A BRIEF NOTE ON TIME FRACTIONAL ORDER THERMAL BEHAVIOR IN A THIN CIRCULAR CYLINDER DUE TO INTERNAL HEAT SOURCE

S. D. Warbhe1, J. Verma2, K. C. Deshmukh3

1Department of Mathematics, Laxminarayan Institute of Technology, Nagpur-440033, Maharastra, India.

2 Department of Applied Mathematics, Pillai HOC College of Engg & Tech , Rasayani, University of

Mumbai, Maharashtra, India. 3Departmentof Mathematics, R.T.M. Nagpur University, Nagpur-440033, Maharashtra, India. e-mail: [email protected] , [email protected], [email protected]

Abstract

The heat conduction equation with the time fractional derivative of order ? due to internal heat generation applied to the two dimensional problem of a thin circular cylinder, whose lower surface is maintained at zero temperature whereas the upper surface is insulated and subjected to a constant temperature distribution on the curved surface and discuss the thermal behavior. The integral transform technique is used to find the temperature in the physical domain with the help of Caputo type fractional derivative. The corresponding thermal stresses are determined by using the displacement function with the help of internal heat generation within the cylinder. A mathematical model is constructed for a copper material. The temperature distribution and the thermal stresses due to internal heat generation are shown graphically for the different values of ?.

Keywords: Quasi-static; thermoelasticity; fractional order; integral transform; thermal stresses; Mittag-Leffler function.

Introduction

The fractional order theory of thermoelasticity is mainly based of space-time nonlocal generalization of Fourier law of heat conduction and space time heat conduction problems. The main object of this problem is to interpolate the classical thermoelasticity.

Hence the fractional order thermoelasticity predict the retarded response in which predicted a physical stimuli which is found in nature and instantaneously opposes the response to a generalized theory of thermoelasticity.

Sherief et al. 2 formulate the fractional order thermoelastic equations. Subsequently , Sherief et al. 1 solved problem on fractional order thermoelasticity of a cylindrical body with infinite length Povstenko 3 – 9 has studied some fractional order thermoelastic problems of various heat sources by quasi static approach. Raslan 10 studied the application of fractional-order theory of thermoelasticity in a thick plate under axisymmetric temperature distribution. Warbhe et al. 12 discussed fractional heat conduction in a thin hollow circular disk and associated thermal deflection. Recently, Warbhe et al. 11 studied the fractional heat conduction in a thin circular plate with constant temperature distribution and discussed the thermal stresses.

In this paper the work of Warbhe et.al 11 is modified and a mathematical model of a fractional order thermoelastic problem for a finite thin circular cylinder under constant temperature distribution and internal heat source is prepared by quasi-static approach. Copper material is chosen for a numerical purposes and the result for temperature and thermal stresses are discussed and illustrated graphically for different values of ? i. e for weak, moderate and strong conductivity for a copper material. This is a new and novel contribution to the field.

Formulation of the problem:

We take the axis of symmetry as the axis and the origin of the system of co-ordinates is at the lower surface of the thin circular cylinder. The problem is studied using the cylindrical polar co-ordinates. Due to the rotational symmetry about the z axis, all quantities are independent of the co-ordinate.

Consider a thin circular cylinder of thickness occupying space defined by,

.

The internal heat source with traction free surface, subjected to the temperature maintained at zero at lower surface whereas the upper surface is insulated. A temperature gradient is applied on the fixed circular boundary at r = b and a mathematical model is prepared considering non-local Caputo type time fractional heat conduction equation of order for a thin circular cylinder.

The differential equation governing the displacement potential function is as

(1)

with at for all time

The stress function and are given by

(2)

(3)

In the plane state of stress within the circular cylinder

(4)

and the time fractional heat conduction equation with the internal heat source given as

(5)

with boundary conditions

(6)

(7)

(8)

and initial conditions

`(9)

(10)

where is the temperature, is the displacement potential function and and are respectively, the Poisson’s ratio and the linear coefficient of thermal expansion of the circular cylinder material.

