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Z. Angew. Math. Phys. (2024) 75:139
c
2024 The Author(s), under exclusive licence to Springer Nature
Switzerland AG
0044-2275/24/040001-20
published online July 5, 2024
https://doi.org/10.1007/s00033-024-02275-y
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Thermoelastic damping analysis for a piezothermoelastic nanobeam resonator using
DPL model under modified couple stress theory
Anjali Srivastava and Santwana Mukhopadhyay
Abstract. The current work investigates the transverse vibration of a piezothermoelastic (PTE) nanobeam in the frame of
dual-phase-lag thermoelasticity theory. Closed-form analytical expression for the thermoelastic damping (TED) in terms
of quality factor for a homogeneous transversely isotropic PTE beam is derived by using Euler–Bernoulli beam theory and
complex frequency approach. The size effect of the nanostructured beam is tackled by applying modified couple stress theory
(MCST). Detailed analysis on damping of vibration owing to thermal fluctuations and electric potential in the present
context under three sets of boundary conditions is attempted to investigate the influences of two-phase-lag parameters,
piezoelectric parameter, thermal effect and size-dependent behaviour on energy dissipation caused by TED in PTE beam
resonators. Analytical results are illustrated with the help of graphical plots on numerical findings for lead zirconate titanate
(PZT-5A) PTE material. The investigation brings out some significant key findings and observations in view of the present
heat conduction model.
Mathematics Subject Classification. Primary 74-10; Secondary 74F05, 74B10.
Keywords. Piezothermoelastic material, Dual-phase-lag heat conduction model, Transversely isotropic beam, Modified couple
stress theory, Thermoelastic damping.
1. Introduction
As we know the classical heat conduction model along with the energy equation implies the instanta-
neous propagation of thermal waves which is physically unacceptable, many non-classical theories have
been proposed to overcome the drawback of infinite propagation of heat conduction. The non-Fourier
heat conduction model has attracted the interest of many researchers as these models offer a mean to
understand the thermal interactions in scenarios involving high-speed energy transport, low temperatures
and extremely high heat fluxes. The thermoelasticity theories have also been modified in many ways by
introducing phase lags and/or by modifying constitutive equations. We must recall the Lord–Shulman
[1] thermoelasticity theory which involves one relaxation parameter with heat flux as a modification to
Fourier’s law which accounts for the finite propagation of thermal waves. In the early twentieth century,
Green–Naghdi [2–4] has proposed three forms of thermoelasticity theory by involving a new constitutive
variable αknown as thermal displacement which follows ˙α=θin which the third one is the generalization
of the first and second forms. Furthermore, Tzou [5] introduced a new constitutive variable known as
phase-lag parameter to generalize the CV heat conduction model [6,7] which is known as single-phase-lag
model. Further, by involving two-phase-lag parameters of heat flux and temperature gradient, Tzou [8,9]
developed dual-phase-lag heat conduction theory.
The dual-phase-lag (DPL) model allows for the inclusion of time-dependent effects in heat conduction,
such as propagation delays and finite thermal relaxation times. It provides a more accurate representation
of heat conduction phenomena in situations where non-Fourier effects are significant, such as in high-
speed heat transfer, transient heat conduction or at small length scales; hence, it is worth recalling some
contributions in this direction. Kang et al. [10] predicted the existence of thermal waves in dual-phase-lag
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