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Weak-form differential quadrature finite elements for functionally graded micro-beams with strain gradient effects

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Abstract

This paper proposes weak-form differential quadrature finite elements for strain gradient functionally graded (FG) Euler–Bernoulli and Timoshenko micro-beams. The elements developed both have six degrees of freedom per node and do not require shape functions. The effective material properties are assumed to change continuously along the thickness direction. To guarantee the inter-element continuity conditions, we construct sixth- and fourth-order differential quadrature-based geometric mapping schemes. The two mapping schemes are combined with the minimum potential energy principle to derive their respective element formulations. Several illustrative examples are presented to demonstrate the convergence and adaptability of our elements. Finally, we utilize the latter element to explore the size-dependent vibration characteristics of multiple-stepped FG micro-beams. Numerical results reveal that our elements have distinct convergence and adaptability advantages over the related standard finite elements. The step location, thickness ratio, power-law index, and material length scale parameter have notable impacts on the structural vibration frequencies and mode shapes.

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Acknowledgements

The work of this paper was financially supported by the National Natural Science Foundation of China (No.11602204), the Fundamental Research Funds for the Central Universities of China (No. 2682020ZT106), and Research Grants Council of Hong Kong (Nos. 15204719 and 15209918). We also thank the anonymous reviewer for her helpful comments and suggestions.

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Authors and Affiliations

  1. Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, People’s Republic of China

    Bo Zhang & Xu Zhang

  2. Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China

    Bo Zhang & Heng Li

  3. Faculty of Engineering, China University of Geosciences, 430074, Wuhan, Hubei, People’s Republic of China

    Liulin Kong

  4. Science and Technology On Reactor System Design Technology Laboratory, Nuclear Power Institute of China, 610213, Chengdu, People’s Republic of China

    Zhipeng Feng

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  1. Bo Zhang
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  2. Heng Li
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  3. Liulin Kong
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  4. Xu Zhang
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  5. Zhipeng Feng
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Correspondence to Heng Li.

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Appendices

Appendix A

The detailed entries of \(\varvec{k}_{e}\) and \(\varvec{m}_{e}\) for the present EB DQFE are as follows:

$$\varvec{k}_{e} {\text{ = }}\left[ {\begin{array}{*{20}l} {\kappa _{{12}} } \hfill & {\kappa _{7} } \hfill & {\kappa _{{14}} } \hfill & {\text{0}} \hfill & { - \kappa _{{18}} } \hfill & { - \kappa _{{23}} } \hfill & { - \kappa _{{12}} } \hfill & {\kappa _{7} } \hfill & { - \kappa _{{14}} } \hfill & {\text{0}} \hfill & {\kappa _{{18}} } \hfill & { - \kappa _{{23}} } \hfill \\ {\kappa _{7} } \hfill & {\kappa _{{15}} } \hfill & {\kappa _{6} } \hfill & {\kappa _{{18}} } \hfill & {\kappa _{2} } \hfill & { - \kappa _{{19}} } \hfill & { - \kappa _{7} } \hfill & { - \kappa _{{13}} } \hfill & {\kappa _{5} } \hfill & { - \kappa _{{18}} } \hfill & {\kappa _{{24}} } \hfill & { - \kappa _{{26}} } \hfill \\ {\kappa _{{14}} } \hfill & {\kappa _{6} } \hfill & {\kappa _{{17}} } \hfill & {\kappa _{{23}} } \hfill & {\kappa _{{19}} } \hfill & {\kappa _{1} } \hfill & { - \kappa _{{14}} } \hfill & { - \kappa _{5} } \hfill & {\kappa _{{16}} } \hfill & { - \kappa _{{23}} } \hfill & {\kappa _{{26}} ,} \hfill & { - \kappa _{8} } \hfill \\ 0 \hfill & {\kappa _{{18}} } \hfill & {\kappa _{{23}} } \hfill & {\kappa _{{20}} } \hfill & {\kappa _{9} } \hfill & {\kappa _{{10}} } \hfill & {\text{0}} \hfill & { - \kappa _{{18}} } \hfill & {\kappa _{{23}} } \hfill & { - \kappa _{{20}} } \hfill & {\kappa _{9} } \hfill & { - \kappa _{{10}} } \hfill \\ { - \kappa _{{18}} } \hfill & {\kappa _{2} } \hfill & {\kappa _{{19}} } \hfill & {\kappa _{9} } \hfill & {\kappa _{{21}} } \hfill & {\kappa _{4} } \hfill & {\kappa _{{18}} } \hfill & { - \kappa _{{24}} } \hfill & {\kappa _{{26}} } \hfill & { - \kappa _{9} } \hfill & {\kappa _{{11}} } \hfill & { - \kappa _{3} } \hfill \\ { - \kappa _{{23}} } \hfill & { - \kappa _{{19}} } \hfill & {\kappa _{1} } \hfill & {\kappa _{{10}} } \hfill & {\kappa _{4} } \hfill & {\kappa _{{22}} } \hfill & {\kappa _{{23}} } \hfill & { - \kappa _{{26}} } \hfill & {\kappa _{8} } \hfill & { - \kappa _{{10}} } \hfill & {\kappa _{3} } \hfill & {\kappa _{{25}} } \hfill \\ { - \kappa _{{12}} } \hfill & { - \kappa _{7} } \hfill & { - \kappa _{{14}} } \hfill & {\text{0}} \hfill & {\kappa _{{18}} } \hfill & {\kappa _{{23}} } \hfill & {\kappa _{{12}} } \hfill & { - \kappa _{7} } \hfill & {\kappa _{{14}} } \hfill & {\text{0}} \hfill & { - \kappa _{{18}} } \hfill & {\kappa _{{23}} } \hfill \\ {\kappa _{7} } \hfill & { - \kappa _{{13}} } \hfill & { - \kappa _{5} } \hfill & { - \kappa _{{18}} } \hfill & { - \kappa _{{24}} } \hfill & { - \kappa _{{26}} } \hfill & { - \kappa _{7} } \hfill & {\kappa _{{15}} } \hfill & { - \kappa _{6} } \hfill & {\kappa _{{18}} } \hfill & { - \kappa _{2} } \hfill & { - \kappa _{{19}} } \hfill \\ { - \kappa _{{14}} } \hfill & {\kappa _{5} } \hfill & {\kappa _{{16}} } \hfill & {\kappa _{{23}} } \hfill & {\kappa _{{26}} } \hfill & {\kappa _{8} } \hfill & {\kappa _{{14}} } \hfill & { - \kappa _{6} } \hfill & {\kappa _{{17}} } \hfill & { - \kappa _{{23}} } \hfill & {\kappa _{{19}} } \hfill & { - \kappa _{1} } \hfill \\ 0 \hfill & { - \kappa _{{18}} } \hfill & { - \kappa _{{23}} } \hfill & { - \kappa _{{20}} } \hfill & { - \kappa _{9} } \hfill & { - \kappa _{{10}} } \hfill & {\text{0}} \hfill & {\kappa _{{18}} } \hfill & { - \kappa _{{23}} } \hfill & {\kappa _{{20}} } \hfill & { - \kappa _{9} } \hfill & {\kappa _{{10}} } \hfill \\ {\kappa _{{18}} } \hfill & {\kappa _{{24}} } \hfill & {\kappa _{{26}} } \hfill & {\kappa _{9} } \hfill & {\kappa _{{11}} } \hfill & {\kappa _{3} } \hfill & { - \kappa _{{18}} } \hfill & { - \kappa _{2} } \hfill & {\kappa _{{19}} } \hfill & { - \kappa _{9} } \hfill & {\kappa _{{21}} } \hfill & { - \kappa _{4} } \hfill \\ { - \kappa _{{23}} } \hfill & { - \kappa _{{26}} } \hfill & { - \kappa _{8} } \hfill & { - \kappa _{{10}} } \hfill & { - \kappa _{3} } \hfill & {\kappa _{{25}} } \hfill & {\kappa _{{23}} } \hfill & { - \kappa _{{19}} } \hfill & { - \kappa _{1} } \hfill & {\kappa _{{10}} } \hfill & { - \kappa _{4} } \hfill & {\kappa _{{22}} } \hfill \\ \end{array} } \right],$$
(A.1)

