Abstract
The current investigation deals with the deformation of a non-homogeneous thermoelastic half space under hydrostatic initial stress for the Green–Naghdi model III. The medium is supposed to be rotating with a constant angular velocity. The non-homogeneous properties of the material are along the x-direction. At the first instance, the problem has been solved analytically to obtain stress and displacement components. Further, the numerical values of these expressions are evaluated using a computer program for a particular medium. The numerical values obtained are then presented graphically to show the effect of initial stress parameter and non-homogeneity parameter on the quantities.
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1 Introduction
The coupled thermoelastic theory was first given by Biot [1] to remove the deficiency of the uncoupled theory which states that the thermal waves propagate with infinite speed. It was observed that the classical theory of thermoelasticity needs modifications. Hence, different theories of thermo-elasticity were established. Lord and Shulman [2] formulated a theory of thermoelasticity by including a flux rate term and another constant called the thermal relaxation time by modifying Fourier’s law of heat conduction. Without affecting Fourier’s law of heat conduction, Green and Lindsay [3] established a theory which depends on the temperature rate by including two constants termed as thermal relaxation times. The two theories mentioned above confirm the finite speed of heat propagation. This wave like thermal disturbance was called as second sound by Chandrasekharaiah [4]. Green and Nagdhi [5,6,7] proposed three new thermoelastic theories known as G-N model of type I, type II, and type III. The type I theory is reduced to the classical theory of thermoelasticity in linearized form. The G-N model of type II predicts the finite velocity for thermal waves without energy dissipation and model of type III allows thermal signals to propagate with finite speed. Barber and Martin-Moran [8] solved the isothermal contact problem of traction free bodies. Marin [9, 10] investigated some important problems in thermoelastic micropolar bodies. Sharma and Marin [11] discussed wave propagation in micropolar thermoelastic solid half space with distinct conductive and thermodynamic temperatures. Marin and Öchsner [12] studied the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. The solution of mixed problems in partial differential equations existing in boundary value problems is given by Marin and Öchsner [13]. Vlase et al. [14] presented a method to simplify the calculus of the eigenmodes of a mechanical system with bars.
Many researchers [15,16,17,18,19] studied the propagation of waves in initially stressed solids for different models. Different problems related to rotating media were investigated by many researchers in the past. Schoenberg and Censor [20], Clarke and Burdness [21], Destrade [22] discussed the effect of rotation on elastic waves. Sharma and Othman [23] discussed the effect of rotation on generalized thermo-viscoelastic Rayleigh waves. Othman and Song [24] studied the effect of rotation in two dimensional problem for a magneto-thermo-elasticity half space under various theories. Ailawalia et al. [25, 26] investigated the effect of rotation on stress and displacement components in thermoelastic half space. Marin et al. [27] constructed numerical estimations of a nonlinear hyperbolic bioheat equation under various boundary conditions for medicinal treatments of tumor cells. Marin et al. [28] studied the mixed initial-boundary value problem in the context of the Moore-Gibson-Thompson theory of thermoelasticity for dipolar bodies. A problem related to Rayleigh waves in an initially stressed two temperature magneto thermoelastic media was studied by Kumari et al. [29]. Hetnarski and Ignaczak [30] reviewed the concept of generalised thermoelasticity. Sharma et al. [31] and Sharma and Chauhan [32] studied the effect of mechanical and thermal sources in an isotropic thermoelastic medium. Sharma et al. [33] discussed the effect of load moving in a homogeneous thermoelastic layer. The problem of moving load response in a thermo-diffusive elastic solid was discussed by Deswal and Choudhary [34].
Over the past few decades, the concept of functionally graded material (FGM) has caught the attention of researchers. FGMs are generally used in high temperature environments. Thermal stresses like sudden heating and cooling can easily being controlled by FGM. Some prominent work [34, 35, 35,36,37,38,39,40,41,42,43] has been done by researchers in the field of functionally graded materials of different types. In the recent couple of years a lot of papers have been published in the field of non-homogeneous thermoelastic media [44,45,46,47,48,49,50].
The present research article deals with the study of deformation of a rotating functionally graded thermoelastic half space under hydrostatic initial stress. The expressions are obtained for displacement, temperature distribution and force stress by using the normal mode technique. The variation of these field variables are calculated numerically. The effect of initial stress and non-homogeneity parameter are shown graphically in the derived expressions using MATLAB.
