Continuum mechanics of two-phase porous media

Ragnar LarssonDivision of material and computational mechanicsDepartment of applied mechanicsChalmers University of TechnologyS-412 96 Göteborg, Sweden20 September, 2006

Course outline

General purpose and contents

The area of multiphase materials modelling is a well established and growing field in the mechani-cal scientific community. There has been a tremendous development in recent years including theconceptual theoretical core of multiphase materials modelling, the development of computationalmethodologies as well as experimental procedures. The applications concern biomechanics, model-ling of structural foams, composites modelling, soils, road mechanics etc. Specific related issuesconcern modelling of: solid-fluid interaction, compressible-incompressible fluids/solids includingphenomena like consolidation, compaction, erosion, growth, wetting, drying etc.

The main purpose of this course is to give an up-to-date account of the fundamental continuummechanical principles pertinent to the theory of porous materials considered as mixture of twoconstituents. The idea of the course will is to provide a framework for the modelling of a “solid”porous material with compressible and incompressible fluid phases. As to constitutive modelling,we restrict to hyper-elasticity and the ordinary Darcy model describing the interaction between theconstituents. Computational procedures associated with the nonlinear response of the coupled two-phase material will be emphasized. The lecture notes [1] focus on the general description of kinemat-ics and material models for FE-modeling of large deformation problems.

Organization of lectures

The course material is defined by [1] plus additional literature references given during the course.

Start: Mon 25/9, 2006, 10.00, Materialtekniks seminarierum.

1. Introduction and applications of the porous media theory, Course outline, [1]:

è The concept of a two-phase mixture: Volume fractions, Effective mass, Effective veloci-ties, Homogenized stress

2. A homogenized theory of porous media

è Kinematics of a two-phase continuum

è Conservation of mass:

è One-phase material, Two-phase material, Mass balance of fluid phase in terms of relative velocity, Mass balance in terms of internal mass supply, Mass balance - final result

3. a) Conservation of momentum changes and b) energy - isothermal case

è a) Total format, Individual phases and transfer of momentum change between phases

è b) Total formulation, Individual phases, Energy equation in localized format, Assumption about ideal viscous fluid and the effective stress of Terzaghi.

4. Conservation of energy (cont’d) and Entropy inequality

è General approach (effective free energy), Localization, Effective drag (or interaction) force

5. Constitutive relations

è Effective stress response, Solid-fluid interaction, Solid densification - reduction of a three phase model, Gas densification - the ideal gas law

6. Summary - Balance relations for different types of porous media

è Classical incompressible solid-fluid medium, Compressible solid-fluid medium, Compress-ible solid-gas medium

è Restriction to small deformations - Compressible solid-fluid medium

7. Assignment: specific model, cont’d

è Modelling of effective solid phase (Hyper-elasticity), Darcy interaction, Issue of incompress-ibility, boundary value problem

è Computational aspectsè Discretization, Set of non-linear FE equations, Solution of coupled problem (monolithic/staggered

solution techniques)

8. Summary of the course

Course work and examination

“Theory questions” involving derivation of continuum mechanical relations related to the modelingof porous materials and “computer implementation” of a chosen specific model” are given. Com-pleted course work gives 5 credit points.

Literature

[1] R. Larsson, Continuum mechanics of two-phase porous media, Lecture notes (2006).

[2] S. Toll, A course on micromechanics (2005).

The lecture notes and overheads will be available in electronic form. The following reference litera-ture is also proposed:

R. de Boer, Theory of Porous Media - Highlights in the Historical Development and Current State,Springer Verlag Berlin - Heidelberg - New York, (2000)

O. Coussy, Mechanics of Porous Media (John Wiley & Sons, Chichester, 1995).

2 Organization of lectures

Applications of the porous media theory

The concept of a two-phase mixture

Volume fractions

In order to formulate a homogenized theory of the two-phase mixture, we shall refrain from thedetailed consideration of the relationships between the constituents. We rather consider the constitu-ents as homogenized with respect to their volume fractions of a Representative Volume Element(RVE) with volume V, as indicated in Fig. 1. To this end, we introduce the (macroscopic) volumefractions na@x, tD as the ratio between the local constituent volume and the bulk mixture volume,i.e. ns = V

sÅÅÅÅÅÅÅV for the solid phase and n

f = Vf

ÅÅÅÅÅÅÅÅV for the fluid phase. Of course, in order to ensure thateach control volume of the solid is occupied with the solid/gas mixture, we have the saturationconstraints

(1)ns + n f = 1 and 0 § ns § 1, 0 § n f § 1

Figure 1. Micro-mechanical consideration of two phase problem

Effective mass

Following the discussion concerning the volume fractions let apply the principle of “mass equiva-lence” to the RVE (with volume V ) in Fig. 1 above. This yields the total mass M of the RVE as:

(2)

= ‡srmic

s d vs + ‡f

rmicf d v f =‡ nmics rmics d v + ‡ nmicf rmicf d v = 1ÅÅÅÅÅÅÅV ‡ nmics d v ‡ rmics d v + 1ÅÅÅÅÅÅÅV ‡ nmicf d v ‡ rmicf d v

where it was assumed in the last equality that e.g. nmics and rmic

s are uncorrelated. According to thePrinciple of Scale Separation (P.S.S.), cf. Toll [2], we now state the relationship between the micro-scopic and the macroscopic fields as

