Refined multi-phase-lags theory for photothermal waves of a gravitated semiconducting half-space

doi:10.1016/j.compstruct.2019.01.015

Composite Structures

Volume 212, 15 March 2019, Pages 346-364

https://doi.org/10.1016/j.compstruct.2019.01.015 Get rights and content

Abstract

A refined multi-phase-lags theory for thermoelastic photothermal response of half-space semiconducting medium is presented. The semiconducting medium is subjected to the internal heat source as well as a gravity effect. The photothermal wave propagation of a gravitated semiconducting half-space has been examined. A fourth equation for the plasma transport is added to the old thermoelastic partial differential equations. All coupled photo-thermoelastic equations have been resolved exactly due to the normal mode model. A harmonic wave solution is adopted to derive the main variables of the medium. The temperature, horizontal and vertical displacements, stresses, and carrier density have been obtained. A comparison is made to show the dependency of all field variables on the internal heat source and the inclusion of gravity. Most variables are very sensitive to the variation of the heat source and gravity factor. Results are tabulated to serve as benchmarks for future comparisons and additional results have been displayed to show the physical meaning of the phenomena.

Introduction

Semiconducting materials have been widely applied in modern engineering applications with the recent development of technologies. Most generalized theories of thermoelasticity dealt with the wave propagation in elastic media without considering the interaction between coupled plasma and thermal effects. The first theory has been presented by Biot [1] in which he developed the Fourier’s law and proposed his theory of coupled thermoelasticity. After that, many authors have investigated Biot’s theory and developed it to get the well-known generalized thermoelasticity theories [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. These generalized theories are more interesting than the original one and have been used in many applications.

However, the problem of wave propagation in a semiconducting medium has become a more important academic and applicable value. The wave propagation problem in semiconducting media during a photothermal process has been investigated by a lot of authors. Some of them are listed here. Song et al. [18] presented the coupled generalized thermoelastic and plasma theories to investigate the reflection problem at the surface of a semi-infinite semiconducting medium during a photothermal process. Abo-Dahab and Lotfy [19] discussed the two-temperature plane-strain problem in a semiconducting medium with the aid of the photothermal theory. Lotfy [20] developed the elastic wave motion for a photothermal medium under an internal heat source in the presence of gravity based on the simple dual-phase-lags (DPL) theory. Othman et al. [21] presented the generalized thermoelasticity model based on the Lord–Shulman (L–S) theory to discuss the photothermal waves in a semiconducting medium.

Lofty [22] presented the DPL model with two different time translations and L–S theory to discuss the effect of hydrostatic initial stress on a medium. He introduced a new model by using two-temperature and photothermal theories. Abbas et al. [23] investigated the photothermal waves in an unbounded semiconducting medium with cylindrical cavity. Lotfy [24] considered a 1D problem of waves in a thermoelastic semiconducting medium with a spherical cavity. Lotfy and Sarkar [25] discussed a new 1D model for an elastic semiconducting medium to describe the interaction between the photothermal excitation and the two-temperature theory. Alzahrani and Abbas [26] presented the fractional order theory for thermal, elastic and plasma waves to obtain different field variables in semiconducting media.

The first purpose of the present paper is to solve a system of four coupled thermoelastic differential equations with photothermal process. The refined multi-phase-lags (RPL) theory for thermoelastic photothermal response of a half-space semiconducting medium is presented. The inclusion of gravity and the internal heat source is discussed. A harmonic wave solution is adopted to derive the main variables of the semiconducting medium. The analytical expressions for the displacements, temperature, carrier density and stresses are obtained. The influences of gravity and multi-phase-lags on the considered field quantities in the absence and presence of internal heat source are tabulated and illustrated graphically.

Section snippets

Basic equations

The generalized homogeneous isotropic thermoelastic medium ( $z \geq 0$ ) is considered with the inclusion of photothermal effect. The coordinates $x$ , $y$ , and $z$ are considered with origin at $y = 0$ , such that $z$ -axis is setting normal to the medium. In here, the medium is subjected to a uniform temperature $T_{0}$ in the undisturbed state. All field quantities are given in terms of the coordinates $x$ , $z$ , and a time $t$ . The fundamental equations for a linear, isotropic and homogeneous thermoelastic medium may be

Formulation of the problem

For the present half-space, all variables depend on the time $t$ and the coordinates $(x, z)$ . For the 2D problem, one suppose that the displacement vector is in the form of $u_{1} = u (x, z, t), u_{2} = 0, u_{3} = w (x, z, t) .$

The above constitutive equations may be simplified as $\begin{matrix} σ_{11} = (λ + 2 μ) \frac{\partial u}{\partial x} + λ \frac{\partial w}{\partial z} - γ (1 + τ_{1} \frac{\partial}{\partial t}) θ - δ_{n} N, \\ σ_{13} = μ (\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}), \\ σ_{33} = λ \frac{\partial u}{\partial x} + (λ + 2 μ) \frac{\partial w}{\partial z} - γ (1 + τ_{1} \frac{\partial}{\partial t}) θ - δ_{n} N . \end{matrix}$

