Magneto-Electro-Thermo-Mechanical Response of a Multiferroic Doubly-Curved Nano-Shell

Document Type : Research Paper


Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran


Free vibration of a simply-supported magneto-electro-elastic doubly-curved nano-shell is studied based on the first-order shear deformation theory in the presence of the rotary inertia effect. To model the electric and magnetic behaviors of the nano-shell, Gauss’s laws for electrostatics and magneto statics are used. By using Navier’s method, the partial differential equations of motion are reduced to a single ordinary differential equation. Then, an analytical relation is obtained for the natural frequency of magneto-electro-elastic doubly-curved nano-shell. Some examples are presented to validate the proposed model. Moreover, the effects of the electric and magnetic potentials, temperature rise, nonlocal parameter, parameters of Pasternak foundation, and the geometry of the nano-shell on the natural frequencies of magneto-electro-elastic doubly-curved nano-shells are investigated. It is found that natural frequency of magneto-electro-elastic doubly-curved nano-shell decreases with increasing the temperature, increasing the electric potential, or decreasing the magnetic potential.                              


[1] Arash B., Wang Q., 2012, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science 51: 303-313.
[2] Pradhan S.C., Kumar A., 2010, Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Computational Materials Science 50: 239-245.
[3] Babaei H., Shahidi A.R., 2013, Free vibration analysis of quadrilateral Nano plates based on nonlocal continuum models using the Galerkin method: the effects of small scale, Meccanica 48(4): 971-982.
[4] Alibeigloo A., 2011, Free vibration analysis of nano-plate using three-dimensional theory of elasticity, Acta Mechanica 222: 149-159.
[5] Hosseini-Hashemi S., Zare M., Nazemnezhad R., 2013, An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity, Composite Structures 100: 290-299.
[6] Daneshmehr A., Rajabpoor A., Hadi A., 2015, Size dependent free vibration analysis of Nano plates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science 95: 23-35.
[7] Malekzadeh P., Shojaee M., 2013, Free vibration of Nano plates based on a nonlocal two-variable refined plate theory, Composite Structures 95: 443-452.
[8] Analooei H.R., Azhari M., Heidarpour A., 2013, Elastic buckling and vibration analyses of orthotropic Nano plates using nonlocal continuum mechanics and spline finite strip method, Applied Mathematical Modelling 37(10-11): 6703-6717.
[9] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of Nano plates using nonlocal elasticity theory, Physica E 43: 954-959.
[10] Tadi Beni Y., Mehralian F., Razavi H., 2015, Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory, Composite Structures 120: 65-78.
[11] Rouhi H., Ansari R., Darvizeh M., 2016, Size-dependent free vibration analysis of Nano shells based on the surface stress elasticity, Applied Mathematical Modelling 40: 3128-3140.
[12] Pouresmaeeli S., Ghavanloo E., Fazelzadeh S.A., 2013, Vibration analysis of viscoelastic orthotropic Nano plates resting on viscoelastic medium, Composite Structures 96: 405-410.
[13] Ghorbanpour Arani A., Khoddami Maraghi Z., Khani Arani H., 2016, Orthotropic patterns of Pasternak foundation in smart vibration analysis of magnetostrictive Nano plate, Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 230(4): 559-572.
[14] Ghorbanpour Arani A, Haghparast E, Rarani MH, Maraghi ZK, 2015, Strain gradient shell model for nonlinear vibration analysis of visco-elastically coupled Boron Nitride nano-tube reinforced composite micro-tubes conveying viscous fluid, Computational Materials Science 96: 448-458.
[15] Liu C., Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2013, Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory, Composite Structures 106: 167-174.
[16] Ke L.L., Liu C., Wang Y.S., 2015, Free vibration of nonlocal piezoelectric Nano plates under various boundary conditions, Physica E 66: 93-106.
[17] Ke L.L., Wang Y.S., Reddy J.N., 2014, Thermo-electro-mechanical vibration of size-dependent piezoelectric cylindrical nanoshells under various boundary conditions, Composite Structures 116: 626-636.
