Journal of Mathematics

Volume 2015 (2015), Article ID 487513, 13 pages

http://dx.doi.org/10.1155/2015/487513

## Two-Temperature Generalized Thermoviscoelasticity with Fractional Order Strain Subjected to Moving Heat Source: State Space Approach

^{1}Department of Mathematics, N.M. Govt. College, Hansi, Haryana 125033, India^{2}Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar, Haryana 125001, India

Received 30 July 2015; Accepted 28 September 2015

Academic Editor: Mario Ohlberger

Copyright © 2015 Renu Yadav et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The theory of generalized thermoelasticity with fractional order strain is employed to study the problem of one-dimensional disturbances in a viscoelastic solid in the presence of a moving internal heat source and subjected to a mechanical load. The problem is in the context of Green-Naghdi theory of thermoelasticity with energy dissipation. Laplace transform and state space techniques are used to obtain the general solution for a set of boundary conditions. To tackle the expression of heat source, Fourier transform is also employed. The expressions for different field parameters such as displacement, stress, thermodynamical temperature, and conductive temperature in the physical domain are derived by the application of numerical inversion technique. The effects of fractional order strain, two-temperature parameter, viscosity, and velocity of internal heat source on the field variables are depicted graphically for copper material. Some special cases of interest have also been presented.

#### 1. Introduction

The classical thermoelasticity theory based on Fourier’s law of heat conduction suffers from the deficiency of admitting thermal signals propagating with infinite speed. Numerous alternative theories of heat conduction have been put forth to overcome this deficiency, allowing heat to propagate as wave at finite speed. Among these theories, the extended theory of thermoelasticity proposed by Lord and Shulman [1] involving one relaxation time and the temperature-rate dependent theory of thermoelasticity propounded by Green and Lindsay [2] involving two relaxation times are the earliest and well established theories. Green and Naghdi [3–5] developed a theory where the characteristics of material response for thermal phenomenon are based on three types of constitutive response functions, labelled as types I, II, and III. When the theory of type I is linearized, we obtain the classical system of thermoelasticity. In model II, the internal rate of production of entropy is taken to be identically zero, implying no dissipation of thermal energy. Model III includes the previous two models as special case and admits dissipation of energy in general.

In the late 1960s, Chen and Gurtin [6] and Chen et al. [7, 8] formulated the two-temperature thermoelasticity theory. In this theory, the classical Clausius Duhamel inequality was replaced by another one depending on two temperatures—conductive temperature and thermodynamical temperature . The first is due to the thermal processes and the second is due to the mechanical processes inherent between the particles and layers of elastic materials. Boley and Tolins [9] found that the two temperatures and the strain have representations in the form of a travelling wave pulse response which occurs instantaneously throughout the body. The key element that sets the two-temperature thermoelasticity apart from the classical theory of thermoelasticity is the material parameter , called the temperature discrepancy. Specifically, if , then and the field equations of classical theory can be fully recovered from the two-temperature thermoelasticity.

Warren and Chen [10] investigated the wave propagation in the two-temperature thermoelasticity. Youssef [11] put forward this theory in the context of generalized theory of thermoelasticity. Youssef and Al-Lehaibi [12] studied a one-dimensional problem of two-temperature generalized thermoelasticity by employing the state-space technique. They showed that the obtained results are qualitatively different as compared to those in case of one-temperature thermoelasticity. Youssef [13] constructed another two-temperature generalized thermoelasticity theory for a homogeneous and isotropic medium in the context of Green and Naghdi model of type II.

Deswal and Kalkal [14] considered a new model of time-fractional derivative in the context of micropolar generalized thermoviscoelasticity theory with two temperatures. Zenkour and Abouelregal [15] employed state-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity and fractional heat conduction. Othman and Hilal [16] studied a two-dimensional problem of thermoelastic rotating material with voids under the effect of gravity and temperature dependent properties employing the two-temperature generalized thermoelasticity in the context of Green-Naghdi theory of types II and III. Bera et al. [17] applied two-temperature generalized thermoelasticity to determine the conductive and thermodynamic temperatures as well as the deformation and stresses in an annular disk.

