Abstract
A new model for the simulation of moisture and thermal diffusivity in a semiconducting solid cylinder according to the Moore-Gibson-Thompson-Photo-Thermal (MGTPT) theory of thermoelasticity has recently been presented. The purpose of this study is to investigate the photo-thermoelasticity of an infinite semiconducting solid cylinder rotating with the boundary surface being subjected to a laser pulse with variable heat flux. For this purpose, the mathematical model is solved by using the Laplace transform technique in the transformed domain. The numerical inversion of the mathematical model yields all the physical parameters in the physical domain, such as displacement components, thermal stresses, and carrier densities. To clearly illustrate the effects of reference moisture, a graphic representation of all these parameters is generated by using the MATLAB software. The results of this study will be useful in further enhancing the behavior of semiconductors under these dynamic loading conditions and hence, improve their performance in various applications.
Thus, the model provides an effective way to model the moisture and thermal diffusivities of the solid cylinder to better understand phenomena occurring in a broad range of semiconductor devices and more effectively design them.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In recent years, researchers have shifted their focus to the study of the behavior of nanostructured semiconductors that are currently developed in the semiconductor industry. It has been revealed that some models can simulate solid behaviors when moisture and heat are taken into account. Putting heat, moisture, and elasticity together in a calculation will provide a more realistic analysis of hygro-thermoelasticity. It is possible in engineering and science that moisture content can influence heat conduction in some cases and vice versa. When analyzing problems where the analyzed domain or medium is subject to the effects of temperature and moisture, it is important to consider the analogy and cross-coupling between heat conduction and moisture diffusion. Indeed, the combination of property and structure plays a crucial role in influencing the physical, electronic, and optical properties of nanostructures. Utilizing experimental techniques and computer simulations, researchers have formulated models to investigate the influence of nano-dimension and structural characteristics on the behavior of nanostructures. This enhanced understanding of the properties and behavior of nanostructures has paved the way for novel opportunities in technological applications. Nanostructured materials have found utility in a multitude of fields, encompassing electronics, optoelectronics, catalysis, energy storage, sensors, and biomedical devices. Their exceptional properties render them highly appealing for the development of advanced materials boasting enhanced performance and functionality.
The Green and Naghdi [1,2,3] hypothesis presents a reliable explanation for heat conduction in a variety of solids and liquids. It is important to further our understanding of the effects of diffusion in addition to thermal effects. Diffusion refers to the movement of particles from areas of high concentration to areas of low concentration. Thermo-diffusion occurs in elastic solids when temperatures, mass diffusions, and strains are coupled. Heat and mass transfer processes at high and low temperatures are crucial for many satellite problems, returning spacecraft, and landing on land or water. Integrated circuits are manufactured by introducing controlled amounts of dopants into the semiconductor substrate through diffusion. Bipolar transistor bases and emitters, embedded resistors, source & drain of metal–oxide–semiconductor transistors, and polysilicon gates are also formed through diffusion. Fick's law is often used to calculate concentration in these applications, but it does not take into consideration the introduction of a substance into a medium or its effect on that medium.
