Stochastic nonlinear bending response of functionally graded material plate with random system properties in thermal environment

Abstract

This study deals with the stochastic nonlinear bending response of functionally graded materials (FGMs) plate with uncertain system properties subjected to transverse uniformly distributed load in thermal environments. The system properties such as material properties of each constituent’s material, volume fraction index and transverse load are taken as independent random input variables. The material properties are assumed to be temperature independent (TID) and temperature dependent (TD). The basic formulation is based on higher order shear deformation theory with von-Karman nonlinear strain kinematics using modified C ⁰ continuity. A direct iterative based nonlinear finite element method in conjunction with first-order perturbation technique developed by last two authors for the composite plate is extended for the FGM plate to compute the second order statistics (mean and standard deviation) of the nonlinear bending response of the FGM plates. Effects of TD, TID material properties, aspect ratios, volume fraction index and boundary conditions, uniform temperature and non-uniform temperature distribution on the nonlinear bending are presented in detail through parametric studies. The present outlined approach has been validated with the results available in the literature and independent Monte Carlo simulation.

Appendix

$$ \left( {A_{ij} ,B_{ij} ,D_{ij} ,E_{ij} ,F_{ij} ,H_{ij} } \right) = \int\limits_{ - h/2}^{h/2} {Q_{ij} \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)} \,dz;\quad \left( {i,j = 1,2,6} \right) $$

$$ \left( {A_{ij} ,D_{ij} ,F_{ij} } \right) = \int\limits_{ - h/2}^{h/2} {Q_{ij} \left( {1,z^{2} ,z^{4} } \right)} \,dz;\quad \left( {i,j = 4,5} \right) $$

$$ \left[ {D_{3} } \right] = \left[ {\begin{array}{*{20}c} {\left[ {A_{1} } \right]} & 0 \\ {\left[ B \right]} & 0 \\ {\left[ E \right]} & 0 \\ 0 & {\left[ {A_{2} } \right]} \\ 0 & {\left[ {C_{2} } \right]} \\ \end{array} } \right],\;\left[ {D_{4} } \right] = \left[ {D_{3} } \right]^{T} \;{\text{and}}\;\left[ {D_{5} } \right] = \left[ {\begin{array}{*{20}c} {\left[ {A_{1} } \right]} & 0 \\ 0 & {\left[ {A_{2} } \right]} \\ \end{array} } \right] $$

$$ \left[ {K_{b} } \right] = \sum\limits_{i = 1}^{n} {\int\limits_{{A^{\left( e \right)} }} {\left[ {B_{b}^{\left( e \right)} } \right]^{T} \left[ {D_{b} } \right]\left[ {B_{b}^{\left( e \right)} } \right]dA} } ; $$

$$ \left[ {K_{s} } \right] = \sum\limits_{i = 1}^{n} {\int\limits_{{A^{\left( e \right)} }} {\left[ {B_{s}^{\left( e \right)} } \right]^{T} \left[ {D_{s} } \right]\left[ {B_{s}^{\left( e \right)} } \right]dA} } $$

$$ \begin{aligned} \left[ {K_{NL} \left\{ q \right\}} \right] = & \int\limits_{A} {\left[ {B_{NL} } \right]^{T} \left[ D \right]\left[ {B_{L} } \right]dA} + \frac{1}{2}\int\limits_{A} {\left[ {B_{L} } \right]^{T} \left[ D \right]\left[ {B_{NL} } \right]dA} + \frac{1}{2}\int\limits_{A} {\left[ {B_{NL} } \right]^{T} \left[ D \right]\left[ {B_{NL} } \right]dA} \\ \left[ {K_{G} } \right] = & \int\limits_{A} {\left[ {B_{NL} } \right]^{T} \left\{ N \right\}dA} = \int\limits_{A} {\left[ G \right]^{T} \left[ {\overline{N} } \right]\left[ G \right]dA} \\ \end{aligned} $$

$$ \left[ {K_{f} } \right] = \frac{1}{2}\int\limits_{A} {\left[ {B_{f} } \right]^{T} \left[ {D_{f} } \right]\left[ {B_{f} } \right]dA} $$

$$ \left\{ q \right\} = \sum\limits_{e = 1}^{NE} {\left\{ \Uplambda \right\}^{\left( e \right)} } $$

$$ \left[ {F^{T} } \right] = \sum\limits_{i = 1}^{n} {\int\limits_{{A^{\left( e \right)} }} {\left[ {\left[ {B_{1i}^{\left( e \right)} } \right]^{T} \left[ {N^{T} } \right] + \left[ {B_{b1i}^{\left( e \right)} } \right]^{T} \left[ {M^{T} } \right] + \left[ {B_{b2i}^{\left( e \right)} } \right]^{T} \left[ {P^{T} } \right]} \right]dA} } $$