So, the above equations of motion may be simplified as $\begin{matrix} μ \nabla^{2} u + (λ + μ) \frac{\partial e}{\partial x} - γ (1 + τ_{1} \frac{\partial}{\partial t}) \frac{\partial θ}{\partial x} - δ_{n} \frac{\partial N}{\partial x} = ρ (\frac{\partial^{2} u}{\partial t^{2}} - g \frac{\partial w}{\partial x}), \\ μ \nabla^{2} w + (λ + μ) \frac{\partial e}{\partial z} - γ (1 + τ_{1} \frac{\partial}{\partial t}) \frac{\partial θ}{\partial z} - δ_{n} \frac{\partial N}{\partial z} = ρ (\frac{\partial^{2} w}{\partial t^{2}} + g \frac{\partial u}{\partial x}), \end{matrix}$ where $g$ is the gravity

Normal mode model

The closed-form solution of the different quantities can be obtained by using the normal modes as expressed here $\begin{matrix} \{ϕ, ψ, θ, N\} (x, z, t) = \{ϕ^{*}, ψ^{*}, θ^{*}, N^{*}\} (z) e^{(i ξ x + ω t)}, \\ Q = Q_{0} e^{(i ξ x + ω t)}, \end{matrix}$ where $ω$ represens the complex frequency, $i = \sqrt{- 1}$ , $ξ$ denotes the wave number in the $x$ -direction, and $ϕ^{*} (z)$ , $ψ^{*} (z)$ , $θ^{*} (z)$ and $N^{*} (z)$ represent amplitudes of all field variables. Using Eq. (20) as well as then the governing equations, Eqs (14), (15), (16), (17), we obtain $(D^{2} - c_{6}) ϕ^{*} + c_{7} ψ^{*} - (1 + ω τ_{1}) θ^{*} - N^{*} = 0,$ $c_{8} ϕ^{*} - (D^{2} - c_{9}) ψ^{*} = 0,$ $c_{1} θ^{*} + (D^{2} - c_{10}) N^{*} = 0,$ $c_{4} {\bar{l}}_{q} ω (D^{2} - ξ^{2}) ϕ^{*} - ({\bar{l}}_{θ} D^{2} - c)$

Boundary conditions

In this section, we will apply the boundary conditions at the surface $z = 0$ to calculate the integration parameters $C_{j}$ ( $j = 1, 2, 3, 4$ ).

Different thermoelasticity theories

The closed-form solution is already given for the generalized multi-phase-lags theory. The heat conduction equation with the carrier density effect appeared in Eq. (3) contains at least four generalized thermoelasticity theories. One of those theories is the dual-phase-lags which is defined by $K (1 + τ_{θ} \frac{\partial}{\partial t}) \nabla^{2} θ + \frac{E_{G}}{τ} N = (1 + τ_{q} \frac{\partial}{\partial t} + \frac{τ_{q}^{2}}{2} \frac{\partial^{2}}{\partial t^{2}}) (ρ C_{e} \frac{\partial θ}{\partial t} + γ T_{0} \frac{\partial e}{\partial t}) .$

The simplest form of the parabolic type of the heat conduction equation with dual-phase-lags is given by neglecting the effect of term $τ_{q}^{2}$ in Eq. (50) as $K (1 +)$

Conclusions

In this paper, the refined multi-phase-lags thermoelasticity theory for a semiconducting half-space medium with and without the inclusion of gravity has been developed. The photothermal wave propagation of a gravitated thermoelastic semiconducting half-space has been taken into consideration. The equation of coupled plasma transport is expressed in the form of coupled thermoelectronic wave equation. So, four partial differential thermoelastic coupled equations have been resolved exactly. Some

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Refined multi-phase-lags theory for photothermal waves of a gravitated semiconducting half-space

Abstract

Introduction

Section snippets

Basic equations

Formulation of the problem

Normal mode model

Boundary conditions

Different thermoelasticity theories

Conclusions

J Mech Phys Solids

Chaos, Solitons Fractals

Results Phys

Optics Laser Techno

Results Phys

Thermoelasticity and irreversible thermodynamics

J Appl Phys

Thermoelasticity

J Elast

Thermoelasticity with second sound: a review

Appl Mech Rev

A re-examination of the basic postulates of thermomechanics

Proceeding Royal Soc Cityplace London A

On undamped heat waves in an elastic solid

J Therm Stresses

Thermoelasticity without energy dissipation

J Elast

A unified filed approach for heat conduction from macro to macroscales

ASME J Heat Transfer

Hyperbolic thermoelasticity: a review of recent literature

Appl Mech Rev

Reflection of generalized thermoelastic waves from the boundary of a half-space

J Therm Stresses

Effect of phase lags on thermoelastic functionally graded microbeams subjected to ramp-type heating

IJST, Trans Mech Eng

Effects of phase-lags in a thermoviscoelastic orthotropic continuum with a cylindrical hole and variable thermal conductivity

Archives Mech

Two-dimensional coupled solution for thermoelastic beams via generalized dual-phase-lags model

Math Model Analys

Effects of phase-lags and variable thermal conductivity in a thermoviscoelastic solid with a cylindrical cavity

Honam Math J