(3)1

ÅÅÅÅÅÅÅV

‡ nmics d v ‡ rmics d v + 1ÅÅÅÅÅÅÅV ‡ nmicf d v ‡ rmicf d v =P.S.S. 1ÅÅÅÅÅÅÅV Ins rs + n f r f M V

3

where the macroscopic intrinsic densities in eq. (3) associated with each constituent are denoted rs

and r f . These effective intrinsic densities are in turn related to the local densities and volumefractions 8nmics , rmics < , 9nmicf , rmicf = of the microstructure (in view of eq. (2)) as

(4)

rs =1

ÅÅÅÅÅÅÅÅÅVs

‡ nmics rmics d v = 1ÅÅÅÅÅÅÅÅÅV s ‡ s rmics d vs , ns = 1ÅÅÅÅÅÅÅV ‡ nmics d v ;r f =

1ÅÅÅÅÅÅÅÅÅÅV f

‡f

rmicf d v f , n f =

1ÅÅÅÅÅÅÅV

‡ nmicf d vIt may be noted that the intrinsic densities directly relates to the issue of compressibility (or incom-pressibility) of the phases. For example, in the case of an incompressible porous mixture the intrin-sic densities are stationary with respect to their reference configurations, i.e. rs = r0

s , r f = r0f .

Let us also introduce the bulk density per unit bulk volume r̀a = na ra , whereby the saturateddensity r̀ = MÅÅÅÅÅÅV becomes

(5)r̀ = r̀s + r̀ f

Effective velocities

In order to motivate the effective kinematical relation already introduced in eq. (16) we consider theequivalence of momentum produced by micro fields and effective kinematical fields of the RVEin Fig. 1:

(6)

= ‡s rmic

s vmics d vs + ‡

f rmic

f vmicf d v f =‡ Inmics rmics vmics + nmicf rmicf vmicf M d v =P.S.S. Ins rs vs + n f r f v f M V

where (again) the last equality follows from the principle of scale separation.Hence, from thisrelation we may choose to specify the effective properties:

(7)‡s rmic

s vmics d vs - ns rs vs V = 0 , ‡

f rmic

f vmicf d v f - n f r f v f V = 0

leading to

(8)vs =

1ÅÅÅÅÅÅÅÅr̀s

1

ÅÅÅÅÅÅÅV

‡s rmic

s vmics d vs , v f =

1ÅÅÅÅÅÅÅÅÅr̀ f

1

ÅÅÅÅÅÅÅV

‡f rmic

f vmicf d v f

Hence, the effective velocity fields vs and v f are considered as mean properties of the momentumof the respective constituents scaled with the bulk densities r̀a . If we in addition assume thatrmic

a ,vmica are completely independent (or uncorrelated) we obtain the direct averaging:

(9)vs =1

ÅÅÅÅÅÅÅÅr̀s

1

ÅÅÅÅÅÅÅV

1

ÅÅÅÅÅÅÅÅÅV s

‡s rmic

s d vs ‡s vmic

s d vs =1

ÅÅÅÅÅÅÅÅÅV s ‡ vmics d vs , v f = 1ÅÅÅÅÅÅÅÅÅÅV f ‡ vmicf d v f

4 Effective mass

Homogenized stress

From homogenization theory and micromechanics of solid materials, cf. Toll [2], let us consider thetotal stress as the volumetric mean value over a representative volume element with the volume Vas

(10)sêêê =1

ÅÅÅÅÅÅÅV

‡ smic d Vwhere smic is the micromechanical variation of the stress field within the RVE in Fig. 1.

Let us next simply generalize this result to the situation of a two-phase mixture of solid (s) and fluid(f) phases where the total (homogenized) stress is obtained as the mean value

(11)sêêê =def

1

ÅÅÅÅÅÅÅV

ikjj‡ s smics d vs + ‡ f smicf d v f y{zz =P.S.S.. ss + s fwhich motivates the introduced homogenized stresses ss and s f of solid and fluid constituentsdefined as

(12)ss =1

ÅÅÅÅÅÅÅV

‡s smic

s d vs , s f =1

ÅÅÅÅÅÅÅV

‡f smic

f d v f

We remark that ss and s f represent the homogenized stress response of the constituents, whichcan be related to the intrinsic stresses upon introducing the fractions ns = V

sÅÅÅÅÅÅÅV of the solid phase and

n f = Vf

ÅÅÅÅÅÅÅÅV of the fluid phase, as defined in (1). The intrinsic stresses are then defined via

(13)ss = ns sins , s f = n f sin

f

with

(14)sins =

1ÅÅÅÅÅÅÅÅÅV s

‡s smic

s d vs , sinf =

1ÅÅÅÅÅÅÅÅÅÅV f

‡f smic

f d v f

As an example, let us consider the important special case (considered later on in this course) of theassumption of an ideal fluid where the intrinsic stress response is defined by the intrinsic fluidpressure p as

(15)sinf = - p 1 fl s f = -n f p 1

where the last expression defines the homogenized fluid stress in the case of and ideal fluid.

