Thermo-elastic-plastic porous material undergoing thermal loading

Thermo-elastic-plastic porous material undergoing thermalloading

Andrzej Søu _zalec *

Technical University of Czestochowa, 42-201 Czestochowa, Poland

Received 8 August 1997; received in revised form 2 December 1998

Abstract

A model for predicting elastic±plastic stresses within a surface-heated porous structure has been devel-oped. The relevant phenomena for the moisture, pressure, temperature and displacement ®elds in thermo-elastic-plastic porous material are analysed. Considering mass and energy transfer processes, a set ofgoverning di�erential equations is presented. The solution of the problem has been obtained with a ®nitedi�erence scheme. The results demonstrate the in¯uence of the evaporation mechanism on pressure andthermal stresses within the porous material. Ó 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

Transport phenomena and elastic±plastic stresses in porous media have recently receivedgrowing attention in light of the common usage of such media in various applications in the ®eldsof energy technology. The physical phenomena of moisture transfer in porous media are usuallyexplained by the following theories: di�usion theory [1]; capillary ¯ow theory [2]; and evaporationcondensation theory [3]. Di�usion theory, which has been used extensively in the past does notgive results in agreement with experimental data. Capillary ¯ow theory, in which the di�usivitydepends on the pore water content, has been employed [4,5] to predict the drying rate in concreteslabs. However, the coe�cient of di�usivity is a complex function of pore moisture, temperatureand other variables of the porous system. It cannot be de®ned as a simple function of poremoisture as done in Refs. [4,5]. Later, drying experiments at elevated temperatures revealed highpore pressure as a consequence of intense water vaporization [6]. The measured pore pressure,determined under equilibrium heating conditions, correspond well to the sum of the calculated

International Journal of Engineering Science 37 (1999) 1985±2005www.elsevier.com/locate/ijengsci

*Corresponding author. Tel.: +48-34-3-250-966; fax: +48-34-3-250-920.

E-mail address: [email protected] (A. Søu_zalec)

0020-7225/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S0020-7225(99)00011-7

saturated water vapor pressure and ambient air pressure. As a consequence of the above models,in a previous work the evaporation mechanism was assumed with concentration and pressuregradient terms [7].

An analysis of poroelastic stresses was developed and improved by Biot [8±10]. In 1976 Riceand Cleary [11] gave a rational reformulation of the Biot theory, and their rationalised version hasproved more convenient for solving relevent problems and in interpreting the solutions obtained(e.g. [12±15]). Modern theories of mixtures, taking into account ®nite deformations and thermale�ects, for most cases have been developed by Crochet and Naghdi [16] and Bowen [17]. In 1971Schi�man [18] developed an extended Biot theory including thermal e�ects. The present paperdescribes equations of heat and mass transfer in a thermo-elasto-plastic material. This theoryshows that the displacement ®eld is, in general, completely coupled with the pore pressure andtemperature ®elds.

2. Assumptions and equation

2.1. Theoretical assumptions and simpli®cations

The following assumptions are made for the theoretical model:1. Heat transfer between the ¯uid and the solid is neglected. The response time for local heat

transfer between the ¯uid and the solid is several orders of magnitude smaller than the timesof interest.

2. Liquid vapor equilibrium exists in the presence of free water, which makes the partial pressureof the vapor equal to the saturation pressure.

3. Movement of the liquid is neglected.4. Darcy's law with a variable coe�cient holds for the gases.5. Air and water vapor are treated as ideal gases.

