The present work is concerned with the solution of a fractional-order thermoelastic problem of a two-dimensional infinite half space under axisymmetric distributions in which lower surface is traction free and subjected to a periodically varying heat source. The thermoelastic displacement, stresses and temperature are determined within the context of fractional-order thermoelastic theory. To observe the variations of displacement, temperature and stress inside the half space, the authors compute the numerical values of the field variables for copper material by utilizing Gaver-Stehfast algorithm for numerical inversion of Laplace transform. The effects of fractional-order parameter on the variations of field variables inside the medium are analyzed graphically. The paper aims to discuss these issues.
Integral transform technique and Gaver-Stehfast algorithm are applied to prepare the mathematical model by considering the periodically varying heat source in cylindrical co-ordinates.
This paper studies a problem on thermoelastic interactions in an isotropic and homogeneous elastic medium under fractional-order theory of thermoelasticity proposed by Sherief (Ezzat and El-Karamany, 2011b). The analytic solutions are found in Laplace transform domain. Gaver-Stehfast algorithm (Ezzat and El-Karamany, 2011d; Ezzat, 2012; Ezzat, El Karamany, Ezzat, 2012) is used for numerical inversion of the Laplace transform. All the integrals were evaluated using Romberg’s integration technique (El-Karamany et al., 2011) with variable step size. A mathematical model is prepared for copper material and the results are presented graphically with the discussion on the effects of fractional-order parameter.
Constructed purely on theoretical mathematical model by considering different parameters and the functions.
The system of equations in this paper may prove to be useful in studying the thermal characteristics of various bodies in real-life engineering problems by considering the time fractional derivative in the field equations.
In this problem, the authors have used the time fractional-order theory of thermoelasticity to solve the problem for a half space with a periodically varying heat source to control the speed of wave propagation in terms of heat and elastic waves for different conductivity like weak conductivity, moderate conductivity and super conductivity which is a new and novel contribution.
Tripathi, J., Warbhe, S., Deshmukh, K. and Verma, J. (2018), "Fractional order generalized thermoelastic response in a half space due to a periodically varying heat source", Multidiscipline Modeling in Materials and Structures, Vol. 14 No. 1, pp. 2-15. https://doi.org/10.1108/MMMS-04-2017-0022Download as .RIS
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The theory of uncoupled thermoelasticity has two major shortcomings. First, the mechanical state of the elastic body and temperature is assumed to be independent quantities contrary to the physical experiments. Second, the heat conduction equation is parabolic, thus predicting infinite speed of thermal signals, which is contrary to the observations. Biot (1956) developed the coupled theory of thermoelasticity to remove the first shortcoming by assuming the elastic changes and the temperature changes are dependent on each other. The heat conduction equation in this theory is still parabolic in nature predicting infinite speed of propagation of thermal and mechanical signals.
Lord and Shulman (1967) introduced the theory of generalized thermoelasticity, also known as extended thermoelasticity theory. In this theory, they modified the Fourier’s law of heat conduction by introducing a relaxation time parameter. The modified equation is hyperbolic in nature, thus predicting finite wave propagation. The equations of generalized thermoelasticity for the anisotropic case with one relaxation time were derived independently by Dhaliwal and Sherief (1980). A thorough discussion on different theories of generalized thermoelasticity can be found in the paper by Hetnarski and Ignaczak (1999). A two dimensional problem for a thick plate with heat sources in generalized thermoelasticity has been solved by El-Maghraby (2005). Tripathi et al. (2015a, b, 2016a) studied problems on generalized thermoelastic diffusion for a thick circular plate and a half space with axisymmetric heat supply within the context of Lord-Shulman theory. Recently, Tripathi et al. (2016b) discussed problems on generalized thermoelastic half space in the influence of a periodically varying heat source.
