Microsystem Technologies

, Volume 25, Issue 1, pp 1–10 | Cite as

Effect of gravity on piezo-thermoelasticity within the dual-phase-lag model

  • Ethar A. A. AhmedEmail author
  • M. S. Abou-Dina
  • A. R. El Dhaba
Review Article


The equations of generalized piezo-thermoelasticity within the frame of dual-phase-lag model are used to study the effect of gravitational force on the behavior of a half-space. Analytic expressions for the displacement components, temperature, stress and strain tensors components are obtained using normal mode analysis. Numerical results for the field quantities of practical interest are given in the physical domain and illustrated graphically. Comparison is made between the results predicted by Lord–Shulman theory and dual-phase-lag (DPL) model. The results obtained by applying both of the L–S theory and DPL model are shown to be very close to each other except in determining one of the components of the electric displacement where the results differ and in general the effect of the presence of gravity is to weaken the absolute values of the physical quantities except in the case of the same component of the electric displacement.


Piezo-thermoelasticity Gravity Dual-phase-lag model Normal modes Generalized thermoelasticity 

List of symbols


The mechanical displacement


Electric potential


Absolute temperature

\(\varepsilon _{ij}\)

Strain tensor

\(\sigma _{ij}\)

Stress tensor

\(\beta _{ij}\)

Thermoelastic tensor


Electric field


Electric displacement


Elastic stiffness tensor


Piezoelectric tensor

\(\in _{ij}\)

The dielectric moduli


Pyroelectric moduli

\(\tau _{\theta }\)

Phase lag of temperature gradient

\(\tau _{q}\)

Phase lag of the heat flux


Heat conduction tensor


Reference temperature


Specific heat at constant strain


Mass density

\(\alpha _{1} ,\alpha _{3}\)

Coefficients of linear thermal expansion

\(v_{p}=\sqrt{\frac{1}{\rho }C_{11}}\)

Longitudinal wave velocity



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ethar A. A. Ahmed
    • 1
    Email author
  • M. S. Abou-Dina
    • 1
  • A. R. El Dhaba
    • 2
  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceDamanhour UniversityDamanhourEgypt

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