2.2. Conservation equations for mass

The conservation equations for mass can be written as

_qi � ÿr�qiwi� � Wi; �1�where qi is the density of species i, wi is the velocity of species i and Wi is the production rate ofspecies i. Since no movement of the liquid (subscript c) is assumed wc � 0. Also, W � ÿWV �ÿWm (v � vapor, m� air±vapor mixture), since the rate of liquid evaporation is the same as therate of vapor production, and since the air does not change phase, the rate of mixture productionequals the rate of vapor production. The above equation thus simpli®es to

_qc � ÿWm: �2�The continuity equation for the air (subscript a) takes the form

_qa � ÿr�qawa� �3�

1986 A. Søu_zalec / International Journal of Engineering Science 37 (1999) 1985±2005

and for the vapor,

_qv � ÿr�qvwv� � Wv: �4�The conservation of gas phase mass gives

_qm � ÿr�qmwm� � Wm: �5�Fick's law allows the ¯uxes to be presented in the forms of Eqs. (6) and (7):

ja � qa�wa ÿ wm� � ÿqmDrqba; �6�where qbi � qi=qm is the mass fraction of species i with respect to the density of the air±vapormixture, and D is the di�usion coe�cient for Fick's law for the air±vapor mixture; and

jv � qv�wv ÿ wm� � ÿqmDrqbv: �7�Finally, we get the following species equations:

qmba � qmwmrqba � r�qmDrqba� ÿ qbaWm �8�and

qmbv � qmwmrqbv � r�qmDrqbv� � �1ÿ qbv�Wm: �9�

2.3. Thermal equations

The ¯uxes of heat q and ¯owing gases r can be expressed as

q � ÿkrh �10�and

r � qawaha � qvwvhv; �11�where k is the thermal conductivity, h is the temperature and hi is the enthalpy of component i perunit mass of component i. Eq. (11) can be transformed [19] to the form

r � qmwmhm ÿ qmDharqba ÿ qmDhvrqbv �12�or

r � qmwmhm � qmD�hv ÿ ha�rqba: �13�Assuming that

qe � qshs � qchc � qmhm ÿ qmRmh; �14�where e is the thermal energy and R is the gas constant, the thermal equations can be ex-pressed as

A. Søu_zalec / International Journal of Engineering Science 37 (1999) 1985±2005 1987

qcp_h � rkrhÿ qm�wmcpm � D�cpv ÿ cpa�rqba�rhÿ ��hv ÿ ha�Wm ÿ _qmrmh�; �15�

where cp is the speci®c heat at constant pressure.

2.4. Darcy's law

The velocity of the air±vapor mixture is given by

wm � ÿkDrp; �16�where kD is Darcy's coe�cient and p is the pressure.

2.5. Thermodynamic relations

Assuming that the vapor and air are ideal gases we have the following relations.

2.5.1. Ideal gas equation for the vapor

pv�V v � qvRvh; �17�

where �V i � qi=qai represents the volume occupied by component i per unit total volume.

2.5.2. Ideal gas equation for the air

pa�V a � qaRah: �18�

2.6. Clausius±Clapeyron equation

Since the liquid and vapor are assumed to be in equilibrium,

pv � psat�h� �19�in the presence of liquid water. An analytic expression for psat is

psat�h� � Chÿ�B=Rv� exp

�ÿ A

RvT

�: �20�

2.7. State equation

Using the notations qbi and �V i, the state equation can be presented as

pv�/ÿ �V c� � qmqbvRvh � �1ÿ qba�qmRvh �21�and


pa�/ÿ �V c� � qmqbaRah; �22�where V v � V a � �/ÿ V c� and / is the porosity.

Combining the above equations we get

p�/ÿ V c� � qmRmh; �23�where

Rm � qbvRv � qbaRa: �24�

2.8. Porosity

In a linearized approach, the current porosity / depends on the current values of ¯uid massdensity, strain and increase in ¯uid mass content according to relation

/ � /o �m

qc;o

ÿ /o

qc ÿ qc;o

qc;o

�� tr e

��25�

where m is the ¯uid mass content, qc;o is the ¯uid mass density in the reference state, /o is theporosity in the reference state. In in®nitesimal transformation, the trace tr e � eii of the linearizedstress tensor e represents the volume change per unit volume in the deformation. It is calledvolume dilatation.

