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Second Strain Gradient Finite Element Analysis of Vibratory Nanostructures Based on the Three-Dimensional Elasticity Theory

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

By introducing the \(C^{2}\) continuous hexahedral element, the free vibration finite element analysis of the nanobeam and nanoplate structures is reported based on the second strain gradient (SSG) theory and three-dimensional elasticity model. The SSG elasticity presents the powerful higher-order continuum theory which can be efficiently used to capture the size-effect on the nano-objects. The finite element discretization procedure is performed within Hamilton’s principle where the quadratic matrix version of the strain and kinetic energies are derived on the basis of the three-dimensional SSG elasticity model. In the proposed \(C^{2}\) continuous hexahedral element, the values of the displacement field and the associated higher-order derivatives are considered as the nodal values to satisfy the continuity conditions. As the case studies, the free vibration of the nanobeams and rectangular nanoplates is investigated. Different results are outlined to show the efficiency and convergence of the present model. The influences of the involved parameters on the natural frequencies of nanobeams and plates are also investigated. It is realized that with the increase in the thickness-to-lattice parameter ratio, the difference of the results related to the SSG theory and classical theory decreases.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    Jalal Torabi, Reza Ansari, Majid Bazdid-Vahdati & Mansour Darvizeh

Authors
  1. Jalal Torabi
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  2. Reza Ansari
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  3. Majid Bazdid-Vahdati
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  4. Mansour Darvizeh
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Corresponding authors

Correspondence to Jalal Torabi or Reza Ansari.

Appendices

Appendix 1

The matrices \({\mathbf{C}}, {\mathbf{D}}, {\mathbf{A}}\) and \({\tilde{\mathbf{B}}}\) introduced in Eq. (14) are defined here. The classical material stiffness matrix \({\mathbf{C}}\) is

$${\mathbf{C}} = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{1} } & 0 \\ 0 & {{\mathbf{C}}_{2} } \\ \end{array} } \right],\;{\mathbf{C}}_{1} = \left[ {\begin{array}{*{20}c} {\lambda + 2\mu } & \lambda & \lambda \\ \lambda & {\lambda + 2\mu } & \lambda \\ \lambda & \lambda & {\lambda + 2\mu } \\ \end{array} } \right],\;{\mathbf{C}}_{2} = \mu {\mathbf{I}}_{3}$$
(53)

Matrix \({\mathbf{D}}\) is presented as follows:

$${\mathbf{D}} = \left[ {\begin{array}{*{20}c} {{\mathbf{D}}_{1} } & {{\mathbf{D}}_{2} } & {{\mathbf{D}}_{3} } \\ \end{array} } \right]$$
(54)
$${\mathbf{D}}_{1} = \left[ {\begin{array}{*{20}c} {\tilde{d}_{1} } & 0 & 0 & 0 & 0 & {\tilde{d}_{3} } & 0 & {\tilde{d}_{3} } & 0 & 0 \\ {d_{1} } & 0 & 0 & 0 & 0 & {\tilde{d}_{2} } & 0 & {d_{1} } & 0 & 0 \\ {d_{1} } & 0 & 0 & 0 & 0 & {d_{1} } & 0 & {\tilde{d}_{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{d_{2} }}{6}} \\ 0 & 0 & {\frac{{d_{3} }}{2}} & 0 & {\tilde{d}_{4} } & 0 & {\frac{{d_{3} }}{2}} & 0 & 0 & 0 \\ 0 & {\frac{{d_{3} }}{2}} & 0 & {\tilde{d}_{4} } & 0 & 0 & 0 & 0 & {\frac{{d_{3} }}{2}} & 0 \\ \end{array} } \right]$$
(55)
$${\mathbf{D}}_{2} = \left[ {\begin{array}{*{20}c} 0 & {d_{1} } & 0 & {\tilde{d}_{2} } & 0 & 0 & 0 & 0 & {d_{1} } & 0 \\ 0 & {\tilde{d}_{1} } & 0 & {\tilde{d}_{3} } & 0 & 0 & 0 & 0 & {\tilde{d}_{3} } & 0 \\ 0 & {d_{1} } & 0 & {d_{1} } & 0 & 0 & 0 & 0 & {\tilde{d}_{2} } & 0 \\ 0 & 0 & {\frac{{d_{3} }}{2}} & 0 & {\frac{{d_{3} }}{2}} & 0 & {\tilde{d}_{4} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{d_{2} }}{6}} \\ {\frac{{d_{3} }}{2}} & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{d_{3} }}{2}} & 0 & 0 \\ \end{array} } \right]$$
(56)
$${\mathbf{D}}_{3} = \left[ {\begin{array}{*{20}c} 0 & 0 & {d_{1} } & 0 & {\tilde{d}_{2} } & 0 & {d_{1} } & 0 & 0 & 0 \\ 0 & 0 & {d_{1} } & 0 & {d_{1} } & 0 & {\tilde{d}_{2} } & 0 & 0 & 0 \\ 0 & 0 & {\tilde{d}_{1} } & 0 & {\tilde{d}_{3} } & 0 & {\tilde{d}_{3} } & 0 & 0 & 0 \\ 0 & {\frac{{d_{3} }}{2}} & 0 & {\frac{{d_{3} }}{2}} & 0 & 0 & 0 & 0 & {\tilde{d}_{4} } & 0 \\ {\frac{{d_{3} }}{2}} & 0 & 0 & 0 & 0 & {\frac{{d_{3} }}{2}} & 0 & {\tilde{d}_{4} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{d_{2} }}{6}} \\ \end{array} } \right]$$
(57)

