Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

Abstract

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.

Appendix

$$\begin{aligned} & \begin{aligned}[b] N_{0}^{T} &= \begin{bmatrix} N_{x}^{T} & M_{x}^{T} & P_{x}^{T} \end{bmatrix} \\ & = \int _{ - h/2}^{h/2} \bigl(1,z,z^{3}\bigr) (Q_{11})\alpha\Delta Tdz \end{aligned} \end{aligned}$$

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$$\begin{aligned} & \begin{aligned}[b] N_{0}^{V} &= \begin{bmatrix} N_{x}^{V} & M_{x}^{V} & P_{x}^{V} \end{bmatrix} \\ &= \sum _{k = 1}^{Na} \int_{Z_{k}}^{Z_{k + 1}} \bigl(1,z,z^{3}\bigr)d_{11}\bigl(Q_{11}^{a} \bigr)\frac{V_{p}}{h_{p}}dz \end{aligned} \end{aligned}$$

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$$\begin{aligned} &[ N ] = \begin{bmatrix} N_{1} & 0 \\ - C_{2}z^{3}\frac{\partial N_{3}}{\partial x} & N_{3} \\ - C_{2}z^{3}\frac{\partial N_{4}}{\partial x} & N_{4} \\ (z - C_{2}z^{3})N_{1} & 0 \\ N_{2} & 0 \\ - C_{2}z^{3}\frac{\partial N_{5}}{\partial x} & N_{5} \\ - C_{2}z^{3}\frac{\partial N_{6}}{\partial x} & N_{6} \\ (z - C_{2}z^{3})N_{2} & 0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} &[ M ] = \begin{bmatrix} pe_{1} & q_{2}e_{2} & q_{2}e_{2} & q_{1}e_{1} \\ q_{2}e_{1} & I_{2}e_{3} + pe_{4} & I_{2}e_{3} + pe_{4} & I_{3}e_{2} \\ q_{2}e_{1} & I_{2}e_{3} + pe_{4} & I_{2}e_{3} + pe_{4} & I_{3}e_{2} \\ q_{1}e_{1} & I_{3}e_{2} & I_{3}e_{2} & I_{1}e_{1} \end{bmatrix} \\ &e_{1} = \int_{0}^{l} N^{T}N dx = \begin{bmatrix} l/3 & l/6 \\ l/6 & l/3 \end{bmatrix} \\ & e_{2} = \int_{0}^{l} N^{T} \bar{N}_{x} dx = \begin{bmatrix} - 1/2 & l/12 & 1/2 & - l/12 \\ - 1/2 & - l/12 & 1/2 & l/12 \end{bmatrix} \\ & e_{3} = \int_{0}^{l} \bar {N}_{x}^{T}\bar {N}_{x} dx \\ &\phantom{e_{3}} = \begin{bmatrix} 6/5l & 1/10 & - 6/5l & 1/10 \\ 1/10 & 2l/15 & - 1/10 & - l/30 \\ - 6/5l & - 1/10 & 6/5l & - 1/10 \\ 1/10 & - l/30 & - 1/10 & 2l/15 \end{bmatrix} \\ & \begin{aligned}[t] e_{4} &= \int_{0}^{l} \bar {N}^{T}\bar {N} dx \\ &= \begin{bmatrix} 13l/35 & 11l^{2}/210 & 9l/70 & - 13l^{2}/420 \\ 11l^{2}/210 & l^{3}/105 & 13l^{2}/420 & - l^{3}/140 \\ 9l/70 & 13l^{2}/420 & 13l/35 & - 11l^{2}/210 \\ - 13l^{2}/420 & - l^{3}/140 & - 11l^{2}/210 & l^{3}/105 \end{bmatrix} \end{aligned} \\ &N = \begin{bmatrix} N_{1} & N_{2} \end{bmatrix} \qquad\bar{N}_{x} = \begin{bmatrix} N_{3} & N_{4} & N_{5} & N_{6} \end{bmatrix} \\ &( p,q_{1},q_{2},I_{1},I_{2},I_{3} ) \\ &\quad = \int_{h/2}^{ - h/2} \bigl(\rho^{c} - \rho^{m}\bigr) \bigl(1,f_{1}(z),f_{2}(z),f_{1}^{2}(z), \\ &\qquad f_{2}^{2}(z),f_{1}(z)*f_{2}(z) \bigr) dz \\ &p = h \biggl( \rho^{m} + \bigl(\rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 1} \biggr] \biggr) \\ & \begin{aligned}[t] q_{1} &= h^{2}C_{1}\bigl(\rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 2} - \frac{1}{2(n + 1)} \biggr] \\ &\quad - C_{2}h^{4}\bigl(\rho^{c} - \rho^{m} \bigr) \biggl[ \frac{1}{n + 4} - \frac{3}{2(n + 3)} \\ &\quad + \frac{3}{4(n + 2)} - \frac{1}{8(n + 1)} \biggr] \end{aligned} \\ & \begin{aligned}[b] q_{2} &= - C_{4}h^{4}\bigl(\rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 4} - \frac{3}{2(n + 3)} \\ &\quad + \frac{3}{4(n + 2)} - \frac{1}{8(n + 1)} \biggr] \end{aligned} \\ & I_{1} = C_{1}^{2}h^{3} \biggl\{ \bigl( \rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 3} - \frac{1}{n + 2} + \frac{1}{4(n + 1)} \biggr] \biggr\} \\ &\qquad + \frac {C_{1}^{2}\rho^{m}h^{3}}{12} + C_{2}^{2}h^{7} \biggl\{ \bigl( \rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 7} - \frac{3}{n + 6} \\ &\qquad + \frac{15}{4(n + 5)}- \frac{5}{2(n + 4)} + \frac{15}{16(n + 3)} \\ &\qquad - \frac{3}{16(n + 2)} + \frac{1}{64(n + 1)} \biggr] \biggr\} + \frac{C_{2}^{2}h^{7}\rho^{m}}{448} \\ &\qquad - 2C_{1}C_{2}h^{5} \biggl\{ \bigl(\rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 5} - \frac{2}{(n + 4)} \\ &\qquad + \frac{3}{2(n + 3)} - \frac{1}{2(n + 2)} + \frac {1}{16(n + 1)} \biggr] \biggr\} \\ &\qquad - \frac{C_{1}C_{2}h^{5}\rho^{m}}{40} \\ & \begin{aligned}[t] I_{2} &= - C_{2}h^{7} \biggl\{ \bigl( \rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 7} - \frac{3}{n + 6} \\ &\quad + \frac{15}{4(n + 5)} - \frac{5}{2(n + 4)} + \frac{15}{16(n + 3)}\\ &\quad - \frac{3}{16(n + 2)} + \frac{1}{64(n + 1)} \biggr] \biggr\} - \frac{C^{2}_{2}h^{7}\rho^{m}}{448} \end{aligned} \\ & \begin{aligned}[t] I_{3} &= C_{2}^{2}h^{7}\bigl( \rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 7} - \frac{3}{n + 6} + \frac{15}{4(n + 5)} \\ &\quad - \frac{5}{2(n + 4)} + \frac{15}{16(n + 3)} - \frac{3}{16(n + 2)}\\ &\quad + \frac{1}{64(n + 1)} \biggr] + \frac{C_{2}^{2}\rho^{m}h^{7}}{448} \\ &\quad - C_{1}C_{2}h^{5}\bigl( \rho^{c} - \rho^{m}\bigr) \biggl[ \frac{1}{n + 5} - \frac{2}{(n + 4)} \\ &\quad + \frac{3}{2(n + 3)} - \frac{1}{2(n + 2)} + \frac {1}{16(n + 1)} \biggr] \\ &\quad - \frac{C_{1}C_{2}h^{5}\rho^{m}}{80} \end{aligned} \end{aligned}$$

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