2D Problem for a Long Cylinder in the Fractional Theory of Thermoelasticity

Abstract

In this manuscript, we solve an asymmetric 2D problem for a long cylinder. The surface is assumed to be traction free and subjected to an asymmetric temperature distribution. A direct approach is used to solve the problem in the Laplace transformed domain. A numerical method is used to invert the Laplace transforms. Graphically results are given and discussed.

Keywords:
Fractional Calculus; Infinitely Long Cylinder; Thermoelasticity

1 INTRODUCTION

In 1967 Lord and Shulman (Lord & Shulman, 1967) were the first to generalize Biot's theory of coupled thermoelasticity. This theory ensures finite speeds of propagation for waves. Sharma and Pathania studied wave propagation(Sharma & Pathania, 2006), Sherief and Anwer solved a two dimensional problem for an infinitly long cylinder (Sherief & Anwar, 1994) , Sherief and Ezzat obtained the fundemental solution in the form of series of functions (Sherief & Ezzat, 1994) and Sherief and Saleh solved a generilized thermoelastic problem for an infinite body with a spherical cavity using complex countor integration (Sherief & Saleh, 1998).

An ingoing process is the use of fractional calculus to create a replacement for many physical models(Hilfer et al., 2000; Machado, Galhano, & Trujillo, 2013). Sherief et al used fractional derivative to generalized Hodgkin and Huxley model (Sherief, El-Sayed, Behiry, & Raslan, 2012). Povstenko used fractional derivatives to derive new models for the conduction of heat(Povstenko, 2009).

The fractional theory of thermoelasticity was introduced in 2010(H. H. Sherief, El-Sayed, & Abd El-Latief, 2010). The main reason behind the introduction of this theory is that it predicts retarded response to physical effects, as is found in nature, as opposed to instantaneous response predicted by the generalized theory of thermoelasticity. This retarded response stems from the fact that fractional derivatives are in fact integrals over time. Physically this results from the weak van der Walles forces.

In the following, some applications of the fractional order theory of thermoelasticity are introduced. Raslan has solved a problem for a cylindrical cavity(Raslan, 2014). El-Karamany and Ezzat applied fractional order theory to perfect conducting thermoelastic medium (El-Karamany & Ezzat, 2011; Ezzat & El-Karamany, 2011) , Sherief and Abd El-Latief studied the effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity (Sherief & Abd El-Latief, 2013), also they applied the theory to a 1D problem for a half-space (Sherief & Abd El-Latief, 2014), Tiwari and Mukhopadhyay introduced Boundary Integral Equations Formulation for Fractional Order Thermoelasticity (Tiwari & Mukhopadhyay, 2014).

2 FORMULATION OF THE PROBLEM

In this manuscript, we consider a homogeneous isotropic cylinder of radius "a" and infinite length. We shall use the cylindrical coordinates (r, Φ,z). The initial conditions are taken to be homogeneous. The surface of the cylinder is assumed to be traction free and subjected to an asymmetric temperature distribution.

The physics of the medium under discussion ensures that all quantities are independent of z. all functions depend on r and Φ. The displacement vector has the non-zero components u and v in r and Φ directions, respectively. The governing equations are

(1)

(2)

where

Applying the divergence operator to both sides of (equation 1), we obtain

(3)

where e is the cubical dilatation given by

(4)

The constitutive equations can be written as

(5a)

(5b)

(5c)

(5d)

(5e)

We shall use the following non-dimensional quantities

where .

These non-dimensional variables were first introduced by (Sherief, 1980) in his PhD thesis. They were obtained by trial and error. They are useful because the solution obtained using these variables does not depend on the units used.

Using the above non-dimensional quantities, (dropping the asterisk for convenience), the governing equations take the form

(6)

where Equation 6) gives the following two equations. (

(7)

(8)

(9)

(10)

(11a)

(11b)

(11c)

(11d)

(11e)

where ε = T0(2 / (( + 2 μ) k η.

We note that in the above transformed equations, all the variables and constants (ε, β, τ0 ) are non dimensional.