where

$$\tilde{d}_{1} = d_{1} + d_{2} + d_{3} , \; \tilde{d}_{2} = \frac{{\left( {d_{1} + d_{2} } \right)}}{3}, \;\tilde{d}_{3} = \frac{{\left( {d_{1} + d_{3} } \right)}}{3}, \;\tilde{d}_{4} = \left( {\frac{{d_{2} }}{3} + \frac{1}{6}d_{3} } \right)$$
(58)

Matrix \({\mathbf{A}}\) is also presented as

$${\mathbf{A}} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1} } & 0 & 0 & 0 \\ 0 & {{\mathbf{A}}_{1} } & 0 & 0 \\ 0 & 0 & {{\mathbf{A}}_{1} } & 0 \\ 0 & 0 & 0 & {{\mathbf{A}}_{4} } \\ \end{array} } \right],$$
(59)
$${\mathbf{A}}_{1} = \left[ {\begin{array}{*{20}c} {\tilde{a}_{1} } & {\tilde{a}_{4} } & {\tilde{a}_{4} } & {\tilde{a}_{5} } & {\tilde{a}_{5} } \\ {} & {\tilde{a}_{2} } & {2a_{3} } & {\tilde{a}_{6} } & {\frac{{a_{2} }}{2}} \\ {} & {} & {\tilde{a}_{2} } & {\frac{{a_{2} }}{2}} & {\tilde{a}_{6} } \\ {} & {} & {} & {\tilde{a}_{3} } & {\frac{{a_{1} }}{2}} \\ {sym.} & {} & {} & {} & {\tilde{a}_{3} } \\ \end{array} } \right],\; {\mathbf{A}}_{4} = \left[ {\begin{array}{*{20}c} {a_{4} } & {\frac{{a_{5} }}{2}} & {\frac{{a_{5} }}{2}} \\ {} & {a_{4} } & {\frac{{a_{5} }}{2}} \\ {{\text{sym}} .} & {} & {a_{4} } \\ \end{array} } \right]$$
(60)

with

$$\begin{aligned} & \tilde{a}_{1} = \mathop \sum \limits_{i = 1}^{5} 2a_{i} , \; \tilde{a}_{2} = 2a_{3} + 2a_{4} ,\;\tilde{a}_{3} = \frac{{a_{1} }}{2} + a_{4} + \frac{{a_{5} }}{2} \\ & \tilde{a}_{4} = a_{2} + 2a_{3} , \;\tilde{a}_{5} = a_{1} + \frac{{a_{2} }}{2} , \;\tilde{a}_{6} = \frac{{a_{2} }}{2} + a_{5} \\ \end{aligned}$$
(61)

