An efficient 3-node triangular plate element for static and dynamic analyses of microplates based on modified couple stress theory with micro-inertia

Abstract

Within the modified couple stress elasticity, a novel nonconforming 3-node triangular plate element is derived from the d’Alembert–Lagrange principle for simulating the size-dependent static and dynamic bending behaviors of thin microplates and studying the effects of micro-inertia. This is accomplished via two main steps. First, the Trefftz functions that are derived from the governing equations of the problem concerned are adopted as the basis functions for constructing the element’s displacement interpolation. Second, according to the generalized conforming theory, the SemiLoof constraints are used to enforce the C¹ compatibility requirement for guaranteeing the computation convergence. The benchmark tests are carried out and the results reveal that the new element exhibits satisfactory numerical accuracy and captures the size dependences effectively in the static, free vibration and forced vibration analyses. Moreover, the findings also show that the micro-inertia affects the dynamic response of the plate mainly through the natural frequency. In general, the influences on higher order modes are more obvious than that on lower order modes.

Appendices

Appendix A

The detailed expression of the matrix ${\mathbf{U}}$ in Eq. (21) is

$${\mathbf{U}} = \left[ {\begin{array}{*{20}c} { - z\psi_{x1}^{0} } & { - z\psi_{x2}^{0} } & { - z\psi_{x3}^{0} } & {...} & { - z\psi_{x12}^{0} } \\ { - z\psi_{y1}^{0} } & { - z\psi_{y2}^{0} } & { - z\psi_{y3}^{0} } & {...} & { - z\psi_{y12}^{0} } \\ {w_{1}^{0} } & {w_{2}^{0} } & {w_{3}^{0} } & {...} & {w_{12}^{0} } \\ \end{array} } \right]$$

(A1)

in which $w_{i}^{0} ,\;\left( {i = 1\sim 12} \right)$ are the 12 functions given in Table 1, while $\psi_{xi}^{0}$ and $\psi_{yi}^{0}$ are the deduced plate rotations in accordance with Eq. (2).

Appendix B

The matrix ${{\varvec{\Lambda}}}$ in Eq. (30) takes the following form:

$${{\varvec{\Lambda}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\Lambda}}}^{w} } \\ {{{\varvec{\Lambda}}}^{\psi } } \\ \end{array} } \right]$$

(B1)

in which ${{\varvec{\Lambda}}}^{w}$ is given by

$${{\varvec{\Lambda}}}^{w} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & 1 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & 1 & 0 & 0 \\ \frac{1}{2} & { - \frac{{x_{12} }}{8}} & { - \frac{{y_{12} }}{8}} & \frac{1}{2} & {\frac{{x_{12} }}{8}} & {\frac{{y_{12} }}{8}} & {} & {} & {} \\ {} & {} & {} & \frac{1}{2} & { - \frac{{x_{23} }}{8}} & { - \frac{{y_{23} }}{8}} & \frac{1}{2} & {\frac{{x_{23} }}{8}} & {\frac{{y_{23} }}{8}} \\ \frac{1}{2} & {\frac{{x_{31} }}{8}} & {\frac{{y_{31} }}{8}} & {} & {} & {} & \frac{1}{2} & { - \frac{{x_{31} }}{8}} & { - \frac{{y_{31} }}{8}} \\ \end{array} } \right]$$

(B2)

with

$$x_{ij} = x_{i} - x_{j} ,\;\;y_{ij} = y_{i} - y_{j} ,\;\;\left( {ij = 12,23,31} \right)$$

(B3)

meanwhile ${{\varvec{\Lambda}}}^{\psi }$ is given by

$${{\varvec{\Lambda}}}^{\psi } = \left[ {\begin{array}{*{20}c} {{{\varvec{\Lambda}}}_{{^{11} }}^{\psi } } & {{{\varvec{\Lambda}}}_{{^{12} }}^{\psi } } & {} \\ {} & {{{\varvec{\Lambda}}}_{{^{22} }}^{\psi } } & {{{\varvec{\Lambda}}}_{{^{23} }}^{\psi } } \\ {{{\varvec{\Lambda}}}_{{^{31} }}^{\psi } } & {} & {{{\varvec{\Lambda}}}_{{^{33} }}^{\psi } } \\ \end{array} } \right]$$

