Nonlocal stress-driven model for functionally graded Mindlin annular plate: bending and vibration analysis

Abstract

This study provides the axisymmetric static and free vibration analysis of moderately thick functionally graded annular nano-plates with different boundary conditions. The displacement components are assumed based on the Mindlin plate approximation. Further, to reflect the small-scale effects, stress-driven nonlocal elasticity is employed. The bending analysis of nano-plates under uniformly distributed loads and natural frequencies of the plate are obtained by implementing a finite element method. The effect of small scales on transversal displacements and the frequencies of the plate is investigated for different parameters, such as material properties, plate aspect ratios, and different boundary conditions. Finally, the nonlocal stress-driven theory based on the Mindlin and Kirchhoff assumptions is compared. It is observed that the Mindlin model demonstrates more capability in detecting the nonlocal effects when the characteristic parameter takes small values.

Appendices

Appendix A

Nonzero arrays of Eq. 39:

(A.1a)

(A.1b)

(A.1c)

(A.1d)

Appendix B

For the axisymmetric bending analysis of the plate with Mindlin’s assumptions, Eqs. 33, 34, and 35 can be reduced to the following equations, respectively.

$$\begin{aligned} \begin{aligned} W^{\left( e\right) } = \left\langle N_{d}\right\rangle ^{\left( e\right) }\left\{ W_{d}\right\} ^{\left( e\right) } \mid _{d=1,2}, \ \ \Phi ^{\left( e\right) }&= \left\langle N_{d}\right\rangle ^{\left( e\right) }\left\{ \Phi _{d}\right\} ^{\left( e\right) } \mid _{d=1,2} \end{aligned} \end{aligned}$$

(B.1)

$$\begin{aligned} \begin{aligned}&\left\{ W_{d}\right\} ^{\left( e\right) }= \left\{ {\begin{aligned}{} W_i \\ W_j \end{aligned}}\right\} _{2 \times 1}, \ \ {}&\left\{ \Phi _{d}\right\} ^{\left( e\right) }= \left\{ {\begin{aligned}{} \Phi _i \\ \Phi _j \end{aligned}}\right\} _{2 \times 1} \end{aligned} \end{aligned}$$

(B.2)

$$\begin{aligned} \begin{aligned}&N_{1}= 1- \dfrac{R}{L}, \ \ N_{2}= \dfrac{R}{L} \end{aligned} \end{aligned}$$

(B.3)