where

\(\kappa _{1} = \frac{{{\mkern 1mu} \Sigma _{4} }}{{\text{2}}}\), \(\kappa _{2} = \frac{{{\mkern 1mu} \Sigma _{5} }}{{\text{2}}}\), \(\kappa _{3} = {\text{8}}\left( {\frac{{{\mkern 1mu} \Sigma _{1} }}{{35}} + \frac{{{\text{6}}\Sigma _{6} }}{{L_{e}^{2} }}} \right)\), \(\kappa _{4} = {\text{2}}\left( {\frac{{{\text{11}}{\mkern 1mu} \Sigma _{1} }}{{35}} + \frac{{{\text{36}}\Sigma _{6} }}{{L_{e}^{2} }}} \right)\), \(\kappa _{5} = \frac{{\Sigma _{3} L_{e}^{2} }}{{{\text{210}}}} - \frac{{8{\mkern 1mu} \Sigma _{2} }}{{35}}\),

\(\kappa _{6} = \frac{1}{5}\left( {\frac{{22{\mkern 1mu} \Sigma _{2} }}{7} + \frac{{\Sigma _{3} L_{e}^{2} }}{{12}}} \right)\), \(\kappa _{7} = \frac{3}{7}\left( {\frac{{{\mkern 1mu} \Sigma _{3} }}{{\text{2}}} + \frac{{40{\mkern 1mu} \Sigma _{2} }}{{{\mkern 1mu} L_{e}^{2} }}} \right)\), \(\kappa _{8} = {\mkern 1mu} \frac{{\Sigma _{4} }}{{\text{2}}} + \frac{{\Sigma _{5} L_{e}^{2} }}{{140}}\), \(\kappa _{9} = \frac{{{\text{120}}}}{{L_{e}^{2} }}\left( {\frac{{\Sigma _{1} }}{{7{\mkern 1mu} }} + \frac{{{\text{6}}\Sigma _{6} }}{{L_{e}^{2} }}} \right)\),