Theory questions

1. Define and discuss the concept of volume fractions in relation to the micro-constituents of a two-phase mixture of solid and fluid phases related to an RVE of the body.

2. Define the effective mass in terms of intrinsic and bulk densities of the phases from equiva-lence of mass. Discuss the issue of (in)compressibility of the basis of this discussion.

3. Define the effective (representative) velocities of the solid and fluid phases related their micro-mechanical variations across an RVE. Discuss also the issue of “direct averaging”.

5

4. Based on the quite general result of stress homogenization of a one-phase material, generalize the result to the two-phase situation. Discuss the partial stresses and their relation to intrinsic stresses and micro-stress fields.

A homogenized theory of porous media

Kinematics of two phase continuum

Let us in the following consider our porous material as a homogenized mixture between solid andfluid (which may be a liquid or a gas depending on the application) phases as motivated in theprevious section. To this end, we denote the phases s, f , where s stands for the solid phase,whereas f stands for the fluid phase. The representation of the porous medium as a mixture ofconstituents, implies that each spatial point x of the current configuration at the time t are simulta-neously occupied by the material particles Xa . We emphasize that the constituents s, f relate todifferent reference configurations, i.e. Xs ∫ X f , cf. Fig. 2. During the deformation these “particles”move to the current configuration via individual deformation maps defined as

(16)x = j@XsD = j f AX f EAs to the associated velocity fields we have in view of the deformation maps j and j f the relations

(17)vs =Ds j@XDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= j° @XD , v f = D f j f AX f EÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t

where Ds è ê D t = è° denotes the material derivative with respect to the solid reference configura-tion, whereas D f è ê D t denotes the material derivative with respect to the fluid reference configura-tion. Please note that the “dot derivative” thus denotes the material derivative with respect to thesolid reference configuration. We remark that indeed vs ∫v f in general. Hence, let us already at thispoint introduce the relative velocity between the phases vr = v f - vs . Related to this, we shall subse-quently also consider Darcian velocity vd = n f vr .

Figure 2. Schematic of basic continuum mechanical transformations for a mixture material

As alluded to above, we focus our attention to the solid reference configuration, and to simplify thenotation we set 0 = 0

s . Additionally, we set X = Xs and v = vs . Hence, the mapping

6 Theory questions

x = j@XD characterizes the motion of the solid skeleton. In accordance with standard notation, weconsider the deformation gradient F and its Jacobian J associated with Xs defined as

(18)F = j ≈ “ with J = det@FD > 0Conservation of mass

One phase material

To warm up for the subsequent formulation of mass conservation of a two phase material, let usconsider the restricted situation of a one-phase material in which case the total mass of the solidmay be written with respect to current and reference 0 configurations as

(19)= ‡ r d v = ‡0

r J d V

where in the last equality we used the substitution d v = J d V .

The basic idea behind the formulation of mass conservation is that the mass of the particles isconserved during deformation, i.e.

(20)m0@XD = m@j@XDD ñ r0 d V = r J d Vwhere it was used that m = r d v .

Let us next apply this basic principle to our one phase solid. First, consider the conservation ofmass from the direct one phase material written as:

(21)

DÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

DÅÅÅÅÅÅÅÅÅÅD t

‡ r d v =‡

0

Jr° J d V + r J d Vêêêêêêêê° N = ·0

Hr° J + r J°êêêêêêêêêêêêêM

°=r J

êêêêê°L dV = lomno J°ÅÅÅÅÅÅJ = v ÿ —|o}~o = ‡ 0 Hr° + r v ÿ —L J d V := 0

or in localized format we obtain

(22)M°

= Hr° + r — ÿ v L J = 0Two phase material

Consider next the situation of two phases of volume fractions ns for the solid phase and n f for thefluid phase. The mass conservation of the two-phase material is now expressed with respect to theconservation of mass for the individual phases, so that the total mass is expressed as

(23)

=

s + f = ‡srs d vs + ‡

fr f d v f = ‡ ns rs d v + ‡ n f r f d v = ‡

0

M s d V + ‡0

M f d V

where M s and M f are the solid and fluid contents, respectively, defined as

(24)M s = J n f r f , M f = J ns rs

7

We thus express the mass conservation for the solid and fluid phases in turn as: The balance ofmass now follows for the solid phase as

(25)Ds

ÅÅÅÅÅÅÅÅÅÅD t

‡srs d vs =

DsÅÅÅÅÅÅÅÅÅÅD t

‡ ns rs d v = DsÅÅÅÅÅÅÅÅÅÅD t ‡0

J r̀s d V = ‡0

M° s

d V = 0

where we used the dot to indicate the rate with respect to the fixed solid reference configuration.This leads to the condition

(26)M° s

= J r̀° s

+ r̀s J°

= J Ir̀° s + r̀s “ ÿ vM = 0Hence, the stationary of the solid content is a representative of mass conservation of the solid phase.

