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Nonlocal vibration analysis of microstretch plates in the framework of space-fractional mechanics—theory and validation

Abstract

In this paper, the nonlocal vibration analysis of plates modeled by generalized microstretch theory using Riesz–Caputo fractional derivative concept is presented. The frequency spectrum and the mode shapes of the microstretch plate with two clamped edges and two free edges for different values of the fractional continua order and the material length scale parameter are carried out. The three-dimensional vibration analysis is obtained by Ritz energy method. Moreover, the mode shapes and the absolute differences between classical and fractional eigenvectors for the first six macrofrequencies and additional microfrequencies between them are presented by using contour plots. The main contribution of the paper is that the nonlocal approach utilizing the fractional calculus gives better results compared to the experimental outcomes than the classical local theory. Besides, defining the nonlocality without using the nonlocal kernels is another advantage of the present approach. The overall conclusion is that the fractional mechanics establishes a new model for the nonlocal vibration analysis of microstretch plates.

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Abbreviations

\(a,b,h \) :

\({\text{Plate}}\,{\text{dimensions}}\,{\text{(m)}}\)

\({\mathbf{u}},{{\varvec{\upphi}}},\theta \) :

\( {\text{Displacement}}\,{\text{vector,}}\,{\text{micro}}\,{\text{rotation}}\,{\text{vector,}}\,{\text{microstretch}}\,{\text{scalar}}\)

\(U,\Phi ,\Theta \) :

\({\text{Amplitude}}\,{\text{functions}}\)

\(\xi {,}\eta ,{\kern 1pt} \zeta \) :

\({\text{Non}}\,{\text{dimensional}}\,{\text{parameters}}\)

\( \kappa \) :

\({\text{Micropolar}}\,{\text{constant (GPa)}}\)

\( \overline{\alpha },\beta ,\gamma \) :

\({\text{Micropolar}}\,{\text{constants}}\,{\text{(GN)}}\)

\( \lambda_{0} ,\lambda_{1} ,a_{0} \) :

\({\text{Microelongational}}\,{\text{constants}}\,{\text{(GN)}}\)

\(\omega \) :

\({\text{Natural}}\,{\text{frequency}}\,({\text{rad/s}})\)

\(V_{{{\text{max}}}} ,T_{{{\text{max}}}} \) :

\( {\text{Maximum}}\,{\text{strain}}\,{\text{and}}\,{\text{kinetic}}\,{\text{energies}}\,({\text{J}}) \)

\(E \) :

\({\text{Young's}}\,{\text{modulus}}\,({\text{Pa}}) \)

\(\nu \) :

\({\text{Poisson's}}\,{\text{ratio}}\)

\(\varepsilon_{ij} \) :

\({\text{Strain}}\,{\text{components}}\)

\(l \) :

\({\text{Length}}\,{\text{scale}}\,{\text{coefficient}} \)

\( l_{x} ,l_{y} ,l_{z} \) :

\({\text{Length}}\,{\text{scale}}\,({\text{m}}) \)

\( j \) :

\( {\text{Micro}}- {\text{inertia}}\,({\text{m}}^{2} )\)

\( \rho \) :

\({\text{Mass}}\,{\text{density}}\,{\text{per}}\,{\text{unit}}\,{\text{volume}}\,{\text{(kg/m}}^{2} {)} \)

\(P_{i} (x) \) :

\( {\text{ith}}\,{\text{Chebyshev}}\,{\text{polynomial}}\)

\(F_{i} (\xi ,\eta )\) :

\( {\text{Boundary}}\,{\text{functions}} \)

\( {\mathbf{K,M}} \) :

\({\text{Stiffness}}\,{\text{and}}\,{\text{mass}}\,{\text{matrices}}\)

\({\mathbf{Z}} \) :

\({\text{Column}}\,{\text{vector}}\)

\(D\) :

\( {\text{Flexural}}\,{\text{rigidity}}\,{\text{of}}\,{\text{the}}\,{\text{plate}}\,{\text{(kg}}/{\text{m}}^{2} {\text{/s}}^{2}) \)

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Acknowledgements

This work was supported by the National Science Centre, Poland, under Grant No. 2017/27/B/ST8/00351.

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Correspondence to Soner Aydinlik.

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Aydinlik, S., Kiris, A. & Sumelka, W. Nonlocal vibration analysis of microstretch plates in the framework of space-fractional mechanics—theory and validation. Eur. Phys. J. Plus 136, 169 (2021). https://doi.org/10.1140/epjp/s13360-021-01110-x

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