generalized porothermoelasticity of asphaltic material

Engineering, 2011, 3, 1102-1114 doi:10.4236/eng.2011.311138 Published Online November 2011 (http://www.SciRP.org/journal/eng)

Copyright © 2011 SciRes. ENG

Generalized Porothermoelasticity of Asphaltic Material

Mohammad H. Alawi Civil Engineering Department, Makkah, Kingdom of Saudi Arabia

E-mail: [email protected] Received September 18, 2011; revised October 11, 2011; accepted October 20, 2011

Abstract In this work, a mathematical model of generalized porothermoelasticity with one relaxation time for poroe-lastic 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 numeri-cally and the numerical values of the temperature, stresses, strains and displacements will be illustrated graphically for the solid and the liquid. Keywords: Porothermoelasticity, Asphaltic Material, Thermal Shock, Tod

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 poro-thermoelasticity has a wide range of applications espe-cially in studying the effect of using the waste materials on disintegration of asphalt concrete mixture.

Porous materials make their appearance in a wide va-riety 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 fluid-saturated 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 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) re-gions. While viscous stresses in the liquid are neglected, the liquid is assumed capable of exerting a velocity- dependent friction force on the skeleton. The mathe-matical 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 meas-ures of each phase caused by temperature changes in the other phase.

M. H. ALAWI 1103

12 ,

As a result of the presence of these “coupling” coeffi-cients, 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 coef-ficient 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 dif-ference 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 Fou-rier law of heat conduction. The most important advan-tage 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 work, a mathematical model of generalized porothermoelasticity with one relaxation time for poroe-lastic half-space saturated with fluid will be constructed in the context of Youssef model. We will obtain the gen-eral 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. 2. Basic Formulations Starting by Youssef model of generalized porothermoe-lasticity [26], the linear governing equations of isotropic, generalized porothermoelasticity in absence of body forces and heat sources, are

1) Equations of motion

, , , 11 ,

11 12 ,

s fi jj j ij i ii i i

i i

u u QU R R

u U

(1)

, , 21 , 22 , 12 22s f

i ii j ij i i i iRU Qu R R u U . (2)

2) Heat equations

2

, 11 12 11 212s s s s f

ii o o ii ok F F T R e T Rt t

(3)

2

, 2

21 22 12 22

f f fii o

s fo ii o

kt t

F F T R e T R

(4)

3) Constitutive equations

11 122 s fij ij kk ij ije e Q R R , (5)

22 21f s

kkR Qe R R . (6)

, ,

1,

2ij i j j i ii i ie u u e e ,u (7)

,i iU . (8)

3. 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:

1 2 3, , , , 0u u x t u x t u x t , (9)

1 2 3, , , , 0U U x t U x t U x t . (10)

Then the governing Equations (1)-(8) will take the forms:

1) Equations of motion

2 211 12

2 2

11 12

2 2 2

,2 2

s fR Ru Q U

x xx x

u U

(11)

2 221 22 21 22

2 2

s fR RU Q uu U

R R x R x R Rx x

.

(12)

2) Equation of heat

2 2

2 2

11 2111 12 ,

sso

s f o os s s s

tx t

T R T RF F u U

x xk k k k

(13)

2 2

2 2

12 2221 22

ff

o

s f o of f f f

tx t

T R T RF F u U

x xk k k k

(14)

3) The constitutive relations


M. H. ALAWI


1104

11

12

2 2

,2

2sxx

f

Ru Q U

x x

R

(15)

2 2

2 2

21 12 22 .

ff

o

fs f

f f f

tx t

F R Ru U

x xk k k

(22)

0 11 0 12

2 2 2s f

xx

T R T Ru Q U

x x

,

(23)

22 21f sR RU Q u

R x R x R R

. (16)

ue

x

(17) 0 22 0 21f sT R T RU Q u

x R x R R

. (24)

U

x

. (18) In the above equation, we dropped the prime for con-venient.