The fractional calculus has attracted intense attention of an increasing number of mathematicians, physicians, and engineers since the early 1990s. Fractional calculus is a branch of mathematical analysis that focuses on the study of differential operators of arbitrary order. Fractional integrals and derivatives extend the well-known definitions of integer-order primitives and derivatives to the ordinary differential calculus to real-order operators. Abel was the first to attack a physical problem using the techniques of fractional calculus. The increased interest in this field is due to the fact that the fractional differential operators are nonlocal and therefore enable us to provide better description of real phenomena. A brief history of the development of fractional calculus can be found in Ross [18] and Miller and Ross [19]. A survey of many emerging applications of the fractional calculus in the area of science and engineering is done in the text by Podlubny [20].

A quasi-static uncoupled theory of thermoelasticity based on fractional heat conduction equation was constructed by Povstenko [21]. The Caputo time-fractional derivative [22] was used by Povstenko [23] to investigate thermal stresses in an infinite body with a circular cylindrical hole. Sherief et al. [24] put forward a new model of generalized thermoelasticity using fractional calculus. A uniqueness theorem, reciprocity relation, and variational principle have also been established in the same article. Youssef [25] also proposed a new theory of generalized thermoelasticity using the methodology of fractional calculus and discussed one-dimensional application. Deswal and Kalkal [26] employed state space approach to study the magneto-thermoelastic interactions in an initially stressed isotropic medium under the purview of two-temperature theory of generalized thermoelasticity. The problem of magnetothermoelastic interactions in an unbounded and perfectly conducting half-space whose surface suffers from a time harmonic thermal shock in the context of micropolar generalized thermoelasticity with fractional heat transfer has been analyzed by Deswal and Kalkal [27]. Abbas [28] considered the problem of fractional order thermoelastic interaction in a material placed in a magnetic field and subjected to moving plane heat source. Santra et al. [29] aimed at studying the effect of rotation on thermoelastic interactions in a homogeneous isotropic three-dimensional medium whose surface is traction-free and is subjected to a time-dependent heat source. The problem has been modelled on the basis of fractional order generalized thermoelasticity. Wang et al. [30] suggested a new theory of generalized thermoelasticity for elastic media with variable properties in the context of fractional order heat conduction equation. They derived the formulations of anisotropic heterogeneous material with temperature dependent material properties by making use of the Clausius inequality and the higher expansions of free energy.

Recently, Youssef [31] derived a new theory of thermoelasticity by modifying the Duhamel-Neumann stress-strain relation. In this theory, this relation depends on the fractional order of strain which adds knowledge about the time history to the deformation of materials after being acted upon by mechanical or thermal loadings. In his work, Youssef constructed a new unified system of differential equations governing seven different models of thermoelasticity in the context of one-temperature type and two-temperature type. A one-dimensional application of the thermoelasticity with fractional order strain for an isotropic and homogeneous medium for some models is also elaborated in the same article.

Due to the extensive engineering applications, such as pulsed-laser cutting and welding and high speed machining and grinding, several research works have been devoted to problems involving a moving heat source or thermal shock. Danilovskaya [32] was the first who solved a dynamical heat source problem under the purview of coupled thermoelasticity. The problems of instantaneous and moving heat sources in infinite and semi-infinite space and static line heat sources in semi-infinite space were considered by Eason and Sneddon [33], Nowacki [34], and others under the coupled theory of thermoelasticity. Sherief and Anwar [35] investigated the thermoelastic interactions due to a continuous line heat source in a linear, homogeneous unbounded solid in the context of the Lord-Shulman model of generalized thermoelasticity.