2 Basic equations
It is considered that the free surface of the initially stressed thermoelastic half space is rotating with a uniform angular velocity of magnitude \(\Omega \). For a rotating media, two additional terms defined as centripetal acceleration \(\vec {\Omega }\times (\vec {\Omega }\times \vec {u})\) and Coriolis acceleration \(2\Omega \times \dot{\vec {u}}\) are included in the displacement equation. All condidered quantities are functions of t, x and z. The thermoelastic medium is under the influence of a mechanical/thermal source of constant magnitude. The model of the problem is such that the free surface lies along x=0 and the x- axis is pointing vertically downwards.
The equation of motion and stress–strain relation for a homogenous, isotropic rotating thermoelastic medium under hydrostatic initial stress and in the absence of body forces and heat sources is given by Montanaro [15], Green and Nagdhi [7] and Schoenberg and Censor [20]:
The heat equation in the context of GN theory of type III is given by:
where \(T=\theta - T_{0}\), \(T_{0}\) is the initial temperature, \(\lambda \), \(\mu \) are the elastic constants termed as Lame’s parameter, U is the specific internal energy measured from the reference state, \(\vec {u}\) is the displacement vector with components \(u_{i}\), \(h_{i}\) are components of heat flux vector, p is the hydrostatic stress parameter, \(\alpha \) represents the volume coefficient of thermal expansion, \(k_{T}\) is the isothermal compressibility, \(\delta _{ij}\) is the Kronecker delta, \(\rho _{0}\) is the density, \(C_{e}\) is the specific heat, \(K(\ge 0)\) is the thermal conductivity and \(k^{*}\) is the material constant of the theory. Letting \(k^{*}\longrightarrow 0\) in Eq. (7), the heat conduction equation is obtained for type II thermoelastic theory under the GN theory.
In case of a non-homogenous medium, the otherwise assumed constants \(\lambda \), \(\mu \), K, \(k^{*}\), \(\rho \), p, \(\upsilon \), \(\alpha \) depend on space variables. Here \(\lambda \), \(\mu \), K, \(k^{*}\), \(\rho \), p, \(\upsilon \), \(\alpha \) are replaced by \(\lambda _{0}\phi (X)\), \(\mu _{0}\phi (X)\), \(K_{0}\phi (X)\), \(k^{*}_{0}\phi (X)\), \(\rho _{0}\phi (X)\), \(p_{0}\phi (X)\), \(\upsilon _{0}\phi (X)\), \(\alpha _{0}\phi (X)\) respectively, \(\phi (X)\) is taken as a dimensionless function which depends on the space variable \(X = (x,y,z)\).
We suppose that the material properties vary along the x-axis only, hence \(\phi (X)\) will be further expressed as \(\phi (x)\), therefore the field Eqs. (6) and (7) can be written as:
The components of the stress tensor are given by:
For the problem under consideration, the initial conditions are:
3 Non-homogeneity variation
The function \(\phi (x)\) is assumed to be of the form \(\phi (x)= e^{-nx} \), where n is a dimensionless constant. Hence, the material properties of the medium are exponentially varying along the x direction.
For numerical computations, the dimensionless parameters are introduced in the equations which are defined below:
where \(c_{0}^{2}=\frac{\lambda _{0}+2 \mu _{0}}{\rho _{0}}\), \(\varpi =\frac{k_0^*}{\rho _0 c_0^2C_e}\).
Substituting the above non-dimensional variables, the field equations and stress components (after suppressing the primes) are obtained in dimensionless form as,
4 Problem solution
The solution of the variables may be considered in the form of modes as:
Here \(\omega \) represents the frequency which is complex, b is the wave number along y direction, and \(u_{i}^{*}\), \(T^*\), \(t_{ij}^{*}\) are the amplitudes of the field quantities.
Using the assumed solutions defined by (21) in (15)–(20), we get
The expressions \(a_{i}(i=1,2,3,4)\), \(d_{j}(j=1\ldots 9)\), \(\beta \), \(\epsilon _{1},\epsilon _{2},\epsilon _{3} \) are given in the appendix.
Eliminating \(u_{2}^{*}\), and \(T^{*}\) from (22)–(24), a non-homogeneous sixth-order differential equation for \(u_{1}^{*}\) is obtained
where \(A_{i}(i=1..6)\) are listed in the appendix.
