ASPHALTIC MATERIAL IN THE CONTEXT OF GENERALIZED POROTHERMOELASTICITY

International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

ASPHALTIC MATERIAL IN THE CONTEXT OF GENERALIZED POROTHERMOELASTICITY Mohammad H. Alawi College of engineering and Islamic architecture, p.o.box:7398 Makkah, Saudi Arabia

ABSTRACT In this work, a mathematical model of generalized porothermoelasticity with one relaxation time for poroelastic half-space saturated with fluid will be constructed in the context of Youssef model (2007). We will obtain the general solution in the Laplace transform domain and apply it in a certain asphalt material which is thermally shocked on its bounding plane. The inversion of the Laplace transform will be obtained numerically and the numerical values of the temperature, stresses, strains and displacements will be illustrated graphically for the solid and the liquid.

KEY-WORDS: Porothermoelasticity; asphaltic Material; Thermal shock.

NOMENCLATURE ui , Ui

The displacements of the skeleton and fluid phases

The poroelastic coefficients λ, µ, R, Q R 11 , R 12 , R 21 , R 22 The mixed and thermal coefficients

θs = T s − T0

The temperature increment of the solid where T s is the solid

θf = T f − T0

The temperature increment of the fluid where T f is the fluid

T0 The reference temperature The porosity of the material β s* f* ρ ,ρ The density of the solid and the liquid phases respectively

ρs = (1 − β ) ρs* f

f*

ρ = βρ ρ11 = ρs − ρ12

The density of the solid phase per unit volume of bulk The density of the solid phase per unit volume of bulk The mass coefficient of solid phase

f

ρ 22 = ρ − ρ12

The mass coefficient of fluid phase

ρ12 k s* , k f * k s = (1 − β ) k s*

The dynamics coupling coefficient

f

k = βk k

f*

The thermal conductivity of the solid and the fluid phases The thermal conductivity of the solid phase The thermal conductivity of the fluid phase The interface thermal conductivity

DOI:10.5121/ijsc.2017.8103

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International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

τso , τof σij

The relaxation time of the solid and the fluid phases The stress components apply to the solid surface

σ eij ε αs , αf α sf , α fs CsE , C fE

The normal stress apply to the fluid surface The strain component of the solid phase The strain component of the fluid phase The coefficients of the thermal expansion of the phases The thermoelastic couplings between the phases The specific heat of the solid and the fluid phases

CsfE

The specific heat couplings between the phases s

s E

ρC ks ρf C f ηf = f E k ρ Csf η = 12 E k P ηs =

The thermal viscosity of the solid The thermal viscosity of the fluid The thermal viscosity couplings between the phases

= 3λ + 2µ

R11

= α s p + α fs Q

R 22

= α f R + 3α sf Q

R12 F11

= α f Q + α sf P = ρs C Es

F22

= ρf C fE

F12

= − ( 3α s R 12 + α fs R 22 ) To

F21

= − ( 3α sf R 11 + α f R 21 ) To

1. INTRODUCTION Due to many applications in the fields of geophysics, plasma physics and related topics, an increasing attention is being devoted to the interaction between fluid such as water and thermo elastic solid, which is the domain of the theory of porothermoelasticity. The field of porothermoelasticity has a wide range of applications especially in studying the effect of using the waste materials on disintegration of asphalt concrete mixture.

Porous materials make their appearance in a wide variety of settings, natural and artificial and in diverse technological applications. As a consequence, a variety of problems arise while dealing with static and strength, fluid flow, heat conduction and the dynamics of such materials. In connection with the later, we note that problems of this kind are encountered in the prediction of behavior of sound-absorbing materials and in the area of exploration geophysics, the steadily growing literature bearing witness to the importance of the subject [1]. The problem of a fluidsaturated porous material has been studied for many years. A short list of papers pertinent to the present study includes Biot [2-3], Gassmann [4], Biot and Willis [5], Biot [6], Deresiewicz and Skalak [7], Mandl [8], Nur and Byerlee [9], Brown and Korringa [10], Rice and Cleary [11], Burridge and Keller [12], Zimmerman et al. [13-14], Berryman and Milton [15], Thompson and Willis [16], Pride et al. [17], Berryman and Wang [18], Tuncay and Corapcioglu [19], Alexander 28

International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

and Cheng [20], Charlez, P. A., and Heugas, O. [21], Abousleiman et al. [22], Ghassemi, A.[23] and Diek, A S. Tod [24].