\(\kappa _{{10}} = \frac{{\text{6}}}{{L_{e} }}\left( {{\mkern 1mu} \frac{{\Sigma _{1} }}{7} + \frac{{20\Sigma _{6} }}{{L_{e}^{2} }}} \right)\), \(\kappa _{{11}} = \frac{{24}}{{L_{e} }}\left( {\frac{{9{\mkern 1mu} \Sigma _{1} }}{{35{\mkern 1mu} }} + {\mkern 1mu} \frac{{14\Sigma _{6} }}{{L_{e}^{2} }}} \right)\), \(\kappa _{{12}} = \frac{{10}}{{7L_{e} }}\left( {\frac{{24{\mkern 1mu} \Sigma _{2} }}{{L_{e}^{2} }} + \Sigma _{3} } \right)\), \(\kappa _{{13}} = \frac{{\Sigma _{3} L_{e} }}{{70}} - \frac{{216{\mkern 1mu} \Sigma _{2} }}{{35L_{e} }}\),

\(\kappa _{{14}} = \frac{{6\Sigma _{2} }}{{7L_{e} }} + \frac{{\Sigma _{3} L_{e} }}{{84}}\), \(\kappa _{{15}} = \frac{8}{{35}}\left( {\frac{{48{\mkern 1mu} \Sigma _{2} }}{{L_{e} }} + \Sigma _{3} L_{e} } \right)\), \(\kappa _{{16}} = \frac{{L_{e} }}{{35}}\left( {\Sigma _{2} + \frac{{\Sigma _{3} L_{e}^{2} }}{{{\text{36}}}}} \right)\), \(\kappa _{{17}} = \frac{{L_{e} }}{{35}}\left( {6{\mkern 1mu} \Sigma _{2} + \frac{{\Sigma _{3} L_{e}^{2} }}{{18}}} \right)\),

\(\kappa _{{18}} = \frac{{\text{3}}}{{L_{e} }}\left( {\frac{{{\text{4}}\Sigma _{4} }}{{L_{e}^{2} }} + \frac{{{\text{3}}{\mkern 1mu} \Sigma _{5} }}{{7{\mkern 1mu} }}} \right)\), \(\kappa _{{19}} = {\text{3}}\left( {{\mkern 1mu} \frac{{\Sigma _{4} }}{{L_{e} }} + \frac{{{\text{3}}{\mkern 1mu} \Sigma _{5} L_{e} }}{{140}}} \right)\), \(\kappa _{{20}} = \frac{{{\text{240}}}}{{L_{e}^{3} }}\left( {\frac{{{\text{6}}\Sigma _{6} }}{{L_{e}^{2} }} + \frac{{\Sigma _{1} }}{{7{\mkern 1mu} }}} \right)\), \(\kappa _{{21}} = \frac{{384}}{{L_{e} }}\left( {{\mkern 1mu} \frac{{\Sigma _{6} }}{{L_{e}^{2} }} + \frac{{{\mkern 1mu} \Sigma _{1} }}{{35{\mkern 1mu} }}} \right)\),

\(\kappa _{{22}} = 6\left( {\frac{{3\Sigma _{6} }}{{L_{e} }} + \frac{{{\mkern 1mu} \Sigma _{1} L_{e} }}{{35}}} \right)\), \(\kappa _{{23}} = \frac{{{\mkern 1mu} \Sigma _{5} }}{7} + {\mkern 1mu} \frac{{6\Sigma _{4} }}{{L_{e}^{2} }}\), \(\kappa _{{24}} = \frac{{11{\mkern 1mu} \Sigma _{5} }}{{14}} + \frac{{12\Sigma _{4} }}{{L_{e}^{2} }}\), \(\kappa _{{25}} = \frac{{\Sigma _{1} L_{e} }}{{35}} - {\mkern 1mu} \frac{{6\Sigma _{6} }}{{L_{e} }}\),

