Adomian’s decomposition method to modeling power functionally graded thermoelastic materials in heat transfer and thermal stress analysis

Hamdy M. Youssef1 , Eman A. N. Al-Lehaibi2

1Mathematics Department, Faculty of Education, Alexandria University, Alexandria, Egypt

1Mechanics Department, Faculty of Engineering, Umm Al-Qura University, Makka, Saudi Arabia

2Mathematics Department, Al-Lith University College, Umm Al-Qura University, Makka, Saudi Arabia

1Corresponding author

Vibroengineering PROCEDIA, Vol. 22, 2019, p. 188-193. https://doi.org/10.21595/vp.2019.20538
Received 21 January 2019; accepted 28 January 2019; published 15 March 2019

Copyright © 2019 Hamdy M. Youssef, 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.
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Abstract.

This work deals with an iteration method for numerical solving the problem of one-dimensional coupled thermoelasticity under given boundary conditions. This iteration based on the Adomian’s decomposition method. All the material properties have been considered variable on position with a power law. The numerical results have been calculated for different cases of the gradient parameter and the gradient index. The numerical results have been shown in figures. The gradient parameter and the gradient index have significant effects on the temperature increment, the strain, the stress, and the displacement.

Keywords: Adomian’s decomposition method, thermoelasticity, iteration method, power law, functionally graded.

1. Introduction

Recently, much attention has been devoted to the numerical methods in which do not require discretization of time-space variables or to the linearization of the non-linear equations [1]. Adomian introduced the decomposition method for solving linear and non-linear ordinary and partial differential equations [2-4]. This method offers accurate and computable with approximately convergent solutions to linear and non-linear partial and ordinary differential equations [5-16]. Adomian got the solutions of many bio-mathematical models of viruses, bacteria, antigens, and tumor tissues [4]. Adomian’s decomposition method (ADM) is to divide the given equation into linear and nonlinear parts of the equation [1, 13].

2. Mathematical idealizations of a FGM by using power law

This particular idealization for FGM modeling is well-known in the fracture mechanics science. The thickness h, the typical material properties Px at any point at a distance x from the reference surface has been modeled to this equation [17]:

(1)
P x = P T o p 1 - R x h n ,       R = 1 - P B o t P T o p ,       P T o p = P x x = 0 ,       P B o t = P x x = L .

R is the gradient parameter when it vanishes all the material properties are in the standard case with constant values. n is the material gradient index which depends upon the design requirements [17].

3. Formulation the problem in general form

Consider an isotropic and thermo-elastic body in one-dimensional fill the region which is defined by Ψ=x: 0x<h  where h is the thickness of the body, and it is initially at rest and has been loaded by the harmonic thermal wave, and the surface is traction free.

The displacement components for one-dimension medium have the form [18]:

(2)
u x , t = u x x , t ,             u y = u z = 0 .

The equation of motion:

(3)
ρ x         2 e   t 2 = λ x + 2   μ x     2 e   x 2 - γ x   2 θ   x 2 .

The generalized equation of heat conduction has the form:

(4)
x K x θ x = ρ x   C E x θ   t + γ x   T 0   e   t ,
(5)
K x 2 θ x 2 + K x x θ x = ρ x   C E x θ   t + γ x   T 0   e   t .

The constitutive relation takes the form:

(6)
σ = λ x + 2 μ x e -   γ x   θ ,       e =   u   x .

In the above equations, θ=T-T0 is the temperature increment, ρx is the density, λx and μx are Lame’s parameters, Kx is the thermal conductivity, γx is a material constant given by γ(x)=3λ(x)+2μ(x)αT(x), αT(x) being the coefficient of linear thermal expansion, and CEx is the specific heat at constant strain.

4. Formulation of the problem by using the exponential law

Substitute from Eq. (7) into Eqs. (4)-(6), we get:

(7)
K x , λ x ,   μ x ,   C E x ,   ρ x ,   α T x = K 0 ,   λ 0 ,   μ 0 , C 0 E ,   ρ 0 , α 0 T 1 - R x h n ,
(8)
ρ 0         2 e   t 2 = λ 0 + 2   μ 0 - γ 0 1 - R x h n   2 θ   x 2 ,
(9)
1 - R x h n 2 θ x 2 - n R h 1 - R x h n - 1 θ x = ρ 0   C 0 E K 0 1 - R x h 2 n θ   t
          + γ 0 T 0 K 0 1 - R x h 2 n e   t ,
(10)
σ = λ 0 + 2 μ 0 1 - R x h n e - γ 0 1 - R x h 2 n θ .

For simplicity, we use the non-dimensional variables (we will drop the primes) [18]:

(11)
x ' , u ' , h ' = c 0 η x , u , h ,       t ' = c 0 2 η t ,       θ ' = θ T 0 ,       σ = σ ' λ 0 + 2 μ 0 ,

where:

η = ρ 0 C 0 E K 0 ,       γ 0 = 3 λ 0 + 2 μ 0 α 0 T ,       c 0 = λ 0 + 2 μ 0 ρ 0 ,       ε 1 = γ 0 T 0 λ 0 + 2 μ 0 ,       ε 2 = γ 0 ρ 0 C 0 E ,

thus, we obtain:

(12)
    2 e   x 2 =     2 e   t 2 + ε 1 1 - R x h n   2 θ   x 2 ,    
(13)
2 θ x 2 = 1 - R x h n θ   t + ε 2 1 - R x h n e   t + n R h 1 - R x h - 1 θ x ,
(14)
σ = 1 - R x h n e - ε 1 1 - R x h 2 n θ .

Eq. (13) has been reduced to the form:

(15)
2 θ x 2 = 1 - R x h n θ   t + ε 2 1 - R x h n e   t + n R h 1 + R x h θ x .

