### Buckling Analysis of Functionally Graded Shallow Spherical Shells Under External Hydrostatic Pressure

Document Type : Research Paper

Authors

Mechanical Engineering Department, Faculty of Engineering, Malayer University, Malayer, Iran

10.22034/jsm.2019.666692

Abstract

The aim of this paper is to determine the critical buckling load for simply supported thin shallow spherical shells made of functionally graded material (FGM) subjected to uniform external pressure. A metal-ceramic functionally graded (FG) shell with a power law distribution for volume fraction is considered, where its properties vary gradually through the shell thickness direction from pure metal on the inner surface to pure ceramic on the outer surface. First, the total potential energy functional is obtained using the first-order shell theory of Love and Kirchhoff, Donnell-Mushtari-Vlasov kinematic equations and Hooke''''''''''''''''s Law. Then, equilibrium equations are derived through the minimization of the total potential energy functional by employing the Euler equations. The stability equations are derived by application of the adjacent - equilibrium criterion. As the nonlinear strain-displacement relations are employed, so the presented analysis is nonlinear with high accuracy. The Galerkin method is used to determine the critical buckling load. The present problem is also analyzed numerically by simulating it in Abaqus software. For validation, the present analytical results are compared with the present numerical results and with the known data in the literature. Also, the effects of some important geometrical and mechanical parameters on the hydrostatic buckling pressure are investigated.

Keywords

[1] Darvizeh M., Darvizeh A., Shaterzadeh A. R., Ansari R., 2010, Thermal buckling of spherical shells with cut-out, Journal of Thermal Stresses 33(5): 441-458.
[2] Von Karman Th., Tsien H. S., 1939, The buckling of spherical shells by external pressure, Journal of the Aeronautical Sciences 7(2): 43-50.
[3] Zang Y.Q., Zhang D., Zhou H.Y., Ma H.Z., Wang T.K., 2000, Non-linear dynamic buckling of laminated composite shallow spherical shells, Composites Science and Technology 60(12-13): 2361-2363.
[4] Nie G.H., 2001, Asymptotic buckling analysis of imperfect shallow spherical shells on non-linear elastic foundation, International Journal of Mechanical Sciences 43(2): 543-555.
[5] Li Q.S., Liu J., Tang J., 2003, Buckling of shallow spherical shells including the effects of transverse shear deformation, International Journal of Mechanical Sciences 45(9): 1519-1529.
[6] Gupta N.K., Mohamed Sheriff N., Velmurugan R., 2008, Experimental and theoretical studies on buckling of thin spherical shells under axial loads, International Journal of Mechanical Sciences 50(3): 422-432.
[7] Sørensen K.D., Jensen H.M., 2008, Buckling-driven delamination in layered spherical shells, Journal of the Mechanics and Physics of Solids 56(1): 230-240.
[8] Hutchinson J.W., 2010, Knockdown factors for buckling of cylindrical and spherical shells subject to reduced biaxial membrane stress, International Journal of Solids and Structures 47(10): 1443-1448.
[9] Wahhaj Uddin M., 1987, Buckling of general spherical shells under external pressure, International Journal of Mechanical Sciences 29(7): 469-481.
[10] Shaterzadeh A. R., Darvizeh M., Darvizeh A., Ansari R., 2011, Thermal post-buckling of shells of revolution, Journal of Thermal Stresses 34(10): 1035-1053.
[11] Sato M., Wadee M.A., Iiboshi K., Sekizawa T., Shima H., 2012, Buckling patterns of complete spherical shells filled with an elastic medium under external pressure, International Journal of Mechanical Sciences 59(1): 22-30.
[12] Sabzikar Boroujerdy M., Eslami M.R., 2014, Axisymmetric snap-through behavior of Piezo-FGM shallow clamped spherical shells under thermo-electro-mechanical loading, International Journal of Pressure Vessels and Piping 120-121: 19-26.
[13] Zhu Y., Wang F., Liu R., 2017, Nonlinear stability of sensor elastic element—corrugated shallow spherical shell in coupled multi-field, Applied Mathematics and Mechanics 38(6): 877-888.
[14] Mirzavand B., Eslami M.R., 2007, Thermal buckling of simply supported piezoelectric FGM cylindrical shells, Journal of Thermal Stresses 30(11): 1117-1135.
[15] Wunderlich W., Albertin U., 2002, Buckling behaviour of imperfect spherical shells, International Journal of Non-Linear Mechanics 37(4-5): 589-604.
[16] Menaa M., Lakis A. A., 2014, Supersonic flutter of a spherical shell partially filled with fluid, American Journal of Computational Mathematics 4: 153-182.
[17] Hutchinson J.W., 1967, Imperfection sensitivity of externally pressurized spherical shells, Journal of Applied Mechanics 34: 49-55.
[18] Nie G.H., 2003, On the buckling of imperfect squarely-reticulated shallow spherical shells supported by elastic media, Thin-Walled Structures 41(1): 1-13.