$$\kappa _{{26}} = {\mkern 1mu} \frac{{3\Sigma _{4} }}{{L_{e} }} + \frac{{11{\mkern 1mu} \Sigma _{5} L_{e} }}{{140}},$$
(A.2)
$$\varvec{m}_{e} = \left[ {\begin{array}{*{20}l} {\Lambda _{{18}} } \hfill & {\Lambda _{2} } \hfill & {\Lambda _{6} } \hfill & {\Lambda _{{16}} } \hfill & { - \Lambda _{{19}} } \hfill & { - \Lambda _{{11}} } \hfill & {\Lambda _{{17}} } \hfill & { - \Lambda _{1} } \hfill & {\Lambda _{3} } \hfill & { - \Lambda _{{16}} } \hfill & {\Lambda _{{19}} } \hfill & { - \Lambda _{{11}} } \hfill \\ {\Lambda _{2} } \hfill & {\Lambda _{4} } \hfill & {\Lambda _{7} } \hfill & {\Lambda _{{19}} } \hfill & 0 \hfill & { - \Lambda _{{13}} } \hfill & {\Lambda _{1} } \hfill & { - \Lambda _{5} } \hfill & {\Lambda _{8} } \hfill & { - \Lambda _{{19}} } \hfill & {\Lambda _{{12}} } \hfill & { - \Lambda _{{14}} } \hfill \\ {\Lambda _{6} } \hfill & {\Lambda _{7} } \hfill & {\Lambda _{{10}} } \hfill & {\Lambda _{{11}} } \hfill & {\Lambda _{{13}} } \hfill & 0 \hfill & {\Lambda _{3} } \hfill & { - \Lambda _{8} } \hfill & {\Lambda _{9} } \hfill & { - \Lambda _{{11}} } \hfill & {\Lambda _{{14}} } \hfill & { - \Lambda _{{15}} } \hfill \\ {\Lambda _{{16}} } \hfill & {\Lambda _{{19}} } \hfill & {\Lambda _{{11}} } \hfill & {\Lambda _{{29}} } \hfill & {\Lambda _{{20}} } \hfill & {\Lambda _{{25}} } \hfill & {\Lambda _{{16}} } \hfill & { - \Lambda _{{19}} } \hfill & {\Lambda _{{11}} } \hfill & {\Lambda _{{22}} } \hfill & { - \Lambda _{{21}} } \hfill & {\Lambda _{{23}} } \hfill \\ { - \Lambda _{{19}} } \hfill & 0 \hfill & {\Lambda _{{13}} } \hfill & {\Lambda _{{20}} } \hfill & {\Lambda _{{30}} } \hfill & {\Lambda _{{27}} } \hfill & {\Lambda _{{19}} } \hfill & { - \Lambda _{{12}} } \hfill & {\Lambda _{{14}} } \hfill & {\Lambda _{{21}} } \hfill & { - \Lambda _{{24}} } \hfill & {\Lambda _{{26}} } \hfill \\ { - \Lambda _{{11}} } \hfill & { - \Lambda _{{13}} } \hfill & 0 \hfill & {\Lambda _{{25}} } \hfill & {\Lambda _{{27}} } \hfill & {\Lambda _{{31}} } \hfill & {\Lambda _{{11}} } \hfill & { - \Lambda _{{14}} } \hfill & {\Lambda _{{15}} } \hfill & {\Lambda _{{23}} } \hfill & { - \Lambda _{{26}} } \hfill & {\Lambda _{{28}} } \hfill \\ {\Lambda _{{17}} } \hfill & {\Lambda _{1} } \hfill & {\Lambda _{3} } \hfill & {\Lambda _{{16}} } \hfill & {\Lambda _{{19}} } \hfill & {\Lambda _{{11}} } \hfill & {\Lambda _{{18}} } \hfill & { - \Lambda _{2} } \hfill & {\Lambda _{6} } \hfill & { - \Lambda _{{16}} } \hfill & { - \Lambda _{{19}} } \hfill & {\Lambda _{{11}} } \hfill \\ { - \Lambda _{1} } \hfill & { - \Lambda _{5} } \hfill & { - \Lambda _{8} } \hfill & { - \Lambda _{{19}} } \hfill & { - \Lambda _{{12}} } \hfill & { - \Lambda _{{14}} } \hfill & { - \Lambda _{2} } \hfill & {\Lambda _{4} } \hfill & { - \Lambda _{7} } \hfill & {\Lambda _{{19}} } \hfill & 0 \hfill & { - \Lambda _{{13}} } \hfill \\ {\Lambda _{3} } \hfill & {\Lambda _{8} } \hfill & {\Lambda _{9} } \hfill & {\Lambda _{{11}} } \hfill & {\Lambda _{{14}} } \hfill & {\Lambda _{{15}} } \hfill & {\Lambda _{6} } \hfill & { - \Lambda _{7} } \hfill & {\Lambda _{{10}} } \hfill & { - \Lambda _{{11}} } \hfill & {\Lambda _{{13}} } \hfill & 0 \hfill \\ { - \Lambda _{{16}} } \hfill & { - \Lambda _{{19}} } \hfill & { - \Lambda _{{11}} } \hfill & {\Lambda _{{22}} } \hfill & {\Lambda _{{21}} } \hfill & {\Lambda _{{23}} } \hfill & { - \Lambda _{{16}} } \hfill & {\Lambda _{{19}} } \hfill & { - \Lambda _{{11}} } \hfill & {\Lambda _{{29}} } \hfill & { - \Lambda _{{20}} } \hfill & {\Lambda _{{25}} } \hfill \\ {\Lambda _{{19}} } \hfill & {\Lambda _{{12}} } \hfill & {\Lambda _{{14}} } \hfill & { - \Lambda _{{21}} } \hfill & { - \Lambda _{{24}} } \hfill & { - \Lambda _{{26}} } \hfill & { - \Lambda _{{19}} } \hfill & 0 \hfill & {\Lambda _{{13}} } \hfill & { - \Lambda _{{20}} } \hfill & {\Lambda _{{30}} } \hfill & { - \Lambda _{{27}} } \hfill \\ { - \Lambda _{{11}} } \hfill & { - \Lambda _{{14}} } \hfill & { - \Lambda _{{15}} } \hfill & {\Lambda _{{23}} } \hfill & {\Lambda _{{26}} } \hfill & {\Lambda _{{28}} } \hfill & {\Lambda _{{11}} } \hfill & { - \Lambda _{{13}} } \hfill & 0 \hfill & {\Lambda _{{25}} } \hfill & { - \Lambda _{{27}} } \hfill & {\Lambda _{{31}} } \hfill \\ \end{array} } \right],$$
(A.3)

where

\(\Lambda _{1} = \frac{{8M_{0} L_{e}^{2} }}{{245}}\), \(\Lambda _{2} = \frac{{33{\mkern 1mu} M_{0} L_{e}^{2} }}{{490}}\), \(\Lambda _{3} = \frac{{23{\mkern 1mu} M_{0} L_{e}^{3} }}{{7056}}\), \(\Lambda _{4} = \frac{{53{\mkern 1mu} M_{0} L_{e}^{3} }}{{3528}}\), \(\Lambda _{5} = \frac{{169{\mkern 1mu} M_{0} L_{e}^{3} }}{{17640}}\), \(\Lambda _{6} = \frac{{179{\mkern 1mu} M_{0} L_{e}^{3} }}{{35280}}\),