The definition of Caputo type fractional derivative is given by 8

(11)

For finding the Laplace transform, the Caputo derivative requires knowledge of the initial values of the function and its integer derivatives of the order

(12)

where the asterisk denotes the Laplace transform with respect to time, is the Laplace transform parameter.

Equations (1) to (10) constitute the mathematical formulation of the problem.

Solution

On applying Fourier, Hankel, and Laplace transforms and their inversions defined as 13, 14 to the initial boundary value problem of heat conduction, one obtains the temperature distribution function as

(13)

where, where are the positive roots of transcendental equation

where are the positive roots of the transcendental equation

. Here represent the Mittag-Leffler function.

Using Eqs. (1) and (13), we get the displacement potential function as follows:

(14)

Using equation (2), (3) and (14), we obtain the radial and angular stresses as follows:

(15)

(16)

where

and

NUMERICAL CALCULATIONS AND GRAPHICAL REPRESENTATION

As a special case, a mathematical model has been constructed with internal heat generation for different values of fractional parameter for Copper material and represented graphically.

Dimensions:

For the sake of simplicity, we choose,

Radius of a thin circular cylinder .

Thickness of a thin circular cylinder .

Central circular paths of circular cylinder in radial and axial direction,, . The heat source is an instantaneous point heat source of strength situated at the center of the thin circular cylinder along the radial ; axial directions releases its heat instantaneously at the time .

We set for convenience,

Material Properties:

The numerical calculation has been carried out for a Copper (Pure) thin circular cylinder with the material properties as,

Thermal diffusivity.Thermal conductivity .

Density .

Specific heat .

Poisson ratio .

Coefficient of linear thermal expansion .

Lamé constant.

Fig. 1: Temperature distribution Function

Fig. 2: Radial Stress Function

Fig. 3: Angular Stress Function

Figures 1-3 depicts the distributions of temperature, radial and angular stresses against the radius of the thin circular cylinder. The curves are plotted for different values of the fractional order parameter. The numerical calculation has been carried out in MATLAB 2013a programming environment. The Mittag–Leffler functions used in the article were evaluated following Podlubny 15.

From Figure 1, because of internal heat generation, it is observed that the temperature is high at the centre of the cylinder along with its edges, whereas the temperature decreases to zero at and then follows a sinusoidal pattern.

From Figure 2, it is seen that the radial stresses are compressive in the range and they are tensile in the region .

From Figure 3, it is observed that the angular stresses are tensile throughout the region follows a sinusoidal pattern and also more prominent at the centre.

The effects of temperature and thermal stresses are observed due to the presence of an instantaneous point heat source situated in close proximity to the center of the thin circular cylinder. Moreover, if the heat source is removed then the results obtained as in Warbhe et al. 11.

Conclusion:

The study is based on time fractional order quasi static thermoelasticity with an instantaneous point heat source of strength situated at the centre of a thin circular cylinder along the radial and axial direction, releases its heat instantaneously at the time t sec. The time fractional differential operator describes memory effects. The heat conduction equation considered here predicts infinite wave propagation in terms of heat energy, whereas hyperbolic wave equation for which predicts a retarded response to physical stimuli as seen in nature. For time fractional equation in the case the fractional heat conduction equation interpolates the Helmholtz equation and the ordinary heat conduction equation and the propose theory of thermal stresses interpolates the thermoelasticity with localized heat conduction equation and classical one.

It is observed that in the case of time fractional equation which interpolates the classical heat conduction equation for and wave equation for and the effect of temperature and thermal stresses are interpolates due to internal heat generation.

From numerical calculations, it is difficult to say whether the solution for appearing, has a jump at the wave front or it is continuous with very fast changes. This aspect invites further investigation which may prove to be useful to the researchers in solid mechanics, design of new materials etc.

References

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