The balance of mass now follows for the fluid phase as

(27)

D fÅÅÅÅÅÅÅÅÅÅD t

‡f

r f d v f =D fÅÅÅÅÅÅÅÅÅÅD t

‡ n f r f d v =PB D fÅÅÅÅÅÅÅÅÅÅD t

‡0fJ f r̀ f d V = ‡

0f

D f J f r̀ fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t d V =

PF ‡ ikjjjj D f r̀ fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + r̀ f “ ÿ v f y{zzzz d v = 0

where the “PF” and the “PB” denotes “Push-Forward” or “Pull-Back” operations. Finally, pull-backto the solid reference configuration yields:

(28)‡0

ikjjjj D f r̀ fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + r̀ f “ ÿ v f y{zzzz J d V = 0

which by localization yields the condition for mass balance of the fluid phase as

(29)Df r̀ f @x, tD

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t

+ r̀ f “ ÿ v f =∑ r̀ fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

∑ t+ I“ r̀ f M ÿ v f + r̀ f “ ÿ v f = 0

Mass balance of fluid phase in terms of relative velocity

Upon introducing the relative velocity vr = v f - v the material time derivative pertinent to the fluidphase may be rewritten as

(30)Df r̀ f

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t

=∑ r̀ fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

∑ t+ I— r̀ f M ÿ v + I— r̀ f M ÿ vr = r̀° f + I— r̀ f M ÿ vr

whereby the balance of mass for the fluid phase in (29) becomes

(31)r̀° f

+ r̀ f — ÿ v f + I— r̀ f M ÿ vr =r̀° f

+ r̀ f — ÿ v + r̀ f vr ÿ —+I— r̀ f M ÿ vr = r̀° f + r̀ f — ÿ v + — ÿ Ir̀ f vrM = 0It is also of interest to consider the fluid mass balance in terms of the fluid content defined as

(32)M f = J r̀ f fl M° f

= J Jr̀° f + r̀ f — ÿ vNwhereby the balance relation (31) is formulated as

8 Conservation of mass

(33)M° f

+ J — ÿ Ir̀ f vrM = 0We thus conclude that the mass balance of the fluid phase may be related to the solid material viathe introduction of the relative velocity. Clearly, the situation of synchronous motion of the phases,i.e. vr = 0 , corresponds to stationarity of both Ms and M f .

Mass balance in terms of internal mass supply

We extend the foregoing discussion by introducing the exchange of mass between the phases viathe internal mass production/exclusion terms Gs and G f , which define the rates of increase of massof the phases due to e.g. an erosion process or chemical reactions. These are in turn related to thelocal counterparts gs and g f defined as

(34)Gs = ‡ gs d v = ‡0

J gs d V , G f = ‡ g f d v = ‡0

J g f d V

Hence, we may extended the balance of mass between the individual solid and fluid phases as:

(35)Ds

ÅÅÅÅÅÅÅÅÅÅD t

s = Gs ñ ‡0

M° s

d V = ‡0

J gs d V

(36)D fÅÅÅÅÅÅÅÅÅÅD t

f = G f ñ ‡0

JM° f + J — ÿ Ir̀ f vrMN d V = ‡0

J g f d V

Let us in addition assume that the total mass total mass is conserved i.e. we have the condition formass production/exclusion

(37)Ds

ÅÅÅÅÅÅÅÅÅÅD t

s +D fÅÅÅÅÅÅÅÅÅÅD t

f = 0 fl Gs + G f = 0 fl gs + g f = 0

Clearly, the last expression represents the case of an erosion process where “the first phase takesmass from the second phase”. The mass balance relationship for the individual phases thus becomes

(38)M

° s= J gs = -J g f

M° f

+ J — ÿ Ir̀ f vrM = J g fwhere it may be noted that erosion normally means g f = -gs > 0 corresponding to the situationthat the fluid phase “gains” mass from solid phase.

Mass balance - final result

Let us summarize the discussion of mass balances in terms of the total mass conservation asexpressed in (37). Hence, in view of (37) we have the local condition for total mass balance as

(39)M° s

+ M° f

+ J — ÿ Ir̀ f vrM = 0where we note that the solid and fluid contents may be expanded in terms of the volume fractions as

(40)M s = J r̀s fl M° s

= J Ir̀° s + r̀s — ÿ vM = J rs ikjjn° s + ns — ÿ v + ns r° sÅÅÅÅÅÅÅÅrs y{zz

Mass balance of fluid phase in terms of relative velocity 9

(41)M f = J r̀f fl M

° f= J Jr̀° f + r̀ f — ÿ vN = J r f ikjjjjn° f + n f — ÿ v + n f r° fÅÅÅÅÅÅÅÅÅr f y{zzzz

In addition, let us next introduce the logarithmic compressibility strains evs for the solid phase

densification and evf for the fluid phase densification expressed in terms of intrinsic densities rs and

r f as

(42)evs = -Log

ÄÇÅÅÅÅÅÅÅ rsÅÅÅÅÅÅÅÅr0s ÉÖÑÑÑÑÑÑÑ fl e° vs = - r° sÅÅÅÅÅÅÅÅrs fl r° sÅÅÅÅÅÅÅÅrs = -e° vs(43)ev

f = -Log

ÄÇÅÅÅÅÅÅÅÅÅ r fÅÅÅÅÅÅÅÅÅr0f ÉÖÑÑÑÑÑÑÑÑÑ fl e° vf = - r° fÅÅÅÅÅÅÅÅÅr f fl r° fÅÅÅÅÅÅÅÅÅr f = -e° vfThe mass balance relations now becomes in the compressibility strains:

(44)M° s

= J rs Hn° s + ns — ÿ v - ns e° vsL = 0(45)M

° f= r f In° f + n f — ÿ v - n f e° vf M = -— ÿ Ir f vdM

where vd = n f vr was introduced as the Darcian velocity. Rewriting once again one obtains

(46)n° f + n f “ ÿ v - n f e° v

f = -1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdM(47)n° s + ns “ ÿ v - ns e° v

s = 0

Combination with due consideration to the saturation constraint, i.e. n f + ns = 1; n° f + n° s = 0, leadsto

(48)n° f + n° s + ns “ ÿ v + n f “ ÿ v - ns e° v

s - n f e° vf = “ ÿ v - ns e° v

s - n f e° vf = -

1ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdMwhere we introduced the Darcian velocity defined as vd = n f vr .

We emphasize that the issue of “compressibily” relates to the changes of the densities rs , r f follow-ing a solid particle j@X , tD . This means, in particular, that r f = r f @j@XD, tD at the assessment offluid compressibility. The reason is that the initial density r0

f relates to 0 (and not 0f ). In this

context, we conclude that evs@x, tD := 0and evf @x, tD := 0 corresponds the important situation of

incompressible solid material and incompressible fluid phase materials, respectively. However, wemay indeed have the situation that e.g. ev

s@x, tD ∫ 0 and evf @x, tD := 0, corresponding a compressiblesolid phase material.

Theory questions

1. Define and discuss the formulation of the kinematics of a two-phase mixture. Introduce the different types of material derivatives and formulate and discuss the velocity fields of the mixture.

2. Prove the formula: J°

= J “ ÿ v .

10 Conservation of mass

3. Formulate the idea of mass conservation pertinent to a one-phase mixture. Generalize this idea of mass conservation to the two-phase mixture. Discuss the main results in terms of the relative velocity between the phases.

4. Describe in words (and some formulas if necessary) the difference between the material deriva-tive r° = D r@x, tD ê D t and the partial derivative ∑ r@x, tD ê ∑ t , where r is the density of the material.

5. Formulate the mass balance in terms of internal mass supply. Discuss a typical erosion process.

6. Formulate the mass balance in terms of the compressibility strains. Set of the total form of mass balance in terms of the saturation constraint. In view of this relationship, introduce the issue of incompressibly/compressibility of the phases in terms of the compressibility strains.

Balance of momentum

Total format

The linear momentum balance of the solid occupying the region as in Fig. 3 may be written interms of the “change” in total momentum in , and the externally and internally applied forces

ext and int . The linear momentum balance relation is specified (as usual) as

(49)DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= ext + int

where the (total) forces of the mixture solid are defined as

(50)ext = ‡∑

tê d G = ‡ sêêê ÿ “d v , int = ‡ Ir̀s g + r̀ f gM d v

Carefully note that tê is the total traction vector acting along the external boundary ∑ and g is the

gravity. The traction vector is related to the total stress tensor sêêê = ss + s f via the outward normalvector n as

(51)tê

= sêêê ÿ n

In addition to linear momentum balance, angular momentum balance should be considered. How-ever, if we restrict to the ordinary non-polar continuum representation the main result from thisconsideration is that the total stress (and also the partial stresses) is symmetric, i.e.

(52)sêêê = sêêêt fl ss = HssLt, s f = Is f Mt

Theory questions 11

Figure 3. Solid in equilibrium with respect to reference and deformed configurations

In view of the fact that the momentum of our solid component, in the present context of a two phasematerial, consists of contributions from the individual phases let us consider the detailed formula-tion of the momentum change “ DÅÅÅÅÅÅÅÅÅÅD t “. To this end, we note in view of the mass balance relation (26) and (27) for the individual phases that the total change of momentum can be written as

(53)DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

:=Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t+

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t

where

(54)Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

DsÅÅÅÅÅÅÅÅÅÅD t

‡0

Ms v d V ,D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

D fÅÅÅÅÅÅÅÅÅÅÅD t

‡0f J f r̀ f v f d V

Note in view of the mass balance relation (26) and (27) for the solid phase

(55)Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= ‡

0

IM s v° + M° s vM d V = ‡0

HM s v° + J gs vL d Vand for the fluid phase we obtain the detailed push-forward - pull-back consideration

(56)

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= ‡

0f ikjjjjJ f r̀ f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + D f J f r̀ fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t v f y{zzzz d V = ‡ 0f ikjjjJ f r̀ f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + J f g f v f y{zzz d V =PF ‡ ikjjj r̀ f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + g f v f y{zzz d v =PB ‡ 0 ikjjjM f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + J g f v f y{zzz d V

where the last equality expresses the momentum change with respect to the solid phase materialreference configuration. In addition, let us also express the fluid acceleration D

f v fÅÅÅÅÅÅÅÅÅÅÅÅÅD t in terms of therelative velocity v f = vr + v , whereby the fluid acceleration can be related to the solid phase mate-rial via a convective term involving the relative vr . This is formulated via the parametrizationv f Aj f AX f E , tE as:

(57)D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

∑ v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

∑ t+ l f ÿ v f =

∑ v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

∑ t+ l f ÿ v + l f ÿ vr = v° f + l f ÿ vr

where l f = v f ≈ “ is the spatial velocity gradient with respect to the fluid motion.