Tang and Song [4] conducted a comprehensive examination of the reflection of waves in nonlocal semiconductor rotating media. Their study employed a plasma diffusion equation to analyze this phenomenon. Alshaikh [5],on the other hand, focused on investigating the process of photothermal wave transmission through semiconducting media, taking into account the effects of diffusion and rotation. To establish a stable equilibrium state when encountering sudden temperature variations, Cattaneo [6] and Vernotte [7, 8] proposed the incorporation of a relaxation time into the heat flux vector:
Green and Naghdi [1,2,3] have made significant contributions to the field of thermo-elasticity, particularly in the areas of energy dissipation and the absence thereof. Their work has also led to advancements in the understanding of Fourier's law
Numerous academic inquiries have been undertaken in recent times to explore the Moore-Gibson-Thompson equation (MGT). Quintanilla [9, 10] employed the MGT equation with 2 T to formulate a novel heat conduction model. Lasiecka and Wang [11] elucidate fluid dynamics through the utilization of a third-order differential equation. The Modified Fourier law, also known as the MGT equation, can be expressed as follows:
\(\mathrm{where }\dot{\vartheta }=T.\) In the case of semiconductor materials exhibiting plasma effect, it is possible to formulate a generalized Fourier law as follows:
\(\mathrm{ where }\dot{\vartheta }=T.\) In Eq. (4), the final term signifies the photo-excitation effect. Upon differentiating Eq. (4) with respect to \(x\), the outcome is
Hence
where
Hence above equation becomes,
By employing Fourier's law, it is possible to express the heat flux in conduction in the following manner:
The moisture flux potential can be defined by employing Fick's law as:
Numerous experiments have revealed a significant correlation between moisture transport and the temperature field, and vice versa. The presence of moisture concentration gradients gives rise to heat fluxes, commonly referred to as Dufour fluxes. These fluxes can be precisely defined as follows [12,13,14]:
The potential of moisture flux, commonly referred to as the Soret flux, may be influenced by variations in temperature gradients:
Heat and moisture flux potentials can be derived based on the coupled diffusion of heat and moisture as
The aforementioned cross-coupling enables the potential influence of material elastic properties on the temperature and moisture concentration field. By integrating the generalized equations of motion with the coupled generalized equations of heat conduction and moisture diffusion, the phenomenon of coupled hygro-thermoelasticity [12,13,14] can be accurately described using these governing equations as
where
The stress–strain constitutive equation can be found as
The terms \({\beta }_{ij}^{m}={\beta }^{m}{\delta }_{ij}\) and \({\beta }_{ij}^{T}={\beta }^{T}{\delta }_{ij}\) are material coefficients responsible for coupling between the stresses and the moisture concentration or temperature fields, respectively.
El-Sapa et al. [15] have analyzed the one-dimensional photothermal MGT using moisture diffusivity to study moisture and heat equations. Hosseini and Ghadiri [16] have developed a model to investigate the propagation of moisture and thermoelastic waves under shock loading. Aouadi et al. [17] have presented a nonlinear theory of thermoelasticity involving diffusion by applying the Green-Naghdi types II and III theories of thermomechanics of continua. Lotfy et al. [18] have investigated a general solution to the problem of wave propagation under H2T theory in a generalized piezo-photo-thermoelastic medium. Alhashash et al. [19] have proposed a mathematical-physical model based on a two-temperature thermoelasticity theory to describe the influence of moisture diffusivity on semiconductor materials. Lotfy et al. [20] have developed a novel stochastic photo-thermoelasticity model by utilizing diffusion interaction processes within an excited semiconductor medium. Lotfy and Tantawi [21] have undertaken a comprehensive investigation of the intricate relationship between thermoelasticity theory and photothermal theory in a semiconductor, widely recognized as photo-thermoelasticity theory. The photothermal diffusion process is employed by Lotfy [22] to investigate the variable thermal conductivity of semiconductor samples by leveraging the temperature gradient. A thorough investigation has been carried out by Allam [23] regarding a matter concerning the theory of generalized thermoelastic diffusion, with a specific emphasis on the impact of an additive Gaussian white noise.
In addition to the aforementioned studies, it is worth noting that other researchers have also conducted investigations on semiconductor media, similar to the works of Kaur et al.[24,25,26,27,28], Lotfy et al. [29], Craciun et al.[30, 31], Lata et al. [32], Jafari et al. [33], Kaur and Singh[34,35,36], Lotfy and Hassan[37], Craciun et al.[38], Malik et al. [39]. However, it is important to highlight that no research has been conducted on the transient investigation of moisture and thermal diffusivity in a semiconducting solid cylinder within the context of the MGTPT thermoelasticity theory. Therefore, our study aims to explore the photo-thermoelasticity of an infinite semiconducting solid cylinder that is rotating, with the boundary surface being subjected to a laser pulse with a variable heat flux. To solve the proposed mathematical model, we employ the Laplace transform in a transformed domain.
Section 1 presents an overview of the development of the MGTPT heat transfer theory, specifically focusing on the GN III model with considerations for moisture and thermal diffusivity. Section 2 highlights the formulation of the equation of motion and the Modified MGT heat conduction equation for semiconducting mediums, incorporating moisture diffusion. In Sect. 3, the mathematical formulation of the research on semiconductor cylinders is outlined, utilizing the MGTPT heat transfer equation. This formulation allows for the derivation of dimensionless expressions for displacement, stress components, and moisture in the transformed domain through the application of Laplace Transforms. Section 4 examines the boundary conditions for the outer surface of the cylinder, which are influenced by time-dependent variable heat. Sections 5 and 6 provide the solution to the problem and explain the method for performing the inverse Laplace Transform. In Sect. 7, the numerical results and the impact of different thermoelasticity theories and Hall current on physical quantities are visually presented using MATLAB software. Finally, Sect. 8 concludes the paper.