The thermo-mechanical coupling in the poroelastic medium turns out to be of much greater complexity than that in the classical case of impermeable elastic solid. In addition to thermal and mechanical interaction within each phase, thermal and mechanical coupling occurs between the phases, thus, a mechanical or thermal change in one phase results in mechanical and thermal changes throughout the aggregate of asphaltic concrete mixtures. Following Biot, it takes one physical model to consist a homogeneous, isotropic, elastic matrix whose interstices are filled with a compressible ideal liquid both solid and liquid form continuous (and interacting) regions.

While viscous stresses in the liquid are neglected, the liquid is assumed capable of exerting a velocity-dependent friction force on the skeleton. The mathematical model consists of two superposed continuous phases each separately filling the entire space occupied by the aggregate.

Thus, there are two distinct elements at every point of space, each one characterized by its own displacement, stress, and temperature. During a thermo-mechanical process they may interact with a consequent exchange of momentum and energy.

Our development Proceeds by obtaining, the stress-strain-temperature relationships using the theory of the generalized thermo elasticity with one relaxation time “Lord-Shulman� [25].

Moreover, to the usual isobaric coefficients of thermal expansion of the single-phase materials, two coefficients appear which represent measures of each phase caused by temperature changes in the other phase.

As a result of the presence of these "coupling" coefficients, it follows that coefficient of thermal expansion of the dry material which differs than that of the saturated ones and the expansion of the liquid in the bulk is not the same as of the liquid phase. Putting into consideration the applications of geophysical interest, it takes the coefficient of proportionality in the dissipation term to be independent of frequency, that is, we confine ourselves to low-frequency motions. The last constituent of the theory is the equations of energy flux. Because the two phases in general, will be at different temperatures in each point of the material, there is a rise of a heat-source term in the energy equations representing the heat flux between the phases. It has been taken this "interphase heat transfer" to be proportional to the temperature difference between the phases.

Finally, by using the uniqueness theorem the proof has been done.

Recently, Youssef has constructed a new version of theory of porothermoelasticity, using the modified Fourier law of heat conduction. The most important advantage for this theory, is predicting the finite speed of the wave propagation of the stress and the displacement as well as the heat [26].

In this paper, a mathematical model of generalized porothermoelasticity with one relaxation time for poroelastic half-space saturated with fluid will be constructed in the context of Youssef model. We will obtain the general solution in the Laplace transform domain and apply it in a certain asphalt material which is thermally shocked on its bounding plane. The inversion of the Laplace transform will be obtained numerically and the numerical values of the temperature, displacement and stress will be illustrated graphically.

BASIC FORMULATIONS Starting by Youssef model of generalized porothermoelasticity [26], the linear governing equations of isotropic, generalized porothermoelasticity in absence of body forces and heat sources, are

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International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

(i)- Equations of motion

&& , µu i, jj + ( λ + µ ) u j,ij + QUi,ii − R11θs,i − R12 θ,if = ρ11&& u i + ρ12 U i s f && .

RU + Qu − R θ − R θ = ρ && u +ρ U i,ii

j,ij

21 ,i

22

,i

12

i

22

i

(1) (2)

(ii)- Heat equations

∂ ∂2  k s θ,iis =  + τso 2  ( F11θs + F12θf + To R11eii + To R 21ε ) ∂t  ∂t ∂ ∂2  k f θ,iif =  + τof 2  ( F21θs + F22θf + To R12 eii + To R 22ε ) ∂t  ∂t

(3)

(4)

(iii)- Constitutive equations

σij = 2µeij + λe kk δij + ( Qε − R 11θs − R 12 θf ) δij ,

(5)

σ = Rε + Qe kk − R 22 θf − R 21θs .