2.9. Thermo-elastic-plastic constitutive equations

2.9.1. Isotropic hardeningThe basic hypothesis of isotropic hardening is the assumption that the shape of the yield surface

is unchanging, and its growth can be described by one scalar parameter which is a function ofplastic deformation. We can describe the plastic potential function F of the porous material by theporous material by the equation [24]

F � F �rij;K; h; p� �26�where rij is the stress tensor, and K is the work ± hardening parameter.

According to the Quinney±Taylor (QT) hypothesis [23]

_K � rij _epij: �27�

If we di�erentiate F by using the chain rule for partial di�erentiation we obtain

_F � oForij

_rij � oFoK

_K � oFoh

_h� oFop

_p: �28�

Using the QT hypothesis, it is seen that the second term in the right-hand side of Eq. (28) can beexpressed in terms of plastic strains rates _ep

ij as


oFoK

_K � oFoK

oKoep

ij

_epij: �29�

Combining Eqs. (28) and (29) gives

_F � oForij

_rij � oFoK

oKoep

ij

_epij �

oFoh

_h� oFop

_p: �30�

Equilibrium conditions require that variation of the plastic energy be stationary

_F � oForij

_rij � oFoK

oKoep

ij

_epij �

oFoh

_h� oFop

_p � 0: �31�

In the approach presented we assume the small strain thermo-elastic-plasticity theory in whichthe total small strain rate is a sum of elastic, thermal, plastic and strain rate due to pressurechanges.

By this assumption strain rate tensor in a thermo-elasto-plastic process in a solid body is of theform

_eij � _eeij � _eT

ij � _ebij � _ep

ij; �32�where _eij; _e

eij; _e

Tij; _e

bij; _e

pij are the rates of the total, elastic, thermal, pressure and plastic strain tensor,

respectively. After rearranging Eq. (32) we can obtain the components of the elastic strain ratetensor

_eeij � _eij ÿ _eT

ij ÿ _ebij ÿ _ep

ij: �33�Making use of Hooke's law the rates of change of the total stress are given as

_rij � Ceijkl�_ee

kl ÿ _eTkl ÿ _eb

kl ÿ _epkl�; �34�

where Ceijkl are components of the elasticity tensor

Ceijkl �

1

4Gdikdjl

�� dildjk ÿ 2

1� mdijdkl

�; �35�

and G is the shear modulus, m is Poisson's ratio, dij is the Kronecker delta.Combining Eqs. (31), (33) and (34) and considering the ¯ow rule

_epij � k

oForij

; �36�

we get

oForij

Ceijkl�_ekl ÿ _eT

kl ÿ _ebkl ÿ _ep

kl� �oFoK

oKoep

ij

oForij

k� oFoh

_h� oFop

_p: �37�


After some calculations we obtain the relation for the proportionality factor k

k �oForij

Ceijkl�_ekl ÿ _eT

kl ÿ _ebkl� � oF

oh_h� oF

op _p� �

oForpq

Cepqrs

oForrsÿ oF

oKoKoep

pq

oFopq

� � : �38�

Introduce the notations

S � oForpq

Cepqrs

oForrsÿ oF

oKoKoep

pq

oForpq

; �39�

_eTij � aij

_h; �40�

aij � ahdij �41�

_ebij � Bij _p; �42�

Bij � 3�mu ÿ m�2GBu�1� m��1� mu� dij; �43�

where mu is the undrained Poisson ratio, ah is the thermal expansion coe�cient and Bu is the in-duced pore pressure parameter. In general,