Using the non-dimensional variables as follows:

20 0

0

, , , , , , , ,

, , , ,2

o o

ijs f s fij

u U x c u U x t c t

TR

4. 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:

0

e ds tf s f t

where t ,

2 12

12

2,

sfE

o

Cc

k

. then, we get 2

11 12 13 142

d d

d dd

ds fuL u L U L L

x xx

, (25)

Then, we get 2

21 22 23 242

d d

d dd

ds fUL u L U L L

x xx

, (26)

2 2

0 11 0 122 2

11

12

2 2 2

,

s fT R T Ru Q U

x xx x

u U

(1

9)

2

31 32 33 342

d d

d dd

ss f u

L L L LdU

x xx

, (27)

2

41 42 43 442

d d

d dd

fs f u

L L L LdU

x xx

, (28)

2 20 21 0 22

2 2

22

12

2 2.

s fT R T RU Q u

R R x Rx x

u UR R

x

(20) 11 12

d d

d ds f

xx

u UA A A

x x , (29)

2 2

2 2

12 11 21 ,

sso

ss f

s s s

tx t

F R Ru U

x xk k k

(21)

21 22

d d

d ds fU u

B A Ax x

, (30)

d

d

ue

x (31)

d

d

U

x . (32)

where

11 21 12 21 11 21 12 2211 12 13 14, , ,

1 1 1 1

C AC C AC A AA A AAL L L L

AB AB AB AB

,

21 11 22 12 21 11 22 1221 22 23 24, , ,

1 1 1 1

C BC C BC A BA A BAL L L L

AB AB AB AB

,

M. H. ALAWI 1105

2 2 212 11 21

31 32 33 34, , ,s s s s so o o

s s

2o

s

s s s s F s s R s s RL L L L

k k

k ,

2 2 221 12 22

41 42 43 44, , ,f f f f

o o o

f f

2fo

f

s s F s s s s R s s RL L L L

k k

k

.

2 20 11 0 12 11

11 12 11 1212

, , , ,2 2 2

T R T RQA A A C s C

s

2 20 21 0 22 2221 22 21 22

12

2 2, , , ,

T R T RQB A A C s C s

R R R R R

,

By using Equations (25)-(28), we get 8 6 4 2 0x x x xD aD bD cD d u , (33)

8 6 4 2 0x x x xD aD bD cD d U , (34)

8 6 4 2 0sx x x xD aD bD cD d , (35)

8 6 4 2 0fx x x xD aD bD cD d , (36)

where

11 13 33 14 43 22 23 34 24 44 31 42a L L L L L L L L L L L L

11 22 23 34 24 44 31 42 12 21 23 33 24 43

13 21 34 22 33 24 34 43 33 44 32 43 33 42

14 21 44 22 43 23 33 44 34 43 31 43 33 41

22 31 42 23 34 42 32 44

( )

( )

b L L L L L L L L L L L L L L

L L L L L L L L L L L L L L


L L L L L L L L

24 31 44 34 41 31 42 32 41 L L L L L L L L L

11 22 31 42 23 34 42 32 44 24 31 44 34 41 31 42 32 41

12 21 31 42 23 33 42 32 43 24 31 43 33 41

13 21 32 44 34 42 22 33 42 32 43

14 21 31 44 34 41 22 33 41 31 43

)

c L L L L L L L L L L L L L L L L L L


L L L L L L L L L L L

L L L L L L L L L L L L

22 31 42 32 41l L L L

11 22 31 42 32 41 12 21 32 41 31 42d L L L L L L L L L L L L ,

and d

d

nnx n

Dx

.

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

4

1

, ie xi

i

u x s

, (37)

4

1

, ie xi

i

U x s

, (38)

4

1

, e i xsi

i

x s

,

4

1

, i

(39)

e xfi

i

x s

where 4

, (40)

, 1, 2,3,i i are the roottic equation of the system (33)-(36) which takes the form

s of the characteris-

8 6a 4 2 0b c d , (41)

To get the relations between the parameters , ,i i i and i , we will use Equations (26)-(28) in form

the following s

222 23 24 21

s fx x xD L U L D L D L u , (42)


M. H. ALAWI 1106

234 31 32 33

s fx xL U D L L L D u , (43)

244 41 42 43

s fx xL D U L D L L D u x

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

4 ,

, (44)

