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The boundary value problem of a threedimensional generalized thermoelastic halfspace subjected to moving rectangular heat source
 Hamdy M. Youssef^{1, 2}Email authorView ORCID ID profile and
 Eman A. N. AlLehaibi^{3}
 Received: 4 September 2018
 Accepted: 2 January 2019
 Published: 15 January 2019
Abstract
This paper deals with a new mathematical model of threedimensional generalized thermoelasticity which has been improved using Lord–Shulman theory. The governing equations on nondimensional forms have been applied to a threedimensional halfspace subjected to a rectangular moving heat source and tractionfree surface by using the Laplace and double Fourier transform techniques. The inverses of the double Fourier and Laplace transforms have been calculated numerically by applying the complex formula of inversion of the transform by of the Fourier expansion method. The numerical results of the temperature increment, strain, stress, and displacement distributions have been represented in graphs for various values of the heat source speed parameter to show its effect on the thermomechanical waves. The heat source speed parameter leads to significant effects on both the thermal and mechanical waves.
Keywords
 Threedimensional modeling
 Thermoelasticity
 Laplace transforms
 Double Fourier transforms
 Moving heat source
1 Nomenclature
 λ, μ:

Lame’s parameters
 ρ :

Density
 \(C_{E}\) :

Specific heat of the material with constant strain
 t :

Time
 T :

Absolute temperature
 \(T_{0}\) :

Reference temperature
 θ :

\(= ( T  T_{0} )\) Temperature increment
 \(\alpha _{T}\) :

Linear thermal expansion coefficient
 γ :

\(= \alpha _{T} ( 3\lambda + 2\mu )\)
 \(\sigma _{ij}\) :

Stress tensor
 \(e_{ij}\) :

Strain tensor
 \(u_{i}\) :

Displacement components
 K :

Thermal conductivity
 \(\tau _{o}\) :

Relaxation time
 \({c_{o}}\) :

\(= \sqrt{\frac{\lambda + 2 \mu }{\rho }}\)
 η :

\(= \frac{\rho C_{E}}{K}\)
 ε :

\(= \frac{\gamma ^{2}T_{0}}{\rho C_{E} (\lambda + 2\mu )}\)
 β :