1 e ij = (u i , j + u j,i ) , e ii = e = u i ,i 2 ε = U i,i .

(6) (7) (8)

FORMULATION THE PROBLEM We will consider one dimensional half-space 0 ≤ x < ∞ is filled with porous, isotropic and elastic material which is considered to be at rest initially. The displacement will be considered to be in one dimensional as follows:

u1 = u ( x, t ) , u 2 ( x, t ) = u 3 ( x, t ) = 0 ,

(9)

U1 = U ( x, t ) , U 2 ( x, t ) = U3 ( x, t ) = 0 .

(10)

Then the governing equations (1)-(8) will take the forms: (a) Equations of motion ∂2u Q ∂2 U R11 ∂ θs R12 ∂ θf ρ11 ρ12 && , && + − − = u + U 2 2 ∂ x ( λ + 2µ ) ∂ x ( λ + 2µ ) ∂ x ( λ + 2µ ) ∂ x ( λ + 2µ ) ( λ + 2µ )

(11)

∂ 2 U Q ∂ 2 u R 21 ∂ θs R 22 ∂ θf ρ21 ρ && . && + − − = u + 22 U ∂ x2 R ∂ x2 R ∂ x R ∂x R R

(12)

(b) Equation of heat 2   F11 s F12 f To R 11 ∂ u To R 21 ∂ U  ∂ 2 θs  ∂ s ∂ = + τ θ + s θ + +  , o 2 2  s ∂x ∂ t  k k ks ∂ x ks ∂ x  ∂t

(13)

30

International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017 2   F21 s F22 f To R 12 ∂ u To R 22 ∂ U  ∂ 2 θf  ∂ f ∂ = + τ θ + f θ + +  . o 2 2  ∂x ∂ t   kf k kf ∂ x kf ∂ x  ∂t

(14)

(c) The constitutive relations

σxx R11 R12 ∂u ∂U Q = + − θs − θf , ( λ + 2µ ) ∂ x ( λ + 2µ ) ∂ x ( λ + 2µ ) ( λ + 2µ )

(15)

σ ∂ U Q ∂ u R 22 f R 21 s = + − θ − θ . (16) R ∂x R ∂x R R ∂u (17) e= ∂x ∂U .

(18) ε= ∂x Using the non-dimensional variables as follows:

( u′, U′, x′) = co η ( u, U, x ) , ( t′, τ′0 ) = co2η ( t, τ0 ) , ( θ′s , θ′f ) = T0 ( θs , θf ) , σ′ij =

σij λ + 2µ

, σ′ =

σ R

where

c2o =

λ + 2µ ρ Csf , η = 12 E . ρ12 k

Then, we get

T0 R 11 ∂ θs T0 R 12 ∂ θf ρ11 ∂2u Q ∂2U && , && + − − = u+U ∂ x 2 ( λ + 2µ ) ∂ x 2 ( λ + 2µ ) ∂ x ( λ + 2µ ) ∂ x ρ12

(19)

∂ 2 U Q ∂ 2 u T0 R 21 ∂ θs T0 R 22 ∂ θf ( λ + 2µ ) ρ ( λ + 2µ ) && && + − − = u + 22 U. 2 2 ∂x R ∂x R ∂x R ∂x R ρ12 R

(20)

2   ηs s F12 f R 11 ∂ u R 21 ∂ U  ∂ 2 θs  ∂ s ∂ = + τ +   θ + s θ + s , o ∂ x2  ∂ t ∂ t2   η kη k η ∂ x ksη ∂ x  2   F21 s ηf f R 12 ∂ u R 22 ∂ U  ∂ 2 θf  ∂ f ∂ = + τ θ + θ + f +   . o ∂ x2  ∂ t ∂ t2   kf η η k η ∂ x kf η ∂ x 

σ xx =

σ=

TR TR ∂u Q ∂U + − 0 11 θs − 0 12 θf , ∂ x ( λ + 2µ ) ∂ x ( λ + 2µ ) ( λ + 2µ )

∂ U Q ∂ u T0 R 22 f T0 R 21 s + − θ − θ .