0 < Bu6 1; 0 < m < mu61

2: �44�

From Eqs. (38)±(42) we get

k � 1

SoForij

Ceijkl�_ekl

�ÿ akl

_hÿ Bkl _p� � oFoh

_h� oFop

_p�: �45�

Next combining Eqs. (34) and (45) we have

_rij � Ceijkl _ekl ÿ Ce

ijkl�akl_h� Bkl _p� � Ce

ijkl

SoForkl

� oForij

Ceijkl�_ekl

�ÿ akl

_hÿ Bkl _p� � oFoh

_h� oFop

_p�: �46�

After rearranging the terms


ijkl�akl_h� Bkl _p� ÿ 1

SCe

ijkl

oForkl

oForij

Ceijkl _ekl

� 1

SCe

ijkl

oForkl

oForij

Ceijkl�akl

_h� Bkl _p� ÿ 1

SCe

ijkl

oForkl

oFoh

_h

�� oF

op_p�: �47�


Assuming the Huber±Mises (HM) condition and making use of the fact

oForij� sij �48�

we can de®ne the so-called plasticity tensor

Cpijkl �

1

SCe

ijkl

oForkl

oForij

Ceijkl �

1

SCe

ijklsklsijCeijkl �49�

where sij are components of the deviatoric stress tensor.

sij � rij ÿ 1

3rkkdij: �50�

By substituting Eq. (49) into Eq. (47) we get


ijkl�akl_h� Bkl _p� ÿ Ce

ijkl _ekl � Cpijkl�akl

_h� Bkl _p�

ÿ 1

SCe

ijkl

oForkl

oFoh

_h

�� oF

op_p�: �51�

De®ning the so-called elasto-plasticity tensor

Cepijkl � Ce

ijkl ÿ Cpijkl �52�

the thermo-elasto-plastic constitutive equation can be expressed as

_rij � Cepijkl _ekl ÿ Cep


ijklskl

SoFoh

_h

�� oF

op_p�: �53�

One may ®nd that during plastic deformation of a solid

Ceijklskl � 2Gsij �54�

due to the fact that the ®rst deviatoric strain invariant vanishes. It leads to the following relationon Cp

ijkl:

CeijklsklsijCe

ijkl � 2Gsijskl2G � 4G2 sijskl: �55�

Now we evaluate the S given by Eq. (39). Since Cepqrssrs � 2Gspq the ®rst term in Eq. (39) can be

expressed as

oForpq

Cepqrs

oForrs� spqCe

pqrssrs � spq2Gspq � 2G2�r2

3� 4

3G�r2; �56�


where

1

3�r2 � 1

2spqspq: �57�

By Eq. (27) and Eq. (36) we have

_K � krijoForij

: �58�

From Eq. (27) we get

rij �_K

_epij

: �59�

During plastic deformation of work-hardening materials, the yield strength increases with load.The de®nitions of the HM plastic potential function can be used to derive the following usefulrelation:

oFoK� o

oKJ2s

�ÿ 1

3�r2

�� ÿ 2

3�r

o�roK

; �60�

where J2s is the second invariant of the devatoric stress.If we refer to a typical ¯ow curve, we have

dK � �rd�ep; �61�or

d�ep

dK� 1

�r: �62�

Since

o�roK� o�r

o�ep

d�ep

dK� H 0

1

�r� H 0

�r; �63�

where H 0 is the plastic modulus of the material in a multiaxial stress state (see Fig. 1)

H 0 � o�ro�ep

: �64�

Eq. (60) will have the form

oFoK� ÿ 2

3�r

H 0

�r

� �� 2H 0

3; �65�

which leads to the following relation:


oFoK

oKoep

pq

oForpq

� ÿ 2H 0

3rpqssq � ÿ 2

3H 0

2

3�r2 � ÿ 4

9�r2H 0: �66�

Then S can be expressed in terms of the material properties and the state of stress as follows:

S � 4

3G�r2 ÿ

�ÿ 4

9�r2H 0

�� ÿ 4

9�r2�3G� H 0�: �67�

2.9.2. Kinematic hardeningIn processes of kinematic hardening, in order to describe the motion of the initial yield surface,

one introduces the translation tensor aij, whose components determine a new position of the yieldsurface centre. Ziegler [20] assumes a motion of the surface in the direction of the di�erence of rand a:

_aij � _l�rij ÿ aij�; �68�where _l is the multiplier. Melan [21] has proposed de®nition of the tensor aij as

_aij � C _epij: �69�

This de®nition was also used by Prager [22].Considering the modi®cation of the yield surface with respect to temperature and pressure we

assume

C � C�h; p�: �70�The plastic potential function F can be expressed as

F � F �rij; aij; h; p�: �71�

Fig. 1. Slope H 0 as a plastic modulus of the material during plastic deformation.