\(\Lambda _{7} = \frac{{11{\mkern 1mu} M_{0} L_{e}^{4} }}{{8820}}\), \(\Lambda _{8} = \frac{{11{\mkern 1mu} M_{0} L_{e}^{4} }}{{11760}}\), \(\Lambda _{9} = \frac{{19{\mkern 1mu} M_{0} L_{e}^{5} }}{{211680}}\), \(\Lambda _{{10}} = \frac{{23{\mkern 1mu} M_{0} L_{e}^{5} }}{{211680}}\), \(\Lambda _{{11}} = \frac{{M_{1} L_{e}^{2} }}{{84}}\), \(\Lambda _{{12}} = \frac{{13{\mkern 1mu} M_{1} L_{e}^{2} }}{{420}}\),

\(\Lambda _{{13}} = \frac{{M_{1} L_{e}^{3} }}{{1008}}\), \(\Lambda _{{14}} = \frac{{13{\mkern 1mu} M_{1} L_{e}^{3} }}{{5040}}\), \(\Lambda _{{15}} = \frac{{M_{1} L_{e}^{4} }}{{5040}}\), \(\Lambda _{{16}} = \frac{{{\mkern 1mu} M_{1} }}{{\text{2}}}\),\(\Lambda _{{17}} = \frac{{53{\mkern 1mu} M_{0} L_{e} }}{{490}}\), \(\Lambda _{{18}} = \frac{{96{\mkern 1mu} M_{0} L_{e} }}{{245}}\),

\(\Lambda _{{19}} = \frac{{11{\mkern 1mu} M_{1} L_{e} }}{{84}}\), \(\Lambda _{{20}} = \frac{{3M_{2} }}{{14}} + \frac{{{\text{33}}M_{0} L_{e}^{2} }}{{{\text{420}}}}\),\(\Lambda _{{22}} = \frac{{53{\mkern 1mu} M_{0} L_{e} }}{{490}} - \frac{{10{\mkern 1mu} M_{2} }}{{7L}}\), \(\Lambda _{{21}} = \frac{{8{\mkern 1mu} M_{0} L_{e}^{2} }}{{{\text{245}}}} - \frac{{{\mkern 1mu} {\text{3}}M_{2} }}{{{\text{14}}}}\),

\(\Lambda _{{23}} = \frac{{L_{e} }}{{84}}\left( {\frac{{23{\mkern 1mu} M_{0} L_{e}^{2} }}{{84}} - M_{2} } \right)\), \(\Lambda _{{24}} = \frac{{L_{e} }}{{70}}\left( {\frac{{169{\mkern 1mu} M_{0} L_{e}^{2} }}{{252}} + M_{2} } \right)\), \(\Lambda _{{25}} = \frac{{L_{e} }}{{84}}\left( {M_{2} + \frac{{179{\mkern 1mu} M_{0} L_{e}^{2} }}{{420}}} \right)\),

\(\Lambda _{{26}} = \frac{{L_{e}^{2} }}{{210}}\left( {M_{2} + \frac{{11{\mkern 1mu} M_{0} L_{e}^{2} }}{{56}}} \right)\), \(\Lambda _{{27}} = \frac{{L_{e}^{2} }}{{60}}\left( {M_{2} + \frac{{11{\mkern 1mu} M_{0} L_{e}^{2} }}{{147}}} \right)\), \(\Lambda _{{28}} = \frac{{L_{e}^{3} }}{{1260}}\left( {M_{2} + \frac{{19{\mkern 1mu} M_{0} L_{e}^{2} }}{{168}}} \right)\),

$$\Lambda _{{29}} = \frac{2}{7}\left( {\frac{{48{\mkern 1mu} M_{0} L_{e} }}{{35}} + \frac{{5M_{2} }}{{L_{e} }}} \right),\Lambda _{{30}} = \frac{{L{}_{e}}}{7}\left( {\frac{{8{\mkern 1mu} M_{2} }}{5} + \frac{{53{\mkern 1mu} M_{0} L_{e}^{2} }}{{504}}} \right),\Lambda _{{31}} = \frac{{L_{e}^{3} }}{{630}}\left( {M_{2} + \frac{{23{\mkern 1mu} M_{0} L_{e}^{2} }}{{336}}} \right).$$
(A.4)