\(= ( \frac{\lambda + 2\mu }{\mu } )^{1 / 2}\)
2 Introduction
The difficulty in applications and in solving problems by using the mode of the Fourier transform is how the boundary conditions will be transformed. The important functions have been adapted by order and integrals of the eigenfunctions, depending on a definite system of coordinates (Morse, Feshbach [12]). To obtain the stresses and displacements, dissimilar derivatives of weight must be suited. Satisfying the limit conditions is a somewhat perplex work due to the appearance of elasticity coefficients in distinct and high powers in the denominators of the equations (Musii [13]). Podil’chuk and Kirichenko used the Fourier method (Podnil’Chuk, Kirichenko [16]) to construct a new approach for calculating the exact solutions to threedimensional thermoelasticity problems in different coordinate systems.
A lot of numerical and computational methods can be found in viscoelasticity and thermoelasticity research (Danyluk et al. [2]; Oza et al. [14]; Vinogradov, Milton [21]). Laplace transform method is one of the wellknown methods for thermoelasticity and viscoelasticity. De Chant used the numerical inversion rule and its limitations in the asymptotic and discontinuities methods (De Chant [3]). Temel got the solutions by applying the numerical method of Durbin using the Laplace transform inversions in the real space (Temel et al. [19]). Because of the intricacy of the determining relations, it is very difficult to gain the exact solutions of thermoelasticity, and the numerical approach has been preferred lately with the advances in the information processing system software incorporating the boundary and finite element methods (Mesquita, Coda [11]).
Youssef and Ezzat solved some models of threedimensional thermoelasticity in the context of a different theory of thermoelasticity (Ezzat, Youssef [5–7]). Youssef studied many models of a thermoelastic material subjected to moving heat source in the context of different theorems of generalized thermoelasticity (Youssef [23, 24]). Youssef and AlLehaibi solved a problem of a threedimensional generalized thermoelastic diffusion for a thermoelastic halfspace subjected to rectangular thermal pulse (Youssef, AlLehaibi [25]). Parnell et al. employed the Wiener–Hopf technique and a modified Cagniard–de Hooptype scheme, a rapidly convergent integral expression has been determined for a class of transient thermal mixed boundary value problems (Parnell et al. [15]). Kumar and Ailawalia studied the moving load response in a micropolar thermoelastic medium without energy dissipation possessing cubic symmetry (Kumar, Ailawalia [9]). Marin constructed an approach of a heatflux dependent theory for a micropolar porous body, including voidage time derivative among the independent constitutive variables (Marin [10]). Sarkar and Lahiri discussed the interactions due to moving heat sources in a generalized thermoelastic halfspace using LS model (Sarkar, Lahiri [17]). Tahouneh and Naei studied the effect of multidirectional nanocomposite materials on the vibrational response of thick shell panels with finite length and rested on twoparameter elastic foundations (Tahouneh, Naei [18]). The asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions has been studied in (Ghisi et al. [8]). The global stability of rarefaction waves for the compressible Navier–Stokes equations has been investigated in (Duan et al. [4]).
In this work, a new mathematical model of threedimensional generalized thermoelasticity will be constructed by using Lord–Shulman model. The governing equations on nondimensional forms will be applied to a threedimensional halfspace subjected to a rectangular moving heat source and tractionfree surface by using the Laplace and double Fourier transform techniques. The numerical results of the temperature increment, strain, stress, and displacement distributions will be represented in figures for various values of the heat source speed parameter to show its effect on the thermomechanical waves.
3 The basic equations
The system of partial differential equations of a homogeneous and isotropic thermoelastic medium based on the generalized thermoelasticity with one relaxation time and without any external body forces in undefined coordinates \(\{ i,j,k = 1,2,3 \}\) takes the following form (Ezzat, Youssef [5]; Youssef, AlLehaibi [25]):
4 Problem formulation
5 Applying the Laplace transform on the governing equations
We assumed the medium is subjected to a rectangular heat source moving with constant speed and constant strength, releasing its energy continuously on a band of constant dimensions \(2a \times 2b\) centered on the yaxis and zaxis, respectively, while moving with constant speed υ along the xaxis and being zero elsewhere as in Fig. 1.
6 Applying the double Fourier transform
7 Inversion of both Fourier and Laplace transforms
To get the final solution in its original variables, we should calculate the inverse of the double Fourier and Laplace transforms in Eqs. (70)–(72). These expressions may be formally written as functions of x, and all the parameters of the Fourier and Laplace transforms, namely p, q, and s, in the form \(\tilde{\bar{f}} ( x,q,p,s )\) (Ezzat, Youssef [5–7]).
8 Numerical results and discussion
The computations were running out for nondimensional time \(t = 0.1\), length \(a = b = 1.0\), and heat source intensity \(Q_{0} = 1.0\). The temperature, strain, stress, and displacement distributions are shown in graphs.
The value of speed of the heat source has significant effects on all the studied functions. Increasing the values of the speed of the heat source leads to decreasing the temperature increment values and the absolute values of the stress before the peaks, while after the peaks, the increase of the speed of the heat source leads to increases in them. Increasing the values of the speed of the heat source leads to decreasing of the strain and the displacement values. The results and the figures have max or min points because the effect of the heat source is working in the interval \(x < \upsilon t\) so the temperature increment increases rapidly, while in the interval \(x \ge \upsilon t\) the heat source disappears, which makes the temperature increment decrease rapidly. Accordingly, all the other functions will be affected by this attitude. In other words, the value of the position \(x = \upsilon t\) is a critical position because it separates between the existence of the heat source and its disappearance. The effects of the moving heat source of the current results agree with the results in (Youssef [22–24]; Youssef, AlLehaibi [25]).
9 Conclusions

The value of the speed of the heat source has significant effects on the temperature increment, stress, strain, and displacement distributions.

The values of the peak points for all the studied functions increase when the speed of the moving heat source increases.

The temperature increment, stress, strain, and displacement have different behavior before and after the critical position \(x = \upsilon t\), which separates between the existence of the heat source and its disappearance.