In addition, by introducing the following coefficients

$$\begin{aligned} & \tilde{b}_{1} = \mathop \sum \limits_{i = 1}^{7} 2b_{i} , \quad \tilde{b}_{2} = \frac{4}{9}b_{2} + \frac{2}{9}b_{3} + \frac{2}{9}b_{5} + \frac{2}{3}b_{6} + \frac{4}{9}b_{7} , \\ & \tilde{b}_{3} = \frac{2}{9}b_{1} + \frac{2}{9}b_{3} + \frac{4}{9}b_{7} , \quad \tilde{b}_{4} = 2b_{5} + 2b_{6} \\ & \tilde{b}_{5} = \frac{2}{9}b_{5} + \frac{2}{3}b_{6} , \quad \tilde{b}_{6} = \frac{2}{3}b_{1} + \frac{{b_{3} }}{3} + \frac{2}{3}b_{4} + \frac{2}{3}b_{5} , \\ & \tilde{b}_{7} = \frac{{b_{3} }}{3} + \frac{2}{3}b_{5} ,\quad \tilde{b}_{8} = \frac{2}{3}b_{1} + \frac{2}{3}b_{2} + \frac{2}{3}b_{3} + \frac{2}{3}b_{4} + \frac{2}{3}b_{5} , \\ & \tilde{b}_{9} = \frac{2}{9}b_{2} + \frac{2}{3}b_{6} + \frac{2}{9}b_{7} ,\quad \tilde{b}_{10} = \frac{4}{9}b_{2} + \frac{2}{9}b_{3} + \frac{2}{9}b_{4} \\ & \tilde{b}_{11} = \frac{2}{9}b_{1} + \frac{2}{9}b_{4} + \frac{2}{9}b_{5} , \quad \tilde{b}_{12} = \frac{{b_{3} }}{3} + \frac{2}{3}b_{4} + \frac{2}{3}b_{7} \\ & \tilde{b}_{13} = \frac{2}{3}b_{1} + \frac{2}{3}b_{2} + \frac{{b_{3} }}{3}, \quad \tilde{b}_{14} = \frac{2}{9}b_{4} + \frac{2}{9}b_{7} \\ & \tilde{b}_{15} = \frac{4}{18}b_{2} + \frac{1}{18}b_{3} ,\quad \tilde{b}_{16} = \frac{2}{9}b_{1} + \frac{1}{9}b_{3} , \\ & \tilde{b}_{17} = \frac{1}{9}b_{3} + \frac{2}{9}b_{4} , \quad \tilde{b}_{18} = \frac{1}{18}b_{3} + \frac{4}{18}b_{7} , \quad \tilde{b}_{19} = \frac{2}{9}b_{1} + \frac{2}{9}b_{2} \\ \end{aligned}$$
(62)