Appendix B

The detailed entries of \(\varvec{k}_{e}\) and \(\varvec{m}_{e}\) for the present TB DQFE are as follows:

$$\varvec{k}_{e} {\text{ = }}\left[ {\begin{array}{*{20}l} {\kappa _{5} } \hfill & {\kappa _{{14}} } \hfill & 0 \hfill & 0 \hfill & {\kappa _{7} } \hfill & {\kappa _{{16}} } \hfill & { - \kappa _{5} } \hfill & {\kappa _{{14}} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{7} } \hfill & {\kappa _{{16}} } \hfill \\ {\kappa _{{14}} } \hfill & {\kappa _{6} } \hfill & 0 \hfill & 0 \hfill & {\kappa _{{16}} } \hfill & {\kappa _{8} } \hfill & { - \kappa _{{14}} } \hfill & { - \kappa _{{15}} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{{16}} } \hfill & { - \kappa _{{17}} } \hfill \\ 0 \hfill & 0 \hfill & {\kappa _{2} } \hfill & {\kappa _{{11}} } \hfill & { - \kappa _{{10}} } \hfill & { - \kappa _{{12}} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{2} } \hfill & {\kappa _{{11}} } \hfill & { - \kappa _{{10}} } \hfill & {\kappa _{{12}} } \hfill \\ 0 \hfill & 0 \hfill & {\kappa _{{11}} } \hfill & {\kappa _{3} } \hfill & {\kappa _{{12}} } \hfill & { - \kappa _{1} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{{11}} } \hfill & { - \kappa _{{13}} } \hfill & { - \kappa _{{12}} } \hfill & {\kappa _{4} } \hfill \\ {\kappa _{7} } \hfill & {\kappa _{{16}} } \hfill & { - \kappa _{{10}} } \hfill & {\kappa _{{12}} } \hfill & {\kappa _{9} } \hfill & {\kappa _{{20}} } \hfill & { - \kappa _{7} } \hfill & {\kappa _{{16}} } \hfill & {\kappa _{{10}} } \hfill & { - \kappa _{{12}} } \hfill & {\kappa _{{18}} } \hfill & { - \kappa _{{19}} } \hfill \\ {\kappa _{{16}} } \hfill & {\kappa _{8} } \hfill & { - \kappa _{{12}} } \hfill & { - \kappa _{1} } \hfill & {\kappa _{{20}} } \hfill & {\kappa _{{21}} } \hfill & { - \kappa _{{16}} } \hfill & { - \kappa _{{17}} } \hfill & {\kappa _{{12}} } \hfill & { - \kappa _{4} } \hfill & {\kappa _{{19}} } \hfill & { - \kappa _{{22}} } \hfill \\ { - \kappa _{5} } \hfill & { - \kappa _{{14}} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{7} } \hfill & { - \kappa _{{16}} } \hfill & {\kappa _{5} } \hfill & { - \kappa _{{14}} } \hfill & 0 \hfill & 0 \hfill & {\kappa _{7} } \hfill & {\kappa _{{19}} } \hfill \\ {\kappa _{{14}} } \hfill & { - \kappa _{{15}} } \hfill & 0 \hfill & 0 \hfill & {\kappa _{{16}} } \hfill & { - \kappa _{{17}} } \hfill & { - \kappa _{{14}} } \hfill & {\kappa _{6} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{{16}} } \hfill & {\kappa _{8} } \hfill \\ 0 \hfill & 0 \hfill & { - \kappa _{2} } \hfill & { - \kappa _{{11}} } \hfill & {\kappa _{{10}} } \hfill & {\kappa _{{12}} } \hfill & 0 \hfill & 0 \hfill & {\kappa _{2} } \hfill & { - \kappa _{{11}} } \hfill & {\kappa _{{10}} } \hfill & { - \kappa _{{12}} } \hfill \\ 0 \hfill & 0 \hfill & {\kappa _{{11}} } \hfill & { - \kappa _{{13}} } \hfill & { - \kappa _{{12}} } \hfill & { - \kappa _{4} } \hfill & 0 \hfill & 0 \hfill & { - \kappa _{{11}} } \hfill & {\kappa _{3} } \hfill & {\kappa _{{12}} } \hfill & {\kappa _{1} } \hfill \\ { - \kappa _{7} } \hfill & { - \kappa _{{16}} } \hfill & { - \kappa _{{10}} } \hfill & { - \kappa _{{12}} } \hfill & {\kappa _{{18}} } \hfill & {\kappa _{{19}} } \hfill & {\kappa _{7} } \hfill & { - \kappa _{{16}} } \hfill & {\kappa _{{10}} } \hfill & {\kappa _{{12}} } \hfill & {\kappa _{9} } \hfill & { - \kappa _{{20}} } \hfill \\ {\kappa _{{16}} } \hfill & { - \kappa _{{17}} } \hfill & {\kappa _{{12}} } \hfill & {\kappa _{4} } \hfill & { - \kappa _{{19}} } \hfill & { - \kappa _{{22}} } \hfill & { - \kappa _{{16}} } \hfill & {\kappa _{8} } \hfill & { - \kappa _{{12}} } \hfill & {\kappa _{1} } \hfill & { - \kappa _{{20}} } \hfill & {\kappa _{{21}} } \hfill \\ \end{array} } \right],$$
(B.1)

where

\(\kappa _{1} = \frac{{\Sigma _{3} }}{2}\), \(\kappa _{2} = \frac{{12}}{L}\left( {\frac{{{\mkern 1mu} \Sigma _{1} }}{{5{\mkern 1mu} }} + \frac{{2\Sigma _{2} }}{{L_{e}^{2} }}} \right)\), \(\kappa _{3} = 4\left( {\frac{{\Sigma _{1} L_{e} }}{{15}} + {\mkern 1mu} \frac{{2\Sigma _{2} }}{{L_{e} }}} \right)\), \(\kappa _{4} = \frac{1}{2}\left( {\frac{{\Sigma _{1} L_{e}^{2} }}{{15}} + {\mkern 1mu} \Sigma _{3} } \right)\),\(\kappa _{5} = \frac{6}{{L_{e} }}\left( {\frac{{4\Sigma _{6} }}{{L_{e}^{2} }} + \frac{{\Sigma _{8} }}{5}} \right)\),