R R ∂x R ∂x

(21) (22) (23) (24)

In the above equation, we dropped the prime for convenient.

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International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

FORMULATION THE PROBLEM IN LAPLACE TRANSFORM DOMAIN Applying the Laplace transform for the both sides of the equations (19)-(24) which is defined as follows: ∞

f ( s ) = ∫ f ( t ) e − s t dt , 0

then, we get

d 2u d θs dθf , = L u + L U + L + L 11 12 13 14 d x2 dx dx

(25)

d2U d θs dθf , = L u + L U + L + L 21 22 23 24 d x2 dx dx

(26)

d 2 θs du dU , = L 31θ s + L 32 θ f + L 33 + L 34 2 dx dx dx

(27)

d2θf du dU , = L 41θ s + L 42 θ f + L 43 + L 44 2 dx dx dx

(28)

du dU +A − A11θs − A12 θf , dx dx

(29)

σ xx =

dU du + B − A 21θ s − A 22 θ f , dx dx du e= dx

σ=

ε=

(30) (31)

dU . dx

(32)

where

 C − AC 21   C12 − AC 21   A11 − AA 21   A12 − AA 22  L11 =  11  , L12 =   , L13 =   , L14 =  ,  1 − AB   1 − AB   1 − AB   1 − AB   C − BC11   C 22 − BC12   A 21 − BA11   A 22 − BA12  L 21 =  21  , L 22 =   , L 23 =   , L 24 =  ,  1 − AB   1 − AB   1 − AB   1 − AB  s + τsos 2 ηs s + τsos 2 F12 s + τsos 2 R 11 s + τsos 2 R 21 L31 = , L32 = , L33 = , L34 = , η ksη ksη ksη

(

L 41

)

(

(s + τ s ) F = f 2 o f

, L 42

(

(s + τ s ) η = f 2 o

η

f

, L 43

)

(s + τ s ) R = f 2 o f

(

12

, L 44

)

(s + τ s ) R =

kη TR TR ρ Q A= , A11 = 0 11 , A12 = 0 12 , C11 = 11 s 2 , C12 = s 2 ρ12 ( λ + 2µ ) ( λ + 2µ ) ( λ + 2µ ) B=

kη

21

)

f 2 o f

kη

22

.

( λ + 2µ ) s 2 , C = ρ22 ( λ + 2µ ) s 2 , TR TR Q , A 21 = 0 21 , A 22 = 0 22 , C 21 = 22 R R R R ρ12 R

By using equations (25)-(28), we get 32

International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

 D8x − aD 6x + bD 4x − cD 2x + d  u = 0 ,  D8x − aD 6x + bD 4x − cD 2x + d  U = 0 ,

(33)

 D8x − aD 6x + bD 4x − cD 2x + d  θ s = 0 ,  D8x − aD 6x + bD 4x − cD x2 + d  θ f = 0 ,

(35)

(34)

(36)

where

a = L11 + L13L33 + L14 L 43 + L 22 + L 23L34 + L 24 L 44 + L 31 + L 42 b = L11 ( L 22 +L 23 L34 +L 24 L 44 +L31 +L 42 ) - L12 ( L 21 + L 23 L33 + L 24 L 43 ) L13 ( L 21L34 - L 22 L33 + L 24 (L34 L 43 - L33 L 44 ) + L32 L 43 - L33 L 42 ) L14 ( L 21L 44 - L 22 L 43 + L 23 ( L33L 44 - L34 L 43 ) - L31L 43 + L33 L 41 ) + L 22 ( L31 + L 42 ) + L 23 (L34 L 42 - L32 L 44 )+ L 24 ( L31L 44 - L34 L 41 ) + L31L 42 - L32 L 41 c = L11 (L 22 (L31 + L 42 ) + L 23 (L34 L 42 - L32 L 44 ) +L 24 (L31L 44 - L34 L 41 ) +L31L 42 - L32 L 41 )