2 Basic equations
The governing equations [40,41,42] for a MGTPT with thermal-plasma-elastic interaction is given by:
1. Constitutive relations
2. Equation of motion
3. Plasma diffusion equation
4. The equation of moisture diffusion for coupled hygro-photo-thermoelasticity is given by
5. Modified MGT heat conduction equation for coupled hygro-photo-thermoelasticity [12,13,14]
\(i\) is not summed.
Also, the subscript followed by ‘,’ a comma denotes the partial derivative w.r.t. space coordinates and a superposed ‘.’ signifies differentiation w.r.t. time variable \(t\).
3 Mathematical model
The semiconductor solid cylinder under consideration possesses the desirable qualities of symmetry, thermal homogeneity, and a radius of \({r}_{0}\). To heat the exterior of the cylinder, an external laser pulse heating device has been employed. Our analysis has been conducted with the \(z\)-axis aligned parallel to the cylinder axis, and a cylindrical coordinate system \(\left(r,\theta ,z\right)\) has been utilized, see Fig. 1. The initial temperatures of the cylinder, denoted as \({T}_{0}\), have been maintained at a constant and uniform.
In addition, it is important to note that, in accordance with the regularity criterion, all fields within a medium are presumed to possess finite characteristics. Moreover, when considering the functions under examination, it is crucial to acknowledge their dependence on both the radial distance \(r\) and time \(t\), particularly in the context of a one-dimensional problem where symmetry is a prevailing factor. Consequently, the expressions representing the relationships between displacement–strain and displacement components can be formulated as follows:
The Eq. (21) using (31) will be the form
Hence, the dynamic motion equation becomes
Using Eqs. (33–36), in Eqs. (37) yields:
Also, Eqs. (26), (28), and (29) can be written as,
Pre-operating both sides of Eq. (38) by \(\left(\frac{1}{r}+\frac{\partial }{\partial r}\right),\) we obtain
The subsequent dimensionless quantities are used to convert the aforementioned equations into dimensionless form:
Applying (43) to Eqs. (39–42) and suppressing the primes, yields
where,
Using Eq. (43) in Eqs. (33–35) and after suppressing the primes, yields
Initial conditions are assumed as
The Laplace transform of a function \(f\) w.r.t. time variable t, is defined as
Using Laplace transforms from Eq. (54) to Eqs. (44–47) yields
where
Using Laplace transforms defined by Eqs. (54) to Eqs. (55–58), we obtain
When Eqs. (55–58) are decoupled, we get
Let \({\lambda }_{i}\), \(i=\mathrm{1,2},\mathrm{3,4}\) be roots of Eq. (62), so we can write
where \({\lambda }_{i}^{2},i=\mathrm{1,2},\mathrm{3,4}\) are the roots of the characteristic equation of Eq. (62)
The general solution of Eq. (63) can be written in the form
where \({I}_{0}\)() specifies the 2.nd form of modified ns of order zero. We get the subsequent expressions by introducing Eq. (65) into Eqs. (55–58)
In the domain of the Laplace transform, the displacement u may be expressed as follows:
where \({I}_{1}\)() specifies the 2nd form of modified Bessel functions of order one. We obtained Eq. (69) using Bessel function relation
Differentiating Eq. (69) in terms of r gives
As a result, the expression for thermal stress in closed form is obtained as follows:
4 Boundary conditions
We assume that the exterior surface of the cylinder is under a force. Because of this, the mechanical boundary condition can be written as
Additionally, a variable heat flux, specifically laser-pulsed heat, is applied at the boundary surface. Consequently, the boundary conditions are
Using dimensionless quantities (43) in Eq. (3) yields
Equations (76) and (77) give the following boundary condition
“The carriers can reach the sample surface during the diffusion phase, with a finite probability of recombination”. The carrier density’s boundary conditions are:
The moisture diffusion boundary condition at the boundary surface at \(\mathrm{ }r={r}_{0}\)
By applying the Laplace transform on the boundary conditions in (75), and after differentiating (68) w.r.t. \(t\) twice and using the value of temperature \(T\), in (78) applying the Laplace transform and also applying Laplace transform in (69) directly yield:
Equations (66) and (69) are substituted into Eq. (81–84), giving
Solving Eqs. (85–88) using Cramer’s rule, we can get the values of \({g}_{i},i=\mathrm{1,2},3\)
\({\Delta }_{i}\) is obtained from \(\Delta\) by replacing its ith column by
And by using the values of \({g}_{i}\left(s\right)\) from (89) in (65), (69), and (72, 73, 74) the expression for displacement, carrier density, stresses, and temperature distribution can be obtained.