\(\kappa _{6} = 2\left( {{\mkern 1mu} \frac{{4\Sigma _{6} }}{{L_{e} }} + \frac{{{\mkern 1mu} \Sigma _{8} L_{e} }}{{15}}} \right)\), \(\kappa _{7} = \frac{6}{{L_{e} }}\left( {\frac{{2\Sigma _{7} }}{{L_{e}^{2} }} + {\mkern 1mu} \frac{{\Sigma _{9} }}{5}} \right)\), \(\kappa _{8} = 2\left( {{\mkern 1mu} \frac{{2\Sigma _{7} }}{{L_{e} }} + \frac{{\Sigma _{9} L_{e} }}{{15}}} \right)\), \(\kappa _{9} = 4\left( {\frac{{14{\mkern 1mu} \Sigma _{1} L_{e} }}{{75}} + \frac{{3{\mkern 1mu} \Sigma _{4} }}{{5{\mkern 1mu} L_{e} }} + \frac{{6\Sigma _{5} }}{{L_{e}^{3} }}} \right)\),

\(\kappa _{{10}} = \Sigma _{1}\), \(\kappa _{{11}} = \frac{{\Sigma _{1} }}{5} + \frac{{12\Sigma _{2} }}{{L_{e}^{2} }}\), \(\kappa _{{12}} = \frac{{\Sigma _{1} L_{e} }}{5} + \frac{{\Sigma _{3} }}{{L_{e} }}\), \(\kappa _{{13}} = \frac{{\Sigma _{1} L_{e} }}{{15}} - {\mkern 1mu} \frac{{4\Sigma _{2} }}{{L_{e} }}\), \(\kappa _{{14}} = \frac{{12\Sigma _{6} }}{{L_{e}^{2} }} + \frac{{\Sigma _{8} }}{8}\),

\(\kappa _{{15}} = \frac{{{\mkern 1mu} \Sigma _{8} L_{e} }}{{30}} - {\mkern 1mu} \frac{{4\Sigma _{6} }}{{L_{e} }}\), \(\kappa _{{16}} = {\mkern 1mu} \frac{{6\Sigma _{7} }}{{L_{e}^{2} }} + \frac{{{\mkern 1mu} \Sigma _{9} }}{{10}}\), \(\kappa _{{17}} = \frac{{{\mkern 1mu} \Sigma _{9} L_{e} }}{{30}} - {\mkern 1mu} \frac{{2\Sigma _{7} }}{{L_{e} }}\), \(\kappa _{{18}} = \frac{{19{\mkern 1mu} \Sigma _{1} L_{e} }}{{75}} - \frac{{12{\mkern 1mu} \Sigma _{4} }}{{5{\mkern 1mu} L_{e} }} - \frac{{24\Sigma _{5} }}{{L_{e}^{3} }}\),

\(\kappa _{{19}} = \frac{{3{\mkern 1mu} \Sigma _{1} L_{e}^{2} }}{{50}} - \frac{{{\mkern 1mu} \Sigma _{4} }}{5} - \frac{{12\Sigma _{5} }}{{L_{e}^{2} }}\), \(\kappa _{{20}} = \frac{{8{\mkern 1mu} \Sigma _{1} L_{e}^{2} }}{{75}} + \frac{{{\mkern 1mu} \Sigma _{4} }}{5} + \frac{{12\Sigma _{5} }}{{L_{e}^{2} }}\), \(\kappa _{{21}} = \frac{{{\mkern 1mu} \Sigma _{1} L_{e}^{3} }}{{50}} + \frac{{4{\mkern 1mu} \Sigma _{4} L_{e} }}{{15}} + {\mkern 1mu} \frac{{8\Sigma _{5} }}{{L_{e} }}\),