The theoretical study and applications in viscoelastic materials have become an important task for solid mechanics with the rapid development of polymer science and plastic industry as well as with the wide use of materials under high temperature in modern technology and application of biology and geology in engineering. Freudenthal [36] commented that most solids, when subjected to loading, exhibit viscous effects, particularly at an elevated temperature. The stress-strain law for many materials such as polycrystalline metals and high polymers can be approximated by the linear viscoelasticity theory. So the consideration of viscoelastic properties of the medium makes the studies more meaningful. The theory of thermoviscoelasticity and the solutions of some boundary value problems of thermoviscoelasticity were explored by Ilioushin and Pobedria [37]. Several investigations relating to thermoviscoelasticity theory have been presented in [38–42].

In the present work, a viscoelastic medium with internal heat source is considered. The governing equations are taken in the context of GN-III model of two-temperature generalized thermoelasticity with fractional order strain. Laplace transform and state-space techniques are adopted to find out the general solution of the problem. The results obtained theoretically have been computed numerically and depicted graphically. Some comparisons are exhibited in figures to demonstrate the effects of fractional order strain, viscosity, two-temperature parameter, and the presence of internal heat source.

#### 2. Basic Equations and Problem Formulation

Following Youssef [31], the constitutive equations and the field equations for generalized viscoelastic two-temperature GN-III model of thermoelasticity with fractional order strain and in the presence of internal heat source may be written as follows:(i)The constitutive relation is(ii)The strain-displacement relation is(iii)Equation of motion is(iv)Two-temperature heat conduction equation is(v)Relation between thermodynamical and conductive temperature isIn the preceding equations, and are the components of stress and strain tensors, respectively, are the components of displacement vector, represents the thermodynamical temperature, being the absolute temperature and denoting the temperature of medium in its natural state, stands for the conductive temperature, is the density of medium, is mechanical relaxation time, is the fractional strain parameter, , , , , , and are Lame’s elastic constants, and are the viscoelastic relaxation times, is the coefficient of linear thermal expansion, represents specific heat at constant strain, is thermal conductivity, stands for the material characteristic of GN theory, is the heat source, a dot over a variable denotes derivative with respect to time , a comma refers to a spatial derivative, and the tensor convention of summing over repeated indices is used.

By taking in the above governing equations, we may resume the equations of two-temperature thermoviscoelasticity with internal heat source in the context of GN-III theory.

We consider an infinite thermoelastic solid body which is unstrained and unstressed initially at uniform temperature occupying the region , whose state depends only on the space variable and time so that the displacement vector , conductive temperature , and thermodynamical temperature can be expressed in the following form:The governing equations (1)–(5) in one-dimensional case assume the shapeEquation (9) may also be expressed asNow we transform the above equations into nondimensional forms by introducing the following dimensionless parameters:where and .

Equations (7) and (10)–(12) may now be reduced to the following system of dimensionless equations (after removing the primes for clarity):where , , and are the coupling parameters and is the temperature discrepancy.

Performing the Laplace transform defined asover (14)–(17), and using the homogeneous initial conditions, we obtain the following system of differential equations:where .

We consider that a moving heat source of constant strength is located at the origin and at time begins moving along the positive direction of -axis with a constant velocity releasing its energy continuously. This moving heat source is considered to be of the following nondimensional form:where is the strength of the heat source, is well-known Dirac-delta function, and stands for Heaviside unit step function. Applying Laplace transform defined in (18) and Fourier transform defined asto (20) and then inverting Fourier transform manually, we getAs in the current problem, the heat source is moving along the positive -axis, and therefore we haveEliminating and from (19), we arrive at the following system of differential equations:where

#### 3. State-Space Formulation

Having chosen the conductive temperature and stress component as state variables, (24) may be recast in matrix form aswhereThe formal solution of the differential equation (26) may be written aswhere and is an identity matrix of second order. The terms containing exponents of growing nature in the space variable have been discarded due to the regularity condition at infinity.

If there is no heat source inside the medium, then (28) assumes the formThe characteristic equation of matrix is obtained aswhere the roots and of (30) must satisfyThe Taylor series expansion of the matrix exponential has the formMaking use of the well-known Cayley-Hamilton theorem, we can express and higher orders of matrix in terms of and .

Thus the infinite series in (32) can be truncated aswhere and are constants depending on and .