- L12 (L 21 (L31 + L 42 ) + L 23 (L33L 42 - L32 L 43 ) + L 24 (L31L 43 - L33 L 41 )) + L13 (L 21 (L32 L 44 - L34 L 42 ) +L 22 (L33 L 42 - L32 L 43 )) - L14 (L 21 (L31L 44 - L34 L 41 ) + L 22 (L33 L 41 - L31L 43 )) + L 22 (l31L 42 - L32 L 41 )

d =L11L 22 (L31L 42 - L32 L 41 ) + L12 L 21 (L32 L 41 - L31L 42 ) , and

D nx =

dn . d xn

According to equations (33)-(36) and to bounded state of functions at infinity, we can consider the following forms 4

u ( x,s ) = ∑ α i e−λi x ,

(37)

i =1

4

U ( x,s ) = ∑ βi e−λi x ,

(38)

i =1 4

θs ( x,s ) = ∑ γ i e −λi x ,

(39)

i =1 4

θ f ( x,s ) = ∑ ωi e−λi x ,

(40)

i =1

where ±λ i , i = 1, 2,3, 4 are the roots of the characteristic equation of the system (33)-(36) which takes the form

λ 8 − aλ 6 + b λ 4 − c λ 2 + d = 0 ,

(41)

To get the relations between the parameters βi , γ i , ωi and α i , we will use equations (26)-(28) in the following forms

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International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

 D 2x − L 22  U − L 23 D x θ s − L 24 D x θ f = L 21u , − L34 U +  D 2x − L31  θ s − L32 θ f = L33 D x u ,

(42)

− L 44 D x U − L 41θ s +  D 2x − L 42  θ f = L 43 D x u ,

(44)

(43)

Inserting the formulas in (37)-(40) into equations (42)-(44), we get

(λ

2 i

− L 22 ) βi + L 23λ i γ i + L 24 λ i ωi = L 21α i , i = 1, 2, 3, 4 ,

(45)

− L34βi + ( λ i2 − L31 ) γ i − L32 ωi = − L33λ i α i , i = 1, 2, 3, 4 ,

(46)

L 44 λ iβi − L 41γ i + ( λ i2 − L 42 ) ωi = − L 43λ i α i , i = 1, 2, 3, 4 ,

(47)

By solving the system in (45)-(47), we obtain

Hi α i , i = 1, 2,3, 4 , Wi G γ i = i α i , i = 1, 2,3, 4 , Wi F γ i = i α i , i = 1, 2,3, 4 , Wi

βi =

Where

Hi = −(λ i4 (L21 + L23L33 + L 24 L 43 ) - λ i2 (L21 (L31 + L42 ) + L23 (L33L 42 - L32 L43 ) + L 24 (L31L 43 - L33L 41 )) + L21 (L31L42 - L32 L 41 ) ) , i = 1, 2,3, 4 G i = L33λ i5 -λi3 (L 22 L33 + L24 L33L 44 - L32 L 43 + L33L 42 ) - L34λ i2 (L21 + L 24 L43 ) + λ(L21L32 L44 + L 22 (L33L 42 - L32 L43 )) + L21L34 L42 , i = 1, 2,3, 4 Fi = L 43λ i5 +λi3 (L21L 44 - L22 L 43 + L 23L33 L 44 - L31L43 +L33L 41 ) + L 23L 43L34λi2 -λi (L 21L31L 44 + L22 (L33L 41 - L31L43 )) - L21L34 L 41 , i = 1, 2,3, 4 Wi = -λ i6 + λ i4 (L 22 + L24 L 44 + L31 + L42 ) - L23L34λ3i - λ i2 (L 22 (L31 + L 42 ) - L 23L32 L 44 +L 24 L31L 44 + L31L 42 - L32 L 41 ) + L34λ i (L 23L 42 - L 24 L41 ) + L 22 (L31L 42 - L32 L 41 ) , i = 1, 2,3, 4 Hence, we have 4