Equations

2 , 1,2,3,L L L L i 22 23 24 21i i i i i i i

(45)

234 31 32 33 , 1,2,3,4i i i i i iL L L L i ,

(46)

244 41 42 43 , 1, 2,3, 4i i i i i i iL L L L i ,

(47)

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

, 1,2,3,4ii i

i

Hi

W ,

, 1, 2,3, 4ii i

i

Gi

W ,

, 1, 2,3, 4ii i

i

Fi

W ,

where

4 221 23 33 24 43 21 31 42 23 33 42 32 43 24 31 43 33 41

21 31 42 32 41 , 1,2,3,4

i i iH L L L L L L L L L L L L L L L L L

L L L L L i

L

5 3 233 22 33 24 33 44 32 43 33 42 34 21 24 43

21 32 44 22 33 42 32 43 21 34 42

, 1, 2,3, 4

i i i iG L L L L L L L L L L L L L L

L L L L L L L L L L L i

5 3 243 21 44 22 43 23 33 44 31 43 33 41 23 43 34

21 31 44 22 33 41 31 43 21 34 41

, 1, 2,3,4

i i i i

i

F L L L L L L L L L L L L L L L

L L L L L L L L L L L i

6 4 3 222 24 44 31 42 23 34 22 31 42 23 32 44 24 31 44 31 42 32 41

34 23 42 24 41 22 31 42 32 41

), 1,2,3,4

i i i i i

i

W L L L L L L L L L L L L L L L L L L L L

L L L L L L L L L L i

Hence, we have

4

1

, e i xii

i i

HU x s

W

, (48)

4

1

, ie xs ii

i i

Gx s

W

, (49)

4

1

, ie xf ii

i i

Fx s

W

To get the values of the parameters

, (50)

i , we have to apply the boundary conditions as follo

1) The thermal conditions

ed by thermal sh

t

e H(t) is the Heaviside unite step function and

ws;

We will consider the bounding plane surface of the medium at x = 0 has been thermally load

ock as follows:

00, 1s t H t , (51)

and

00,f t H , (52)

wher 0 is constant which gives after using ththe following conditions

e Laplace transform

010,s s

s

, (53)

and

00,f ss

, (54)

2) The mechanical conditions We will consider the bounding plane surface of the

um at x = 0 has been connected to a rigid surface h prevents any displacement to accrue on that sur-

face, i.e.,

mediwhic

0, 0u t , (55)

and

0, 0U t , (56)

which gives after using the Laplace transform the fol-lowing conditions

0, 0u s , (57)

and

0, 0U s . (58)


M. H. ALAWI 1107

oundary conditions in (53), (54), (57) and (58), we get the following system

After using the b

4

1

0ii

, (59)

4

1

0ii iW

, (60)

4

iH

0

1

1ii

i i

G

W s

, (61)

40

1

ii

i i

F

W s

,

Then we get

(62)

0 11 2 3 4 4 3 3 4 4 3 3 4G H G H F H 4 3

3 2 4 4 2 2 4 4 2 2 4 4 2 4 2 3 3 2 2 3 3 2 2 3 3 2 ,

WW F H F H F H

s

W F H F H G H G H F H F H W F H F H G H G H F H F H

0 22 1 3 4 4 3 3 4 4 3 3 4 4 3

3 1 4 4 1 1 4 4 1 1 4 4 1 4 1 3 3 1 1 3 3 1 1 3 3 1

W=

,

W F H F H G H G H F H F Hs


0 33 1 2 4 4 2 2 4 4 2 2 4 4 2

2 1 4 4 1 1 4 4 1 1 4 4 1 4 1 2 2 1 1 2 2 1 1 2 2 1

W=

,



0 44 1 2 3 3 2 2 3 3 2 2 3 3 2

2 1 3 3 1 1 3 3 1 1 3 3 1 3 1 2 2 1 1 2 2 1 1 2 2 1

W=

,



where

1 2 3 4 4 3 3 4 2 2 4 4 2 3 3 2

2 1 3 4 4 3 3 4 1 1 4 4 1 3 3 1

3 1 2 4 4 2 2 4 1 1 4 4 1 2 2 1

4 1 2 3 3 2 2 3 1 1 3 3 1 2 2 1

,

W F G H G H F G H G H F G H G H

W F W H G H F G H G H F G H G H



Those complete the solution in the Laplace transform

domain.

