Study of transversely isotropic nonlocal thermoelastic thin nano-beam resonators with multi-dual-phase-lag theory
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O R I G I NA L
Iqbal Kaur
· Parveen Lata · Kulvinder Singh
Study of transversely isotropic nonlocal thermoelastic thin nano-beam resonators with multi-dual-phase-lag theory
Received: 8 May 2020 / Accepted: 28 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This study deals with a novel model of forced vibrational analysis of nonlocal transversely isotropic thermoelastic nanobeam resonators due to ramp-type heating and due to time varying exponentially decaying load with multi-dual-phase-lag theory of thermoelasticity. The mathematical model is prepared for the nanobeam in a closed form with the application of Euler–Bernoulli (E–B) beam theory using nonlocal generalized thermoelasticity with multi-dual phase lags. The nonlocal nanobeam theory has a nonlocal parameter to depict small-scale effect. The Laplace Transform technique has been used to find the expressions for lateral deflection, conductive temperature, thermal moment, nonlocal axial stress and thermodynamic temperature for (i) clamped–clamped, (ii) simply supported–simply supported, (iii) clamped–simply supported, (iv) clamped–free, and (v) free–free nanobeam in the transformed domain. The general algorithm of the inverse Laplace Transform is developed to compute the results numerically in physical domain. The results exhibit that the amplitude of deflection and thermal moment is attenuated and depends upon the schematic design of the nanobeam being considered. Also, it can be found from both the numerical calculations and the analytic results that nonlocal multi-dual-phase-lag theory of thermoelasticity with two temperatures due to time varying exponentially decaying load has significant effect on deflection and thermal moment. The effect of different theories of nonlocal thermoelasticity, due to time varying exponentially decaying load, has been depicted on the various quantities. Some particular cases have also been deduced. Keywords Transversely isotropic thermoelastic · Nanobeam · Multi-dual-phase-lag theory of thermoelasticity · Time varying load · Nonlocal nanobeam · Laplace transform Mathematics Subject Classification
74Axx · 80A20 · 74H15 · 74Kxx
List of symbols δi j T0 βi j
Kronecker delta Reference temperature Thermal elastic coupling tensor
I. Kaur (B) · P. Lata Department of Basic and Applied Sciences, Punjabi University, Patiala, Punjab, India E-mail: [email protected] P. Lata E-mail: [email protected] K. Singh Kurukshetra University, Kurukshetra, India E-mail: [email protected]
I. Kaur et al.
τq ci jkl ϕ T ei j CE ρ ui ai j αi j Ki j u δ(x) ξ ti j τθ τ0 I
Phase lag of heat flux Elastic parameters conductive temperature Absolute temperature Strain tensors Specific heat Medium density Components of displacement Two temperature parameters Linear thermal expansion coefficient Thermal conductivity Frequency of the applied load Displacement vector Dirac delta function Nonlocal parameter Stress tensors Phase lag of temperature gradient Relaxation time Moment of inertia
1 Introduction In la
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