Hi αi e−λi x , i =1 Wi 4 G θs ( x,s ) = ∑ i αi e−λi x , i =1 Wi U ( x,s ) = ∑

(48) (49)

4

Fi αi e −λi x , (50) W i =1 i To get the values of the parameters α i , we have to apply the boundary conditions as follows; θ f ( x,s ) = ∑

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International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

(1)

The thermal conditions

We will consider the bounding plane surface of the medium at x = 0 has been thermally loaded by thermal shock as follows:

θs ( 0, t ) = (1 − β ) θ0 H ( t ) ,

(51)

and

θf ( 0, t ) = β θ0 H ( t ) , (52) where H(t) is the Heaviside unite step function and θ0 is constant which gives after using the Laplace transform the following conditions

θs ( 0,s ) =

(1 − β ) θ0 s

,

(53)

and

θ f ( 0,s ) = (2)

βθ0 , s

(54)

The mechanical conditions

We will consider the bounding plane surface of the medium at x = 0 has been connected to a rigid surface which prevents any displacement to accrue on that surface, i.e.

u ( 0, t ) = 0 ,

(55)

U ( 0, t ) = 0 ,

(56)

and

which gives after using the Laplace transform the following conditions

u ( 0,s ) = 0 ,

(57)

U ( 0,s ) = 0 .

(58)

and After using the boundary conditions in (53), (54), (57) and (58), we get the following system 4

∑α

= 0,

i

(59)

i =1

4

Hi

∑W α i =1 4

i

= 0,

i

=

(1 − β ) θ0 ,

i

=

βθ0 , s

(60)

i

Gi

∑W α i =1 4

i

Fi

∑W α i =1

i

s

(61) (62)

Then we get 35

International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

 W2 ( β(F3H 4 - F4 H 3 +G 3H 4 − G 4 H 3 )-F3H 4 + F4 H3 ) -   θ0 W1  α1 =  W3 ( β(F2 H 4 - F4 H 2 +G 2 H 4 − G 4 H 2 )-F2 H 4 + F4 H 2 ) +  , s∆    W4 ( β(F2 H 3 - F3H 2 +G 2 H 3 − G 3H 2 )-F2 H 3 + F3H 2 )   W1 ( β(F3H 4 - F4 H3 +G 3H 4 − G 4 H 3 )-F3H 4 + F4 H3 ) -  θ0 W2  α2 = −  W3 ( β(F1H 4 - F4 H1 +G1H 4 − G 4 H1 )-F1H 4 + F4 H1 ) +  , s∆    W4 ( β(F1H 3 - F3H1 +G1H3 − G 3H1 )-F1H3 + F3H1 )   W1 ( β(F2 H 4 - F4 H 2 +G 2 H 4 − G 4 H 2 )-F2 H 4 + F4 H 2 ) -  θ0 W3  α3 =  W2 ( β(F1H 4 - F4 H1 +G1H 4 − G 4 H1 )-F1H 4 + F4 H1 ) +  , s∆    W4 ( β(F1H 2 - F2 H1 +G1H 2 − G 2 H1 )-F1H 2 + F2 H1 )   W1 ( β(F2 H3 - F3H 2 +G 2 H3 − G 3H 2 )-F2 H 3 + F3H 2 ) -   θW  α 4 = − 0 4  W2 ( β(F1H3 - F3H1 +G1H3 − G 3H1 )-F1H 3 + F3H1 ) +  , s∆    W3 ( β(F1H 2 - F2 H1 +G1H 2 − G 2 H1 )-F1H 2 + F2 H1 )  where

∆ = - W1 (F2 (G 3H 4 - G 4 H3 ) + F3 (G 4 H 2 - G 2 H 4 ) + F4 (G 2 H3 - G 3H 2 )) + W2 (F1 (W3H 4 - G 4 H 3 ) + F3 (G 4 H1 - G1H 4 ) + F4 (G1H 3 - G 3H1 )) W3 (F1 (G 2 H 4 - G 4 H 2 ) + F2 (G 4 H1 - G1H 4 ) + F4 (G1H 2 - G 2 H1 )) + , W4 (F1 (G 2 H 3 - G 3H 2 ) + F2 (G 3H1 - G1H 3 ) + F3 (G1H 2 - G 2 H1 )) Those complete the solution in the Laplace transform domain.

