A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source

Applied Mathematical Modelling xxx (2014) xxx–xxx

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A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source Allam A. Allam ⇑ Department of Basic and Applied Science, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Abu Quir Campus, Alexandria, Egypt

a r t i c l e

i n f o

Article history: Received 20 October 2012 Received in revised form 16 October 2013 Accepted 25 March 2014 Available online xxxx Keywords: Diffusion Half-space Second sound Stochastic simulation Thermoelasticity White noise

a b s t r a c t A stochastic half-space problem, driven by an additive Gaussian white noise, is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. The mean and variance are derived and analyzed for temperature, displacement, stress, strain, concentration and chemical potential. The asymptotic behavior for the solution is discussed. Numerical results are carried out and represented graphically. The second sound effect is observed in the simulation. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Thermoelasticity describes the behavior of elastic bodies under the influence of nonuniform temperature fields. Its foundations have been laid in the first half of the nineteenth century. Wide-spread interest in this field did not develop until the years subsequent to World War Two. There are good reasons for this. First, in the field of aeronautics, the high velocities of modern aircraft have been found to give rise to aerodynamic heating; in turn, this produces intense thermal stresses and reduces the strength of the aircraft structure. Secondly, in the nuclear field, the extremely high temperatures and temperature gradients originating in nuclear reactors influence their design and operation. Likewise, in the technology of modern propulsive systems, such as jet and rocket engines, the high temperatures associated with combustion processes are the origin of unwelcome thermal stresses. Similar phenomena are encountered in the technologies of space vehicles and missiles, in the mechanics of large steam turbines, and even in shipbuilding, where, strangely enough, ship fractures are often attributed to thermal stresses of moderate intensities, see Nowinski [1]. In 1837, Duhamel [2] was the first to consider elasticity problems with heat changes. In 1841, Neumann [3] had rederived the same set of equations obtained by Duhamel earlier. Their theory is known as the classical or uncoupled theory of thermoelasticity. One defect of this theory is that it does not consider elastic changes due to temperature and vice versa. Another defect is that the equation of temperature has a parabolic type that predicts an infinite speed of propagation for temperature which contradicts physical observations.

⇑ Tel.: +20 01001859657. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apm.2014.03.044 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

In 1956, Biot [4] formulated the theory of coupled thermoelasticity to eliminate the paradox inherent in the classical uncoupled theory that elastic changes have no effect on the temperature, but still shares the uncoupled theory of thermoelasticity in that the temperature has infinite speed. The concept of the second sound that the hyperbolic nature involving finite speeds of thermal disturbance is reported by Maxwell [5]. The fact that the so-called second sound must exist in any solid is justified by Chester [6]. Most of the approaches that came out to overcome the unacceptable prediction of the classical theory are based on the general notion of relaxing the heat flux in the classical Fourier heat conduction equation. One of the simplest forms of these equation is due to the work of Lord and Shulman [7]. Lord and Shulman theory, L–S Theory, involves one relaxation time for the special case of an isotropic body. In 1980, this theory was extended by Sherief [8] and Dhaliwal and Sherief [9] to include the anisotropic case. In this theory a modified law of heat conduction including both the heat flux and its time derivative replaces the conventional Fourier’s law. Sherief et al. [10] extended this generalized theory to include micropolar materials. Lately, Sherief et al. [11] developed a new theory using fractional calculus. Another thermoelasticity theory that admits the second sound effect is the temperature rate dependent theory of thermoelasticity that takes into account two relaxation times. This theory was developed by Green and Lindsay [12] and is known as G–L Theory. An alternative approach in the formulation of a theory predicting the finite propagation speed of the thermal disturbances is due to Green and Naghdi (G–N), where they formulated three models of thermoelasticity for homogeneous and isotropic materials labeled as models I, II, and III, see [13,14]. These theories of thermoelasticity (L–S, G–L, and G–N theories) are known as the generalized theories, or thermoelasticity theories with the second sound effect or with finite thermal wave speed. Ignaczak [15] suggested a combined system of coupled equations for the L–S and G–L theories. Also, the same author reported a survey of the domain of influence for the results of the L–S and G–L theories, see [16]. Francis [17], Ignaczak [15] and Chandrasekharaiah [18,19] have reported brief reviews of these theories. Bagri and Eslami [20] has proposed a new unified formulation for the generalized coupled thermoelasticity theories based on the L–S, G–L, and G–N models. It was shown by Maurer and Thompson [21] that, by submitting a thin slab to an intense thermal shock, its surface temperature is 300 °C larger than the value predicted by the classical theory. During the early works on generalized thermoelasticity, authors used to make comparisons with the coupled theory. They have found that for large values of time the solutions obtained from either theory are almost identical. For short times though the generalized theory gives markedly larger values for the temperature and stresses than the coupled theory, see [22]. Design and building of nuclear reactors is one of the most common examples given for the importance of using the generalized theory of thermoelasticity which gives more accurate results for the resulting stresses during the initial stages of running the reactor. The concept of thermodiffusion is used to describe the process of thermo-mechanical treatment of metals. The study of this phenomenon is of great concern due to its many geophysical and industrial applications. For example, oil companies are interested in the process of thermodiffusion for more efficient extraction of oil from oil deposits. The first critical review on this phenomenon was presented by Oriani [23]. Using the coupled thermoelastic model, Nowacki [24–26] developed the theory of coupled thermoelastic diffusion and studied some dynamical problems of diffusion in solids. The theory developed by Nowacki uses what is called Fick’s law. It has a major drawback that it predicts infinite speeds for wave propagation. Sherief et al. [27] developed the theory of generalized thermoelastic diffusion, which allows the finite speeds of propagation of waves. In this theory, the Fick’s law is modified to include the time derivative of the flux of the diffusive mass. It is referred to as a non-Fickian theory. This theory is used in the current work. Recently, Aouadi [28] developed a theory of thermoelastic diffusion materials with voids. This theory is modified in the context of Lord and Shulman theory of generalized thermoelasticity by Singh [29]. Lately, Ezzat and Fayik [30] has derived a new theory of thermodiffusion in elastic solids using the methodology of fractional calculus. A three-dimensional thermoelastic problem for a half-space without energy dissipation has been solved by Sarkar and Lahir [31]. Partial differential equations with random parameters, such as the coefficient or the forcing term, have begun to attract the attention of many researchers science 1960s. Most of them were motivated by applications to physical and biological problems. Notable examples are turbulent flow in fluid dynamics, diffusion and wave in random media, see [32,33]. In general the random parameters or random fields involved need not be of white-noise type, but, in many applications, models with white noises provide reasonable approximation, see [34]. The analysis of thermal problems subjected to external loads is developed under the assumption that the structure’s parameters are deterministic quantities. For a significant number of circumstances, this assumption is not valid, and the probabilistic aspects of the thermal problem need to be taken into account, [35]. There are several reasons that suggest the replacement of deterministic models with stochastic ones. The following reasons can be mentioned [36–38]: Table 1 Material constants. T 0 ¼ 293 K