the asymmetric matrix \({\tilde{\mathbf{B}}}\) is introduced as

$$\widetilde{{\mathbf{B}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{B}}_{11} } & {{\mathbf{B}}_{12} } & {{\mathbf{B}}_{13} } \\ {{\mathbf{B}}_{21} } & {{\mathbf{B}}_{22} } & {{\mathbf{B}}_{23} } \\ {{\mathbf{B}}_{31} } & {{\mathbf{B}}_{32} } & {{\mathbf{B}}_{33} } \\ \end{array} } \right]$$
(63)
$${\mathbf{B}}_{11} = \left[ {\begin{array}{*{20}l} {\tilde{b}_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{4} } \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{4} } \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & {\tilde{b}_{2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & {\tilde{b}_{2} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\tilde{b}_{6} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{11} + \tilde{b}_{9} } \hfill & 0 \hfill & {\tilde{b}_{11} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & {\widetilde{{b_{5} }}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\tilde{b}_{6} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{11} } \hfill & 0 \hfill & {\tilde{b}_{9} + \tilde{b}_{11} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\widetilde{{b_{5} }}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{9} }}{2}} \hfill \\ \end{array} } \right]$$
(64)
$${\mathbf{B}}_{22} = \left[ {\begin{array}{*{20}l} {\tilde{b}_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{1} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{6} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{6} }}{3}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{4} } \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & {\tilde{b}_{9} + \tilde{b}_{11} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{11} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & {\tilde{b}_{5} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\tilde{b}_{7} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{2} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & {\tilde{b}_{2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{2}{3}b_{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & {\tilde{b}_{1} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & {\tilde{b}_{11} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{11} + \tilde{b}_{9} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{9} }}{3}} \hfill \\ \end{array} } \right]$$
(65)
$${\mathbf{B}}_{33} = \left[ {\begin{array}{*{20}l} {\tilde{b}_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{4} } \hfill & 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{1} } \hfill & 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{6} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\frac{2}{3}b_{5} } \hfill & 0 \hfill & {\tilde{b}_{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & {\tilde{b}_{9} + \tilde{b}_{11} } \hfill & 0 \hfill & {\tilde{b}_{11} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{2}{3}b_{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{5} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{6} } \hfill & 0 \hfill & {\tilde{b}_{11} } \hfill & 0 \hfill & {\tilde{b}_{9} + \tilde{b}_{11} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\tilde{b}_{7} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & {\tilde{b}_{2} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\tilde{b}_{7} } \hfill & 0 \hfill & {\frac{{\tilde{b}_{7} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{2} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{\tilde{b}_{9} }}{3}} \hfill \\ \end{array} } \right]$$
(66)
$${\mathbf{B}}_{12} = \left[ {\begin{array}{*{20}l} 0 \hfill & {2b_{1} } \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill \\ {2b_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{6}} \hfill \\ {\tilde{b}_{12} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{10} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill \\ 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & {\tilde{b}_{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill \\ 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{19} } \hfill & 0 \hfill \\ {\frac{2}{3}b_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & {\tilde{b}_{14} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{6}} \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]$$
(67)
$${\mathbf{B}}_{13} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & {2b_{1} } \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{6}} \hfill \\ {2b_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill \\ {\tilde{b}_{12} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & {\tilde{b}_{10} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & {\tilde{b}_{19} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{2}{3}b_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{14} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & {\tilde{b}_{3} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill \\ 0 \hfill & {\frac{1}{6}b_{3} } \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill & 0 \hfill \\ \end{array} } \right]$$
(68)
$${\mathbf{B}}_{23} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{6}} \hfill \\ 0 \hfill & 0 \hfill & {2b_{1} } \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {2b_{4} } \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{4}{3}b_{1} } \hfill & 0 \hfill & {\tilde{b}_{19} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & {\tilde{b}_{14} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill \\ 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{10} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & {\tilde{b}_{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{1}{6}b_{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]$$
(69)
$${\mathbf{B}}_{21} = \left[ {\begin{array}{*{20}l} 0 \hfill & {2b_{4} } \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill \\ {2b_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{6}} \hfill \\ {\tilde{b}_{13} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{3} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill \\ 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & {\tilde{b}_{10} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill \\ 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{14} } \hfill & 0 \hfill \\ {\frac{2}{3}b_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & {\tilde{b}_{19} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{1}{6}b_{3} } \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]$$
(70)
$${\mathbf{B}}_{31} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & {2b_{4} } \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{3}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {2b_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill \\ {\tilde{b}_{13} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & {\widetilde{{b_{3} }}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & {\tilde{b}_{14} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{2}{3}b_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{19} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & {\tilde{b}_{10} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill \\ 0 \hfill & {\frac{1}{6}b_{3} } \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill & 0 \hfill \\ \end{array} } \right]$$
(71)
$${\mathbf{B}}_{32} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{{b_{3} }}{3}} \hfill \\ 0 \hfill & 0 \hfill & {2b_{4} } \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {2b_{1} } \hfill & 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{2}{3}b_{4} } \hfill & 0 \hfill & {\tilde{b}_{14} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\frac{2}{3}b_{1} } \hfill & 0 \hfill & {\tilde{b}_{19} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill \\ 0 \hfill & {\tilde{b}_{13} } \hfill & 0 \hfill & {\tilde{b}_{16} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{3} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill \\ 0 \hfill & 0 \hfill & {\tilde{b}_{12} } \hfill & 0 \hfill & {\tilde{b}_{17} } \hfill & 0 \hfill & {\tilde{b}_{10} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{1}{6}b_{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\tilde{b}_{15} } \hfill & 0 \hfill & {\tilde{b}_{18} } \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]$$
(72)