(B4)

with

$${{\varvec{\Lambda}}}_{{^{ii} }}^{\psi } = \left[ {\begin{array}{*{20}c} 0 & { - \frac{1 + 1/\sqrt 3 }{{2L_{ij} }}y_{ij} } & {\frac{1 + 1/\sqrt 3 }{{2L_{ij} }}x_{ij} } \\ 0 & { - \frac{1 - 1/\sqrt 3 }{{2L_{ij} }}y_{ij} } & {\frac{1 - 1/\sqrt 3 }{{2L_{ij} }}x_{ij} } \\ \end{array} } \right],\;\;\;\left( {ij = 12,23,31} \right)$$

(B5)

$${{\varvec{\Lambda}}}_{{^{ij} }}^{\psi } = \left[ {\begin{array}{*{20}c} 0 & { - \frac{1 - 1/\sqrt 3 }{{2L_{ij} }}y_{ij} } & {\frac{1 - 1/\sqrt 3 }{{2L_{ij} }}x_{ij} } \\ 0 & { - \frac{1 + 1/\sqrt 3 }{{2L_{ij} }}y_{ij} } & {\frac{1 + 1/\sqrt 3 }{{2L_{ij} }}x_{ij} } \\ \end{array} } \right],\;\;\;\left( {ij = 12,23,31} \right)$$

(B6)

Besides, the matrix ${{\varvec{\uplambda}}}$ in Eq. (30) can be calculated by

$${{\varvec{\uplambda}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\uplambda}}}^{w} } \\ {{{\varvec{\uplambda}}}^{\psi } } \\ \end{array} } \right]$$

(B7)

where ${{\varvec{\uplambda}}}^{w}$ is described as

$${{\varvec{\uplambda}}}^{w} = \left[ {\begin{array}{*{20}c} {w_{1}^{0} \left( {x_{1} ,y_{1} } \right)} & {w_{2}^{0} \left( {x_{1} ,y_{1} } \right)} & {...} & {w_{12}^{0} \left( {x_{1} ,y_{1} } \right)} \\ {w_{1}^{0} \left( {x_{2} ,y_{2} } \right)} & {w_{2}^{0} \left( {x_{2} ,y_{2} } \right)} & {...} & {w_{12}^{0} \left( {x_{2} ,y_{2} } \right)} \\ {w_{1}^{0} \left( {x_{3} ,y_{3} } \right)} & {w_{2}^{0} \left( {x_{3} ,y_{3} } \right)} & {...} & {w_{12}^{0} \left( {x_{3} ,y_{3} } \right)} \\ {w_{1}^{0} \left( {x_{4} ,y_{4} } \right)} & {w_{2}^{0} \left( {x_{4} ,y_{4} } \right)} & {...} & {w_{12}^{0} \left( {x_{4} ,y_{4} } \right)} \\ {w_{1}^{0} \left( {x_{5} ,y_{5} } \right)} & {w_{2}^{0} \left( {x_{5} ,y_{5} } \right)} & {...} & {w_{12}^{0} \left( {x_{5} ,y_{5} } \right)} \\ {w_{1}^{0} \left( {x_{6} ,y_{6} } \right)} & {w_{2}^{0} (x_{6} ,y_{6} )} & {...} & {w_{12}^{0} \left( {x_{6} ,y_{6} } \right)} \\ \end{array} } \right]$$

(B8)

and ${{\varvec{\uplambda}}}^{\psi }$ is determined by

$${{\varvec{\uplambda}}}^{\psi } = {\mathbf{T}}_{n} {{\varvec{\Phi}}}$$

(B9)

with

$${{\varvec{\Phi}}} = \left[ {\begin{array}{*{20}c} {\psi_{x1}^{0} \left( {x_{{A{1}}} ,y_{{A{1}}} } \right)} & {\psi_{x2}^{0} \left( {x_{{A{1}}} ,y_{{A{1}}} } \right)} & {...} & {\psi_{x12}^{0} \left( {x_{{A{1}}} ,y_{{A{1}}} } \right)} \\ {\psi_{y1}^{0} \left( {x_{{A{1}}} ,y_{{A{1}}} } \right)} & {\psi_{y2}^{0} \left( {x_{{A{1}}} ,y_{{A{1}}} } \right)} & {...} & {\psi_{y12}^{0} \left( {x_{{A{1}}} ,y_{{A{1}}} } \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {\psi_{x1}^{0} \left( {x_{{B{3}}} ,y_{{B{3}}} } \right)} & {\psi_{x2}^{0} \left( {x_{{B{3}}} ,y_{{B{3}}} } \right)} & {...} & {\psi_{x12}^{0} \left( {x_{{B{3}}} ,y_{{B{3}}} } \right)} \\ {\psi_{y1}^{0} \left( {x_{{B{3}}} ,y_{{B{3}}} } \right)} & {\psi_{y2}^{0} \left( {x_{{B{3}}} ,y_{{B{3}}} } \right)} & {...} & {\psi_{y12}^{0} \left( {x_{{B{3}}} ,y_{{B{3}}} } \right)} \\ \end{array} } \right]$$

