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Study of frequency shift and thermoelastic damping in transversely isotropic nano-beam with GN III theory and two temperature

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Abstract

In this study, a nanoscale beam of transversely isotropic thermoelastic (TIT) medium with two temperature and with Green–Naghdi (GN) III theory of thermoelasticity for free vibrations with simply supported boundaries have been examined. Euler–Bernoulli (EB) beam theory is used to formulate a mathematical model of the nanoscale beam in a closed form. The lateral deflection, frequency shift, thermal moment and thermoelastic damping have been solved. A program in MATLAB software is developed to find the numerical values for different physical quantities. The lateral deflection, frequency shift, thermal moment and thermoelastic damping for the varying two temperature, different modes, frequency of time harmonic sources, thickness of nano-beam, and material medium of nano-beam has been represented graphically and discussed. The results for some particular cases have also been compared to some earlier work done.

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  • 12 January 2021

    Journal abbreviated title on top of the page has been corrected to “Arch Appl Mech”

Abbreviations

\(\delta_{ij}\) :

Kronecker delta

\(C_{ijkl}\) :

Elastic parameters (Nm−2)

\(\beta_{ij}\) :

Thermal elastic coupling tensor (Nm−2K−1)

T :

Absolute temperature (K)

\(T_{0}\) :

Reference temperature (K)

\(\varphi\) :

Conductive temperature (K)

\(t_{ij}\) :

Stress tensors (Nm−2)

\(e_{ij}\) :

Strain tensors (m m−1)

\(u_{i}\) :

Components of displacement (m)

\(\rho\) :

Medium density (kgm−3)

\(C_{E}\) :

Specific heat (J kg−1 K−1)

\(a_{ij}\) :

Two temperature parameters,

\(\alpha_{ij}\) :

Linear thermal expansion coefficient (K−1)

\(K_{ij}\) :

Thermal conductivity (Wm−1K−1)

\(K_{ij}^{*}\) :

Materialistic constant (Ns−2K−1)

\(\omega\) :

Frequency (Hz)

I :

Moment of inertia (m4)

\(C_{11} I\) :

Flexural rigidity of the nano-beam (Nm2)

s :

Laplace transform parameter

\(\varepsilon_{T}\) :

Thermoelastic coupling

A :

Area of cross section(m2)

\(M_{T}\) :

Thermal moment (Km2)

\(M\left( {x,t} \right)\) :

Flexural moment

\(w\left( {x,t} \right)\) :

Lateral deflection (m)

t :

Time (s)

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Authors and Affiliations

  1. Department of Basic and Applied Science, Punjabi University, Patiala, Punjab, India

    Iqbal Kaur & Parveen Lata

  2. Kurukshetra University, Kurukshetra, India

    Kulvinder Singh

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  1. Iqbal Kaur
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  2. Parveen Lata
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Kaur, I., Lata, P. & Singh, K. Study of frequency shift and thermoelastic damping in transversely isotropic nano-beam with GN III theory and two temperature. Arch Appl Mech 91, 1697–1711 (2021). https://doi.org/10.1007/s00419-020-01848-3

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