The positions along the y and z axes have significant effects on all the studied functions.
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
This work has not been supported by any fund from anywhere.
Authors’ contributions
HY constructed the problem and the governing equations and advised the coauthor on getting the numerical solutions. HY wrote the following sections: abstract, introduction, the basic equation, the formulation of the problem, conclusions, and reviewed all the work. EA solved the problem, applied the Laplace transform on the governing equations and its inversions, prepared figures of the numerical results, and wrote the remaining sections. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Abbas, I.A., Youssef, H.M.: Twodimensional fractional order generalized thermoelastic porous material. Lat. Am. J. Solids Struct. 12(7), 1415–1431 (2015) View ArticleGoogle Scholar
 Danyluk, M., Geubelle, P., Hilton, H.: Twodimensional dynamic and threedimensional fracture in viscoelastic materials. Int. J. Solids Struct. 35(28–29), 3831–3853 (1998) MathSciNetView ArticleGoogle Scholar
 De Chant, L.: Impulsive displacement of a quasilinear viscoelastic material through accurate numerical inversion of the Laplace transform. Comput. Math. Appl. 43(8–9), 1161–1170 (2002) MathSciNetView ArticleGoogle Scholar
 Duan, R., Liu, H., Zhao, H.: Nonlinear stability of rarefaction waves for the compressible Navier–Stokes equations with large initial perturbation. Trans. Am. Math. Soc. 361(1), 453–493 (2009) MathSciNetView ArticleGoogle Scholar
 Ezzat, M.A., Youssef, H.M.: Threedimensional thermal shock problem of generalized thermoelastic halfspace. Appl. Math. Model. 34(11), 3608–3622 (2010) MathSciNetView ArticleGoogle Scholar
 Ezzat, M.A., Youssef, H.M.: Twotemperature theory in threedimensional problem for thermoelastic half space subjected to ramp type heating. Mech. Adv. Mat. Struct. 21(4), 293–304 (2014) View ArticleGoogle Scholar
 Ezzat, M.A., Youssef, H.M.: Threedimensional thermoviscoelastic material. Mech. Adv. Mat. Struct. 23(1), 108–116 (2016) View ArticleGoogle Scholar
 Ghisi, M., Gobbino, M., Haraux, A.: A concrete realization of the slowfast alternative for a semilinear heat equation with homogeneous Neumann boundary conditions. Adv. Nonlinear Anal. 7(3), 375–384 (2018) MathSciNetView ArticleGoogle Scholar
 Kumar, R., Ailawalia, P.: Moving load response in micropolar thermoelastic medium without energy dissipation possessing cubic symmetry. Int. J. Solids Struct. 44(11–12), 4068–4078 (2007) View ArticleGoogle Scholar
 Marin, M.: An approach of a heatflux dependent theory for micropolar porous media. Meccanica 51(5), 1127–1133 (2016) MathSciNetView ArticleGoogle Scholar
 Mesquita, A., Coda, H.: A boundary element methodology for viscoelastic analysis: part I with cells. Appl. Math. Model. 31(6), 1149–1170 (2007) View ArticleGoogle Scholar
 Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. II (2010) MATHGoogle Scholar
 Musii, R.: Equations in stresses for twoand threedimensional dynamic problems of thermoelasticity in spherical coordinates. Mater. Sci. 39(1), 48–53 (2003) View ArticleGoogle Scholar
 Oza, A., Vanderby, R., Lakes, R.S.: Generalized solution for predicting relaxation from creep in soft tissue: application to ligament. Int. J. Mech. Sci. 48(6), 662–673 (2006) View ArticleGoogle Scholar
 Parnell, W.J., Nguyen, V.H., Assier, R., Naili, S., Abrahams, I.D.: Transient thermal mixed boundary value problems in the halfspace. SIAM J. Appl. Math. 76(3), 845–866 (2016) MathSciNetView ArticleGoogle Scholar
 Podnil’Chuk, I.N., Kirichenko, A.: Thermoelastic deformation of a parabolic cylinder. Vychisl. Prikl. Mat. 67, 80–88 (1989) Google Scholar
 Sarkar, N., Lahiri, A.: Interactions due to moving heat sources in generalized thermoelastic halfspace using LS model. Int. J. Appl. Mech. Eng. 18(3), 815–831 (2013) View ArticleGoogle Scholar
 Tahouneh, V., Naei, M.H.: The effect of multidirectional nanocomposite materials on the vibrational response of thick shell panels with finite length and rested on twoparameter elastic foundations. Int. J. Adv. Struct. Eng. 8(1), 11–28 (2016) View ArticleGoogle Scholar
 Temel, B., Çalim, F.F., Tütüncü, N.: Quasistatic and dynamic response of viscoelastic helical rods. J. Sound Vib. 271(3), 921–935 (2004) View ArticleGoogle Scholar
 Tzou, D.: Macro to Micro Heat Transfer. Taylor & Francis, Washington (1996) Google Scholar
 Vinogradov, V., Milton, G.: The total creep of viscoelastic composites under hydrostatic or antiplane loading. J. Mech. Phys. Solids 53(6), 1248–1279 (2005) MathSciNetView ArticleGoogle Scholar
 Youssef, H.: A twotemperature generalized thermoelastic medium subjected to a moving heat source and ramptype heating: a statespace approach. J. Mech. Mater. Struct. 4(9), 1637–1649 (2010) View ArticleGoogle Scholar
 Youssef, H.M.: Generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Mech. Res. Commun. 36(4), 487–496 (2009) MathSciNetView ArticleGoogle Scholar
 Youssef, H.M.: Twotemperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Arch. Appl. Mech. 80(11), 1213–1224 (2010) View ArticleGoogle Scholar
 Youssef, H.M., AlLehaibi, E.A.: Threedimensional generalized thermoelastic diffusion and application for a thermoelastic halfspace subjected to rectangular thermal pulse. J. Therm. Stresses 41(8), 1008–1021 (2018) View ArticleGoogle Scholar