A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected
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A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load P. Desai • T. Kant
Received: 21 July 2011 / Accepted: 5 January 2012 / Published online: 20 January 2012 Ó Springer Science+Business Media, B.V. 2012
Abstract A simplified and accurate analytical cum numerical model is presented here to investigate the behavior of functionally graded (FG) cylinders of finite length subjected to thermal load. A diaphragm supported FG cylinder under symmetric thermal load which is considered as a two dimensional (2D) plane strain problem of thermoelasticity in (r, z) direction. The boundary conditions are satisfied exactly in axial direction (z) by taking an analytical expression in terms of Fourier series expansion. Fundamental (basic) dependent variables are chosen in the radial coordinate of the cylinder. First order simultaneous ordinary differential equations are obtained as mathematical model which are integrated through an effective numerical integration technique by first transforming the boundary value problem into a set of initial value problems. For FG cylinders, the material properties have power law dependence in the radial coordinate. Effect of non homogeneity parameters and orthotropy of the materials on the stresses and displacements of FG cylinder are studied. The numerical results obtained are also first validated with existing literature for their accuracy. Stresses and displacements in axial and radial directions in cylinders having various l/ri and ro/ri ratios parameter are presented for future reference. P. Desai (&) T. Kant Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India e-mail: [email protected]
Keywords Functionally graded materials Numerical Integration Boundary value problems Thick cylinder List of symbols r, h, z Cylindrical coordinates u, v, w Displacement components rr, rh, rz Normal stress components on planes normal to r, h, and z axis szr Shearing stress component in cylindrical coordinates e r , e h, e z Unit elongations (normal strain) components in cylindrical coordinates czr Shearing strain component in cylindrical coordinates Cij Material constants for orthotropic materials ai Coefficient of thermal expansion per degree centigrade for orthotropic materials T Temperature rise at any point in a cylinder m Poisson’s ratio ri Inner radius of the cylinder ro Outer radius of the cylinder l Length of the cylinder Tm Initial reference temperature u; w Nondimensionalized displacement components rr ; rh ; rz Nondimensionalized normal stress components srz Nondimensionalized shearing stress component in cylindrical coordinates
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r R
P. Desai, T. Kant
Nondimensionalized radius Mean radius (ro/ri)/2
1 Introduction The demand for improved structural efficiency in space structures and nuclear reactors has resulted in the development of a new class of materials, called functionally graded materials (FGMs). FGMs have become one of the major resear
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