12 Balance of momentum

Hence, the momentum change may be written as

(58)Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= ‡

0

HM s v° + J gs vL d V = ‡ Hr̀s v° + gs vL d v(59)

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= ‡

0

IM f Iv° f + Iv f ≈ “M ÿ vrM + J g f v f M d V = ‡ Ir̀ f Iv° f + l f ÿ vrM + g f v f M d vIn view of the relations (49-59), we may finally localize the momentum balance relation for the twophase mixture as

(60)DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= ext + int ñ sêêê ÿ “+ r̀ g = r̀s v° s + r̀ f Iv° f + l f ÿ vrM + g f vr " x œ

where r̀ = r̀s + r̀g and we made use of the fact that gs + g f = 0 and vr = v f - v .

Individual phases and transfer of momentum change between phases

As alluded to in the previous sub-section we establish the resulting forces and momentum changesin terms of contributions from the individual phases, i.e. we have that

(61)DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

=Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t+

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t, ext = ext

s + extf , int = int

s + intf

where

(62)exts = ‡ ss ÿ “d v , extf = ‡ s f ÿ “d v , ints = ‡ r̀s g d v , intf = ‡ r̀ f g d v

Hence, we are led to subdivide the momentum balance relation (49) as

(63)

Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t = ints + ext

s + s

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t = intf + ext

f + f

|oo}~oowhere s and f are interaction forces due to drag interaction between the phases. These aredefined in terms of the local interaction forces hs and h f

(64)s = ‡ hs d v , f = ‡ h f d vIn addition, it is assumed that the total effect of the interaction forces (or internal rate of momentumsupply) will result in no change of the total momentum, i.e. we have that

(65)s + f := 0 fl hs + h f = 0

Hence, we may localize the relation (63) along with (62) and (64) as

(66)ss ÿ “ + r̀s g + hs = r̀s v° + gs v

s f ÿ “ + r̀ f g + h f = r̀ f Iv° f + l f ÿ vrM + g f v fNote that the summation of these relation indeed corresponds to the total form of the momentumbalance specified in eq. (60).

Total format 13

Theory questions

1. Formulate the principle of momentum balance in total format. Formulate also the contribution of momentum from the different phases. Consider, in particular, the conservation of mass in the formulation.

2. Express the principle of momentum balance with respect to the individual phases. Focus on the solid phase reference configuration via the introduction of the relative velocity. In this, develop-ment express the localized equations of equilibrium for the different phases. Formulate your interpretation of the local interaction forces.

Conservation of energy

Total formulation

Let us first establish the principle of energy conservation (=first law of thermodynamics) written asthe balance relation applied to the mixture of solid and fluid phases as

(67)DÅÅÅÅÅÅÅÅÅÅÅÅD t

+DÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= W + Q

where = s + f is the total internal energy and = s + f is the total kinetic energy of themixture solid and the total material velocity with respect to the mixture material is defined asD èÅÅÅÅÅÅÅÅÅD t :=

Ds èsÅÅÅÅÅÅÅÅÅÅÅÅD t +D f è fÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t . Moreover, W is the mechanical work rate of the solid and Q is the heat sup-

ply to the solid. In the following we shall not disregard Q in the energy balance although we willconfine ourselfes to isothermal conditions subsequently. The different “elements” involved in theenergy balance are depicted in Fig. 4 below.

Figure 4. Elements involved in the formulation of the principle of energy conservation.

Formulation in contributions from individual phases

The individual contributions a and a to the total internal and kinetic energies , are definedas

14 Balance of momentum

(68)s = ‡ r̀s es d v = ‡

0

Ms es d V , f = ‡ r̀ f e f d v = ‡0fJ f r̀ f e f d V

(69)s = ‡ r̀s ks d v = ‡

0

Ms ks d V , f = ‡ r̀ f k f d v = ‡0fJ f r̀ f k f d V

where es is the internal energy density (per unit mass) pertinent to the solid phase and e f is theinternal energy density pertinent to the fluid phase. Moreover, we introduced ks = 1ÅÅÅÅ2 v ÿ v andk f = 1ÅÅÅÅ2 v

f ÿ v f . The material time derivatives of the involved contributions thus becomes

(70)

Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅD t =DsÅÅÅÅÅÅÅÅD t Ÿ 0 M s es d V = Ÿ 0 HM s e° s + J gs esL d V

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t =D fÅÅÅÅÅÅÅÅD t Ÿ 0f J f r̀ f e f d V =PF Ÿ Ir̀ f D f e fÅÅÅÅÅÅÅÅÅÅÅÅÅD t + g f e f M d v =PB Ÿ 0 IM f D f e fÅÅÅÅÅÅÅÅÅÅÅÅÅD t + J g f e f M d V

(71)

Ds sÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t =DsÅÅÅÅÅÅÅÅD t Ÿ 0 Ms ks d V = Ÿ 0 HMs v ÿ v° + J gs ksL d V

D f fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t =D fÅÅÅÅÅÅÅÅD t Ÿ 0f J f r̀ f k f d V =PF,PB Ÿ 0 IM f v f ÿ D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅD t + J g f e f M d V

where the mass balance relations (35) and (36) were used.

