Abstract
In this work, taking into account the influences of thermal and diffusion, a new four-phase-lag model comprising the macroscopic and microscopic is constructed. The introduced model is an extension of the papers of Nowacki (Bull Acad Pol Sci Ser Sci Tech 22:55–64, 1974; Bull Acad Pol Sci Ser Sci Tech 22:129–135, 1974; Bull Acad Pol Sci Ser Sci Tech 22:257–266, 1974; Proc Vib Prob 15:105–128, 1974), Sherief et al. (Int J Eng Sci 42:591–608, 2004) and Aouadi (J Therm Stress 30:665–678, 2007). In this model, the Fourier’s and the Fick’s laws have been modified to include higher-order time derivatives of heat flow vector, the gradient of temperature, diffusing mass flux and gradient of chemical potential. Many models in the thermoelastic-diffusion field have been deduced as special cases from the current investigation. Using the resulting formulation, we have studied a thermoelastic-diffusion interaction in a half-space exposed to thermal and chemical shock. Also, the sensitivity of the studied field variables to the variation of the parameters of higher-order time derivative has been investigated.
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Abbreviations
- \(\lambda , \mu \) :
-
Lamé’s constants
- \(\alpha _{t}\) :
-
Thermal expansion coefficient
- \(\alpha _{c}\) :
-
Coefficient of linear diffusion
- \(\beta _{1}=(3\lambda +2\mu )\alpha _{t}\) :
-
Thermal coupling parameter
- \(T_{0}\) :
-
Environmental temperature
- \(\theta =T-T_{0}\) :
-
Temperature increment
- T :
-
Absolute temperature
- \(C_{e}\) :
-
Specific heat
- \(e=\mathrm{div}\,{{\varvec{u}}}\) :
-
Cubical dilatation
- \(\sigma _{ij}\) :
-
Stress tensor
- \(e_{ij}\) :
-
Strain tensor
- \({{\varvec{u}}}\) :
-
Displacement vector
- \({{\varvec{q}}}\) :
-
Heat flux vector
- \({\varvec{\eta }}\) :
-
Flow of diffusing mass vector
- l, m, n, h :
-
Higher-order time derivatives
- K :
-
Thermal conductivity
- \(\rho \) :
-
Material density
- Q :
-
Heat source
- \(\beta _{2}=(3\lambda +2\mu )\alpha _{c}\) :
-
Diffusion coupling parameter
- \(\delta _{ij}\) :
-
Kronecker’s delta function
- \(\nabla ^{2}\) :
-
Laplacian operator
- \(\tau _{q}\) :
-
Phase lag of heat flux
- \(\tau _{\theta }\) :
-
Phase lag of temperature gradient
- \(\tau _{\eta }\) :
-
Phase lag of diffusing mass
- \(\tau _{p}\) :
-
Phase lag of chemical potential gradient
- \(\alpha \) :
-
Fractional order
- D :
-
Diffusion coefficient
- P :
-
Chemical potential
- C :
-
Concentration of diffusion material
- a :
-
Thermoelastic diffusion effect
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Abouelregal, A.E. Generalized mathematical novel model of thermoelastic diffusion with four phase lags and higher-order time derivative. Eur. Phys. J. Plus 135, 263 (2020). https://doi.org/10.1140/epjp/s13360-020-00282-2
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DOI: https://doi.org/10.1140/epjp/s13360-020-00282-2