cE ¼ 383:1 J=ðkg KÞ

at ¼ 1:78ð10Þ5 K1

ac ¼ 1:98ð10Þ4 m3 =kg l ¼ 3:86ð10Þ10 kg=ðm s2 Þ

k ¼ 386 W=ðm KÞ

k ¼ 7:76ð10Þ10 kg=ðm s2 Þ

D ¼ 0:85ð10Þ8 kg s=m3 s0 ¼ 10 s ¼ 0:02 s

a ¼ 1:2ð10Þ4 m2 =ðs2 KÞ q ¼ 8954 kg=m3

b ¼ 0:9ð10Þ6 m5 =ðkg s2 Þ

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

3

(1) The system is not fully isolated, thus background fields give rise to additional noise. (2) Not all the variables that characterize the system are included in the model thus these variables give rise to additional noise. (3) The accuracy of the measuring devices for the temperature, etc. are not 100%. In the topic of stochastic simulation techniques for analysis of heat conduction and thermoelastic problems, some works have been developed by many researchers. Ahmadi [39] and Tzou [40] considered uncertainty in the thermal conductivity, while Val’kovskaya and Lenyuk [41] considered problems with stochastic internal heat generation. Stochastic analysis have also been performed for cases in which the boundary conditions or the initial condition varies randomly, see [42–45]. Barrett [46] studied problems where the shape and material properties are random. This paper is devoted to study the influence of an additive noise on heat, elastic and diffusion waves. The existence of such noise is reasonable due to the reasons mentioned above and the way how linearized theory of generalized thermoelasticity is derived, where the second and higher order terms are neglected. The noise is assumed to be Gaussian white noise, since white noise is often taken as a mathematical idealization of phenomena involving sudden and extremely large fluctuations which occurs in diffusion. An analytical technique using Laplace transform is presented to obtain the strong solutions of temperature, displacement, stress, strain, chemical potential and concentration of the diffusive material in the Laplace transform domain, where Laplace transform of the white noise is defined formally as an Itô integral. As a statistical measurements, we drive the mean and the variance of the distributions. 2. Mathematical formulation of the problem In this work we consider a one-dimensional problem for an isotropic half-space x P 0 with a permeating substance in contact with the upper plane of the half-space x ¼ 0. The x-axis is taken perpendicular to the upper plane pointing inwards. The surface of the bounding plane of the half-space is taken to be traction free subjected to a time dependent thermal shock and a time dependent chemical potential. All considered functions are assumed to be bounded and vanish as x ! 1. From [27], the governing equations for generalized thermoelastic diffusion in the absence of body forces consist of 1. The equation of motion

q u€i ¼ l ui;jj þ ðk þ lÞ uj;ij  b1 T;i  b2 C;i ;

ð1Þ

where T is the absolute temperature, C is the concentration of the diffusive material, q is the density, k and l are Lamé’s constants and b1 and b2 are material constants given by b1 ¼ ð3k þ 2lÞ at and b2 ¼ ð3k þ 2lÞ ac , where at is the coefficient of linear thermal expansion, and ac is the coefficient of linear diffusion expansion. 2. The energy equation

      k T;ii ¼ q cE T_ þ s0 T€ þ b1 T 0 ðe_ þ s0 €eÞ þ a T 0 C_ þ s0 C€  q Q þ s0 Q_ ;

ð2Þ

where k is the thermal conductivity of the medium, cE is the specific heat at constant strain, s0 is the thermal relaxation time, T 0 is the reference temperature chosen such that jðT  T 0 Þ=T 0 j  1, a is a measure of thermodiffusion effect, e ¼ ui;i is the cubical dilatation, where ui ; i ¼ 1; 2; 3 are the components of the displacement vector and Q is the amount of heat resulted from the heat source. 3. The equation of mass diffusion

€  D b C; ¼ 0; D b2 e;ii þ D a T;ii þ C_ þ s C ii where D is the diffusion coefficient, b is a measure of diffusive effect and 4. The constitutive equations

ð3Þ

s is the diffusion relaxation time.

rij ¼ 2 l ij þ dij ðk e  b1 ðT  T 0 Þ  b2 CÞ ;

ð4Þ

P ¼ b2 e þ b C  a ðT  T 0 Þ ;

ð5Þ

where rij ; i; j ¼ 1; 2; 3 are the components of the stress tensor, P is the chemical potential of the material per unit mass, and i; j ¼ 1; 2; 3 are the components of the strain tensor, given by

ij ;

ij ¼

1 ðui;j þ uj;i Þ; 2

i; j ¼ 1; 2; 3:

ð6Þ

For the considered problem (one-dimensional problem), the displacement vector u has the form u ¼ ðu; 0; 0Þ. Therefore, the components of the strain tensor of (6) can be written in the form

xx ¼ D u

and

yy ¼ zz ¼ xy ¼ yz ¼ zx ¼ 0;

ð7Þ

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

where D denotes a partial derivative with respect to x, hence the cubical dilatation e ¼ div u has the following form

e ¼ D u:

ð8Þ

Also the governing equations (1)–(5) will have the following forms

q u€ ¼ ðk þ 2 lÞ D2 u  b1 D T  b2 D C;

ð9Þ

      k D2 T ¼ q cE T_ þ s0 T€ þ b1 T 0 ðe_ þ s0 €eÞ þ a T 0 C_ þ s0 C€  q Q þ s0 Q_ ;

ð10Þ

€  D b D2 C ¼ 0; D b2 D2 e þ D a D2 T þ C_ þ s C

ð11Þ

rxx ¼ ðk þ 2 lÞ D u  b1 ðT  T 0 Þ  b2 C; ryy ¼ rzz ¼ k D u  b1 ðT  T 0 Þ  b2 C; rxy ¼ rzy ¼ rxz ¼ 0;

ð12aÞ ð12bÞ ð12cÞ

P ¼ b2 e þ b C  a ðT  T 0 Þ:

ð13Þ

Introducing the following non-dimensional variables

x ¼ c1 g x;

t  ¼ c21 g t;

u ¼ c1 g u;

s0 ¼ c21 g s0 ;

b1 ðT  T 0 Þ b C ; C ¼ 2 ; k þ 2l k þ 2l P cq Q P ¼ ; Q  ¼ ; b2 k T 0 c21 g2

s ¼ c21 g s; h ¼ rij ¼

rij k þ 2l

;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g ¼ q cE =k and c1 ¼ ðk þ 2 lÞ=q is the speed of propagation of isothermal elastic waves. Under these non-dimensional variables, (9)–(13) can be written as (dropping the asterisks for convenience)

€ ¼ D2 u  D h  D C; u

ð14Þ

  €  Q þ s0 Q_ ; D2 h ¼ h_ þ s0 €h þ e ðe_ þ s0 €eÞ þ e a1 ðC_ þ s0 CÞ

ð15Þ

€  a3 D2 C ¼ 0; D2 e þ a1 D2 h þ a2 ðC_ þ s CÞ

ð16Þ

rxx ¼ e  h  C; ryy ¼ rzz

ð17aÞ

  2 ¼ 1  2 e  h  C; b

ð17bÞ

P ¼ a3 C  e  a1 h;

ð18Þ

where

e¼

b21 T 0 ; q cE ðk þ 2 lÞ

a3 ¼

b ðk þ 2 lÞ b22

;

a1 ¼

a ðk þ 2 lÞ ; b1 b2

and b2 ¼

ðk þ 2 lÞ

l

a2 ¼

ðk þ 2 lÞ b2 D g

;

:

The derivation of the governing equations for the thermoelastic diffusion theory and the reasons mentioned earlier in Section 1 lead us to consider an additive noise on our system. Writing the energy equation (15) as

  € ¼ Q þ s0 Q_ þ rnðtÞ em x ; h_ þ s0 €h  D2 h þ e ðe_ þ s0 €eÞ þ e a1 ðC_ þ s0 CÞ

ð19Þ

where m is a constant and nðtÞ denotes a Gaussian white noise processes with real valued parameter r P 0 controls the strength of the induced noise. The initial conditions of the problem are taken to be homogenous, while the boundary conditions are assumed to be

hðx; tÞjx¼0 ¼ h0 ðtÞ hðx; tÞjx¼1 ¼ 0; rðx; tÞjx¼0 ¼ 0 rðx; tÞjx¼1 ¼ 0;

ð20Þ

Pðx; tÞjx¼0 ¼ P0 ðtÞ Pðx; tÞjx¼1 ¼ 0 for known functions of t; h0 ðtÞ and P 0 ðtÞ. Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

1.2 1

t = 0.050

0.8 0.6

t = 0.100

t = 0.300

0.4 0.2 0 −0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.4

1.6

1.8

2

x

0.2 0 −0.2 −0.4 −0.6

t = 0.300

−0.8 t = 0.100 −1

t = 0.050

−1.2 0

0.2

0.4

0.6

0.8

1

1.2

x

1.5

1

t = 0.050

0.5

t = 0.300

0 t = 0.100 −0.5 0

0.2

0.4

0.6

0.8

1

1.2

x

Fig. 1. The mean of temperature, stress and strain for different values of times in the context of coupled and generalized theories of thermoelasticity without heat source at h0 ¼ P 0 ¼ m ¼ 1.