Let us next reformulate the material time derivatives relative to the solid reference configurationusing the relative velocity vr . To this end, it is first noted that

(72)Ds esÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= e° s ,

D f e fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

∑ e fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

∑ t+ I—e f M ÿ v + I—e f M ÿ vr = e° f + I —e f M ÿ vr

which leads to

(73)

DÅÅÅÅÅÅÅÅÅÅÅÅD t

+DÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= ‡ Ir̀s e° s + r̀ f e° f + r̀ f I —e f M ÿ vr + gs es + g f e f M d v +‡ ikjjjr̀s v ÿ v° + r̀ f v f ÿ D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + gs ks + g f k f y{zzz d v = ‡ Iè° + r̀ f I— e f M ÿ vrM d v +‡ ikjjjr̀s v ÿ v° + r̀ f v f ÿ D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz d v + ‡ g f I e f + k f - H es + ksLM d vIn the last equality we introduced the (solid) material change of internal energy of the mixturedefined as

(74)è°

= r̀s e° s + r̀ f e° f

The mechanical work rate and heat supply to the mixture solid

In view of Fig. 4 we may “work out” the mechanical work rate produced by the gravity forces in and the forces acting on the external boundary G . This is formulated as

(75)

W = ‡ Ir̀s g ÿ v + r̀ f g ÿ v f M d v + ‡G

Iv ÿ Hss ÿ nL + v f ÿ Is f ÿ nMM d G =DIV‡ I H— ÿ ss + r̀s gL ÿ v + Ir̀ f g + — ÿ s f M ÿ v f + ss : l + s f : l f M d v

Formulation in contributions from individual phases 15

where the last expression was obtained using the divergence theorem “DIV” (and some additonalderivations). Combination of this last relationship with the equilibrium relations of the phases ineqs. (65) and (66) yields the work rates of our mixture continuum as

(76)

W = ‡ ikjjjr̀s ÿ v° ÿ v + r̀ f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t ÿ v f y{zzz d v +‡ I ss : l + s f : l f - h f ÿ vrM d v + ‡ Igs v ÿ v + g f v f ÿ v f M d vLet us next consider the heat supply Q , cf. Fig. 4 , to the solid formulated as

(77)Q = -‡G

n ÿ h d G =DIV

-‡ — ÿ h d v = -‡ q d vwhere the last equality was obtained by using the divergence theorem. We note that h is the thermalflux vector that transports heat energy from the mixture solid, and q = — ÿ h is the divergence ofthermal flow. In particular, we have that q > 0 corresponding to production of energy at the mate-rial point

Energy equation in localized format

Hence the balance of energy stated in (67) along with (73), (76) and (77) can now be formulated as

(78)‡ è° d v + ‡ r̀ f I e f —M ÿ vr d v + ‡ g f I e f + k f - H es + ksLM d v =‡ I ss : l + s f : l f - h f ÿ vrM d v + ‡ g f Iv f ÿ v f - v ÿ vM d v - ‡ q d vIn view of (78), let us immediately specify the localized format of the energy equation (while notingthat ks = 1ÅÅÅÅ2 v ÿ v and k

s = 1ÅÅÅÅ2 vf ÿ v f ) as

(79)è°

+ r̀ f I—e f M ÿ vr + g f I e f - esM = ss : l + s f : l f - h f ÿ vr + g f Ik f - ksM - qOnce again, let us represent the fluid phase term s f : l f in (79) to the motion of the solid phase.Hence, we rewrite the term s f : l f as

(80)

s f : l f =

s f : Hl + lrL = Is f : lr = — ÿ I vr ÿ s f M - vr ÿ — ÿ s f M = s f : l + — ÿ I vr ÿ s f M - vr ÿ — ÿ s f =s f : l + — ÿ I vr ÿ s f M + vr ÿ ikjjjh f + r̀ f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz - g f v f y{zzz

where the the equilibrium relation (66) was used once again. Hence, the relation (79) is re-estab-lished in terms of the total stress sêêê of the mixture as

(81)

è°

+ q + g f I e f - esM =Iss + s f M : l + — ÿ I vr ÿ s f M + r̀ f vr ÿ ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz - r̀ f vr ÿ I— e f M - g f 1ÅÅÅÅÅ2 vr ÿ vr =sêêê : l + — ÿ I vr ÿ s f M + r̀ f vr ÿ ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t - —e f y{zzz - g f 1ÅÅÅÅÅ2 vr ÿ vr

16 Conservation of energy

Assumption about ideal viscous fluid and the effective stress of Terzaghi

Using the assumption about the ideal viscous fluid, the fluid stress may be represented as

(82)s f = n f s f - n f p 1

where s f is the intrinsic (normally viscous) portion of the fluid stress and p is the intrinsic (non-viscous) fluid pressure. We note already at this point that s f is completely deviatoric, and the non-viscous fluid pressure is equal to the total fluid pressure. Nevertheless, we formulate the term— ÿ I vr ÿ s f M of (81) as

