Bars, Plates, and Shells, Bull Ridge Corporation.
[50] Ventsel E., Krauthammer T., Carrera E., 2002, Thin Plates and Shells: Theory, Analysis, and Applications, CRC Press.
[51] Murphy K.D., Virgin L.N., Rizzi S.A., 1997, The effect of thermal prestress on the free vibration characteristics of clamped rectangular plates: Theory and experiment, Journal of Vibration and Acoustics 119: 243.
[52] Huang C.S., Leissa A.W., Chan C.W., 2011, Vibrations of rectangular plates with internal cracks or slits International Journal of Engineering Science 53: 436-445.
[53] Bose T., Mohanty A.R., 2013, 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: 7123-7141.