(B10)

$${\mathbf{T}}_{n} = \left[ {\begin{array}{*{20}c} {{\mathbf{T}}_{{1}} } & {} & {} \\ {} & {{\mathbf{T}}_{{2}} } & {} \\ {} & {} & {{\mathbf{T}}_{{3}} } \\ \end{array} } \right],\;\;\; {\mathbf{T}}_{i} = \left[ {\begin{array}{*{20}c} { - \frac{{y_{ij} }}{{L_{ij} }}} & {\frac{{x_{ij} }}{{L_{ij} }}} & {} & {} \\ {} & {} & { - \frac{{y_{ij} }}{{L_{ij} }}} & {\frac{{x_{ij} }}{{L_{ij} }}} \\ \end{array} } \right],\;\;\;\left( {ij = 12,\;23,\;31} \right)$$

(B11)

Appendix C

The matrices ${\mathbf{N}}_{\theta }$, ${\mathbf{B}}_{\varepsilon }$ and ${\mathbf{B}}_{\chi }$ in Eq. (33) take the following forms, respectively:

$${\mathbf{N}}_{\theta } = {\mathbf{{\rm M}\lambda }}^{ - 1} {{\varvec{\Lambda}}},\;\;\;{\mathbf{B}}_{\varepsilon } = {\mathbf{E\lambda }}^{ - 1} {{\varvec{\Lambda}}},\;\;\;{\mathbf{B}}_{\chi } = {\mathbf{{\rm X}\lambda }}^{ - 1} {{\varvec{\Lambda}}}$$

(C1)

where

$${\mathbf{\rm M}} = \left[ {\begin{array}{*{20}c} {\psi_{y1}^{0} } & {\psi_{y2}^{0} } & {...} & {\psi_{y11}^{0} } & {\psi_{y12}^{0} } \\ { - \psi_{x1}^{0} } & { - \psi_{x2}^{0} } & {...} & { - \psi_{x11}^{0} } & { - \psi_{x12}^{0} } \\ \end{array} } \right]$$

(C2)

$${\mathbf{E}} = \left[ {\begin{array}{*{20}c} {\varepsilon_{x1}^{0} } & {\varepsilon_{x2}^{0} } & {...} & {\varepsilon_{x11}^{0} } & {\varepsilon_{x12}^{0} } \\ {\varepsilon_{y1}^{0} } & {\varepsilon_{y2}^{0} } & {...} & {\varepsilon_{y11}^{0} } & {\varepsilon_{y12}^{0} } \\ {2\varepsilon_{xy1}^{0} } & {2\varepsilon_{xy2}^{0} } & {...} & {2\varepsilon_{xy11}^{0} } & {2\varepsilon_{xy12}^{0} } \\ \end{array} } \right]$$

(C3)

$${\mathbf{\rm X}} = \left[ {\begin{array}{*{20}c} {\chi_{x1}^{0} } & {\chi_{x2}^{0} } & {...} & {\chi_{x11}^{0} } & {\chi_{x12}^{0} } \\ {\chi_{y1}^{0} } & {\chi_{y2}^{0} } & {...} & {\chi_{y11}^{0} } & {\chi_{y12}^{0} } \\ {2\chi_{xy1}^{0} } & {2\chi_{xy2}^{0} } & {...} & {2\chi_{xy11}^{0} } & {2\chi_{xy12}^{0} } \\ \end{array} } \right]$$

(C4)

in which the components of the matrices ${\mathbf{\rm M}}$, ${\mathbf{E}}$ and ${\mathbf{\rm X}}$ are calculated, respectively, by substituting the deflection $w_{i}^{0} ,\;\left( {i = 1\sim 12} \right)$ listed in Table 1 into Eqs. (3) to (5).