Email Alert    RSS

Applied Mathematics and Mechanics >

2023 , Vol. 44 >Issue 5: 773 - 786

DOI: https://doi.org/10.1007/s10483-023-2981-7

Articles

Nonlocal stress gradient formulation for damping vibration analysis of viscoelastic microbeam in thermal environment

Expand
  • State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received date: 2022-05-16

  Revised date: 2022-11-25

  Online published: 2023-04-24

Supported by

the National Natural Science Foundation of China (No.12172169)

Fold

Abstract

An integral nonlocal stress gradient viscoelastic model is proposed on the basis of the integral nonlocal stress gradient model and the standard viscoelastic model, and is utilized to investigate the free damping vibration analysis of the viscoelastic Bernoulli-Euler microbeams in thermal environment. Hamilton's principle is used to derive the differential governing equations and corresponding boundary conditions. The integral relations between the strain and the nonlocal stress are converted into a differential form with constitutive constraints. The size-dependent axial thermal stress due to the variation of the environmental temperature is derived explicitly. The Laplace transformation is utilized to obtain the explicit expression for the bending deflection and moment. Considering the boundary conditions and constitutive constraints, one can get a nonlinear equation with complex coefficients, from which the complex characteristic frequency can be determined. A two-step numerical method is proposed to solve the elastic vibration frequency and the damping ratio. The effects of length scale parameters, viscous coefficient, thermal stress, vibration order on the vibration frequencies, and critical viscous coefficient are investigated numerically for the viscoelastic Bernoulli-Euler microbeams under different boundary conditions.

Key words： damping vibration; size effect; integral nonlocal stress gradient model; standard viscoelastic model; Laplace transformation

Cite this article

Hai QING, Huidiao SONG . Nonlocal stress gradient formulation for damping vibration analysis of viscoelastic microbeam in thermal environment[J]. Applied Mathematics and Mechanics, 2023 , 44(5) : 773 -786 . DOI: 10.1007/s10483-023-2981-7

References

[1] GUTIERREZ-LEMINI, D. Engineering Viscoelasticity, Springer, New York (2014)
[2] ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710(1983)
[3] KOITER, W. T. Couple stresses in the theory of elasticity, I and II. Koninklijke Nederlandse Akademie van Wetenschappen, 67, 17-44(1964)
[4] YANG, F., CHONG, A. C. M., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731-2743(2002)
[5] GHAYESH, M. H. Dynamics of functionally graded viscoelastic microbeams. International Journal of Engineering Science, 124, 115-131(2018)
[6] GHAYESH, M. H. Viscoelastic dynamics of axially FG microbeams. International Journal of Engineering Science, 135, 75-85(2019)
[7] ERINGEN, A. C. Theory of nonlocal elasticity and some applications. Res Mechanica, 21(4), 313-342(1987)
[8] LEI, Y., MURMU, T., ADHIKARI, S., and FRISWELL, M. I. Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beams. European Journal of Mechanics-A/Solids, 42, 125-136(2013)
[9] LEI, Y., ADHIKARI, S., and FRISWELL, M. I. Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams. International Journal of Engineering Science, 66-67, 1-3(2013)
[10] LIM, C. W., ZHANG, G., and REDDY, J. N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298-313(2015)
[11] SHARIFI, Z., KHORDAD, R., GHARAATI, A., and FOROZANI, G. An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory. Applied Mathematics and Mechanics (English Edition), 40(12), 1723-1740(2019) https://doi.org/10.1007/s10483-019-2545-8
[12] LU, L., ZHU, L., GUO, X., ZHAO, J., and LIU, G. A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells. Applied Mathematics and Mechanics (English Edition), 40(12), 1695-1722(2019) https://doi.org/10.1007/s10483-019-2549-7
[13] ZENG, S., WANG, K., WANG, B., and WU, J. Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory. Applied Mathematics and Mechanics (English Edition), 41(6), 859-880(2020) https://doi.org/10.1007/s10483-020-2620-8
[14] SONG, R., SAHMANI, S., and SAFAEI, B. Isogeometric nonlocal strain gradient quasi-three-dimensional plate model for thermal postbuckling of porous functionally graded microplates with central cutout with different shapes. Applied Mathematics and Mechanics (English Edition), 42(6), 771-786(2021) https://doi.org/10.1007/s10483-021-2725-7
[15] ZAERA, R., SERRANO, O., and FERNANDEZ-SAEZ, J. On the consistency of the nonlocal strain gradient elasticity. International Journal of Engineering Science, 138, 65-81(2019)
[16] LI, C., QING, H., and GAO, C. Theoretical analysis for static bending of Euler-Bernoulli using different nonlocal gradient models. Mechanics of Advanced Materials and Structures, 88, 1965-1977(2021)
[17] BIAN, P. L. and QING, H. On bending consistency of Timoshenko beam using differential and integral nonlocal strain gradient models. Zeitschrift für Angewandte Mathematik und Mechanik, 101, 202000132(2021)
[18] BARRETTA, R. and DE SCIARRA, F. M. Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. International Journal of Engineering Science, 130, 187-198(2018)
[19] BIAN, P. L. and QING, H. Elastic buckling and free vibration of nonlocal strain gradient Euler-Bernoulli beams using Laplace transform. Zeitschrift für Angewandte Mathematik und Mechanik, 102, 202100152(2022)
[20] ZHANG, P. and QING, H. The consistency of the nonlocal strain gradient integral model in size-dependent bending analysis of beam structures. International Journal of Mechanical Sciences, 189, 105991(2021)
[21] ZHANG, P. and QING, H. Exact solutions for size-dependent bending of Timoshenko curved beams based on a modified nonlocal strain gradient model. Acta Mechanica, 231, 5251-5276(2020)
[22] BARRETTA, R., FAGHIDIAN, S. A., DE SCIARRA, F. M., PENNA, R., and PINNOLA, F. P. On torsion of nonlocal Lam strain gradient FG elastic beams. Composite Structures, 233, 111550(2020)
[23] BARRETTA, R. and DE SCIARRA, F. M. Variational nonlocal gradient elasticity for nano-beams. International Journal of Engineering Science, 143, 73-91(2019)
[24] FANG, Y., LI, P., ZHOU, H., and ZUO, W. Thermoelastic damping in flexural vibration of bilayered microbeams with circular cross-section. Applied Mathematical Modelling, 77, 1129-1147(2020)
[25] ZHANG, P., SCHIAVONE, P., and QING, H. Local/nonlocal mixture integral models with bi-Helmholtz kernel for free vibration of Euler-Bernoulli beams under thermal effect. Journal of Sound and Vibration, 525, 116798(2022)
Options
Outlines

/

〈 〉

APS Journals | CSTAM Journals | AMS Journals | EMS Journals | ASME Journals
Total visitors:
Copyright © Applied Mathematics and Mechanics (English Edition)
Address：P. O. Box 122, Shanghai University, 99 Shangda Road, Shanghai 200444, China
E-mail： amm@department.shu.edu.cn
Technical support: Beijing Magtech Co.ltd support@magtech.com.cn