.

opt a ries

x

5 Numerical Inversion of the Laplace

Transforms In order to invert the Laplace transforms, we adnumerical inversion method based on a Fourier se

pansion [27]. eBy this method the inverse f t of the Laplace

transform f s is approximated by

11 1 1

1 exp ,2

0 2

k

f t f c R f ct t t

t t

1

π π

,

N ik i k t

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

such that

e 1ct

11 1

π πexp 1 exp

iN iN tc t R f c

t t

,

where 1 is a prescribed small positive number that cor-responds to the degree of accuracy required. The pa-rameter 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 in [27]. 6. Numerical Results and Discussion The Ferrari’s method has been constructed by using the FORTRAN program to solve Equation (41). The mate-rial properties of asphaltic material saturated by water have been taken as follow [28,29].


M. H. ALAWI 1108

Figure 1. Asphalt temperature distribution.

Figure 2. Water temperature distribution.


M. H. ALAWI 1109

Figure 3. Asphalt stress distribution.

Figure 4. Water stress distribution.


M. H. ALAWI 1110

F

igure 5. Asphalt strain distribution.

Figure 6. Water strain distribution.


M. H. ALAWI 1111

Figure 7. Asphalt displacement distribution.

Figure 8. Water displacement distribution.


M. H. ALAWI


1112

We will take the non-dimensional x variable to be in interval

*

11 20

11 2

11 2

11 2 5 1

* 3 311

1 1 1 1

27 C, 0.4853 10 dyne cm ,

0.0362 10 dyne cm ,

0.2160 10 dyne cm

0.0926 10 dyne cm , 2.16 10 C ,

2.35 gm cm , 0.002gm cm

0.8W m k , 800 J. kg C ,

0.02 s, 0.

s

s

s sE

so

T Q

R

k C

k

*

1 1

1 * 3

1 1 1 1

001W m k

0.0001 C , 0.82 gm cm ,

0.3 W m k , 1.9 .gm C ,

0.00001 s,

f sf fs f

f fE

fo

k C cal

0,1x same instance

sity

and all the results will be calculated at the for two different values of the poro

0.1t of th terial whene ma 0.25 and

0.35 . The temperature, the stress, the strain and the dis-

placement 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. 7. Conclusions

his work was dealing with studyingporosity of isotropic and poroelastic one dimensional half-space which is saturated with fluid. The mathemati-cal model of generalized porothermoelasticity with one relaxation time has been constructed in the context of Youssef model. The general solution has been obtained in the Laplace transform domain and applied it in a cer-tain asphalt material which is thermally shocked on bounding plane. The inversion of the Laplace transform has been obtained numerically and the numerical values of the temperature, stresses, strains and displacements have been presented graphically for the solid and the liquid and the graphs shown the significant effect of the porosity value. 8.

eresiewof Acta M

nica 16

T the effect of the

its

References [1] C. Pecker and H. D iez, “Thermal Effects on Wave

in Liquid-Filled Porous Media,” Journal echa- , Vol. , No. 1-2, 1973, pp. 45-64.

doi:10.1007/BF01177125

[2] M. A. Biot, “General Theory of Three-Dimensional Con-solidation,” Journal of Applied Physics, Vol. 12, No. 2, 1941, pp. 155-164. doi:10.1063/1.1712886

[3] M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range,” Journal of Acoustical Society America, Vol. 28, No. 2,

1956, pp. 168-178. doi:10.1121/1.1908239

[4] F. Gassmann, “Uber die Elastizitat Poroser Medien,” Veirteljahrsschrift der Naturforschenden Gesellschaft in Zzirich, Vol. 96, 1951, pp. 1-23.

[5] M. A. Biot and D. G. Willis, “The Elastic Coefficients of the Theory of Consolidation,” Journal of Applied Me-chanics, Vol. 24, No. 4, 1957, pp. 594-601.