5 Inversion of the transforms
The solution in the physical domain is obtained by inverting the transforms in Eqs. (65) and (69) so obtained, by using
Finally, apply Romberg's integration [43] with an adaptive step size to evaluate the integral in Eq. (92).
6 Particular cases
-
If \({K}^{*}\ne 0, {D}_{m}^{*}\ne 0, {D}_{m}\ne 0,K\ne 0\,and\,{\tau }_{0}\ne 0,\) in Eqs. (65), (69), and (72–74) so obtained, the results for the MGTPT for a coupled hygro-photo-thermoelastic system can be obtained.
-
If \({K}^{*}\ne 0,{D}_{m}^{*}\ne 0, {D}_{m}\ne 0, K\ne 0\,and\,{\tau }_{0}=0,\) in Eqs. (65), (69), and (72–74) so obtained, the results for a coupled hygro-photo-thermoelastic system with the GN III model can be acquired.
-
If \(K\ne 0, {D}_{m}\ne 0, and\,{\tau }_{0}=0,\) in Eqs. (65), (69), and (72–74) so obtained, the outcomes of for coupled hygro-photo-thermoelastic system with GN II model can be acquired.
-
If \({\tau }_{0}=0,{K}^{*}=0,and\,{D}_{m}^{*}= 0,\) in Eqs. (65), (69), and (72–74) so obtained, the outcomes of classic coupled photo-hygro-thermoelasticity theory (CPTE) are achieved.
-
If \({K}^{*}=0,{D}_{m}^{*}= 0,\) in equation ((65), (69) and (72)-(74) so obtained, the outcomes of generalized Lord and Shulman photo-hygro-thermoelasticity model (PLS) are achieved.
7 Numerical results and discussion
The following physical data of silicon (Si) material is used to illustrate the theoretical findings and to graphically depict the effects of reference moisture parameter and the MGTPT using MATLAB software:
\(\lambda =3.64\times {10}^{10} N{m}^{-2},\) | \({\mathrm{T}}_{0} = 300\mathrm{ K},\) |
\(\mu =5.46\times {10}^{10} N{m}^{-2},\) | \({\mathrm{H}}_{0} = 1\mathrm{ J}{\mathrm{m}}^{-1}{\mathrm{nb}}^{-1},\) |
\(\beta =7.04\times {10}^{6}N{m}^{-2}{deg}^{-1},\) | \(\uptau =5\times {10}^{-5}\mathrm{ s},\) |
\({\delta }_{n}=-9\times {10}^{-31} {m}^{-3},\) | \({N}_{0}={10}^{20}{m}^{-3},\) |
\(\rho =2.33\times {10}^{3}K{gm}^{-3},\) | \({\upvarepsilon }_{0}= 8.838 \times {10}^{-12}{\mathrm{Fm}}^{-1},\) |
\({C}_{E}=695 JK{g}^{-1}{K}^{-1},\) | \({E}_{g}=1.11eV,\) |
\(K=150 W{m}^{-1}{K}^{-1}\) | \({\alpha }_{T}=3\times {10}^{-6}{K}^{-1},\) |
\({K}^{*}=1.54\times {10}^{2}Ws,\) | \({s}_{v}=2m{s}^{-1}.\) |
\({D}_{E}=2.5\times {10}^{-3}{m}^{2}{s}^{-1},\) | \({H}_{0}={10}^{8}Col.{cm}^{-1}{s}^{-1},\) |
\({\mu }_{0}=4\pi \times {10}^{-7}H{m}^{-1}\) | \({\sigma }_{0}=9.36\times {10}^{5}{Col}^{2}{C}^{-1}{m}^{-1}{s}^{-1}.\) |
The figures analyze four cases of the reference moisture parameter, represented by solid black lines (\(m=20\%\)), solid red lines (\(m=40\%\)), solid blue lines (\(m=80\%\)), and solid purple lines (\(m=100\%\)). The variation in the radial displacement \(u\) for the MGTPT with reference moisture is depicted in Fig. 2, starting from a positive zero value and rapidly increasing until reaching a peak maximum value near half of the radius due to photo-excitation and moisture diffusivity. It has been noted that the radial displacement decreases as the reference moisture increases. For the parameter value \(m=20\%\) of the reference moisture, the radial displacement exhibits the greatest deviation at the center of the cylinder. At \(r=0\) and \(r={r}_{0}\), the radial displacement is zero, satisfying the boundary conditions stated in Eqs. (56) and (82).