Again by Cayley-Hamilton theorem the characteristic roots and of the matrix must satisfy (33). Therefore, we haveOn solving the above linear system of equations, we obtainSubstituting the values of and along with and into (33), we havewhere the components are given byHence solution (28) can be written asPlugging the values of , , and into (38) and after some straightforward calculation, the expressions for conductive temperature and stress are evaluated aswhereInserting the expression of into (16), the expression for can be derived as

#### 4. Application

We consider a homogeneous isotropic viscoelastic medium occupying the region with quiescent initial state and boundary conditions in the following forms.

##### 4.1. Mechanical Boundary Condition

We will suppose that the medium is subjected to a mechanical shock at as follows:where is a constant.

By applying Laplace transform defined in (18), we obtain

##### 4.2. Thermal Boundary Condition

The medium at is kept at reference temperature ; that is,Operating Laplace transform on the above equation, one can obtainHence, we can utilize the values of and from (43) and (45) in (39) and (41) to finally achieve the solutions in the Laplace transform domain asUsing dimensionless variables and Laplace transform in (9), the displacement component may be evaluated aswhere .

Substitution of from (47) into the above equation yields

#### 5. Limiting Cases

##### 5.1. Without Viscous Effect

If we neglect the effect of viscosity, then we will be left with the corresponding problem in generalized two-temperature thermoelasticity with fractional order strain. In this case, we put which implies , , , and . By implementing the above changes, the corresponding expressions of the physical fields can be obtained from (46)–(48) and (50).

##### 5.2. With One Temperature

By setting and consequently in the governing equations, we get the expressions for different field variables from (46)–(48) and (50) for one-temperature case, that is, the case when conductive temperature coincides with thermodynamical temperature.

##### 5.3. Without Internal Heat Source

Neglecting the influence of internal heat source, that is, (), the expressions of conductive temperature, stress, thermodynamical temperature, and displacement are obtained in a viscothermoelastic medium with two-temperature and fractional order strain as

#### 6. Numerical Inversion of the Transform

Equations (46)–(48) and (50) provide the expressions for conductive temperature, stress, thermodynamical temperature, and displacement in Laplace transform domain. To determine these in physical domain, Laplace inversion is applied with the help of numerical technique based on Fourier expansion of functions performed by Honig and Hirdes [43].

Let be the Laplace transform of function . The inversion formula of Laplace transform states thatwhere is an arbitrary real number greater than all the real parts of singularities of . Taking and using Fourier series in the interval , we get the approximate formulawhereand is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such thatwhere is a prescribed small positive value that corresponds to the degree of accuracy to be achieved.

#### 7. Numerical Results and Discussion

With an aim of illustrating the contribution of fractional strain parameter, mechanical relaxation time, two-temperature parameter, viscosity coefficients, and heat source on field quantities, a numerical analysis is carried out. For this purpose, we have taken the following values of relevant parameters:

Making use of the above mentioned numerical values, we have computed the dimensionless values of displacement, stress, thermodynamical temperature, and conductive temperature with distance and shown them graphically in four groups. In the first group (Figures 1–4), we have shown the effects of mechanical relaxation time and parameter on the considered physical variables with location . For this, three different sets of values of and are considered: (i) and , (ii) and , and (iii) and . The case leads to Green-Naghdi model of type III while leads to the same model in context of fractional order strain. The influence of the velocity of internal heat source on the considered physical variables for three values of , namely, , and , is shown graphically in the second group (Figures 5–8). Attention is paid to the investigation of effects of viscosity and two-temperature parameter on the physical quantities in the third group (Figures 9–12). For this, all the considered field variables are examined for three different cases: (i) thermoviscoelastic solid with two-temperature under fractional order strain and heat source (TV2T), (ii) thermoelastic solid with two-temperature under fractional order strain and heat source (T2T), and (iii) thermoviscoelastic solid with one temperature under fractional order strain and heat source (TV1T). Pattern of different fields in the presence and absence of heat source has been observed in the fourth group (Figures 13–16).