3. Solution of the problem in the Laplace transform domain In this section, we are going to formulate and solve our problem in the Laplace transform domain. The Laplace transform for a function f ðx; tÞ is defined by the formula

f ðx; sÞ ¼ Lff ðx; tÞg ¼

Z

t

es t f ðx; tÞ dt

0

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

0.1

0

t = 0.300

−0.1

t = 0.100 −0.2

t = 0.050

−0.3 0

0.2

0.4

0.6

0.8

1

1.2

x

1.2 1 0.8 0.6 0.4 t = 0.050

t = 0.300

0.2 0

t = 0.100

−0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.2

1.4

1.6

1.8

2

x

0.3 0.25 t = 0.300

0.2 t = 0.100 0.15 0.1 0.05 0

t = 0.050

−0.05 0

0.2

0.4

0.6

0.8

1

x Fig. 2. The mean of displacement, Chemical potential and concentration for different values of times in the context of coupled and generalized theories of thermoelasticity without heat source at h0 ¼ P 0 ¼ m ¼ 1.

with its inverse transform

1 f ðx; tÞ ¼ L1 ff ðx; sÞg ¼ 2pj

Z

dþj 1

est f ðx; sÞ ds:

dj 1

From now on, we take Q ðx; tÞ in the form

Qðx; tÞ ¼ Q 0 HðtÞem x ;

ð21Þ

where Q 0 is the strength of the heat source and HðtÞ is the Heaviside function. By the transformation of (14), (19), (16), (17) and (18), and applying the homogenous initial conditions, we can obtain Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

1 0.8 t = 0.050 t = 0.075 t = 0.100

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

1

0.5

0

−0.5

−1 0

t = 0.050 t = 0.075 t = 0.100 0.5

1

1.5

2

2.5

3

x 2

1.5 t = 0.050 t = 0.075 t = 0.100

1

0.5

0 0

0.5

1

1.5

2

2.5

3

x Fig. 3. The mean of temperature, stress and strain for different values of times at h0 ¼ P 0 ¼ Q 0 ¼ m ¼ 1.



 2

 ¼ 0;   D h  D C D2  s u

ð22Þ 



 þ rnðsÞ em x ; e ðs þ s0 s2 Þ e þ ðs þ s0 s2  D2 Þ h þ ea1 ðs þ s0 s2 Þ C ¼ FðsÞ

ð23Þ



 ¼ 0; D2 e þ a1 D2 h þ a2 ðs þ s s2 Þ  a3 D2 C

ð24Þ

 r xx ¼ e  h  C;

ð25aÞ

r yy ¼ r zz

ð25bÞ

  2  ¼ 1  2 e  h  C; b

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

0.1 0 −0.1 −0.2

t = 0.050 t = 0.075 t = 0.100

−0.3 −0.4 −0.5 −0.6 −0.7 0

0.5

1

1.5

2

x

2.5

3

3.5

4

1.5

1 t = 0.050 t = 0.075 t = 0.100

0.5

0

−0.5 0

0.5

1

1.5

2

x

2.5

3

3.5

4

0.25 0.2 0.15

t = 0.050 t = 0.075 t = 0.100

0.1 0.05 0 −0.05 0

0.2

0.4

0.6

0.8

1

x

1.2

1.4

1.6

1.8

2

Fig. 4. The mean of displacement, chemical potential and concentration for different values of times at h0 ¼ P 0 ¼ Q 0 ¼ m ¼ 1.

  e  a1 h;  ¼ a3 C P

ð26Þ

where FðtÞ ¼ Q 0 ð HðtÞ þ s0 dðtÞ Þ. The divergence of (22) gives

ðD2  s2 Þ e  D2 h  D2 C ¼ 0 :

ð27Þ

 sÞ from (23), (24) and (27), we obtain the following six Eliminating the transformed strain  eðx; sÞ and concentration Cðx; hðx; sÞ order partial differential equation for the transformed temperature 

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 2 Values of the distributions for different times at x ¼ 0:0234 and x ¼ 0:1953. x ¼ 0:0234

hðx; t Þ uðx; tÞ C ðx; tÞ

x ¼ 0:1953

t ¼ 0:050

t ¼ 0:075

t ¼ 0:100

t ¼ 0:050

t ¼ 0:075

t ¼ 0:100

0:9526 0:2884 0:2066

0:9607 0:4459 0:2079

0:9665 0:6040 0:2089

0:6047 0:2239 0:1229

0:6685 0:3372 0:1353

0:7135 0:4531 0:1452

0.025 0.02 0.015

t = 0.050 t = 0.075 t = 0.100

0.01 0.005 0 0

0.5

1

1.5

2

x

2.5

3

1.5

t = 0.050 t = 0.075 t = 0.100

1

0.5

0 0

0.5

1

x

1.5

2

2.5

2

1.5 t = 0.050 t = 0.075 t = 0.100

1

0.5

0 0

0.5

1

x

1.5

2

2.5

Fig. 5. The dispersion of temperature, stress and strain about its mean for different values of times at h0 ¼ P 0 ¼ Q 0 ¼ r ¼ m ¼ 1.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

ðD6  a1 D4 þ a2 D2  a3 Þ h ¼



1

a3  1

        þ r nðsÞ em x ; ð1  a3 Þ D4 þ a3 s2 þ a2 s þ s s2 D2  a2 s2 s þ s s2 FðsÞ ð28Þ

where the coefficients a1 ; a2 and a3 are given in the appendix. In the same manner, we can show that the transformed strain  sÞ satisfy the following ordinary differential equations eðx; sÞ and concentration Cðx;

2.5 2 1.5

t = 0.050 t = 0.075 t = 0.100

1 0.5 0 0

0.5

1

1.5

x

2

2.5

0.8

0.6 t = 0.050 t = 0.075 t = 0.100

0.4

0.2

0 0

0.5

1

1.5

2

2.5

x

3

3.5

4

−3

3.5

x 10

3 2.5 2

t = 0.050 t = 0.075 t = 0.100

1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

x

1

1.2

1.4

1.6

1.8

Fig. 6. The dispersion of displacement, chemical potential and concentration about its mean for different values of times at h0 ¼ P 0 ¼ Q 0 ¼ r ¼ m ¼ 1.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 3 Values of the variance for different times at x ¼ 0:1953 and x ¼ 0:6328. x ¼ 0:1953

hðx; t Þ rxx ðx; tÞ eðx; tÞ uðx; tÞ C ðx; tÞ P ðx; tÞ

x ¼ 0:6328

t ¼ 0:050

t ¼ 0:075

t ¼ 0:100

t ¼ 0:050

t ¼ 0:075

t ¼ 0:100

0:0099 0:3157 0:4746 0:4736

0:0126 0:6976 0:9644 0:9588

0:0141 1:1580 1:5230 1:5080

0:0065 0:1278 0:1974 0:1989

0:0128 0:2445 0:3994 0:4026

0:0197 0:3755 0:6329 0:6330

0:09ð10Þ5 0:1247

0:18ð10Þ5 0:1410

0:28ð10Þ5 0:1627

0:01ð10Þ5 0:49499

0:10ð10Þ5 0:5430

0:23ð10Þ5 0:6970

ðD6  a1 D4 þ a2 D2  a3 Þ e ¼



1











a3 ð1 þ a1 Þ D4 þ a1 a3 s2 þ a2 s þ s s2 þ a1 a2 s þ s s2 D2 a3  1      þ r nðsÞ em x ; a1 a2 s2 s þ s s2 FðsÞ