The temperature distribution variance is illustrated in Fig. 3 as the reference moisture change, starting for each of the four reference moisture parameter from a positive maximum value and rapidly decrease until reaching the zero value near half of the radius due to photo-excitation and moisture diffusivity. It has been observed that the inner core of the cylinder displays a higher temperature distribution compared to the outer core. Furthermore, an increase in reference moisture leads to an increase in temperature distribution.
The moisture diffusion of the MGTPT at various values of reference moisture is illustrated in Fig. 4. It has been observed that a lower value of reference moisture results in greater variability in moisture diffusion compared to a higher value of reference moisture. The moisture diffusion for each of the for each of the four reference moisture parameter situations starts at zero, reaches maximum values near the surface, and then decreases exponentially to zero as it propagates. At \(r=0\) and \(r={r}_{0}\), the moisture diffusion is zero, satisfying the boundary condition stated in Eq. (87). The variation in the dilatation distributions for the MGTPT at different values of reference moisture is shown in Fig. 5. It has been noticed that lower values of reference moisture exhibit more variability in dilatation distributions compared to higher values of reference moisture. As the radial distance r increases, it has been observed that the variability in dilatation distributions significantly decreases. The figure presented in Fig. 6 illustrates the fluctuation in carrier density for the MGTPT at various reference moisture. It has been observed that an increase in reference moisture values corresponds to an increase in carrier density. Furthermore, it has been noted that as the radial distance r expands, the variance in carrier density significantly diminishes.
Figures 7, 8and9 depict the variation in stress components for the MGTPT with reference moisture. It is noticeable that there is a decrease in the stress components as the values of the reference moisture increase. Furthermore, as the radial distance r grows, there is a significant reduction in the variation of the stress components.
8 Conclusions
In this study, we employ a distinctive methodology to investigate the impact of photo-thermoelasticity theory on moisture diffusivity in an elastic semiconductor cylinder. Specifically, we focus on the generalized Moore-Gibson-Thompson photothermal (MGTPT) model, while also incorporating the GN-III model in our investigation. Our study aims to examine the effects of applying an exponential laser pulse on the surface of an infinite semiconducting solid cylinder. The governing equations are obtained with the use of the MGTPT with thermal and moisture diffusivity.
-
The graphical illustrations provided in this study showcase the impact of reference moisture on various components such as displacement, thermal stresses, dilation distributions, moisture diffusivity, carrier density, and temperature field. These graphs clearly demonstrate that reference moisture factors play a significant role in the examined domains. The thermal effect occurs when semiconductor materials come into contact with concentrated laser beams or beams of sunlight. These materials have extensive applications in the renewable energy sector, particularly in the solar cell industry, where semiconductor materials are heavily relied upon.
-
It has been observed that temperature and moisture greatly influence mechanical deformation and diffusion. The relationship between deformation, temperature, and humidity is evident in a wide range of engineering problems. Therefore, it is crucial to closely monitor moisture and temperature levels in semiconductor fabrication to prevent any issues related to product quality and yield.