The total di�erential of F is

_F � oForij

_rij � oFoaij

_aij � oFoh

_h� oFop

_p � 0: �72�

The condition, Eq. (72), is sometimes called the compatibility equation for the plastic yieldingcondition. On the basic of Eq. (68) we can obtain the following relations:

oFoaij� ÿ oF

orij; �73�

orij

oaij� ÿ1: �74�

If one substitutes the above expressions into Eq. (72) the following relations is obtained:

oForij� _rij ÿ _aij� � oF

oh_h� oF

op_p � 0 �75�

or

oForij

_aij � oForij

_rij � oFoh

_h� oFop

_p: �76�

Further substitution of Eq. (68) into the above expression yields

_l �oForij

_rij � oFoh

_h� oFop _p

� ��rkl ÿ akl� oF

orkl

� � : �77�

In kinematic hardening theory we de®ne the so-called translated stress tensor

r�ij � rij ÿ aij; �78�and translated stress deviators

s�ij � r�ij ÿ1

3r�kkdij: �79�

The yield function becomes

F � 2

3s�ijs�ij; �80�

which leads to the expression

oForij� 3s�ij

os�ijorij� 6s�ij: �81�


By substituting Eq. (81) into Eqs. (77) and (78) we get

_u �6s�ij _rij � oF

oh_h� oF

op _p

6r�pqs�pq

; �82�

k � 1

C�h; p�6s�ij _rij � oF

oh_h� oF

op _p

36s�pqs�pq

; �83�

_epij � 6s�ijk: �84�

Assuming the decomposition (32)

_eij � _eeij � _eT

ij � _ebij � _ep

ij; �85�and using Hooke's law one obtains

_rij � Ceijkl�_ekl ÿ _eT

kl ÿ _ebkl ÿ _ep

kl� �86�or


ijkl�akl_h� Bkl _p � 6s�klk�: �87�

Substituting Eq. (69) and Eq. (81) into Eq. (36) gives

_aij � 6C�h; p�s�ijk: �88�

The above expression, after substituting into Eq. (72), gives

_F � � _rij ÿ 6C�h; p�s�ijk�oForij� oF

oh_h� oF

op_p � 0: �89�

By Eqs. (89) and (83) we get

k �oForij�Ce

ijkl _ekl ÿ Ceijkl�akl

_h� Bkl _p�� oFoh

_h� dFop

_p

6�C�h; p�s�pq � Cepqrss�rs�

oForpq

: �90�

Substituting Eq. (90) into Eq. (87) we get


ijkl�akl_h� Bkl _p�

ÿ Ceijkls

�kl

oForij�Ce

ijkl _ekl ÿ Ceijkl�akl

_h� Bkl _p�� oFoh

_h� oFop _p

6�C�h; p�s�pq � C�pqrss�rs� oForpq

: �91�


Let

S � �C�h; p�s�pq � Cepqrss

�rs�

oForpq

� 6s�pqs�pqC�h; p� � 6s�pqCepqrss

�rs: �92�

With the aid of Eq. (92) the relation Eq. (91) becomes



ijkls�kl

oForij

1

S�Ce

ijkl _ekl�akl_h� Bkl _p��

ÿ 1

SCe

ijkls�kl

oFoh

_h

�� oF

op_p�� Ce

ijkl

�ÿ 1

SCe

ijkls�kl

oForij

Ceijkl

� ��_ekl

ÿ Ceijkl

�ÿ 1

SCe

ijkls�kl

oForij

Ceijkl

� ��akl

_h� Bkl _p� ÿ 1

SCe

ijkls�kl

oFoh

_h

�� oF

op_p�: �93�

Assuming that

1

sCe

ijkls�kl

oForij

Ceijkl

� �� 1

S�6Ce

ijkls�kls�ijC

eijkl� � Cp

ijkl �94�

and

Cepijkl � Ce

ijkl ÿ Cpijkl; �95�

the constitutive equation for kinematic hardening material subject to thermo-elasto-plastic de-formation takes the form