¼ ðD6  a1 D4 þ a2 D2  a3 Þ C



1

a3  1

ða1 þ 1Þ D4 þ a1 s2 D2



  þ r nðsÞ em x : FðsÞ

ð29Þ ð30Þ

hðx; sÞ can be obtained by writing (28) as The transformed temperature  2 2 2 ðD2  k1 ÞðD2  k2 ÞðD2  k3 Þ hðx;sÞ ¼

1

a3  1



        þ r nðsÞ em x ; ð1  a3 ÞD4 þ a3 s2 þ a2 s þ s s2 D2  a2 s2 s þ s s2 FðsÞ ð31Þ

that has a complementary solution in the form

hc ðx; sÞ ¼

3 X

 i ðsÞ e ki x ; A

ð32Þ

i¼1

where k1 ; k2 and k3 are the roots with positive real parts of the characteristic equation

K 6  a1 K 4 þ a2 K 2  a3 ¼ 0:

ð33Þ

The roots with negative real parts are neglected since the exponential terms in (32) will be unbounded as x ! 1. The particular solution is given by

   h ðsÞ FðsÞ  þ r nðsÞ em x ; hp ðx; sÞ ¼ M

ð34Þ

where

 h ðsÞ ¼ M

    

1  ð1  a3 Þ m4 þ a3 s2 þ a2 s þ s s2 m2  a2 s2 s þ s s2 : ða3  1Þ ðm6  a1 m4 þ a2 m2  a3 Þ

ð35Þ

The final form for the transformed temperature,  hðx; sÞ, can be obtained by adding (32) and (34) as

hðx; sÞ ¼

3 X

   i ðsÞ e ki x þ M  h ðsÞ FðsÞ  þ r nðsÞ em x : A

ð36Þ

i¼1

 sÞ can be obtained from (29) and (30) as eðx; sÞ and concentration Cðx; Similarly, the transformed strain 

eðx; sÞ ¼

3 X

   i ðsÞ e ki x þ M  e ðsÞ FðsÞ  þ rnðsÞ em x ; B

ð37Þ

i¼1

 sÞ ¼ Cðx;

3 X

   i ðsÞ e ki x þ M  C ðsÞ FðsÞ  þ rnðsÞ em x ; C

ð38Þ

i¼1

where

 e ðsÞ ¼ M

 C ðsÞ ¼ M

1 ða3  1Þ ðm6  a1 m4 þ a2 m2  a3 Þ       

 a3 ð1 þ a1 Þ m4 þ a1 a3 s2 þ a2 s þ s s2 þ a1 a2 s þ s s2 m2  a1 a2 s2 s þ s s2 ;

1 ða1 þ 1Þ m4 þ a1 s2 m2 ; ða3  1Þ ðm6  a1 m4 þ a2 m2  a3 Þ

ð39Þ ð40Þ

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

12

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

2.5 6

t = 37 x 10

t = 3 x 10

8

14

15

t = 3 x 10 & t = 3 x 10

9

t = 3 x 10

t = 3 x 10

2 1.5 1 0.5 0 t = 30 x 10 −0.5 0

3

2

1

t = 3 x 10

t = 3 x 10

t = 3 x 10 5

10

15

20

t = 35 x 10

4

t = 3 x 10 25

x

30

35

40

45

50

40 30 t=2 t = 22

20

t = 23

5

t=2

4

t=2

6

7

t=2 & t=2

10 0 −10 0

5

10

15

20

25

x

30

35

40

45

50

14 12

t = 213 & t = 214

t=2

10

t = 25

t = 23

t = 27

6

t=2

8 6 4

t = 22

2 0 0

5

10

15

20

25

x

30

35

t = 28

40

45

50

Fig. 7. Asymptotic behavior for the mean of temperature, stress and strain at h0 ¼ P 0 ¼ Q 0 ¼ m ¼ 1.

 i ðsÞ; B  i ðsÞ; i ¼ 1; 2; 3 are parameters depending on s only. Substituting from (36), (37) and (38) into (23),  i ðsÞ and C where A  i ðsÞ as  i ðsÞ; i ¼ 1; 2; 3 can be written in terms of A  i ðsÞ and C (24) and (27), the parameters B

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

13

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx 2

2

2  i ðsÞ;  i ðsÞ ¼ ki ½ki  ð1  e a1 Þ ðs þ s0 s Þ A B 2 e ðs þ s0 s2 Þ ½ð1 þ a1 Þ ki  a1 s2  4

i ¼ 1; 2; 3;

ð41Þ

2

2 2 3  i ðsÞ;  i ðsÞ ¼ ki  ki ½s þ ðe þ 1Þ ðs þ s0 s Þ þ s ð1 þ s0 sÞ A C 2 e ðs þ s0 s2 Þ ½ð1 þ a1 Þ ki  a1 s2 

i ¼ 1; 2; 3:

ð42Þ

 ðx; sÞ in the form Applying Laplace transform to Eq. (8) and integrating both sides, we get the transformed displacement u

 ðx; sÞ ¼ u

3 X

   i ðsÞ e ki x þ M  u ðsÞ FðsÞ  þ r nðsÞ em x ; D

ð43Þ

i¼1

 i ðsÞ ¼  1 B  ðsÞ, i = 1,2,3 and M  u ðsÞ ¼  1 M  e ðsÞ. Also the transformed normal stress r  xx ðx; sÞ and the chemical where D ki i m  potential Pðx; sÞ are obtained by substituting (36), (37) and (38) into (25a) and (26),

r xx ðx; sÞ ¼

3 X

  i ðsÞ e ki x þ M  r ðsÞ FðsÞ  þ rnðsÞ em x ; E x

ð44Þ

i¼1

 sÞ ¼ Pðx;

3 X

   P ðsÞ FðsÞ  þ r nðsÞ em x ; Fi ðsÞ e ki x þ M

ð45Þ

i¼1

s sÞ  i ðsÞ ¼ s22 B  i ðsÞ; Fi ðsÞ ¼ a2 s ð1þ where E C i ðsÞ, i = 1,2,3 and k2 k i

i

2 1 0 −1 t = 23

−2

t = 26

t = 25

t = 24

−3 −4 −5

t = 28 & t = 29

−6 0

5

10

15

20

25

30

x

0.5

35

t = 216

14

t=2

40

20

t=2

45

50

21

& t=2

0.4 0.3 0.2 0.1 3 0 t=2 4

−0.1 −0.2 0

t=2

6

t = 25

t=2

5

10

t = 27

t = 28 15

t = 29 20

t = 210 25

x

12 t = 211 t = 2

30

35

40

45

50

Fig. 8. Asymptotic behavior for the mean of chemical potential and concentration at h0 ¼ P 0 ¼ Q 0 ¼ m ¼ 1.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

14

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

 r ðsÞ ¼ M x

      

1  a1 m2  s2 ða3  1Þ m2  a2 s þ s s2  s2 a3 m2 þ a2 s þ s s2 ; ða3  1Þ ðm6  a1 m4 þ a2 m2  a3 Þ ð46Þ

 P ðsÞ ¼ M

1 ða3  1Þ ðm6  a1 m4 þ a2 m2  a3 Þ      

 a1 ð1  a3 Þ m4  2 a2 s2 s þ s s2 þ m2 s ð2 a2 þ a3 s þ 2 a2 s sÞ þ a2 m2 s þ s s2 :