-
The model used in renewable energy research and engineering is highly beneficial for improving the performance of semiconductors, including diodes, triodes, modern electronic devices, solar cells, electrical circuits, and computer processors.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
- \({\delta }_{ij}\) :
-
Kronecker delta
- \({e}_{ij}\) :
-
Strain tensors(mm−1)
- \({d}_{n}\) :
-
Coefficient of electronic deformation
- \({D}_{E}\) :
-
Carrier diffusion coefficients
- \({N}_{0}\) :
-
Carrier concentration at equilibrium position
- \({D}_{m}\) :
-
Moisture diffusivity coefficient
- \(\lambda ,\mu\) :
-
Lame’s elastic constants
- \({D}_{T}\) :
-
Temperature diffusivity
- \({\beta }_{ij}\) :
-
Thermal elastic coupling tensor
- \(m\) :
-
Moisture concentration
- \(\kappa\) :
-
Coupling parameter for thermal activation
- \({\tau }_{0}\) :
-
Thermal relaxation parameter
- \({C}_{E}\) :
-
Specific heat at constant strain
- \({\alpha }_{t}\) :
-
Linear thermal expansion coefficient
- \(N\) :
-
Carrier density
- \({a}_{ij}\) :
-
Two temperature parameter
- \({m}_{0}\) :
-
Reference moisture
- \(T\) :
-
Thermodynamic temperature,
- \({s}_{v}\) :
-
Surface recombination velocity
- \({\delta }_{n}\) :
-
Electronic deformation coefficient
- \({u}_{i}\) :
-
Components of displacement(m)
- \({K}^{*}\) :
-
Materialistic constant,
- \({e}_{kk}\) :
-
Cubical dilatation,
- \(\rho\) :
-
Medium density(Kgm−3)
- \({D}_{TM}\),\({D}_{MT}\) :
-
Cross-coupled diffusivities
- \(t\) :
-
Time
- \({K}_{m}\) :
-
Moisture diffusivity
- \(\Omega\) :
-
Angular frequency
- \(K\) :
-
Coefficient of Thermal conductivity
- \({T}_{0}\) :
-
Reference temperature s.t. \(\left|\frac{T}{{T}_{0}}\right|<<1\)
- \({\epsilon }_{ijk}\) :
-
Permutation symbol
- \({E}_{g}\) :
-
Energy gap of the semiconductor parameter
- \(\tau\) :
-
Photo-generated carrier lifetime
- \({D}_{E}\) :
-
Carrier diffusion coefficient
- \({\sigma }_{ij}\) :
-
Stress tensors(Nm−2)
- \(c\) :
-
Heat capacity
References
Green AE, Naghdi PM. A re-examination of the basic postulates of thermomechanics. Proc R Soc London Ser A Math Phys Sci. 1991;432:171–94. https://doi.org/10.1098/rspa.1991.0012.
Green AE, Naghdi PM. On undamped heat waves in an elastic solid. J Therm Stress. 1992;15:253–64. https://doi.org/10.1080/01495739208946136.
Green AE, Naghdi PM. Thermoelasticity without energy dissipation. J Elast. 1993;31:189–208. https://doi.org/10.1007/BF00044969.
Tang F, Song Y. Wave reflection in semiconductor nanostructures. Theor Appl Mech Lett. 2018;8:160–3. https://doi.org/10.1016/j.taml.2018.03.003.
Alshaikh F. Mathematical modeling of photothermal wave propagation in a semiconducting medium due to L-S theory with diffusion and rotation effects. Mech Based Des Struct Mach. 2020. https://doi.org/10.1080/15397734.2020.1776620.
Cattaneo C. A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus. Acad Sci Paris Ser. 1958;II(247):431–3.
Vernotte P. Les paradoxes de la theorie continue de l’equation de lachaleur, Comptes Rendus. Acad Sci Paris Ser. 1958;II(246):3154–5.
Vernotte P. Some possible complications in the phenomena of thermal conduction, Comptes Rendus. Acad Sci Paris Ser. 1961;II(252):2190–1.
Quintanilla R. Moore–Gibson–Thompson thermoelasticity. Math Mech Solids. 2019;24:4020–31. https://doi.org/10.1177/1081286519862007.
Quintanilla R. Moore-Gibson-Thompson thermoelasticity with two temperatures. Appl Eng Sci. 2020;1: 100006. https://doi.org/10.1016/j.apples.2020.100006.
Lasiecka I, Wang X. Moore-Gibson-Thompson equation with memory, part II: general decay of energy. Anal PDEs. 2015. https://doi.org/10.48550/arXiv.1505.07525.
Szekeres A. Analogy between heat and moisture. Comput Struct. 2000;76:145–52. https://doi.org/10.1016/S0045-7949(99)00170-4.