Appendix 2

The formulation for the finite element vibration analysis of SSG Timoshenko beam theory is presented in this section. On the basis of Timoshenko beam theory, the displacement field is introduced as

$${\mathbf{U}} = \left[ {\begin{array}{*{20}c} {u_{1} \left( {x_{1} ,x_{2} ,x_{3} } \right)} \\ {u_{3} \left( {x_{1} ,x_{2} } \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {u\left( {x_{1} ,x_{2} } \right) + x_{3} \psi \left( {x_{1} ,x_{2} } \right)} \\ {w\left( {x_{1} ,x_{2} } \right)} \\ \end{array} } \right] = {\mathbf{P}}_{0} \left( {x_{3} } \right){\mathbf{q}}\left( {x_{1} ,x_{2} } \right),$$
(73)
$${\mathbf{P}}_{0} \left( {x_{3} } \right) = \left[ {\begin{array}{*{20}c} 1 & 0 & {x_{3} } \\ 0 & 1 & 0 \\ \end{array} } \right], {\mathbf{q}}\left( {x_{1} ,x_{2} } \right) = \left[ {\begin{array}{*{20}c} u \\ w \\ \psi \\ \end{array} } \right],$$
(74)

The strain vector and vectors of second- and third-order derivatives of displacement field are defined as

$${\varvec{\upvarepsilon}}_{1} = \left[ {\begin{array}{*{20}c} {\varepsilon_{11} } \\ {2\varepsilon_{13} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {u_{,1} + x_{3} \psi_{,1} } \\ {w_{,1} + \psi } \\ \end{array} } \right] = {\mathbf{P}}_{1} {\mathbf{E}}_{1} {\mathbf{q}},$$
(75)
$${\mathbf{P}}_{1} \left( {x_{3} } \right) = \left[ {\begin{array}{*{20}c} 1 & {x_{3} } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right], {\mathbf{E}}_{1} = \left[ {\begin{array}{*{20}c} {\partial_{1} } & 0 & 0 \\ 0 & 0 & {\partial_{1} } \\ 0 & {\partial_{1} } & 1 \\ \end{array} } \right],$$
(76)
$${\varvec{\upvarepsilon}}_{2} = \left[ {\begin{array}{*{20}c} {\varepsilon_{111} } \\ {\varepsilon_{113} } \\ {2\varepsilon_{311} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {u_{,11} + x_{3} \psi_{,11} } \\ {w_{,11} } \\ {2\psi_{,1} } \\ \end{array} } \right] = {\mathbf{P}}_{2} {\mathbf{E}}_{2} {\mathbf{q}},$$
(77)
$${\mathbf{P}}_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & {x_{3} } \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right], {\mathbf{E}}_{2} = \left[ {\begin{array}{*{20}c} {\partial_{,11} } & 0 & 0 \\ 0 & {\partial_{,11} } & 0 \\ 0 & 0 & {2\partial_{,1} } \\ 0 & 0 & {\partial_{,11} } \\ \end{array} } \right],$$
(78)
$${\varvec{\upvarepsilon}}_{3} = \left[ {\begin{array}{*{20}c} {\varepsilon_{1111} } \\ {3\varepsilon_{1131} } \\ {\varepsilon_{1113} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {u_{,111} + x_{3} \psi_{,111} } \\ {3\psi_{,11} } \\ {w_{,111} } \\ \end{array} } \right] = {\mathbf{P}}_{3} {\mathbf{E}}_{3} {\mathbf{q}}$$
(79)
$${\mathbf{P}}_{3} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & {x_{3} } \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right], {\mathbf{E}}_{3} = \left[ {\begin{array}{*{20}c} {\partial_{,111} } & 0 & 0 \\ 0 & 0 & {3\partial_{,11} } \\ 0 & {\partial_{,111} } & 0 \\ 0 & 0 & {\partial_{,111} } \\ \end{array} } \right]$$
(80)