ð47Þ

Taking the Laplace transform to the boundary conditions and applying them to (36), (44) and (45), we obtain 3 X

   i ðsÞ ¼ h0 ðsÞ  M  h ðsÞ FðsÞ  þ r nðsÞ ; A

ð48Þ

i¼1 2 3 X   s2 ½ki  ð1  e a1 Þ ðs þ s0 s2 Þ   r ðsÞ FðsÞ  þ r nðsÞ ; Ai ðsÞ ¼ M x 2 2 2 i¼1 e ðs þ s0 s Þ½ð1 þ a1 Þ ki  a1 s 

  4 2 3 a2 ð1 þ s sÞ k  k ½s2 þ ðe þ 1Þ ðs þ s0 s2 Þ þ s3 ð1 þ s0 sÞ X i i

e ð1 þ s0 sÞ k2i ½ð1 þ a1 Þ k2i  a1 s2 

i¼1

   i ðsÞ ¼ P0 ðsÞ  M  P ðsÞ FðsÞ  þ r nðsÞ A

ð49Þ

ð50Þ

 i ðsÞ; i ¼ 1; 2; 3. This completes the solution of the problem in the Laplace transfrom which we can obtain the parameters A formed domain. 4. Temperature distribution Due to the additive noise on our system, the temperature becomes a stochastic process. Here, we obtain the solution in  i ðsÞ, can be written in the folterms of general boundary conditions h0 ðtÞ and P0 ðtÞ. The solution of the system (48)–(50), A lowing form

 i ðsÞ ¼ A

3 X

 ij ðsÞ f j ðsÞ ; C

i ¼ 1; 2; 3

ð51Þ

j¼1

 ij ðsÞ that can be obtained using a mathematical software like matlab or mathematica and f j ðsÞ are the left for some functions C hand sides of the system (48)–(50), namely

  f 1 ðsÞ ¼ h0 ðsÞ  M  h ðsÞ FðsÞ  þ rnðsÞ ;   f 2 ðsÞ ¼ M  r ðsÞ FðsÞ  þ rnðsÞ ; x   f 3 ðsÞ ¼ P 0 ðsÞ  M  P ðsÞ FðsÞ  þ rnðsÞ :

ð52Þ

hc ðx; sÞ, can be written in the following form Therefore, the complementary part for the transformed temperature, 

hc ðx; sÞ ¼

3 X

   j ðx; sÞ f j ðsÞ ¼ c  1 ðx; sÞ h0 ðsÞ þ X  3 ðx; sÞ P0 ðsÞ;  þ r nðsÞ þ X ðx; sÞ FðsÞ X

ð53Þ

j¼1

where





 1 ðx; sÞ M  2 ðx; sÞ M  3 ðx; sÞ M  h ðsÞ þ X  r ðsÞ þ X  P ðsÞ ; cðx; sÞ ¼  X x  j ðx; sÞ ¼ X

3 X

 ij ðsÞ eki x ; C

ð54Þ

j ¼ 1; 2; 3 :

ð55Þ

i¼1

The inverse Laplace transform of (53) and (34) yields the temperature hðx; tÞ in the following form

hðx; tÞ ¼

Z

t

½ X1 ðx; t  uÞ h0 ðuÞ þ X3 ðx; t  uÞ P0 ðuÞ þ ð cðx; t  uÞ þ M h ðt  uÞ em x Þ ð FðuÞ þ r nðuÞ Þ  du:

ð56Þ

0

Recalling the fact that FðtÞ ¼ Q 0 ð HðtÞ þ s0 dðtÞ Þ, the temperature hðx; tÞ is given by

hðx; tÞ ¼

Z

t

½ X1 ðx; t  uÞ h0 ðuÞ þ X3 ðx; t  uÞ P0 ðuÞ  du þ Q 0

0

þ s0 Q 0 ð cðx; tÞ þ M h ðtÞ em x Þ þ r

Z

Z

t

½ cðx; t  uÞ þ Mh ðt  uÞ em x  du

0 t

½ cðx; t  uÞ þ M h ðt  uÞ em x  dWðuÞ ;

ð57Þ

0

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

15

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

1.4 σ = 0.0 σ = 1.0 σ = 0.5 σ = 0.1

1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

x

2.5

3

3.5

4

3.5

4

3.5

4

2.5 σ = 0.0 σ = 1.0 σ = 0.5 σ = 0.1

2 1.5 1 0.5 0 −0.5 −1 0

0.5

1

1.5

2

x

2.5

3

4 σ = 0.0 σ = 1.0 σ = 0.5 σ = 0.1

3

2

1

0 0

0.5

1

1.5

2

x

2.5

3

Fig. 9. The effect of different values for the noise intensity on a single sample path of temperature, stress and strain at t ¼ 0:075 and h0 ¼ P 0 ¼ Q 0 ¼ m ¼ 1.

where we have used the fact that the white noise process, nðtÞ, is the formal derivative of the Wiener process, WðtÞ. Since the Wiener process, WðtÞ, has infinite variation over every time interval, the last integral in (57) cannot be defined in the usual way (Riemann–Stieltjes or Lebesgue integral). This integral is of a stochastic type that can be solved in the sense of Itô calculus, see [47–50]. Since the temperature becomes stochastic, it has an infinite number of solutions (sample pathes). All these sample pathes are considered as solutions for our problem, but some sample pathes may be more probable than others. Thus, instead of dealing with only one of these sample pathes, we drive mathematically the mean and variance of the stochastic temperature to get a better insight of the overall behavior of the solution.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

16

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

0.5 0 −0.5 −1 −1.5

σ = 0.0 σ = 1.0 σ = 0.5 σ = 0.1

−2 −2.5 −3 0

0.5

1

1.5

2

2.5

x

3

3.5

4

1.5 σ = 0.0 σ = 1.0 σ = 0.5 σ = 0.1

1 0.5 0 −0.5 −1 −1.5 0

0.5

1

1.5

2

2.5

x

3

3.5

4

4.5

5

0.3 σ = 0.0 σ = 1.0 σ = 0.5 σ = 0.1

0.25 0.2 0.15 0.1 0.05 0 −0.05 0

0.2

0.4

0.6

0.8

x

1

1.2

1.4

1.6

Fig. 10. The effect of different values for the noise intensity on a single sample path of displacement, chemical potential and concentration at t ¼ 0:075 and h0 ¼ P 0 ¼ Q 0 ¼ m ¼ 1.

The mean of the temperature can be obtained by taking the expectation of (57) and using the property that the expectation of an Itô integral equal zero, see [51]

hðx; tÞ ¼ E½hðx; tÞ ¼ e

Z

t

½ X1 ðx; t  uÞ h0 ðuÞ þ X3 ðx; t  uÞ P0 ðuÞ  du þ Q 0

0

Z

t

½ cðx; t  uÞ þ M h ðt  uÞ em x  du

0

þ s0 Q 0 ð cðx; tÞ þ Mh ðtÞ em x Þ :

ð58Þ

hðx; sÞ, be written in the form In order to obtain the variance of the temperature, let the transformed temperature, 

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

17

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 4 Values of the distributions for different values of noise intensity at x ¼ 0:1953. x ¼ 0:1953

hðx; t Þ rxx ðx; tÞ eðx; tÞ uðx; tÞ C ðx; tÞ P ðx; tÞ

r ¼ 1:0

r ¼ 0:5

r ¼ 0:1

r ¼ 0:0

0.8767 1.0660 2.1630 2.1990 0.2208 1.0770

0.7726 0.2789 1.2300 1.2680 0.1780 1.0270

0.6896 0.3504 0.4828 0.5234 0.1439 0.9874

0.6685 0.5077 0.2961 0.3372 0.1353 0.9774

3.5 3

θ =1

2.5

θ0 = 2

0

θ =3 0

2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

x

1.2

1.4

1.6

1.8

2

1

0

−1 θ0 = 1 θ =2

−2

0

θ0 = 3 −3 0

0.5

1

1.5

x

2

2.5

3

2.5

3

4 θ0 = 1 θ0 = 2

3

θ0 = 3 2

1

0 0

0.5

1

1.5

x

2

Fig. 11. The mean of temperature, stress and strain for different values of temperature boundary h0 at t ¼ 0:075 and P 0 ¼ Q 0 ¼ m ¼ 1.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