Szekeres A. Cross-coupled heat and moisture transport: part 1—theory. J Therm Stress. 2012;35:248–68. https://doi.org/10.1080/01495739.2012.637827.
Szekeres A, Engelbrecht J. Coupling of generalized heat and moisture transfer. Period Polytech Mech Eng. 2000;44:161–70.
El-Sapa S, Becheikh N, Chtioui H, Lotfy K, Seddeek MA, El-Bary AA, El-Dali A. Moore–Gibson–Thompson model with the influence of moisture diffusivity of semiconductor materials during photothermal excitation. Front Phys. 2023. https://doi.org/10.3389/fphy.2023.1224326.
Hosseini SM, GhadiriRad MH. Application of meshless local integral equations for two-dimensional transient coupled hygrothermoelasticity analysis: Moisture and thermoelastic wave propagations under shock loading. J Therm Stress. 2017;40:40–54. https://doi.org/10.1080/01495739.2016.1224134.
Aouadi M, Lazzari B, Nibbi R. A theory of thermoelasticity with diffusion under Green-Naghdi models. ZAMM J Appl Math Mech Zeitschrift Für Angew Math Und Mech. 2014;94:837–52. https://doi.org/10.1002/zamm.201300050.
Lotfy K, Elidy ES, Tantawi RS. Piezo-photo-thermoelasticity transport process for hyperbolic two-temperature theory of semiconductor material. Int J Mod Phys C. 2021;32:2150088. https://doi.org/10.1142/S0129183121500881.
Alhashash A, Elidy ES, El-Bary AA, Tantawi RS, Lotfy K. Two-temperature semiconductor model photomechanical and thermal wave responses with moisture diffusivity process. Crystals. 2022;12:1770. https://doi.org/10.3390/cryst12121770.
Lotfy K, Ahmed A, El-Bary A, El-Shekhipy A, Tantawi RS. A novel stochastic photo-thermoelasticity model according to a diffusion interaction processes of excited semiconductor medium. Eur Phys J Plus. 2022;137:972. https://doi.org/10.1140/epjp/s13360-022-03185-6.
Lotfy K, Tantawi RS. Photo-thermal-elastic interaction in a functionally graded material (FGM) and magnetic field. SILICON. 2020;12:295–303. https://doi.org/10.1007/s12633-019-00125-5.
Lotfy K. Effect of variable thermal conductivity during the photothermal diffusion process of semiconductor medium. SILICON. 2019;11:1863–73. https://doi.org/10.1007/s12633-018-0005-z.
Allam AA. A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source. Appl Math Model. 2014;38:4995–5021. https://doi.org/10.1016/j.apm.2014.03.044.
Kaur I, Singh K, Craciun E-M. A mathematical study of a semiconducting thermoelastic rotating solid cylinder with modified moore–gibson–thompson heat transfer under the hall effect. Mathematics. 2022;10:2386. https://doi.org/10.3390/math10142386.
Kaur I, Lata P, Singh K. Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures. Int J Mech Mater Eng. 2020. https://doi.org/10.1186/s40712-020-00122-2.
Kaur I, Singh K, Craciun E-M. New modified couple stress theory of thermoelasticity with hyperbolic two temperature. Mathematics. 2023;11:432. https://doi.org/10.3390/math11020432.
Kaur I, Singh K, Craciun E-M. Recent advances in the theory of thermoelasticity and the modified models for the nanobeams: a review. Discov Mech Eng. 2023;2:2. https://doi.org/10.1007/s44245-023-00009-4.
Kaur I, Singh K, Marius G, Ghita D, Craciun EM. Modeling of a magneto-electro-piezo-thermoelastic nanobeam with two temperature subjected to ramp type heating. 2022;23:141–149.
Lotfy K, El-Bary AA, Tantawi RS. Effects of variable thermal conductivity of a small semiconductor cavity through the fractional order heat-magneto-photothermal theory. Eur Phys J Plus. 2019;134:280. https://doi.org/10.1140/epjp/i2019-12631-1.
Craciun EM, Rabaea A, Das S. Cracks interaction in a pre-stressed and pre-polarized piezoelectric material. J Mech. 2020;36(2):177–82. https://doi.org/10.1017/jmech.2019.57.