Fractional-order differential equations have been the forefront of research due to their applications in many real-life problems of fluid mechanics, viscoelasticity, biology, physics, and engineering. It is a well-known fact that the integer-order differential operator is a local operator but the fractional-order differential operator is non-local. Hence, the next state of a system depends not only upon its current state, but also upon all of its historical states. This is much more realistic and due to this reason, fractional derivative is also known as memory dependent derivative. In recent times, various types of definition and approaches of fractional-order derivatives have become popular amongst many researchers. The reason behind introduction of the fractional theory is that it predicts retarded response to physical stimuli, as is found in nature, as opposed to instantaneous response predicted by the generalized theory of thermoelasticity. The first application of fractional derivatives was given by Abel, who applied fractional calculus in the solution of an integral equation that arises in the formulation of the tautochrone problem. Caputo (1967, 1974) and Caputo and Mainardi (1971a, b) used fractional derivatives and their results found good agreement with the empirical evidences for description of viscoelastic materials. Study on thermoelasticity with fractional calculus was initiated by Povstenko (2004, 2005). A review of thermoelasticity with fractional-order heat conduction equation was given by Povstenko (2009). Povstenko (2010) investigated thermal stresses in an infinite body with a circular cylindrical hole using the heat conduction equation based on Caputo time fractional derivative. Recently, Sherief et al. (2010) developed a new theory of thermoelasticity using fractional calculus and proved the uniqueness theorem. Ezzat and El-Karamany (2011a) solved a problem on fractional-order thermoelasticity for a perfectly conducting medium. Ezzat and El-Karamany (2011b, c) derived theories of fractional-order electro-thermoelasticity and electro-thermoelasticity. Ezzat and El-Karamany (2011d) and Ezzat (2012) solved problems on fractional-order magneto-thermoelasticity involving two temperatures and thermoelectric fluid with fractional-order heat transfer using state space approach. Many researchers (Ezzat, El Karamany, and Ezzat, 2012; El-Karamany et al., 2011; Ezzat, El Karamany, and Fayik, 2012; Ezzat et al., 2013, 2015a, b; Ezzat and El-Bary, 2016; Sherief et al., 2010) studied various problems on fractional-order thermoelasticity. Recently, Tripathi et al. (2016c) have studied a dynamic problem on fractional-order thermoelaticity for a thick circular plate with finite wave speeds. Kanoria and Mallik (2010) discussed a generalized thermo-viscoelastic problem due to application of a periodically varying heat source in the context of three-phase-lag theory. Roychoudhuri and Dutta (2005) studied the thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources. Shaw and Mukhopadhyay (2012) studied a problem of a periodically varying heat source in a functionally graded microelongated medium.
In the present paper, we study a problem on thermoelastic interactions in an isotropic and homogeneous elastic medium under the fractional-order theory of thermoelasticity proposed by Sherief et al. (2010). The analytic solutions are found in Laplace transform domain. Gaver-Stehfast algorithm (Gaver, 1966; Stehfast, 1970a, b) is used for numerical inversion of the Laplace transform. All the integrals were evaluated using Romberg’s integration technique (Press et al., 1986) with variable step size. A mathematical model is prepared for copper material and the results are presented graphically with the discussion on the effects of fractional-order parameter.
Formulation of the problem
We shall consider a homogeneous isotropic thermoelastic solid half space occupying the region z⩾0. The z-axis is taken perpendicular to the bounding plane pointing inwards. The problem is considered within the context of fractional-order Lord-Shulman theory of generalized thermoelasticity with one relaxation time. We shall assume that the initial state of the medium is quiescent at a temperature T0. The surface of the medium is traction free and subjected to a known temperature distribution. A periodically varying heat source is applied on the domain. Cylindrical polar co-ordinates (r, ϕ, z) are used. Due to the rotational symmetry about the z-axis, all quantities are independent of the co-ordinate φ.
The problem is thus two-dimensional with all functions considered depending on the spatial variables r and z as well as on the time variable t.
The displacement vector, thus, has the form .
The equations of motion can be written as (Hetnarski and Ignaczak, 1999):
The generalized equation of heat conduction has the form:
The following constitutive relations supplement the above equations:
We shall use the following non-dimensional variables:
Using the above non-dimensional variables, the governing equations take the form (dropping the primes for convenience):
We assume that the initial state is quiescent, that is, all the initial conditions of the problem are homogeneous.
The thermal and mechanical boundary conditions of the problem at z=0 are taken as:
Solution of the problem
Applying the Laplace transform defined by the relation:
After factorization the above equation becomes:
The solution of Equation (30) is written in the form:
and is a particular integral of Equation (30).