18

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

 ðx; sÞ þ r U  ðx; sÞ nðsÞ; hðx; sÞ ¼ W

ð59Þ

where

Wðx; tÞ ¼

Z

t

½ X1 ðx; t  uÞ h0 ðuÞ þ X3 ðx; t  uÞ P0 ðuÞ  du þ Q 0

Z

0

t

½ cðx; t  uÞ þ M h ðt  uÞ em x  du

0

þ s0 Q 0 ð cðx; tÞ þ M h ðtÞ em x Þ ;

ð60Þ

Uðx; tÞ ¼ cðx; tÞ þ Mh ðtÞ em x :

ð61Þ

0 −0.1 −0.2 −0.3 −0.4

θ =1 0

−0.5

θ0 = 2

−0.6

θ0 = 3

−0.7 0

0.5

1

1.5

2

2.5

3

3.5

4

0.8

1

1.2

1.4

1.6

1.4

1.6

x

1.5 1 0.5 0 −0.5

θ0 = 1

−1

θ0 = 3

−1.5 0

θ =2 0

0.2

0.4

0.6

x

0.6 0.5

θ0 = 1

0.4

θ0 = 2 θ0 = 3

0.3 0.2 0.1 0 −0.1 0

0.2

0.4

0.6

0.8

x

1

1.2

Fig. 12. The mean of displacement, chemical potential and concentration for different values of temperature boundary h0 at t ¼ 0:075 and P 0 ¼ Q 0 ¼ m ¼ 1.

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

19

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

1.2 1

P =1 0

P0 = 31

0.8

P0 = 61

0.6 0.4 0.2 0 −0.2 0

0.2

0.4

0.6

0.8

1

x

1.2

1.4

1.6

1.8

2

0.5

0

P0 = 1

−0.5

P0 = 4 P0 = 7 −1 0

0.5

1

1.5

x

2

2.5

3

2.5

3

3.5 3

P0 = 1 P0 = 31

2.5

P = 61 0

2 1.5 1 0.5 0 0

0.5

1

1.5

x

2

Fig. 13. The mean of temperature, stress and strain for different values of chemical potential boundary P 0 at t ¼ 0:075 and h0 ¼ Q 0 ¼ m ¼ 1.

Also, we need the following properties for Laplace transform, see [1]. (a) For a stochastic process xðtÞ, we have

E ½ LfxðtÞg  ¼ LfE ½xðtÞg:

ð62Þ

(b) Let Rxx ðt 1 ; t 2 Þ ¼ E½xðt1 Þ xðt2 Þ denotes the auto-correlation function of a stochastic process xðtÞ. Introduce the designation

 ðs ; s Þ ¼ E ½ xðs Þ xðs Þ  ; R xx 1 2 1 2

ð63Þ

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

20

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

0.1 0 −0.1 −0.2 −0.3 −0.4

P0 = 1

−0.5

P0 = 31

−0.6

P0 = 61

−0.7 0

0.5

1

1.5

2

x

2.5

3

3.5

4

4 P0 = 1

3

P0 = 2 P0 = 3

2 1 0 −1 0

0.5

1

1.5

2

x

2.5

3

0.3 0.25

P0 = 1

0.2

P0 = 2 P0 = 3

0.15 0.1 0.05 0 −0.05 0

0.2

0.4

0.6

0.8

x

1

1.2

1.4

1.6

Fig. 14. The mean of displacement, chemical potential and concentration for different values of chemical potential boundary P 0 at t ¼ 0:075 and h0 ¼ Q 0 ¼ m ¼ 1.

we can see that

 ðs ; s Þ ¼ L fR ðt ; t Þg ; R xx 1 2 2 xx 1 2

ð64Þ

where L2 denotes the double Laplace transform defined by

L2 ff ðt 1 ; t2 Þg ¼

Z 0

1

es1 t1

Z

1

es2 t2 f ðt 1 ; t2 Þ dt 2

 dt 1 :

ð65Þ

0

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

21

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

1.2 1

Q =0

0.8

Q0 = 1

0

Q0 = 2

0.6 0.4 0.2 0 −0.2 0

0.5

1

1.5

2

x

2.5

3

1 0.5 0 −0.5

Q =0 0

Q0 = 1

−1 −1.5 0

Q0 = 2 0.5

1

1.5

x

2

2.5

3

3.5

3 2.5

Q0 = 0

2

Q0 = 2 Q0 = 4

1.5 1 0.5 0 −0.5 0

0.5

1

1.5

2

x

2.5

3

3.5

4

Fig. 15. The mean of temperature, stress and strain for different values of heat source strength Q 0 at t ¼ 0:075 and h0 ¼ P 0 ¼ m ¼ 1.

(c)

L2 ff1 ðt 1 Þ f 2 ðt 2 Þg ¼ Lff1 ðt1 Þg Lff2 ðt2 Þg :

ð66Þ

(d)

8 nR o > < L2 0t1 f ðt 1  u; t 2  uÞ du for t1 6 t 2 1 nR o L2 ff ðt1  u; t 2  uÞg ¼ : t >L s1 þ s2 2 : f ðt 1  u; t 2  uÞ du for t 2 < t1 2 0

ð67Þ

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

22

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

0.1 0.05 0 −0.05 −0.1

Q0 = 0

−0.15

Q0 = 2

−0.2

Q0 = 4

−0.25 0

0.5

1

1.5

x

2

2.5

3

3.5

1.5 Q0 = 0 1

Q0 = 1 Q0 = 2

0.5

0

−0.5 0

0.5

1

1.5

2

x

2.5

3

3.5

4

4.5

1.4

1.6

0.3 0.25

Q0 = 0

0.2

Q0 = 1 Q0 = 2

0.15 0.1 0.05 0 −0.05 0

0.2

0.4

0.6

0.8

x

1

1.2

Fig. 16. The mean of displacement, chemical potential and concentration for different values of heat source strength Q 0 at t ¼ 0:075 and h0 ¼ P 0 ¼ m ¼ 1.

Using property (b), for the temperature process we have

 

 ðx; s ; s Þ ¼ E hðx; s Þ hðx; s Þ ¼ E  W  ðx; s2 Þ þ r nðs2 Þ U  ðx; s1 Þ þ r nðs1 Þ U  ðx; s1 Þ W  ðx; s2 Þ R hh 1 2 1 2  ðx; s Þ W  ðx; s Þ þ r2 R  ðx; s Þ U  ðx; s Þ ;  ðs ; s Þ U ¼W

ð68Þ

 ðs ; s Þ ¼ E ½ nðs Þ nðs Þ  : R nn 1 2 1 2

ð69Þ

1

2

nn

1

2

1

2

where

The auto-correlation function of the Gaussian white noise process is given by the following formula, see [45]. Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

23

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

Rnn ðt 1 ; t 2 Þ ¼ dðsÞ ;

ð70Þ

where dðtÞ is the Dirac delta function and

 ðs ; s Þ ¼ R nn 1 2

s ¼ t2  t1 is the time span. The double Laplace transform of (70) is given by

1 : s1 þ s2

ð71Þ

Substituting (71) into (68), we get

 ðx; s ; s Þ ¼ W  ðx; s1 Þ W  ðx; s2 Þ þ r2 R hh 1 2

1  ðx; s1 Þ U  ðx; s2 Þ : U s1 þ s2

ð72Þ

Using properties (c) and (d), Eq. (72) can be written as

L2 fRhh ðx; t1 ; t 2 Þg ¼ L2 fWðx; t1 Þ Wðx; t 2 Þg þ r2 L2

Z

t1

Uðx; t1  uÞ Uðx; t 2  uÞ du

ð73Þ

0

from which we can obtain the auto-correlation function of the temperature in the following form

