Free vibration analysis of sigmoid functionally graded nanobeams based on a modified couple stress theory with general shear deformation theory

Abstract

Free vibration of nano-scale beams made of sigmoid functionally graded material is studied based on the modified couple stress theory. In the context of the modified couple stress theory, a material length scale parameter is used to describe the size effect. The classical, first-order and some higher order beam theories are considered to assess the effect of shear deformation. Differential equations of motion and boundary conditions are derived using Hamilton’s principle. The equations of motion are discretized by generalized differential quadrature method. Finally, a comprehensive solution for the present problem with general shear deformation theory and with different boundary conditions is provided. Then effects of the shear deformation, boundary conditions, Poisson’s ratio, and material composition are studied.

Appendix

The component of matrices ${\bar{\mathbf{A}}}$, ${\bar{\mathbf{B}}},$ and ${\bar{\mathbf{C}}}$ can be represented as follows, when the boundary conditions do not apply. If the boundary conditions are considered, rows 1, N, N + 1, 2 N, 2 N + 1, and 3 N change according to the corresponding boundary conditions.

$${\bar{\mathbf{A}}} = \left[ {\bar{a}_{ij} } \right]_{3N \times 3N} ,\,\,\,\,\,\,\,\,{\bar{\mathbf{B}}} = \left[ {\bar{b}_{ij} } \right]_{3N \times 3N} \,,\,\,\,\,\,\,\,\,{\bar{\mathbf{C}}} = \left[ {\bar{c}_{ij} } \right]_{3N \times 3N}$$

$$\bar{a}_{ij} = \left\{ {\begin{array}{*{20}c} {A_{1} c_{ij}^{(2)} } & {i,j = 1, \ldots ,N} \\ { - A_{2} c_{ij}^{(3)} } & {i = 1, \ldots ,N;\,j = N + 1, \ldots ,2N} \\ {A_{4} c_{ij}^{(2)} } & {i = 1, \ldots ,N;\,j = 2N + 1, \ldots ,3N} \\ {A_{2} c_{ij}^{(3)} } & {i = N + 1, \ldots ,2N;\,j = 1, \ldots ,N} \\ { - A_{3} c_{ij}^{(4)} } & {i = N + 1, \ldots ,2N;\,j = N + 1, \ldots ,2N} \\ {A_{5} c_{ij}^{(3)} } & {i = N + 1, \ldots ,2N;\,j = 2N + 1, \ldots ,3N} \\ {A_{4} c_{ij}^{(2)} } & {i = 2N + 1, \ldots ,3N;\,j = 1, \ldots ,N} \\ { - A_{5} c_{ij}^{(3)} } & {i = 2N + 1, \ldots ,3N;\,j = N + 1, \ldots ,2N} \\ {A_{6} c_{ij}^{(2)} } & {i = 2N + 1, \ldots ,3N;\,j = 2N + 1, \ldots ,3N} \\ \end{array} } \right.$$

(38)

$$\bar{b}_{ij} = \left\{ {\begin{array}{*{20}c} {B_{1} } & {i,j = 1, \ldots ,N} \\ { - B_{2} c_{ij}^{(1)} } & {i = 1, \ldots ,N;\,j = N + 1, \ldots ,2N} \\ {B_{4} } & {i = 1, \ldots ,N;\,j = 2N + 1, \ldots ,3N} \\ {B_{2} c_{ij}^{(1)} } & {i = N + 1, \ldots ,2N;\,j = 1, \ldots ,N} \\ { - B_{3} c_{ij}^{(2)} } & {i = N + 1, \ldots ,2N;\,j = N + 1, \ldots ,2N} \\ {B_{5} c_{ij}^{(1)} } & {i = N + 1, \ldots ,2N;\,j = 2N + 1, \ldots ,3N} \\ {B_{4} } & {i = 2N + 1, \ldots ,3N;\,j = 1, \ldots ,N} \\ { - B_{5} c_{ij}^{(1)} } & {i = 2N + 1, \ldots ,3N;\,j = N + 1, \ldots ,2N} \\ {B_{6} } & {i = 2N + 1, \ldots ,3N;\,j = 2N + 1, \ldots ,3N} \\ \end{array} } \right.$$

(39)

$$\bar{c}_{ij} = \left\{ {\begin{array}{*{20}c} 0 & {i = 1, \ldots ,N;\,j = 1, \ldots ,3N} \\ 0 & {i = 1, \ldots ,3N;\,j = 1, \ldots ,N} \\ { - \xi^{2} C_{1} c_{ij}^{(4)} } & {i = N + 1, \ldots ,2N;\,j = N + 1, \ldots ,2N} \\ {\frac{1}{2}\xi^{2} C_{2} c_{ij}^{(3)} } & {i = N + 1, \ldots ,2N;\,j = 2N + 1, \ldots ,3N} \\ { - \frac{1}{2}\xi^{2} C_{2} c_{ij}^{(3)} } & {i = 2N + 1, \ldots ,3N;\,j = N + 1, \ldots ,2N} \\ {\frac{1}{4}\xi^{2} \left( {C_{3} c_{ij}^{(2)} - C_{4} } \right)} & {i = 2N + 1, \ldots ,3N;\,j = 2N + 1, \ldots ,3N} \\ \end{array} } \right.$$

(40)