$$\kappa _{{22}} = \frac{{\Sigma _{1} L_{e}^{3} }}{{75}} + \frac{{{\mkern 1mu} \Sigma _{4} L_{e} }}{{15}} - {\mkern 1mu} \frac{{4\Sigma _{5} }}{{L_{e} }},$$
(B.2)
$$\varvec{m}_{e} {\text{ = }}\left[ {\begin{array}{*{20}l} {\Lambda _{2} } \hfill & {\Lambda _{4} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{8} } \hfill & {\Lambda _{{10}} } \hfill & {\Lambda _{1} } \hfill & { - \Lambda _{3} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{7} } \hfill & { - \Lambda _{9} } \hfill \\ {\Lambda _{4} } \hfill & {\Lambda _{5} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{10}} } \hfill & {\Lambda _{{11}} } \hfill & {\Lambda _{3} } \hfill & { - \Lambda _{6} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{9} } \hfill & { - \Lambda _{{12}} } \hfill \\ 0 \hfill & 0 \hfill & {\Lambda _{2} } \hfill & {\Lambda _{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\Lambda _{1} } \hfill & { - \Lambda _{3} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\Lambda _{4} } \hfill & {\Lambda _{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\Lambda _{3} } \hfill & { - \Lambda _{6} } \hfill & 0 \hfill & 0 \hfill \\ {\Lambda _{8} } \hfill & {\Lambda _{{10}} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{14}} } \hfill & {\Lambda _{{16}} } \hfill & {\Lambda _{7} } \hfill & { - \Lambda _{9} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{13}} } \hfill & { - \Lambda _{{15}} } \hfill \\ {\Lambda _{{10}} } \hfill & {\Lambda _{{11}} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{16}} } \hfill & {\Lambda _{{17}} } \hfill & {\Lambda _{9} } \hfill & { - \Lambda _{{12}} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{15}} } \hfill & { - \Lambda _{{18}} } \hfill \\ {\Lambda _{1} } \hfill & {\Lambda _{3} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{7} } \hfill & {\Lambda _{9} } \hfill & {\Lambda _{2} } \hfill & { - \Lambda _{4} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{8} } \hfill & { - \Lambda _{{10}} } \hfill \\ { - \Lambda _{3} } \hfill & { - \Lambda _{6} } \hfill & 0 \hfill & 0 \hfill & { - \Lambda _{9} } \hfill & { - \Lambda _{{12}} } \hfill & { - \Lambda _{4} } \hfill & {\Lambda _{5} } \hfill & 0 \hfill & 0 \hfill & { - \Lambda _{{10}} } \hfill & {\Lambda _{{11}} } \hfill \\ 0 \hfill & 0 \hfill & {\Lambda _{1} } \hfill & {\Lambda _{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\Lambda _{2} } \hfill & { - \Lambda _{4} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & { - \Lambda _{3} } \hfill & { - \Lambda _{6} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - \Lambda _{4} } \hfill & {\Lambda _{5} } \hfill & 0 \hfill & 0 \hfill \\ {\Lambda _{7} } \hfill & {\Lambda _{9} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{13}} } \hfill & {\Lambda _{{15}} } \hfill & {\Lambda _{8} } \hfill & { - \Lambda _{{10}} } \hfill & 0 \hfill & 0 \hfill & {\Lambda _{{14}} } \hfill & { - \Lambda _{{16}} } \hfill \\ { - \Lambda _{9} } \hfill & { - \Lambda _{{12}} } \hfill & 0 \hfill & 0 \hfill & { - \Lambda _{{15}} } \hfill & { - \Lambda _{{18}} } \hfill & { - \Lambda _{{10}} } \hfill & {\Lambda _{{11}} } \hfill & 0 \hfill & 0 \hfill & { - \Lambda _{{16}} } \hfill & {\Lambda _{{17}} } \hfill \\ \end{array} } \right],$$
(B.3)

where

\(\Lambda _{1} = \frac{{19M_{0} L_{e} }}{{150}}\), \(\Lambda _{2} = \frac{{28{\mkern 1mu} M_{0} L_{e} }}{{75}}\), \(\Lambda _{3} = \frac{{3M_{0} L_{e}^{2} }}{{100}}\), \(\Lambda _{4} = \frac{{4M_{0} L_{e}^{2} }}{{75}}\), \(\Lambda _{5} = \frac{{M_{0} L_{e}^{3} }}{{100}}\), \(\Lambda _{6} = \frac{{M_{0} L_{e}^{3} }}{{150}}\),

\(\Lambda _{7} = \frac{{19{\mkern 1mu} M_{1} L_{e} }}{{150}}\), \(\Lambda _{8} = \frac{{28{\mkern 1mu} M_{1} L_{e} }}{{75}}\), \(\Lambda _{9} = \frac{{3{\mkern 1mu} M_{1} L_{e}^{2} }}{{100}}\), \(\Lambda _{{10}} = \frac{{4{\mkern 1mu} M_{1} L_{e}^{2} }}{{75}}\), \(\Lambda _{{11}} = \frac{{M_{1} L_{e}^{3} }}{{100}}\), \(\Lambda _{{12}} = \frac{{M_{1} L_{e}^{3} }}{{150}}\),

$$\Lambda _{{13}} = \frac{{19{\mkern 1mu} M_{2} L_{e} }}{{150}},\Lambda _{{15}} = \frac{{3{\mkern 1mu} M_{2} L_{e}^{2} }}{{100}},\Lambda _{{16}} = \frac{{4{\mkern 1mu} M_{2} L_{e}^{2} }}{{75}},\Lambda _{{17}} = \frac{{M_{2} L_{e}^{3} }}{{100}},\Lambda _{{18}} = \frac{{M_{2} L_{e}^{3} }}{{150}}.$$
(B.4)

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Zhang, B., Li, H., Kong, L. et al. Weak-form differential quadrature finite elements for functionally graded micro-beams with strain gradient effects. Acta Mech 232, 4009–4036 (2021). https://doi.org/10.1007/s00707-021-03028-y

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