Rhh ðx; t1 ; t 2 Þ ¼ Wðx; t 1 Þ Wðx; t 2 Þ þ r2

Z

t1

Uðx; t 1  uÞ Uðx; t2  uÞ du :

ð74Þ

0

Let t 1 ¼ t2 ¼ t, we have

Rhh ðx; tÞ ¼ W2 ðx; tÞ þ r2

Z

t

U2 ðx; t  uÞ du :

ð75Þ

0

Putting t  u ¼ u in (75), we obtain

Rhh ðx; tÞ ¼ W2 ðx; tÞ þ r2

Z

t

U2 ðx; uÞ du :

ð76Þ

0

Finally the variance of the temperature can be obtained form (76) and (58) as

Var½hðx; tÞ ¼ r2

Z

t

U2 ðx; uÞ du :

ð77Þ

0

In the same manner, we can discuss the stochastic distributions for the other physical quantities. 5. Inversion of the Laplace transform In order to invert the Laplace transform, we shall use a numerical technique based on Fourier expansions of functions. Let f ðx; sÞ be the Laplace transform of the function f ðx; tÞ. The inversion formula for the Laplace transform can be written in the form

f ðx; tÞ ¼

1 2pi

Z

dþi 1

es t f ðx; sÞ ds ;

di 1

where d is an arbitrary real number greater than all the real parts of the singularities of f ðx; sÞ. Taking s ¼ d þ i y, the above integral takes the form

f ðx; tÞ ¼

ed t 2p

Z

1

ei t y f ðx; d þ i yÞ dy:

1

The following results is due to Honig and Hirdes [52]. Expanding the function hðx; tÞ ¼ ed t f ðx; tÞ in a Fourier series in the interval ½ 0; 2T , we obtain the approximate formula

f ðx; tÞ ¼ f1 ðx; tÞ þ ED ;

ð78Þ

where

f1 ðx; tÞ ¼

ck ¼

1 X 1 c0 þ ck ; 2 k¼1

for 0 6 t 6 2 T ;

ed t Re ½ ei k p t=T f ðx; d þ i k p=TÞ  T

ð79Þ

ð80Þ

and ED is the discretization error. This error can be made arbitrary small by choosing d large enough. From Eq. (78), we have the approximate formula

f ðx; tÞ ¼ fN ðx; tÞ þ ET þ ED ;

ð81Þ

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

where

fN ðx; tÞ ¼

N X 1 c0 þ ck ; 2 k¼1

for 0 6 t 6 2 T

ð82Þ

and ET is the truncation error. There are two methods to reduce the total error. First, the e-algorithm is used to reduce the truncation error and hence to P accelerate convergence. Let M ¼ 2 q þ 1, where q is a natural number, and let sm ¼ m k¼1 c k be the sequence of partial sums of PN the summation k¼1 ck . We define the e-algorithm by e0;m ¼ 0; e1;m ¼ sm ; m ¼ 1; 2; 3; . . . and

epþ1;m ¼ ep1;mþ1 þ

1

ep;mþ1  ep;m

;

p ¼ 1; 2; 3; . . . :

ð83Þ

The results in Honig and Hirdes [52], shows that the sequence e1;1 ; e3;1 ; e5;1 ; . . . eN;1 converges to f ðx; tÞ þ ED  c0 =2 faster than the sequence of partial sums sm ; m ¼ 1; 2; 3; . . .. Next the Korrecktur method is used to reduce the discretization error. The Korrecktur uses the following formula to evaluate the function f ðx; tÞ:

f ðx; tÞ ¼ f1 ðx; tÞ  e2 d T f 1 ðx; 2 T þ tÞ þ E0D ;

ð84Þ

where E0D  ED . From Eqs. (82) and (84), we have the following approximate formula for the function f ðx; tÞ

f ðx; tÞ ¼

! N N0 X X 1 1 c0 þ ck  e2 d t c0 þ c0k ; 2 2 k¼1 k¼1

ð85Þ

where ck is given by Eq. (80), N 0 is an integer less than N, and

c0k ¼

ed ð2 TþtÞ Re ½ ei k p ð2 TþtÞ=T f ðx; d þ i k p=TÞ : T

ð86Þ

6. Numerical results and discussion In this section, we present numerical results to illustrate the behavior of our physical properties, namely, temperature, stress, strain, displacement, chemical potential and concentration using the numerical technique outlined above. These quantities are studied as functions of time t and under the effect of different strengths of the thermal shock, chemical potential and heat source. Also, we study the effect of the noise intensity on the behavior of the system. Due to the stochastic nature of the problem, we mainly examine the mean and variance. Without loss of generality, we take the boundary conditions in the form:

h0 ðtÞ ¼ h0 HðtÞ and P0 ðtÞ ¼ P0 HðtÞ ;

ð87Þ

where h0 and P 0 represent the strengths of the temperature and the chemical potential on the boundary, respectively. It should be noted that a unit of non-dimensional time corresponds to 6:5ð10Þ12 s, while a unit of non-dimensional length corresponds to 2:7ð10Þ8 m. The problem was solved numerically for the special case of a copper material. The constants of the problem (in SI units) are taken as (see Table 1): 6.1. Physical properties of the problem-evolution in time t In Figs. 1 and 2, we study the solution of our problem in the context of two different theories of thermoelasticity, coupled and generalized theories. In these figures, the computations were carried out for three values of time, namely, t ¼ f0:050; 0:100; 0:300g; h0 ¼ P 0 ¼ m ¼ 1 and Q 0 ¼ 0 (without heat source). Also, in these figures, solid lines represent the solution corresponding to using the coupled theory due to [4], while dashed lines represent the solution corresponding to using the generalized theory due to Lord and Shulman [7]. It is clear that for large values of time, the coupled and the generalized theories give almost identical results. The case is quite different when we consider small values of time. The coupled theory predicts infinite speeds of wave propagation. This is manifested by the fact that for any small value of time, the solution of the temperature, say, is not identically zero but fades gradually to very small values at points far removed from the source of heating. However, the solution obtained using the equations of generalized theory justify the so called second sound effect, that is all the solution of the problem have finite speed of thermal disturbance. For small values of time, the solution has a nonzero value only in a bounded region of space and vanish identically outside this region. This region expands with the passage of time, see [53]. The wave propagation of the medium states under the evolution of time is studied in Figs. 3 and 4. It is clear that, the magnitude values of the temperature, the displacement and the concentration of the diffusive material inside the medium increase as the time increases, see Table 2 and Figs. 3(a), 4(a) and 4(c). At a given instant, these values decrease from the free Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

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A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

surface as the distance, x, from the free boundary increases. Moreover, Fig. 4(a) shows that the medium always undergoes expansion deformation which agree with those results in Fig. 3(c) where the medium is always stretched. Fig. 3(b) represent the distribution of the thermal stress, from which we can see that the stress at the free surface is always zero which coincide with the mechanical boundary condition that the free surface is traction free. For a given time, we have three wave fronts. The medium between the first and the second wave fronts suffers from compressive stress while the other parts of the medium suffers from tensile stress. The presence of compressive stress may be due to the influences of cross effects arising from the coupling of the fields of temperature and mass diffusion. The chemical potential is the change in energy due to the absorbed material. The results shown in Fig. 4(b) illustrates that the part of the medium adjacent to the free surface gain energy due to the chemical potential boundary. With the passage of time, this part becomes larger and larger. Moreover, as the time increase, the wave fronts move to the right and the magnitude of the jumps in all the physical properties decreases. In Figs. 5 and 6, we study the dispersion of our physical properties around its mean. It can be seen from these figures that the variance increases as the time increase, see also Table 3. In addition, the variance travel like a wave ended with a diffusive part. Moreover, the variance on the free surface is equal to zero except for the displacement which agree with Figs. 3 and 4, where all the physical properties are constant on the boundary except the displacement. The maximum dispersion for all the properties move to the right as the time increase except for the displacement, where the maximum occurs on the free surface. The asymptotic behavior for the solution is studied in Figs. 7 and 8. In these Figures, it can be seen that the thermal stress and the chemical potential reach their steady state more faster than the temperature, strain and the concentration of the diffusive material. Moreover, it can be observed that for x P 4:25, the steady state of the medium is approximately at e ðx; tÞ ¼ 0:3895. Unfortunately, we could not reach a e tÞ ¼ 4:581 and C e e xx ðx; tÞ ¼ 5:43; e hðx; tÞ ¼ 1:993; r e ðx; tÞ ¼ 7:795; Pðx; steady state for the displacement. 6.2. Effect of noise intensity