The fractional derivative used in (equation 10) is the Caputo derivative. The boundary conditions can be expressed as:

(12a)

(12b)

(12c)

The boundary conditions on the stress components means that the component of the stress in the normal direction (r direction) are zero. This follow from the fact that there are no surface forces affecting the boundary so the normal component of stress that act to neutralize these forces are also zero. The stress component σΦΦ is the resultant of internal forces and not necessarily zero.

3 SOLUTION IN THE TRANSFORM DOMAIN

Applying the Laplace transform with parameter s (denoted by an over bar) to both sides of (equations 7), (9-11), we get the following equations

(13)

(14)

(15)

(16a)

(16b)

(16c)

(16d)

(Equations 12) transform to:

(17a)

(17b)

(17c)

Applying the operator (∇2 - s2) to both sides of (equation 15) and multiplying both sides of (equation 14) by (s + τ0sα + 1)ε and subtracting, we obtain

(18)

(Equation 18) can be written in the form:

(19)

where k21 and k22 are the complex roots which have positive real parts of the following characteristic equation

(20)

The solution of (equation 19) can be written in the form

(21)

where is the solution of

(22a)

(22b)

The solution of (equation 19), bounded at the origin, may be written as

(23)

where A_in are some parameters depends on s only and I_n (k_ir) is the modified Bessel function of first kind of order n. In a similar manner, the solution for ē compatible with (equation 14) can be written as

(24)

The Laplace transforms of (equations 4) and (7) can be combined to give

(25)

Substituting from (23) and (24) into (25), we obtain

(26)

We have used the following relations of the modified Bessel functions (Bell, 1986)

(27)

After some manipulations, the solution of equation (26) takes the form

(28)

where B_n (s) are some parameters depending on s only. We note that we have set B₀ = 0 because is not bounded.

Substituting from equations (24) and (28) into (3), and integrating with respect to Φ, we obtain

(29)

We expand the function in a Fourier cosine series in Φ as

where F_n (s)are the Fourier coefficient given by

We have chosen to expand the function in a cosine series to facilitate the computations. This means that we take the temperature as an even function of Φ. A full expansion in terms of sine and cosine will add nothing to the physical meaning of the problem considered.

Substituting from (equations 23), (24), and (28) into (Equation 16a), and applying the boundary condition (17a), we get for n = 0

(30)

while for n = 1,2,3,...

(31)

Similarly, boundary (equation 17b) yields for n = 1,2,3,...

(32)

Finally the boundary condition (17c) leads to for n = 0

(33)

and for n = 1,2,3,...

(34)

(Equations 30) and (33) can be solved to obtain A ₁₀ and A ₂₀,

where

Γ = (αβ² I ₀(k ₁ a)I ₀(k ₂ a)s ₂(k ₁ ² - k ₂ ²) = (I ₀(k ₁ a)I ₁(k ₂ a)k ₂(k ₁ ² - s ²) + I ₀(k ₂ a)I ₁(k ₁ a)k ₁(s ² - k₂ ² )))

(Equations 31), (32) and (34), can be written as:

a₁₁A_1n + a ₁₂ A _{2 n} + a ₁₃ B_n = 0

a₂₁A_1n+a₂₂A_2n+a₂₃B_n = 0

a ₃₁ A _{1 n +} a ₃₂ A _{2 n +} F_n

where

Solving the above equations, we obtain

where ( = a ₂₃(a ₁₂ a ₃₁ - a ₁₁ a ₃₂) + a ₁₃(a ₂₁ a ₃₂ - a ₂₂ a ₃₁)

4 NUMERICAL RESULTS AND DISCUSSION

We shall apply our results to a medium composed of the copper material. The parameters of the problem are k = 386 W/(m K), αt = 1.78 (10)-5 K-1, cE = 381 J/(kg K), ( = 8886.73, μ = 3.86 (10)¹⁰ kg/(m s²), ( = 7.76 (10)¹⁰ kg/(m s²), ρ = 8954 kg/m³, T₀ = 293 K, Φ0= π/12, a = 1 m, (₀ = 0.025 s( and ( = 0.0168.

The above values were obtained from ((Thomas, 1980) except for (₀ which was assumed.