The associated stress vectors are

$${\varvec{\uptau}}_{1} = \left[ {\begin{array}{*{20}c} {\tau_{11} } \\ {\tau_{13} } \\ \end{array} } \right], {\varvec{\uptau}}_{2} = \left[ {\begin{array}{*{20}c} {\tau_{111} } \\ {\tau_{113} } \\ {\tau_{311} } \\ \end{array} } \right], {\varvec{\uptau}}_{3} = \left[ {\begin{array}{*{20}c} {\tau_{1111} } \\ {\tau_{1131} } \\ {\tau_{1113} } \\ \end{array} } \right],$$
(81)

The material stiffness matrices are

$$\begin{aligned} & {\mathbf{C}} = \left[ {\begin{array}{*{20}c} E & 0 \\ 0 & {k_{s} G} \\ \end{array} } \right] \\ & {\mathbf{A}} = \left[ {\begin{array}{*{20}c} {\mathop \sum \limits_{i = 1}^{5} 2a_{i} } & 0 & 0 \\ {} & {2a_{3} + 2a_{4} } & {\frac{{a_{2} }}{2} + a_{5} } \\ {{\text{sym}} .} & {} & {\frac{{a_{1} }}{2} + a_{4} + \frac{{a_{5} }}{2}} \\ \end{array} } \right], \\ & \widetilde{{\mathbf{B}}} = \left[ {\begin{array}{*{20}c} {\widetilde{{b_{1} }}} & 0 & 0 \\ 0 & {\widetilde{{b_{2} }}} & {\widetilde{{b_{12} }}} \\ 0 & {\widetilde{{b_{12} }}} & {\widetilde{{b_{4} }}} \\ \end{array} } \right], {\mathbf{D}} = \left[ {\begin{array}{*{20}c} {\widetilde{{d_{1} }}} & {\widetilde{{d_{3} }}} & 0 \\ 0 & {\widetilde{{d_{4} }}} & {\frac{{d_{3} }}{2}} \\ \end{array} } \right] \\ \end{aligned}$$
(82)

where the total strain energy can be further presented as

$${\mathcal{U}} = \mathop \int \limits_{V} {\tilde{\mathcal{U}}}{\text{d}}V = \mathop \int \limits_{V} \frac{1}{2}\left( {{\varvec{\upvarepsilon}}_{1}^{\text{T}} {\mathbf{C\varepsilon }}_{1} + {\varvec{\upvarepsilon}}_{2}^{\text{T}} {\mathbf{A\varepsilon }}_{2} + {\varvec{\upvarepsilon}}_{3}^{\text{T}} {\mathbf{B\varepsilon }}_{3} +\, {\varvec{\upvarepsilon}}_{3}^{\text{T}} {\mathbf{D}}^{\text{T}} {\varvec{\upvarepsilon}}_{1} + {\varvec{\upvarepsilon}}_{1}^{\text{T}} {\mathbf{D\varepsilon }}_{3} } \right){\text{d}}V,$$
(83)

Also, the kinetic energy is defined as

$$T = \frac{1}{2}\mathop \int \limits_{\text{V}} {\dot{\mathbf{U}}}^{\text{T}} \rho {\dot{\mathbf{U}}}{\text{d}}V = \frac{1}{2}\mathop \int \limits_{\text{V}} {\dot{\mathbf{q}}}^{\text{T}} {\mathbf{P}}_{0}^{\text{T}} \rho {\mathbf{P}}_{0} {\dot{\mathbf{q}}}{\text{d}}V.$$
(84)

Now, considering the defined shape functions of one-dimensional element in Eq. (49) and following the same procedure explained in Sect. 4, the finite element governing equations can be obtained as follows:

(85)

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Torabi, J., Ansari, R., Bazdid-Vahdati, M. et al. Second Strain Gradient Finite Element Analysis of Vibratory Nanostructures Based on the Three-Dimensional Elasticity Theory. Iran J Sci Technol Trans Mech Eng 44, 631–645 (2020). https://doi.org/10.1007/s40997-019-00298-9

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