Craciun E-M, Baesu E, Soós E. General solution in terms of complex potentials for incremental antiplane states in prestressed and prepolarized piezoelectric crystals: application to Mode III fracture propagation. IMA J Appl Math. 2004;70:39–52. https://doi.org/10.1093/imamat/hxh060.
Lata P, Kaur I, Singh K. Deformation in transversely isotropic thermoelastic thin circular plate due to multi-dual-phase-lag heat transfer and time-harmonic sources. Arab J Basic Appl Sci. 2020;27:259–69. https://doi.org/10.1080/25765299.2020.1781328.
Jafari M, Chaleshtari MHB, Abdolalian H, Craciun E-M, Feo L. Determination of forces and moments per unit length in symmetric exponential FG plates with a Quasi-Triangular Hole. Symmetry (Basel). 2020;12:834–50. https://doi.org/10.3390/sym12050834.
Kaur I, Singh K. A study of influence of hall effect in semiconducting spherical shell with moore-gibson-thompson-photo-thermoelastic model. Iran J Sci Technol Trans Mech Eng. 2022. https://doi.org/10.1007/s40997-022-00532-x.
Kaur I, Singh K. Plane wave in non-local semiconducting rotating media with Hall effect and three-phase lag fractional order heat transfer. Int J Mech Mater Eng. 2021;16:1–16. https://doi.org/10.1186/S40712-021-00137-3/FIGURES/16.
Kaur I, Singh K. The two-temperature effect on a semiconducting thermoelastic solid cylinder based on the modified Moore – Gibson – Thompson heat transfer St. Petersbg. Polytech Univ J Phys Math. 2023;16:65–81. https://doi.org/10.18721/JPM.16106.
Lotfy K, Hassan W. Normal mode method for two-temperature generalized thermoelasticity under thermal shock problem. J Therm Stress. 2014;37:545–60. https://doi.org/10.1080/01495739.2013.869145.
Craciun EM, Carabineanu A, Peride N. Antiplane interface crack in a pre-stressed fiber-reinforced elastic composite. Comput Mater Sci. 2008;43:184–9. https://doi.org/10.1016/j.commatsci.2007.07.028.
Malik S, Gupta D, Kumar K, Sharma RK, Jain P. Reflection and transmission of plane waves in nonlocal generalized thermoelastic solid with diffusion. Mech Solids. 2023;58:161–88. https://doi.org/10.3103/S002565442260088X.
Mahdy AMS, Lotfy K, Ahmed MH, El-Bary A, Ismail EA. Electromagnetic Hall current effect and fractional heat order for microtemperature photo-excited semiconductor medium with laser pulses. Results Phys. 2020;17: 103161. https://doi.org/10.1016/j.rinp.2020.103161.
Abouelregal AE, Atta D. A rigid cylinder of a thermoelastic magnetic semiconductor material based on the generalized Moore–Gibson–Thompson heat equation model. Appl Phys A Mater Sci Process. 2022;128:1–14. https://doi.org/10.1007/S00339-021-05240-Y/TABLES/7.
Youssef HM, El-Bary AA. Theory of hyperbolic two-temperature generalized thermoelasticity. Mater Phys Mech. 2018. https://doi.org/10.18720/MPM.4022018_4.
Press WH, Teukolsky SA, Flannery BP. Numerical recipes in Fortran. Cambridge: Cambridge University Press; 1980.
Acknowledgments
Not applicable.
Funding
No fund /grant / scholarship has been taken for the research work.
Author information
Authors and Affiliations
Contributions
KS: Conceptualization, Effective literature review, Experiments and Simulation, Investigation, Methodology, Software, Supervision, Validation, Visualization, Writing—original draft. IK: Idea formulation, Conceptualization, Formulated strategies for mathematical modelling, methodology refinement, Formal analysis, Validation, Writing—review & editing. E-MC: Idea formulation, Formulated strategies for mathematical modelling, methodology refinement, Formal analysis, Validation, Writing- review & editing. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Singh, K., Kaur, I. & Craciun, EM. Hygro-photo-thermoelastic solid cylinder under moisture and thermal diffusivity with Moore-Gibson-Thompson theory. Discov Mechanical Engineering 2, 21 (2023). https://doi.org/10.1007/s44245-023-00028-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s44245-023-00028-1