The Hankel transform of a function is defined by the relation:
The inversion of Hankel transform is given by the relation:
On applying the Hankel transform to Equation (33), we get:
The solution of Equation (36) which is bounded at infinity, can be expressed as follows:
On applying the Hankel transform to Equation (30), we get:
The periodically varying heat source Q(r, z, t) in cylindrical co-ordinates is taken in the following form:
On applying Laplace transform and Hankel transforms to Equation (39), we get:
The solution of the Equation (38) has the form:
Then the complete solution in the transformed domain is obtained as follows:
On applying the inverse Hankel transform to Equation (42), we get:
On applying Hankel transform to Equation (44), we get:
Applying the inverse Hankel Transform to Equation (46), one obtains:
On applying the inverse Hankel transform to Equation (48), we get:
On applying inverse Hankel transform to Equation (50), one obtains:
Inversion of double transforms
The Laplace transform of a continuous function f(t) is given by:
for t>0 and s=x+iy.
If the solution is given in the Laplace domain, the inversion integral is used to find the original function f(t):
By this method, the inverse f(t) of the Laplace transform is approximated by:
Numerical results and discussion
To illustrate the above results graphically, the axisymmetric function f(r, t), i.e. the value of the temperature on the surface z=0 of the thermoelastic medium was chosen to be 0 except for the inside of the circular region r⩽a where it has a fixed constant value of θ0, that is:
On taking Hankel and Laplace transform of the above function, we get:
Copper material was chosen for purposes of numerical computations, with the physical data given as (Ezzat and El-Karamany, 2011d):
Figures 1-3 exhibit the variations of θ, the radial displacement component u and the axial stress component σzz considered as functions of radial distance r at the boundary of the half space (z=0) for different time instants t=0.05,0.1. The value of α in these figures was taken equal to 0.98. In these figures, solid line represents the solutions for t=0.05 and dashed line represents the solutions for t=0.1.
Figures 4-6 depict the behavior of θ, u and σzz along the radial direction for the different values of fractional-order parameter α and hence shows the variations between the generalized and fractional-order thermoelasticity theories. In all these figures, the solid line represents the solutions for α=0 and dashed lines represent the solutions for α=0.5 and 0.1, respectively.
Figure 1 shows the variation of temperature with radial distance. It is observed that the temperature decreases with radial distance and finally becomes identically 0 at r=7. It is also observed that the values of temperature at t=0.05 are more as compared to its values at t=0.1.
Figure 2 depicts the variation of u with the radial distance. It is observed that u increases with the radial distance upto r=3 and then gradually decreases till r=7. In the complete region, the values of u for t=0.1 are less than its values at t=0.05.
Figure 3 shows the variation of axial stress σzz with radial distance. It is observed that the axial stresses are tensile in the region 0⩽r⩽3.9 then the axial stress component values are compressive in the region 3.9⩽r⩽7.
Figures 4-6 show the variations of θ, u and σzz along the radial direction for different values of the fractional-order parameter α. One can clearly observe that for different values of α, the velocity of the wave propagation changes and an inference can be drawn that the speed of waves is directly proportional to the values of fractional-order parameter α. Hence, an increase in the conductivity of energy in the material is directly related with the fractional-order parameter. Forα ≅ 1, the solutions behave like the generalized theory of thermoelasticity.
In the present work, a mathematical model of fractional-order generalized thermoelasticity with one relaxation time has been used to solve the problem for a half space with a periodically varying heat source. An unbounded isotropic medium is considered which is subjected to a periodically varying heat source in the context of time fractional generalized thermoelastic model in which the thermo-physical properties are space and temperature dependent. Because of the presence of periodically varying heat source with time, the variations are shown in the temperature, displacement and the stresses. It is also observed that the thermal wave is the faster wave. Due to the presence of one relaxation time in the field equations the heat wave assumes finite speed of propagation. It is concluded that for different values of the fractional-order parameter α, the velocity of the wave changes. The fractional-order parameter seems to be directly proportional to the conductivity of the material. The system of equations in this paper may prove to be useful in studying the thermal characteristics of various bodies in real-life engineering problems by considering the time fractional derivative in the field equations.
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The authors sincerely thank the referees and the editor for their constructive comments which have improved the manuscript greatly.