Large deflection analysis of functionally graded circular microplates with modified couple stress effect

Abstract

This paper addresses the microstructure dependent axisymmetric large deflection bending of pressure loaded circular microplates made of functionally graded materials in two cases of boundary conditions. The modulus of elasticity is assumed to vary across the thickness direction so the power-law distribution is used to describe the constituent components, while Poisson’s ratio depends weakly on position and is assumed to be a constant. The underlying higher order continuum theory behind the proposed approach is the modified couple stress theory, which can take into account the small size effects through using one scale parameter. The governing differential equations with nonlinearity and coupling between the deflection and radial membrane force are developed following the von Kármán assumption. By employing the orthogonal collocation point technique, the governing equations are transformed into systems of algebraic equations. The numerical solutions are generated through the Newton–Raphson method. Selected results are presented for the bending deflection of FGM circular microplates and influences played by the important parameters including length scale parameter, functionally graded variation, boundary conditions and the symmetrical transverse loadings are discussed.

Appendix 1

Definition of variables $\gamma_{1} \sim \gamma_{4}$

$$ \gamma_{1} = \frac{{n + E_{r} }}{n + 1} $$

(40)

$$ \gamma_{2} = \frac{{n\left( {1 - E_{r} } \right)}}{{2\left( {n + 1} \right)\left( {n + 2} \right)}} $$

(41)

$$ \gamma_{3} = \frac{{3E_{r} \left( {n^{2} + n + 2} \right) + n^{3} + 3n^{2} + 8n}}{{12\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)}} $$

(42)

$$ \gamma_{4} = \frac{{n + E_{r} }}{{2\left( {n + 1} \right)}} $$

(43)