[6] M. A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous Media,” Journal of Applied Phys-ics, Vol. 33, No. 4, 1962, pp. 1482-1498. doi:10.1063/1.1728759

[7] H. Deresiewicz and R. Skalak, “On Uniqueness in Dy-namic Poroelasticity,” Bulletin of the Seismological Soci-ety of America, Vol. 53, No. 4, 1963, pp. 783-788.

[8] G. Mandl, “Change in Skeletal Volume of a Fluid-Filled Porous Body under Stress,” Journal of Applied Physics, Vol. 12, No. 5, 1964, pp. 299-315. doi:10.1016/0022-5096(64)90027-4

[9] A. Nur and J. D. Byerlee, “An Exact Effective Stress Law for Elastic Deformation of Rock with Fluids,” Journal of Geophysical Research, Vol. 76, No. 26, 1971, pp. 6414- 6419. doi:10.1029/JB076i026p06414

[10] R. J. S. Brown and J. Korringa, “On the Dependence of the Elastic Properties of a Porous Rock on the Com-pressibility of the Pore Fluid,” Geophysics, Vol. 40, 1975, pp. 608-616. doi:10.1190/1.1440551

[11] J. R. Rice and M. P. Cleary, “Some Basic, Stress Diffu-sion Solutions for Fluid-Saturated Elastic Porous Media

ible Constituents,” Review of Geophysics, , No. 2, 1976, pp. 227-241.

014i002p00227

with CompressSpace Physics, Vol. 14doi:10.1029/RG

[12] R. Burridge and J. B. Keller, “Poroelasticit, y Equations Derived from Microstructure,” Journal of Acoustical So-ciety America, Vol. 70, No. 4, 1981, pp. 1140-1146. doi:10.1121/1.386945

[13] R. W. Zimmerman, W. H. Somerton and M. S. King, “Compressibility of Porous Rocks,” Journal of Geo-physical Research, Vol. 91, No. B12, 1986, pp. 12765- 12777. doi:10.1029/JB091iB12p12765

[14] R. W. Zimmerman, L. R. Myer and N. G. W. Cook, “Grain and Void Compression in Fractured and Porous Rock,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 31, No. 2, 1994, pp. 179-184. doi:10.1016/0148-9062(94)92809-6

[15] J. G. Berryman and G. W. Milton, “Exact Results for Generalized Gassmann’s Equations in Composite Porous

wo Constituents,” Geophysics, Vol. 56, -1960. doi:10.1190/1.1443006

Media with T1991, pp. 1950

[16] M. Thompson and J. R. Willis, “A Reformation of the Equations of Anisotropic Poroelasticity,” Journal of Ap-plied Mechanics, Vol. 58, No. 3, 1991, pp. 612-616. doi:10.1115/1.2897239

[17] S. R. Pride, A. F. Gangi and F. D. Morgan, “Deriving the Equations of Motion for Porous Isotropic Media,” Jour-nal of Acoustical Society America, Vol. 92, No. 6, 1992, pp. 3278-3290. doi:10.1121/1.404178

http://dx.doi.org/10.1007/BF01177125

http://dx.doi.org/10.1063/1.1712886

M. H. ALAWI 1113 [18] J. G. Berryman and H. F. Wang, “The Elastic Coeffi-

cients of Double-Porosity Models for Fluid Transport in Jointed Rock,” Journal of Geophysical Research, Vol. 100, No. B12, 1995, pp. 24611-24627. doi:10.1029/95JB02161

[19] K. Tuncay and M. Y. Corapcioglu, “Effective Stress Prin-ciple for Saturated Fractured Porous Media,” Water Re-sources Research, Vol. 31, No. 12, 1995, pp. 3103-3106. doi:10.1029/95WR02764

[20] A. H. D. Cheng, “Integral Equation for Poroelasticity in Frequency Domain with BEM solution,” Journal of En-gineering Mechanics, Vol. 117, No. 5, 1991, pp. 1136- 1157. doi:10.1061/(ASCE)0733-9399(1991)117:5(1136)

[21] P. A.Charlez and O. Heugas, “Measurement of Ther-moporoelastic Properties of Rocks: Theory and Applica-tions,” In: J. A. Hudson, Ed., International Society for Rock Mechanics, Tour Total, Paris, 1992, pp. 42-46.