The bounded solution of (28), after applying radiation conditions as \(x \rightarrow \infty \), is given by
The roots \(k_{i}\) \((i=1,2,3)\) are obtained by solving Eq. (28) and \(P_{i}(a,\omega ) (i=1,2,3)\) are the parameters which depend on a and \(\omega \), and the coupling constants \(T_{1i}\) and \(T_{2i}\) are given by,
Under the assumed solution, the stress components are given by:
where
5 Boundary conditions
In order to evaluate the constants \( P_{i}\) (i =1,2,3), the following boundary conditions are applied along the surface \(x = 0\),
Here \(F_{1}\) and \(F_{2}\) are the magnitude of the applied mechanical and thermal source respectively on the free surface.
Using Eqs. (29)–(31) and (32)–(34) in boundary conditions (35)–(37), we get a set of three non-homogeneous equations in three unknowns as:
Solving Eqs. (38)–(40), which can be written in the matrix form as:
The parameters \(P_{i} (i = 1,2,3) \) are defined as follows:
where
6 Different cases of non-homogeneous medium
6.1 Rotating thermoelastic solid without hydrostatic initial stress
The expressions (29)–(31) and (32)–(34) in the absence of initial stress are obtained by substituting \(p_0=0\).
6.2 Initially stressed thermoelastic medium without rotation
When the medium under consideration is not rotating, the corresponding expressions are reduced by taking \(\Omega =0\) in the above mentioned expressions.
7 Homogeneous thermoelastic medium
Taking \(n=0\) in particular cases (6.1) and (6.2), the analytical expressions are reduced for:
-
(i)
Rotating thermoelastic solid without hydrostatic initial stress,
-
(ii)
Initially stressed thermoelastic medium without rotation.
8 Mechanical and thermal sources
8.1 Mechanical force
For \(F_1\ne 0\) and \(F_2=0\), the expressions given by (29)–(31), and (32)–(34) are obtained for a mechanical force.
8.2 Thermal source
For \(F_1=0\) and \(F_2\ne 0\), the expressions given by (29)–(31), and (32)–(34) are obtained for a thermal point source applied along the free surface of the thermoelastic medium.
9 Numerical results
To compliment the theoretical results derived earlier, we now present a numerical example for a particular medium. The numerical results in graphical form represent the variations of physical quantities. The following values of physical constants are considered for numerical computations:
The numerical computations are carried out at the instance \(t=0.1\). The results for tangential displacement \(u_2\), normal displacement \(u_1\), tangential force stress \(t_{xy}\), normal force stress \(t_{xx}\) and temperature distribution T are presented graphically in Figs. 1, 2, 3, 4 and 5 to show the effect of non-homogeneity \((n=0,1,2)\) and hydrostatic initial stress \((p_0=0,1.5)\). Here \(\omega =\omega _0+\iota \omega _1\) with \(\omega _0=1.5\), \(\omega _1=0.2\), \(b=1.6\) and \(\Omega =1.5\).
10 Discussions
The variations of all the quantities are oscillatory in nature and the magnitude of oscillations decrease as we move away from the point of source. In the absence of hydrostatic initial stress, the magnitude of displacement components is very small and hence, the oscillations are less oscillatory. Very close to the point of application of source, the values of displacement components (tangential and normal), for a fixed value of hydrostatic initial stress parameter \(p_0\), increase with increase in the non-homogeneity parameter n. These variations of displacement components are shown in Figs. 1 and 2 respectively. The variations of tangential stress and normal stress are significantly oscillatory for a homogeneous thermoelastic medium with hydrostatic initial stress. When the thermoelastic medium is not initially stressed (\(p_0=0\)), the values of tangential stress increase sharply in the range \(0\le x\le 1.0\) and henceforth, these values are close to zero. These variations of stress components are shown in Figs. 3 and 4 respectively. It can be observed from Fig. 5, that the values of the temperature field lie in a large range when the medium is initially stressed. The difference in values of temperature distribution in the thermoelastic without initial stress is observed in the range \(0\le x\le 1.5\) but in the remaining range, these values are quite close to each other.
11 Conclusion
The graphical results obtained for the present problem are similar in nature for both type of sources represented by \(P_1\) and \(P_2\) in magnitude. The authors have presented the results for the mechanical force which conclude that:
-
1.
The effect of initial stress and non-homogeneity can be observed on all the quantities.
-
2.
Close to the point where the source is applied, the values of displacement components is directly proportional to non-homogeneity parameter n.
-
3.
The variations of mechanical stress are more oscillatory for an initially stresses homogeneous thermoelastic medium.
-
4.
The values of displacement components and temperature distribution for an initially stressed medium are highly significant.
-
5.
As expected from the analytical expressions, the graphical results prove that due to initial stress the body is deformed to a more extent.
