Vibration study of micromorphic annular sector plates using a 3D finite element analysis

Abstract

Based on the micromorphic theory (MMT), the vibrational behavior of annular sector plates with different angles and subjected to various boundary conditions (BCs) is studied. To this end, the linear formulation of three-dimensional (3D) MMT is first presented and the matrix representation of this formulation is given. Then, a 3D size-dependent 8-node brick element is developed by the user element (UEL) subroutine within the commercial finite element software Abaqus to investigate the effects of microstructures on the vibration response of micromorphic structures. The effects of thickness-to-length scale parameter ratio and sector angle on the vibration behavior of the micromorphic annular sector plates are studied. Also, the results obtained by MMT are compared to those of the classical elasticity theory.

Appendices

Appendix A

$$ {\mathbf{A}}\, = \,\left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{{11}} \quad } & 0 \\ {0\quad } & {{\mathbf{A}}_{{22}} } \\ \end{array} } \right]_{{27 \times 27}}, $$

$$ {\mathbf{A}}_{{11}} \, = \,{\mathbf{I}}_{3} \otimes \left[ {\quad \begin{array}{*{20}c} {a\quad } & {{\mathbf{a}}\quad } & {\mathbf{a}} \\ \quad & {\overline{{\mathbf{A}}} _{{11}} \quad } & {\overline{{\overline{{{\mathbf{A}}_{{11}} }} }} } \\ {{\text{sym}}.} & \quad & {\overline{{\overline{{{\mathbf{A}}_{{11}} }} }} } \\ \end{array} } \right], $$

$$ a = 2\left( {a_{1} + a_{2} + a_{5} + a_{8} } \right) + a_{3} + a_{4} + a_{6} + a_{7} + a_{9} + a_{10} + a_{11}, $$

$$ {\mathbf{a}} = \left[ {\begin{array}{*{20}c} {a_{1} + a_{2} + a_{3} } & {a_{1} + a_{4} + a_{5} } & {a_{2} + a_{5} + a_{6} } \\ \end{array} } \right], $$

$$ \overline{{\mathbf{A}}} _{{11}} \, = \,\left[ {\begin{array}{*{20}c} {a_{3} + a_{7} + a_{{10}} \quad } & {a_{1} + a_{8} + a_{{11}} \,} & {a_{2} + a_{8} + a_{9} } \\ \quad & {a_{4} + a_{7} + a_{9} \,} & {a_{5} + a_{8} + a_{{10}} } \\ {{\text{sym}}.\quad } & \quad & {a_{6} + a_{7} + a_{{11}} } \\ \end{array} } \right], $$

(A.1)

$$ \overline{{\overline{{{\mathbf{A}}_{{11}} }} }} \, = \,\left[ {\begin{array}{*{20}c} {a_{3} \quad } & {a_{1} \quad } & {a_{2} } \\ \quad & {a_{4} \quad } & {a_{5} } \\ {{\text{sym}}.\quad } & \quad & {a_{6} } \\ \end{array} } \right], $$

$$ {\mathbf{A}}_{{22}} \, = \,\left[ {\begin{array}{*{20}c} {a_{7} \quad } & {a_{{10}} \quad } & {a_{8} \quad } & {a_{{11}} \quad } & {a_{8} \quad } & {a_{9} } \\ \quad & {a_{7} \quad } & {a_{9} \quad } & {a_{8} \quad } & {a_{{11}} \quad } & {a_{8} } \\ \quad & \quad & {a_{7} \quad } & {a_{{10}} \quad } & {a_{8} \quad } & {a_{{11}} } \\ \quad & \quad & \quad & {a_{7} \quad } & {a_{9} \quad } & {a_{8} } \\ \quad & \quad & \quad & \quad & {a_{7} \quad } & {a_{{10}} } \\ {{\text{sym}}.\quad } & \quad & \quad & \quad & \quad & {a_{7} } \\ \end{array} } \right], $$

$$ {\mathbf{B}} = \left[ {\begin{array}{*{20}c} {{\mathbf{B}}_{{11}} \quad } & 0 \\ {0\quad } & {{\mathbf{B}}_{{22}} } \\ \end{array} } \right] ,$$

$$ {\mathbf{B}}_{{11}} \, = \,\left[ {\begin{array}{*{20}c} {b_{1} + b_{2} + b_{3} \quad } & {b_{1} \quad } & {b_{1} } \\ \quad & {b_{1} + b_{2} + b_{3} \quad } & {b_{1} } \\ {{\text{sym}}.} & \quad & {b_{1} + b_{2} + b_{3} } \\ \end{array} } \right],\,{\mathbf{B}}_{{22}} \, = \,{\mathbf{I}}_{3} \otimes \left[ {\begin{array}{*{20}c} {b_{2} \quad } & {b_{3} } \\ {b_{3} \quad } & {b_{2} } \\ \end{array} } \right] ,$$

(A.2)

$$ {\mathbf{C}}\, = \,\left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{{11}} \quad } & 0 \\ {0\quad } & {{\mathbf{C}}_{{22}} } \\ \end{array} } \right],\,{\mathbf{C}}_{{11}} \, = \,\left[ {\begin{array}{*{20}c} {c_{1} + 2c_{2} \quad } & {c_{1} \quad } & {c_{1} } \\ \quad & {c_{1} + 2c_{2} \quad } & {c_{1} } \\ {{\text{sym}}.\quad } & \quad & {c_{1} + 2c_{2} } \\ \end{array} } \right],\,{\mathbf{C}}_{{22}} \, = \,c_{2} {\mathbf{I}}_{3}, $$

(A.3)

$$ {\mathbf{D}} = \left[ {\begin{array}{*{20}c} {{\mathbf{D}}_{{11}} }\quad & 0 \\ 0\quad & {{\mathbf{D}}_{{22}} } \\ \end{array} } \right],{\mathbf{D}}_{{11}} = \left[ {\begin{array}{*{20}c} {d_{1} + 2d_{2} }\quad & {d_{1} }\quad & {d_{1} } \\ {}\quad & {d_{1} + 2d_{2} }\quad & {d_{1} } \\ {sym.}\quad & {}\quad & {d_{1} + 2d_{2} } \\ \end{array} } \right], $$

(A.4)

$$ {\mathbf{D}}_{22} = d_{2} {\mathbf{I}}_{3} \otimes \left[ {\begin{array}{*{20}c} 1 & 1 \\ \end{array} } \right] ,$$

$$ c_{1} = \lambda , c_{2} = \mu , b_{1} = \eta - \tau , b_{2} = \kappa - \sigma , b_{3} = \chi - \sigma , d_{1} = \tau , d_{2} = \sigma.$$

(A.5)

Appendix B

$$ \begin{aligned} N_{1} & = \frac{1}{8}(1 + \xi )(1 + \eta )(1 + \zeta ) ,\\ N_{2} & = \frac{1}{8}(1 + \xi )(1 + \eta )(1 + \zeta ), \\ N_{3} & = \frac{1}{8}(1 - \xi )(1 + \eta )(1 - \zeta ), \\ N_{4} & = \frac{1}{8}(1 + \xi )(1 + \eta )(1 - \zeta ) ,\\ N_{5} & = \frac{1}{8}(1 + \xi )(1 - \eta )(1 + \zeta ), \\ N_{6} & = \frac{1}{8}(1 - \xi )(1 - \eta )(1 + \zeta ), \\ N_{7} & = \frac{1}{8}(1 - \xi )(1 - \eta )(1 - \zeta ), \\ N_{8} & = \frac{1}{8}(1 + \xi )(1 - \eta )(1 - \zeta ). \\ \end{aligned} $$

(B.1)