We shall consider two cases of the applies heating

Case 1

F_n are thus given by

Case 2

The Fourier coefficients F_n are thus given by

Two methods were tried to solve the problem. Firstly, the Laplace transform of the terms of the series were inverted term by term and then summed up as a series of real numbers. Secondly, the series was summed up as a complex-term series and then the inverse Laplace transform was applied. It was found that the first method is better. It achieves higher order of convergence. (Figure 1) shows the solution for different values of N (maximum number of terms taken in the series). It was found that the solution stabilized after N = 9.The programming was done using the Fortran language on an I7 core computer. The numerical inversion of the Laplace transform was done using a method outlined in(Honig & Hirdes, 1984).

Figure 1
Convergente graph for temperatura at α = 0.5 (case 1) for t = 0.1

Figs 2 to 5 represent case 1 while figures 6 to 9 represent case 2. We did the evaluations using 3 values of α, which are: α = 0.5, 0.95 and 1 for t = 0.06.

Figure 2
Temperature distribution for different α (case 1) for t = 0.06

Figure 3
Radial displacement distribution for different α (case 1) for t = 0.06

Figure 4
Tangential displacement distribution for different α (case 1) for t = 0.06

Figure 5
Radial stress distribution for different α (case 1) for t = 0.06

Figure 6
Temperature distribution for different α (case 2) for t = 0.06

Figure 7
Radial displacement distribution for different α (case 2) for t = 0.06

Figure 8
Tangential displacement for different α (case 2) for t = 0.06

Figure 9
Radial stress distribution for different α (case 2) for t = 0.06

The results are shown in Fig. 2, 6 for the temperature distribution, Fig. 3, 7 for the radial displacement distribution, Fig. 4, 8 for the tangential displacement distribution and Fig. 5, 9 For stress distribution.

Figs 10 to 17 represent the time evolution of the different functions when t = 0.1, 0.2 and 0.3 for α = 0.5. Case 1 is shown in figures 10 to 13 while figures 14 to 17 represent case 2.

Figure 10
Temperature distribution for different t (case 1) for α = 0.5

Figure 11
Radial displacement distribution for different t (case 1) for α = 0.5

Figure 12
Tangential distribution for different t (case 1) for α = 0.5

Figure 13
Radial stress distribution for different t (case 1) for α = 0.5

Figure 14
Temperature distribution for different t (case 2) for α = 0.5

Figure 15
Radial displacement distribution for different t (case 2) for α = 0.5

Figure 16
Tangential distribution for different t (case 2) for α = 0.5

Figure 17
Radial stress distribution for different t (case 2) for α = 0.5

Figures 18 and 19 represent temperature versus φ for case 1 and 2 respectively at t = 0.1 and r = 0.4.

Figure 18
Temperature vs Φ for different α (case 1) for t = 0.1, r = 0.4

Figure 19
Temperature vs Φ for different α (case 2) at t = 0.1, r = 0.4

All these figures represent the functions as functions of r on the diagonal φ = π/15 and φ = - 14π/15.

The computations show that:

1. For α = 0.5, we can see from the graphs that the waves in the medium propagate with infinite speeds like the coupled theory of thermoelasticity. The program was run with α = 0 (corresponding to the coupled theory of thermoelaciticity), the results were almost identical to those when α = 0.5. For α = 1, the solution exhibits finite speeds since it is that of the generalized theory. The heat Equation associated with the couple theory of thermoelasticity (α = 0) is of parabolic type and predicts infinite speed of propagation for heat waves. The solution is nonzero (though it may be very small) at points far removed from the source of heating. The heat equation of the generalized theory of thermoelasticity (α = 1) is of hyperbolic type and predicts finite speed for heat waves. This means that heat propagates from the source of heating with a finite velocity. The solution is identically zero at points farther than the wave front. The location of the wave fronts and the value of the velocities for heat and elastic waves were discussed in (Sherief, H., & Hamza, 1994).

2. For α ≈ 1, the situation is somewhat difficult to determine. The solution seems to travel with finite speeds. Of course, this is based on numerical evaluations only. This aspect would be very important when proved theoretically. The same conjecture was expressed in(Povstenko, 2011).

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Publication Dates

• Publication in this collection
  Aug 2016

History

• Received
  01 Sept 2015
• Reviewed
  27 Feb 2016
• Accepted
  04 Apr 2016