[22] Y. Abousleiman and L. Cui, “Poroelastic Solutions in Transversely Isotropic Media for Wellbore and Cylin-der,” International Journal of Solids Structures, Vol. 35, No. 34-35, 1998, pp. 4905-4903. doi:10.1016/S0020-7683(98)00101-2

[23] A. Ghassemi and A. Diek, “Porothermoelasticity for Swelling Shales,” Journal of Petroleum Science & Engi-neeing, Vol. 34, 2002, pp. 123-135.

[24] S. R. Tod, “An Anisotropic Fractured Poroelastic Effec-

of the Mechanics

tive Medium Theory,” Geophysical Journal International, Vol. 155, No. 3, 2003, pp. 1006-1020.

[25] H. Lord and Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” Journal and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5

[26] H. M. Youssef, “Theory of Generalized Porothermoelas-ticity,” International Journal of Rock Mining Sciences, Vol. 44

Mechanics and , No. 2, 2007, pp. 222-227.

doi:10.1016/j.ijrmms.2006.07.001

[27] G. Honig and U. Hirdes, “A Method for the Numerical Inversion of Laplace Transform,” Journal of Computa-tional and Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132. doi:10.1016/0377-0427(84)90075-X

[28] G.-B. Liu, S.-R. Ding, R.-H. Ye and X.-H. Liu, “Relaxa-tion Effects of a Saturated Porous Media Using the Two- Dimensional Generalized Thermoelastic Theory,” Trans-

port in Porous Media, Vol. 86, No. 1, 2011, pp. 283-303. doi:10.1007/s11242-010-9621-9

[29] Y. Zhong and L. Geng, “Thermal Stresses of Asphalt Pavement under Dependence of Material Characteristics on Reference Temperature,” Mechanics of Time-De-pendent Materials, Vol. 13, No. 1, 2009, pp. 81-91. doi:10.1007/s11043-008-9073-6


http://dx.doi.org/10.1063/1.1728759

http://dx.doi.org/10.1063/1.1728759

http://dx.doi.org/10.1029/JB076i026p06414

http://dx.doi.org/10.1029/95JB02161




http://dx.doi.org/10.1029/95WR02764

http://dx.doi.org/10.1061/(ASCE)0733-9399(1991)117:5(1136)




http://dx.doi.org/10.1016/S0020-7683(98)00101-2

M. H. ALAWI 1114

Nomenclature

,iu Ui The displacements of the skeleton and fluid phases

, , ,R Q The poroelastic coefficients

11 12 21 22, , ,R R R Rs sT T

The mixed and thermal coefficients

0 The temperature increment of the solid where sT

f fis the solid

0 The temperature increment of the fluid where

T T fT is the fluid

0T The reference temperature The porosity of the material

* , *s f The density of the solid and the liquid phases respectively

*1s s The density of the solid phase per unit volume of bulk

*f f The density of the solid phase per unit volume of bulk

11 12s f

The mass coefficient of solid phase

22 12 The mass coefficient of fluid phase

12 The dynamics coupling coefficient * , *s fkk The thermal conductivity of the solid and the

fluid phases *1s sk k The thermal conductivity of the solid

phase *f fk k The thermal conductivity of the fluid

phase k The interface thermal conductivity

,s fo o The relaxation time of the solid and the fluid

phases

ij The stress components apply to the solid surface

The normal stress apply to the fluid surface

ije The strain component of the solid phase The strain component of the fluid phase

,s f The coefficients of the thermal expansion of the phases

,sf fs The thermoelastic couplings between the phases

,s fE EC C The specific heat of the solid and the fluid

phases sfEC The specific heat couplings between the phases

s ss E

s

C

k

The thermal viscosity of the solid

f ff E

f

C

k

The thermal viscosity of the fluid

12sfEC

k

The thermal viscosity couplings between

the phases 3 2P

11s fsR p Q

Q

P

22 3f sfR R

12f sfR Q

11s s

EF C

22f f

EF C

12 12 223 s fsoF R R T

21 11 213 sf foF R R T


generalized porothermoelasticity of asphaltic material

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