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Effect of the gradient on the deflection of functionally graded microcantilever beams with surface stress

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Abstract

The surface stress-induced deflection of a microcantilever beam with arbitrary axial nonhomogeneity and varying cross section is investigated. The surface stresses are assumed to be uniformly distributed on the upper surface of the beam. Based on the small deformation and Euler–Bernoulli beam theory, the second-order integral–differential governing equation is derived. A simple Taylor series expansion method is proposed to calculate the static deformation. The approximate solution of functionally graded microbeams can degenerate into the solution of homogeneous microbeams, and the explicit expressions for the static deflection, slope angle curvature, and surface stress are derived. Particularly, the influence of the gradient parameters on the static deformation of functionally graded rectangular and triangular microbeams is presented by Figures primarily. Obtained results indicate that choosing an appropriate gradient parameter is beneficial for different surface stresses. The proposed method and derived solution can be used as a theoretical benchmark for validating the obtained results of microcantilever beams as micro-mechanical sensors and atomic force microscopy.

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Acknowledgements

This work was supported by the National Nature Science Foundation of China (Grant No. 11202038) and the China Scholarship Council (201908430028), the State Key Development Program for Basic Research of China (Grant No. 015CB057705), the Civil Engineering Key Subject Foundation of Changsha University of Science and Technology (No. 18ZDXK04), and the Postgraduate research innovation project of Changsha University of Science & Technology (Grant No. CX2020SS22).

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Authors and Affiliations

  1. School of Civil Engineering, Changsha University of Science & Technology, Changsha, 410114, Hunan, People’s Republic of China

    Xu-Long Peng, Li Zhang, Zi-Xuan Yang, Zhan-Yong Feng & Bing Zhao

  2. School of Civil Engineering, Central South University, Changsha, 410075, Hunan, People’s Republic of China

    **an-Fang Li

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  1. Xu-Long Peng
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Correspondence to **an-Fang Li.

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Appendix

Appendix

When using the fourth-order Taylor expansion, the matrixes H, Y, and B in Eq. (26) are replaced by the following ones:

$$\begin{aligned} {\mathbf {H}}= & {} \left[ {{\begin{array}{c} {h_{\left( 2 \right) } (\xi )} \\ {h_{\left( 1 \right) } (\xi )} \\ {h\left( \xi \right) } \\ {-\frac{1}{E(\xi )}} \\ {-\frac{{b}'\left( \xi \right) }{E(\xi )b\left( \xi \right) }} \\ \end{array} }} \right] ,\nonumber \\ Y= & {} \left[ {{\begin{array}{c} {Y\left( \xi \right) } \\ {{Y}'\left( \xi \right) } \\ {{Y}''\left( \xi \right) } \\ {{Y}'''\left( \xi \right) } \\ {Y^{IV}\left( \xi \right) } \\ \end{array} }} \right] , \end{aligned}$$
(A.1)
$$\begin{aligned} B= & {} \left[ {\begin{array}{l} 0 \quad B_{\left( 2 \right) } \left( {\xi ,1} \right) \quad B_{\left( 2 \right) } \left( {\xi ,2} \right) \qquad \qquad \quad B_{\left( 2 \right) } \left( {\xi ,3} \right) \quad B_{\left( 2 \right) } \left( {\xi ,4} \right) \\ 0 \quad B_{\left( 1 \right) } \left( {\xi ,1} \right) \quad B_{\left( 1 \right) } \left( {\xi ,2} \right) \qquad \qquad \quad B_{\left( 1 \right) } \left( {\xi ,3} \right) \quad B_{\left( 1 \right) } \left( {\xi ,4} \right) \\ 0\quad B_{\left( 0 \right) } \left( {\xi ,1} \right) \quad B_{\left( 0 \right) } \left( {\xi ,2} \right) \qquad \qquad \quad B_{\left( 0 \right) } \left( {\xi ,3} \right) \quad B_{\left( 0 \right) } \left( {\xi ,4} \right) \\ 0\quad -h(\xi ) \qquad \quad \frac{\left[ {{\overline{E}} \left( \xi \right) b\left( \xi \right) } \right] ^{\prime }}{q{\overline{E}} \left( \xi \right) b\left( \xi \right) }\quad \quad \quad \quad \quad 0\quad \quad \quad \quad \quad \quad 0 \\ 0\quad \frac{1}{{\overline{E}} (\xi )} \qquad \qquad \frac{\left[ {{\overline{E}} \left( \xi \right) b\left( \xi \right) } \right] ^{\prime \prime }}{q{\overline{E}} \left( \xi \right) b\left( \xi \right) } -h\left( \xi \right) \quad \frac{2\left[ {{\overline{E}} \left( \xi \right) b\left( \xi \right) } \right] ^{\prime }}{q{\overline{E}} \left( \xi \right) b\left( \xi \right) } \quad 0 \\ \end{array}} \right] . \end{aligned}$$
(A.2)

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Peng, XL., Zhang, L., Yang, ZX. et al. Effect of the gradient on the deflection of functionally graded microcantilever beams with surface stress. Acta Mech 231, 4185–4198 (2020). https://doi.org/10.1007/s00707-020-02759-8

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