References
Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27(3), 240–253 (1956)
Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phy. Solids 15(5), 299–309 (1967)
Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2(1), 1–7 (1972)
Chandrasekharaiah, D.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(3), 355–376 (1986)
Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 432(1885), 171–194 (1991)
Green, A., Naghdi, P.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15(2), 253–264 (1992)
Green, A., Naghdi, P.: Thermoelasticity without energy dissipation. J. Elast. 31(3), 189–208 (1993)
Barber, J., Martin-Moran, C.: Green’s functions for transient thermoelastic contact problems for the half-plane. Wear 79(1), 11–19 (1982)
Marin, M.: Some results on eigenvalue problems in the theory of piezoelectric porous dipolar bodies. Comptes Rendus de l’Acad. des Sci. Srie II 321(12), 475–480 (1995)
Marin, M.: Some basic theorems in elastostatics of micropolar materials with voids. J. Comput. Appl. Math. 70(1), 115–126 (1996)
Sharma, K., Marin, M.: Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space. Univ. Politeh. Buch. Sci. Bull. Ser. A 75(2), 121–132 (2013)
Marin, M., Ochsner, A.: The effect of a dipolar structure on the holder stability in Green–Naghdi thermoelasticity. ZAMM 100(12), e202000090 (2020)
Marin, M., Ochsner, A.: Essentials of Partial Differential Equations. Springer, Cham
Sorin, V., Cristi, N., Marin, M., Mihalcica, M.: A method for the study of the vibration of mechanical bars systems with symmetries. Acta Tech. Napoc. Ser. Appl. Math. Mech. Eng. 60(4), 539–544 (2017)
Montanaro, A.: On singular surfaces in isotropic linear thermoelasticity with initial stress. J. Acoust. Soc. Am. 106(3), 1586–1588 (1999)
Singh, B.: Reflection of SV waves from the free surface of an elastic solid in generalized thermoelastic diffusion. J. Sound Vib. 291(3–5), 764–778 (2006)
Singh, B.: Effect of hydrostatic initial stresses on waves in a thermoelastic solid half-space. Appl. Math. Comput. 198(2), 494–505 (2008)
Othman, M.I., Song, Y.: Reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation. Int. J. Solids Struct. 44(17), 5651–5664 (2007)
Ailawalia, P., Kumar, S., Khurana, G.: Deformation in a generalized thermoelastic medium with hydrostatic initial stress subjected to different sources. Mech. Mech. Eng. 13(1), 5–24 (2008)
Schoenberg, M., Censor, D.: Elastic waves in rotating media. Q. Appl. Math. 31(1), 115–125 (1973)
Clarke, N., Burdess, J.: Rayleigh waves on a rotating surface. J. Appl. Math. 61(3), 724–726 (1994)
Destrade, M.: Surface acoustic waves in rotating orthorhombic crystals. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2042), 653–665 (2004)
Sharma, J., Othman, M.I.: Effect of rotation on generalized thermoviscoelastic Rayleigh-lamb waves. Int. J. Solids Struct. 44(13), 4243–4255 (2007)
Othman, M., Song, Y.Q.: The effect of rotation on 2-d thermal shock problems for a generalized magneto–thermo–elasticity half-space under three theories. Multidiscip. Mod. Mat. Struct. 5(1), 43–58 (2009)
Ailawalia, P., Narah, N., Kumar, R.: Effect of rotation due to various sources at the interface of elastic half space and generalized thermoelastic half space. Int. J. Appl. Math. Mech. 5(1), 68–88 (2009)
Ailawalia, P., Budhiraja, S., Singh, B.: Effect of hydrostatic initial stress and rotation in Green–Naghdi (type III) thermoelastic half-space. Multidiscip. Mod. Mat. Struct. 7(2), 131–145 (2011)
Marin, M., Hobiny, A., Abbas, I.: Finite element analysis of nonlinear bioheat model in skin tissue due to external thermal sources. Mathematics 9(13), 1459 (2021)
Marin, M., Öchsner, A., Bhatti, M.M.: Some results in Moore–Gibson thompson thermoelasticity of dipolar bodies. J. Taibah Univ. Sci. 16(1), 1264–1274 (2022)
Kumari, S., et al.: Effects of two-temperature on Rayleigh wave in generalized magneto-thermoelastic media with hydrostatic initial stress. J. Heat Trans. 141(7), 072002 (2019)
Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stress. 22(4–5), 451–476 (1999)