5. Adomian’s decomposition method (ADM)

The differential operator L is defined as following [8, 10, 15]:

(16)
L x x e x , t = L t t e x , t + ε 1 1 - R x h n L x x θ x , t ,
(17)
L x x θ x , t = 1 - R x h n L t θ x , t + ε 2     L t e x , t + n R h 1 + R x h L x θ x , t .

The appeared operators in the above equations are defined as:

(18)
L t = t ,       L t t = 2 t 2 ,         L x = x ,         L x x = 2 x 2 .

Assuming that the inverse of the operator Lx-1 , Lxx-1 exists in the forms [8, 10, 15]:

(19)
L x - 1 f x = 0 x f x 1 d x 1 ,         L x x - 1 f x = 0 x 0 x 2 f x 1 d x 1 d x 2 .

Thus, applying the inverse operator on both the sides of Eqs. (17) and (18), we obtain:

(20)
e x , t = e 0 , t + e x , t x x = 0 + L x x - 1 L t t e x , t + ε 1 1 - R x h n L x x θ x , t   ,
(21)
θ x , t = θ 0 , t + θ x , t x x = 0
          + L x x - 1 1 - R x h n L t θ x , t + ε 2 1 - R x h n L t e x , t + n R h 1 + R x h L x θ x , t .

We decompose the functions θx,t and ex,t as following [8, 10, 15]:

(22)
θ x , t = k = 0 θ k x , t = θ 0 + k = 1 θ k x , t ,         e x , t = k = 0 e k x , t = e 0 + k = 1 e k x , t ,
(23)
e 0 = e 0 , t + e x , t x x = 0 ,         θ 0 = θ 0 , t + θ x , t x x = 0 ,
(24)
k = 0 e k x , t = e 0 , t + e x , t x x = 0
          + L x x - 1 L t t k = 0 e k x , t + ε 1 1 - R x h n   L x x k = 0 θ k x , t   ,
(25)
k = 0 θ k x , t = θ 0 , t + θ x , t x x = 0
          + L x x - 1 1 - R x h n   L t k = 0 θ k x , t + ε 2 1 - R x h n L t k = 0 e k x , t + n R h 1 + R x h L x k = 0 θ k x , t .

We obtain these components by ekx,t and θkx,t the recursive formulas [8, 10, 15]:

(26)
e k + 1 x , t = L x x - 1 L t t e k x , t + ε 1 1 - R x h n L x x θ k x , t   ,         k 1 ,
(27)
θ k + 1 x , t = L x x - 1 1 - R x h n L t θ k x , t + ε 2     L t e k x , t
          + n R h L x x - 1 1 + R x h L x θ k x , t ,         k 1 .

The bounding plane x= 0 is thermally loaded by harmonic heat and traction free as follows:

(28)
θ x , t x = 0 = θ 0 s i n ω t ,       θ x , t x x = 0 = 0 ,         σ x , t x = 0 = 0 ,         e x , t x x = 0 = 0 .

Moreover, the series solutions of Eq. (22) are convergent very rapidly in real physical problems as in [12, 13]. The convergence of the series has investigated by several authors in [1, 5, 8, 9, 12-15]. In an algorithmic form, the suitable value for the tolerance Tol=  10-6 [10]:

6. The numerical results

The constants of the material properties were taken as follows [18]: K0= 386 W/(mK), α0T= 1.78×10-5K-1, C0E= 383.1 J/(kgK), η= 8886.73 s/m2, T0= 293 K, μ0=  3.86×1010 N/m2, λ0=  7.76×1010 N/m2, ρ0=  8954 kg/m3, ε1= 0.0104443, ε2= 1.60862, ω=π, θ0= 1.0, h= 1.0.

Fig. 1. θ x , 2.0 distribution for R= 0.0, 0.1

θx,2.0 distribution for R= 0.0, 0.1

Fig. 2. e x , 2 . 0 distribution when R= 0.0, 0.1

ex,2.0 distribution when R= 0.0, 0.1

Fig. 3. σ x , 2.0 distribution when R= 0.0, 0.1

σx,2.0 distribution when R= 0.0, 0.1

Fig. 4. u x , 2.0 distribution when R= 0.0, 0.1

ux,2.0 distribution when R= 0.0, 0.1

Fig. 5. θ x , 2.0 distribution when R= 0.0, 0.5

θx,2.0 distribution when R= 0.0, 0.5

Fig. 6. e x , 2.0 distribution when R= 0.0, 0.5

ex,2.0 distribution when R= 0.0, 0.5

Fig. 7. σ x , 2.0 distribution when R= 0.0, 0.5

σx,2.0 distribution when R= 0.0, 0.5

Fig. 8. u x , 2.0 distribution when R= 0.0, 0.5

ux,2.0 distribution when R= 0.0, 0.5

7. Conclusions

Figs. 1-8 represent the temperature increment, the strain, the stress, and the displacement distribution with various values of the gradient parameter. R= 0.0 gives the normal case of non-functionally graded material, while R0.0 performs a functionally graded material of power law with different valuesR= (0.1, 0.2). From the consideration in Eq. (1), R= 0.1 means that the ratio PBot= 90 % PTop, and R= 0.2 gives that the ratio PBot= 80 % PTop. According to the results and the figures, the parameter R has significant effects on all the stat-functions θ=θx,t, e=ex,t, σ=σxxx,t, and u=ux,t. Moreover, n has been assumed with different values through the calculations (n= 1, 5, 10) which give various materials’ designs. Figs.  1, 3, 5 and 7 show that when the value of the parameters n and R increase, the value of the temperature increment and the absolute value of the stress increase, while Figs. 2, 4, 6, and 8 show that when the value of the parameters n and R increases the value of the strain and the displacement decrease.

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