_rij � Cepijkl _ekl ÿ Cep

ijkl�akl_h� Bkl _p� ÿ 1

SCe

ijkls�kl

oFoh

_h

�� oF

op_p�: �96�

2.10. Momentum balance

If no body force exists, the momentum balance in the quasi-static nonisothermal context is

rij;j � 0: �97�

2.11. Strain±displacement relation

The deformation of the material is described by the strain tensor eij, which is de®ned in terms ofthe displacement ui of the solid constituent as follows:

eij � 1

2�ui;j � uj;i�: �98�


3. Example

3.1. Simpli®ed equations for a 1D axisymmetrical problem

One of the simple examples of the thermo-poro-elasticity theory presented in this paper is 1Daxisymmetrical problem. In such a case the set of governing equations and suitable boundaryconditions in a cylindrical coordinate system take the form as follows.

3.1.1. Thermal equations

qcp

ohot� k

o2hor2� 1

roor

r�

ÿ qmcpm wm

�� D�cpr ÿ cpa�

cpm

oqba

or

�ohor

�ÿ �hv

�ÿ ha�Wm ÿ o

ot�qmRmh�

�; �99�

h�r; 0� � h0�r�; �100�

ohor�0; t� � 0; �101�

and

ÿkohor�R; t� � h�h�R; t� ÿ hf�t�� f r�h4�R; t� ÿ h4

f �t��; �102�

where R is the radius of the element, hf is the environmental temperature, h is the convective heattransfer coe�cient, f is the e�ective shape factor for radiation and r is the Stefan±Boltzmanconstant.

3.1.2. Species equations

oqba

ot� D

o2qba

or2� 1

qmroor�rqmD�

�ÿ wm

�oqba

orÿ qbaWm

qm

; �103�

qba�r; 0� � qba;0�r�; �104�and

oqba

or�0; t� � 0: �105�


3.1.3. Continuity equations

oqm

ot� 1

roor�rqmwm� � Wm; �106�

wm�0; t� � 0; �107�

oqc

ot� ÿWm; �108�

and

qc�r; 0� � qc;0�t�: �109�

3.1.4. Field equations

drrr

drhh

( )� 2G

1ÿm1ÿ2m

m1ÿ2m

m1ÿ2m

1ÿm1ÿ2m

" #derr

dehh

( )ÿ 3�ml ÿ m�

Bl�1� ml��1ÿ 2m� dp

ÿ 2G�1� m�1ÿ 2m

adhÿ 2GSo

s2rr srrshh

srrshh s2hh

" #d 2rr

d 2hh

( )

� 2GSo

s2rr srrshh

srrs2hh s2

hh

" #adh

�� 3�ml ÿ m�

2GBl�1� m��1� ml� dp�

ÿ 2GSo

1ÿm1ÿ2m

m1ÿ2m

m1ÿ2m

1ÿm1ÿ2m

" #srr

shh

( )oFoh

dh

�� oF

opdp�

�110�

err

ehh

� ��

our

orur

r

� ��111�

where

So � 2

3�r2 1

�� H 0

3G

�: �112�

The plastic potential function is assumed in the form

F ��1

2�rrr ÿ rhh�2

r� �Bp � chÿ k� � 0 �113�

where b; c and k are material parameters.


The thermodynamics relations are given by Eq. (17) and Eq. (18), the Clausius±Clapeyronequation by Eq. (20) and Darcy's law by Eq. (16).