Sharma, J., Chauhan, R.S., Kumar, R.: Time-harmonic sources in a generalized thermoelastic continuum. J. Therm. Stress. 23(7), 657–674 (2000)
Sharma, J., Chauhan, R.S.: Mechanical and thermal sources in a generalized thermoelastic half-space. J. Therm. Stress. 24(7), 651–675 (2001)
Sharma, J., Sharma, P., Gupta, S.: Steady-state response to moving loads in thermoelastic solid media. J. Therm. Stress. 27(10), 931–951 (2004)
Deswal, S., Choudhary, S.: Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion. Appl. Math. Mech. 29(2), 207–221 (2008)
Reddy, J., Chin, C.: Thermomechanical analysis of functionally graded cylinders and plates. J. Therm. Stress. 21(6), 593–626 (1998)
Sankar, B.V., Tzeng, J.T.: Thermal stresses in functionally graded beams. AIAA 40(6), 1228–1232 (2002)
Wang, B.-L., Mai, Y.-W.: Transient one-dimensional heat conduction problems solved by finite element. Int. J. Mech. Sci. 47(2), 303–317 (2005)
Pelletier, J.L., Vel, S.S.: An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int. J. Solids Struct. 43(5), 1131–1158 (2006)
Mallik, S.H., Kanoria, M.: Generalized thermoelastic functionally graded solid with a periodically varying heat source. Int. J. Solids Struct. 44(22–23), 7633–7645 (2007)
Abbas, I.A., Zenkour, A.M.: LS model on electro–magneto–thermoelastic response of an infinite functionally graded cylinder. Comp. Struct. 96, 89–96 (2013)
Sur, A., Kanoria, M.: Thermoelastic interaction in a viscoelastic functionally graded half-space under three-phase-lag model. Eur. J. Comput. Mech. 23(5–6), 179–198 (2014)
Gunghas, A., Kumar, R., Deswal, S., Kalkal, K. K.: Influence of rotation and magnetic fields on a functionally graded thermoelastic solid subjected to a mechanical load. J. Math. (2019) (Article ID 1016981)
Meshkini, M., Ghafari, A.S., Firoozbakhsh, K., Jabbari, M.: Steady-state thermal and mechanical stresses in two-dimensional functionally graded piezoelectric materials (2D-FGPMS) for a hollow infinite cylinder. Sci. Iran. B 26(1), 428–444 (2019)
Kumar Kalkal, K., Gunghas, A., Deswal, S.: Two-dimensional magnetothermoelastic interactions in a micropolar functionally graded solid. Mech. Based Des. Struct. Mach. 48(3), 348–369 (2020)
Kumar, R., Thakran, S., Gunghas, A., Kalkal, K.: Transient disturbances in a nonlocal functionally graded thermoelastic solid under Green–Lindsay model. Int. J. Numer. Meth. Heat Fluid Flow 31(7), 2288–2307 (2021)
Barak, M.S., Dhankar, P.: Effect of inclined load on a functionally graded fiber-reinforced thermoelastic medium with temperature dependent properties. Act. Mech. 233, 3645–3662 (2022)
Sheokhand, S.K., Kalkal, K.K., Deswal, S.: Thermoelastic interactions in a functionally graded material with gravity and rotation under dual phase-lag heat conduction. Mech. Based Des. Struct. Mach. 51(6), 3026–3045 (2023)
Boora, K., Deswal, S., Kadian, A.: Thermo-mechanical interactions in a functionally graded orthotropic thermoelastic medium with rotation and gravity. Mech. Based Des. Struct. Mach. (In Press)
Sheoran, D., Yadav, K., Punia, B.S., Kalkal, K.K.: Thermodynamical interactions in a rotating functionally graded semiconductor material with gravity. multidiscip. Model. Mat. Struct. 19(2), 226–252 (2023)
Purkait, P., Kanoria, M.: Thermoelastic interaction in a functionally graded medium due to refined three-phase-lag Green–Naghdi model under gravitational field. J. Multiscale Model. (In Press)
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“P.A., A.S., M.M. and A.O. wrote the main manuscript text and P.A. and A.S. prepared figures. All authors reviewed the manuscript.”
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Ailawalia, P., Sharma, A., Marin, M. et al. Analysis of an initially stressed functionally graded thermoelastic medium (type III) without energy dissipation. Continuum Mech. Thermodyn. (2024). https://doi.org/10.1007/s00161-024-01315-2
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DOI: https://doi.org/10.1007/s00161-024-01315-2