r on the physical properties of the problem

Referring to Figs. 9 and 10, we study the effect of the noise intensity on a single trajectory as a solution of the system. From these Figures, it can be observed that a decrease in the noise intensity leads to a convergence of the sample path to the mean solution for all the properties. Moreover, it can be deduced form Eq. (77) that the variance is directly proportional with the square of the noise intensity (r2 ). That is, the dispersion of all the trajectories about their mean decreases rapidly as the noise intensity decrease. Table 4 shows that as the noise intensity, r, converges to zero, the sample path converges to the mean solution. 6.3. Effect of the strength h0 of the temperature boundary The effect of various values for the mean temperature on the boundary is studied in Figs. 11 and 12 at t ¼ 0:075 and P 0 ¼ Q 0 ¼ m ¼ 1. As the mean temperature increases on the free surface, it increases at all the points inside the medium, see Fig. 11(a). The results in Fig. 12(a) illustrate that small changes in the temperature boundary lead to a slight differences in the displacement values. For t ¼ 0:075, we have three wave fronts at positions x ¼ 7:813ð10Þ2 ; x ¼ 5:313ð10Þ1 and x ¼ 1:398, approximately. It can be seen from Fig. 11(b) that the thermal stress increases in all tensile and compressive parts of the medium. The results shown in Fig. 11(c) illustrates that the part of the medium between the first and the second wave fronts is stretched with decreasing values as the mean temperature on the boundary increases while the other parts of the medium are stretched with increasing values. Also, it can be observed from Fig. 12(c) that, as the mean temperature increases on the free boundary, the number of particles absorbed by the medium from the diffusive material on the free surface increase in the part of the medium between the free surface and the second wave front while in the other part, this number decreases. Due to this increase in the concentration of the diffusive material, the medium gain a chemical energy. Fig. 12(b) illustrates that this increase in energy is very small and that is because the chemical potential has a constant value on the free surface. Moreover, as the value of the temperature at the boundary increases, the thermal elastic properties (stress, displacement and strain) change by very small values in the interval that lies between the second and the third wave fronts. However during this interval the chemical Table 5 Values of the distributions for different heat source intensity at x ¼ 0:1953 and x ¼ 0:6328. x ¼ 0:1953

x ¼ 0:6328

Q0 ¼ 0

Q0 ¼ 1

Q0 ¼ 2

0:6338 0:8040 0:0493

0:6685 0:5077 0:6415

0:7033 0:2115 1:3320

uðx; tÞ

0:0070

0:0618

0:1307

C ðx; tÞ P ðx; tÞ

0:1208 0:9864

0:1353 0:9774

0:1498 0:9685

hðx; t Þ rxx ðx; tÞ eðx; tÞ

Q0 ¼ 0 0:0020 0:0190 6:3ð10Þ5 3:1ð10Þ5 0:0210 0:7733

Q0 ¼ 1

Q0 ¼ 2

0:0377 0:1531 0:4442

0:0774 0:3258 0:8885

0:0444

0:0888

0:0312 0:7067

0:0415 0:6402

Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

26

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

properties (chemical potential and concentration) are clearly changed. In addition, it is clear from Figs. 11 and 12 that the magnitude of the wave fronts increase as the temperature boundary increase. 6.4. Effect of the strength P 0 of the chemical potential boundary The thermal elastic and chemical properties of the problem under the effect of different chemical potential on the free surface are studied in Figs. 13 and 14 at t ¼ 0:075 and h0 ¼ Q 0 ¼ m ¼ 1. It can be observed from Figs. 13(c) and 14(a) that small changes for the energy on the boundary remain the strain and the displacement waves approximately constant. For large values of chemical potential on the boundary, the whole medium suffers from a compressive stress except the part adjacent to the free surface that lies between the free boundary and the first wave front which suffers from a tensile stress, see Fig. 13(b). In spite of the existence of heat source and temperature on the boundary, for larger values of chemical potential on the free surface, the part of the medium between the second and the third wave fronts becomes cooler and cooler, see Fig. 13(a). The results in Figs. 14(b) indicate that the increase of the chemical potential at the boundary increases the energy inside the medium which agree with Fig. 14(c) where the concentration of the diffusive material increases. 6.5. Effect of the strength Q 0 of the heat source The effect of the amount of heat produced by the heat source is presented in Figs. 15 and 16 and Table 5. It can be seen that the temperature, displacement, strain and concentration increase as the strength of the heat source increases while the energy decreases along the whole medium. For large values of the strength of the heat source, the whole medium suffers from a tensile stress. Moreover, it can be seen that at Q 0 ¼ 0, the mean solution coincide with the deterministic solution obtained in [54]. 7. Conclusion In this work, a problem in the theory of generalized thermoelastic diffusion is investigated under the effect of an additive Gaussian white noise. The transform method is used to obtain the analytical solution for the temperature, stress, strain, displacement, concentration and chemical potential. The second sound effect is observed in the numerical simulation. The coupled and the generalized theories give almost identical results for large values of time, while the case is quite different when we consider small values. Due to the stochastic nature of the solution, we derived mathematically the mean and variance of the stochastic thermoelastic and chemical properties of the problem. It was deduced from the mathematical derivation of the variance that its value is directly proportional to the square noise intensity. Moreover, it was deduced that, as the noise intensity converges to zero, the trajectories of the solution converges to its expected sample path. It was found that, as the time increase, the wave fronts move to the right and the magnitude of the jumps in all the physical properties decreases. In addition, the maximum dispersion for all the properties move to the right as the time increase except for the displacement, where the maximum occurs on the free surface. The asymptotic behavior of the system is investigated. It was observed that the thermal stress and the chemical potential reach their steady state more faster than the temperature, strain and the concentration of the diffusive material. It was observed that small changes in the temperature or chemical potential boundary lead to a slight differences in the displacement values. It was found that the temperature, displacement, strain and concentration increase as the strength of the heat source increases. Appendix A The coefficients a1 ; a2 and a3 are given by

a1 ¼ a10 s2 þ a11 s ; a2 ¼ a20 s4 þ a21 s3 þ a22 s2 ; a3 ¼ a30 s6 þ a31 s5 þ a32 s4 ; 



a10 ¼

a3 þ a2 s þ a3 þ a3   1 þ 2 a1  þ a21  s0 ; a3  1

a11 ¼

1 þ a2 þ a3 þ 2 a1  þ a21  þ a3  ; a3  1

a20 ¼

a2 s þ a3 þ a21  þ a2 s þ a2  s s0 ; a3  1

a21 ¼

a2 þ a3 þ a21  þ a2 s þ a2  s þ a2 s0 þ a2  s0 ; a3  1





Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044

A.A. Allam / Applied Mathematical Modelling xxx (2014) xxx–xxx

a22 ¼

a2 ð1 þ Þ ; a3  1

a30 ¼

a2 ss0 ; a3  1

a31 ¼

a2 ðs þ s0 Þ ; a3  1

a32 ¼

a2 : a3  1

27

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Please cite this article in press as: A.A. Allam, A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.044