3.2. The solution

The solution is obtained by an implicit ®nite di�erence technique with a constant grid andvariable time step sizes. Since the equations describing heat and mass transfer in porousmaterials are partial di�erential equations of parabolic type, the solutions of such equationsare well known and can be found in many textbooks on numerical analysis. These data do notprovide any special information regarding this paper and have been omitted. For most heatingsituations with rapid heating a time step size of 1 s is found to be satisfactory for the cal-culations. A mesh size of slightly less than 2� 10ÿ3 m was found to be adequate for theexample.

3.3. Temperature, pressure and stresses in a 1D axisymmetrical element

The speci®c case of a 1D axisymmetrical structural element with a uniform initial temperatureis considered to illustrate the results of the analysis (Fig. 2). The radius of the considered cylin-drical element is 0.12 m. The initial moisture content is from 0 to 0.108 m and the remaining 0.108to 0.12 m is supposed to be dry. The assumption of the existence of a dry region close to thesurface is often validated. The thermal and mechanical parameters used are presented in Table 1.Fig. 3 presents the temperature in the porous element as a function of time for the assumedheating curve for a � 10 J=s m2K, kD � 10ÿ11 m3 s=kg and qc;o � 70 kg=m3. In Fig. 4 one canobserve maximum pressures for various values of kD and in Fig. 5 changes of pressure with time.

Fig. 2. Cylinder under consideration.


Fig. 6 shows the distribution of stresses for maximum pressures (kD � 10ÿ12 m3 s=kg andt � 3� 103 s).

Figs. 4 and 5 show the pressure pro®les and pressure histories respectively for the several valuesof Darcy's coe�cient. Fig. 4 indicates that for lower values of Darcy's coe�cients pressures inelements increases. So for very porous material such as sand the pressures are very low and we donot observe signi®cant deformations due to pressure e�ect during heating.

Increasing of Darcy's coe�cient shifts the point of maximum pressure toward the inside. This isconsistent with the preceding argument since to have reasonable ¯ow resistance the point should

Fig. 3. Temperatures in the structural element for a � 10 =sm2 K, kD � 10ÿ11 m3s=kg and qc;0 � 70 kg=m3: (a)

r � 120 mm; (b) r � 100 mm; (c) r � 80 mm; (d) r � 60 mm.

Table 1

Thermal and mechanical parameters used in this study

cp � 1040 J=kg K kD � 5� 10ÿ12 to 1 m3 s=kg

D � 2:142� 10ÿ5 m2=s qba1 � 1:0f � 0:9 a � 5:22� 10ÿ7 m2=s

a � 0 to A / � 0:2aD � 1 q � 2400 kg=m

3

A � 3:18� 106 J=kg qc;o � 0:200 kg=m3

B � 2470J=kg K Bu � 0:7C � 6:05� 1026 N=m2 m � 0:16

ah � 10� 10ÿ6 1=K mu � 0:17

b � 0:1 E � 2� 1010 N=m2

c � 0:1K � 5� 106 N=m2


Fig. 5. Changes of pressure as a function of time for various kD values: (a) kD � 10ÿ12 m3s=kg; (b) kD � 10ÿ10 m3s=kg.

Fig. 4. Maximum pressures as a function of the distance from the heating surface for various kD values: (a)

kD � 10ÿ12 m3 s=kg; (b) kD � 10ÿ10 m3 s=kg.


Fig. 6. Radial (a) and circumferential (b) stresses in an element for kD � 10ÿ12 m3s=kg and t � 3� 103 s.


be farther inside the medium for a higher value of Darcy's coe�cient. The pressure term in¯uenceson the stresses pro®les in the element.

4. Concluding remarks

An analysis is developed for heat and mass transfer in a wet porous medium subject to un-steady, nonlinear boundary conditions. The simpli®ed equations have been solved simultaneouslyby an implicit ®nite di�erence technique. The assumption of no liquid movement was made in thedevelopment of the present theory. This assumption is probably valid for low and mediumpressures. However, for high pressures it is questionable.

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Thermo-elastic-plastic porous material undergoing thermal loading