NUMERICAL INVERSION OF THE LAPLACE TRANSFORMS In order to invert the Laplace transforms, we adopt a numerical inversion method based on a Fourier series expansion [27].

By this method the inverse f ( t ) of the Laplace transform f (s ) is approximated by

f (t) =

e ct t1

N 1 f c + R1 f ( )  ∑ k =1 2

  i k π t  ikπ c +  exp   , t1    t1  

0 < t1 < 2t,

Where N is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such that

   i N π t  iNπ exp ( c t ) R1  f  c +  exp   ≤ ε 1 , t t  1   1    where ε1 is a prescribed small positive number that corresponds to the degree of accuracy required. The parameter c is a positive free parameter that must be greater than the real part of all the singularities of f (s ) . The optimal choice of c was obtained according to the criteria described 36

International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017

in [27].

NUMERICAL RESULTS AND DISCUSSION The Ferrari's method has been constructed by using the FORTRAN program to solve equation (41). The material properties of asphaltic material saturated by water have been taken as follow [28], [29]: T0 = 27 o C,

Q = 0.4853 × 1011 dyne.cm −2 , R = 0.0362 × 1011 dyne.cm −2 , λ = 0.2160 × 1011 dyne.cm −2

µ = 0.0926 × 1011 dyne.cm −2 , α s = 2.16 × 10-5 o C −1 , ρs* = 2.35 gm.cm −3 , ρ11 = 0.002 gm.cm −3 *

k s = 0.8 W m −1 o k −1 , CsE = 800 J.kg −1. o C −1 , τso = 0.02 s, k = 0.001 W m −1 o k −1 *

α f = α sf = α fs = 0.0001 o C −1 , ρf * = 0.82 gm.cm −3 , k f = 0.3 W m −1 o k −1 , CfE = 1.9 cal.gm −1. o C −1 , τfo = 0.00001s, We will take the non-dimensional x variable to be in interval 0 ≤ x ≤ 1 and all the results will be calculated at the same instance t = 0.1 for two different values of the porosity β of the material when β = 0.25 and β = 0.35 .

The temperature, the stress, the strain and the displacement for the solid and the liquid have been shown in figures 1-8 respectively. We can see that, the value of the porosity has a significant effect on all the studied fields.

Figure 1 shows the temperature increment distribution of the solid with two different values of the porosity; β = 0.25 and β = 0.35 . It shows that the porosity parameter has a significant effect.

Figure 2 shows the temperature increment distribution of the liquid with two different values of the porosity; β = 0.25 and β = 0.35 . We can see that the porosity parameter has a significant effect where the liquid temperature increases when the porosity increases.

Figure 3 shows the stress distribution of the solid with two different values of the porosity; β = 0.25 and β = 0.35 . We can see that the porosity parameter has a significant effect where the absolute value of the stress acts on the solid increases when the porosity increases.

Figure 4 shows the stress distribution of the liquid with two different values of the porosity; β = 0.25 and β = 0.35 . We can see that the porosity parameter has a significant effect where the absolute value of the stress acts on the liquid increases when the porosity increases for wide range of x.

Figures 5-8, show that the porosity parameter has significant effects on the deformation and the displacement for both medium solid and liquid. The absolute value of the peak points (sharp points) increase when the value of the porosity parameter increases for the both medium solid and water.

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CONCLUSION In studying a mathematical model of generalized porothermoelasticity with one relaxation time for poroelastic half-space saturated with fluid in the context of Youssef model we found that: 1- The porosity parameter of the poroelastic material has significant effects on the temperature, the stress, the deformation and the displacement distributions for the both medium the solid and the liquid.

2- Youssef model of porothermoelasticity with one relaxation time introduce finite speed of thermal wave propagation which agree with realistic physical behavior for the solid and the liquid.

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