[18] Vaezi M., Moory Shirbany M., Hajnayeb A., 2016, Free vibration analysis of magneto-electro-elastic micro beams subjected to magneto-electric loads, Physica E 75: 280-286.
[19] Amiri A., Pournaki I.J., Jafarzadeh E., Shabani R., Rezazadeh G., 2016, Vibration and instability of fluid‑conveyed smart micro‑tubes based on magneto‑electro‑elasticity beam model, Micro Fluids Nano Fluids 20: 38-48.
[20] Ebrahimi F., Barati M.R., 2016, A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures, International Journal of Engineering Science 107: 183-196.
[21] Li Y.S., Ma P., Wang W., 2016, Bending, buckling, and free vibration of magnetoelectroelastic nanobeam based on nonlocal theory, Journal of Intelligent Material Systems and Structures 27(9): 1139-1149.
[22] Ansari R., Hasrati E., Gholami R., Sadeghi F., 2015, Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto-electro-thermo elastic nanobeams, Composites Part B 83: 226-241.
[23] Ansari R., Gholami R., Rouhi H., 2015, Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko Nano beams based upon the nonlocal elasticity theory, Composite Structures 126: 216-226.
[24] Ke L.L., Wang Y.S., 2014, Free vibration of size-dependent magneto-electro-elastic nanobeams based on the nonlocal theory, Physica E 63: 52-61.
[25] 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 30(4): 516-525.
[26] Wang W., Li P., Jin F., 2016, Two-dimensional linear elasticity theory of magneto-electro-elastic plates considering surface and nonlocal effects for nanoscale device applications, Smart Materials and Structures 25: 095026-095041.
[27] Li Y.S., Cai Z.Y., Shi S.Y., 2014, Buckling and free vibration of magnetoelectroelastic Nano plate based on nonlocal theory, Composite Structures 111: 552-529.
[28] Pan E., Waksmanski N., 2016, Deformation of a layered magnetoelectroelastic simply-supported plate with nonlocal effect, an analytical three-dimensional solution, Smart Materials and Structures 25: 095013-095030.
[29] Ansari R., Gholami R., 2016, Nonlocal free vibration in the pre- and post-buckled states of magneto-electro-thermo elastic rectangular Nano plates with various edge conditions, Smart Materials and Structures 25: 095033-095050.
[30] Farajpour A., Hairi Yazdi M.R., Rastgoo A., Loghmani M., Mohammadi M., 2016, Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic Nano plates, Composite Structures 140: 323-336.
[31] Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2014, The size-dependent vibration of embedded magneto-electro-elastic cylindrical Nano shells, Smart Materials and Structures 23: 125036-125053.
[32] Ghadiri M., Safarpour H., 2016, Free vibration analysis of embedded magneto-electro-thermo-elastic cylindrical Nano shell based on the modified couple stress theory, Applied Physics A 122: 833-844.
[33] Mohammadimehr M., Okhravi S.V., Akhavan Alavi S.M., 2016, Free vibration analysis of magneto-electro-elastic cylindrical composite panel reinforced by various distributions of CNTs with considering open and closed circuits boundary conditions based on FSDT, Journal of Vibration and Control 24: 1551-1569.
[34] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press.
[35] Hosseini-Hashemi S., Atashipour S.R., Fadaee M., Girhammar U.A., 2012, An exact closed-form procedure for free vibration analysis of laminated spherical shell panels based on Sanders theory, Archive of Applied Mechanics 82: 985-1002.
[36] Fadaee M., Atashipour S.R., Hosseini-Hashemi S., 2013, Free vibration analysis of Lévy-type functionally graded spherical shell panel using a new exact closed-form solution, International Journal of Mechanical Sciences 77: 227-238.
[37] Khare R.K., Kant T., Garg A.K., 2004, Free vibration of composite and sandwich laminates with a higher-order facet shell element, Composite Structures 65: 405-418.
[38] Chern Y.C., Chao C.C., 2000, Comparison of natural frequencies of laminates by 3D theory-part II: curved panels, Journal of Sound and Vibration 230: 1009-1030.
[39] Pouresmaeeli S., Fazelzadeh S.A., Ghavanloo E., 2012, Exact solution for nonlocal vibration of double-orthotropic Nano plates embedded in elastic medium, Composites Part B 43: 3384-3390.