(83)— ÿ I vr ÿ s f M = — ÿ I vd ÿ s f M - — ÿ ikjj r̀ f pÅÅÅÅÅÅÅÅÅr f vry{zz = — ÿ I vd ÿ s f M - pÅÅÅÅÅÅÅÅÅr f — ÿ Ir f vdM - r f vd — ÿ ikjj pÅÅÅÅÅÅÅÅÅr f y{zzwhere the introduction of the Darcian velocity vd = n f vr is noteworthy. As a result the energybalance in (81) becomes

(84)

è°

+ g f I e f - esM =sêêê : l + — ÿ I vd ÿ s f M - pÅÅÅÅÅÅÅÅÅ

r f— ÿ Ir f vdM + r f vd ÿ ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅDt - — e f - — ÿ ikjj pÅÅÅÅÅÅÅÅÅr f y{zzy{zzz - g f 1ÅÅÅÅÅ2 vr ÿ vr

Upon combining the term - pÅÅÅÅÅÅÅr f — ÿ Ir f vdM in (84) with the balance of mass of the mixture materialin eq. (48) we obtain

(85)-p

ÅÅÅÅÅÅÅÅÅr f

— ÿ Ir f vdM = p — ÿ v - ns p e° vs - n f p e° vfleading to

(86)

è°

+ q + g f I e f - esM = s :l + — ÿ I vd ÿ s f M - ns p e° vs - n f p e° vf + r f vd ÿ ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅDt - — e f - — ÿ ikjj pÅÅÅÅÅÅÅÅÅr f y{zzy{zzz - g f 1ÅÅÅÅÅ2 vr ÿ vr

with the effective stress s of Terzaghi defined as

(87)s = sêêê + p 1

In order to consider the influence of the viscous contribution to the fluid stress we develop the term— ÿ I vr ÿ s f M in its indices as

(88)“k J IvdM j Is f MjkN = IldMkj Is f MkjN + J IvdM j “k Is f MjkN = s f : ld + vd ÿ I— ÿ s f Mwhich leads to

(89)

è°

+ q + g f I e f - esM = Is - n f s f M : l + n f s f : l f - ns p e° vs - n f p e° vf +r f vd ÿ

ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅDt - — e f - — ÿ ikjj pÅÅÅÅÅÅÅÅÅr f y{zz + 1ÅÅÅÅÅÅÅÅÅr f — ÿ s f y{zzz - g f 1ÅÅÅÅÅ2 vr ÿ vr

Assumption about ideal viscous fluid and the effective stress of Terzaghi 17

We thus conclude that the effective stress “felt” by the continuum may be represented by s - n f s f .It should be emphasized, however, the it is normally assumed that the viscous portion s f of thefluid stress may be neglected, i.e. s f º 0 .

Theory questions

1. Establish the principle of energy conservation of the mixture material. Discuss the involved “elements” and formulate the internal energy and kinetic energy with respect to the solid refer-ence configuration using the relative velocity.

2. “Work out” the mechanical work rate of the mixture solid.

3. Derive the energy equation using the assumption about an ideal viscous fluid. In this context, define effective stress of Terzaghi. Discuss the interpretation of the Terzaghi stress.

Entropy inequality

Formulation of entropy inequality

Let us next consider the second law of thermodynamics formulated in terms of the total entropy written as

(90)= ‡ Ir̀s ss + r̀ f s f M d vwhere ss and s f are the local entropies per unit mass of the solid and fluid phases, respectively. Asto the temperature, it is assumed that it is constant, i.e. it is stationary, and in common for bothphases, i.e. we have that q = qs = q f .

The second law of thermodynamics (which is sometimes also named the "Clausius Duhems Inequal-ity" (CDI), or simply the “entropy inequality”) is now stated as

(91)DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

- Qq r 0

where " DÅÅÅÅÅÅÅÅÅÅD t " is the total time derivative of our two-phase porous material as we have discussedextensively (so far) during the course. Moreover, Qq is the net heat thermal supply/per temperatureunit defined as

(92)Qq =

-‡G

n ÿ hÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

qd G =

DIV-‡ — ÿ K hÅÅÅÅÅÅq O d v = -‡ 1ÅÅÅÅÅq H— ÿ h - h ÿ — qL d v = -‡ 1ÅÅÅÅÅq Hq - h ÿ — qL d v

where q > 0 is the absolute temperature of the medium

With due consideration to the balance of mass stated in eqs. (65) and (66) (involving the masstransfer factor g f ), we develop the total material derivative of the total entropy stated in eq. (90).Thereby, the total material change of the entropy is obtained as

(93)DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= ‡ ikjjjr̀s Ds ssÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + gs ss + r̀ f D f s fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + g f s f y{zzz d v

18 Conservation of energy

In particular, the material time derivative of the fluid is represented in terms of that of the solidphase (in the usual way), i.e.:

(94)Ds ssÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= s° s

(95)D f s fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

∑ s fÅÅÅÅÅÅÅÅÅÅÅÅÅ

∑ t+ I—s f M ÿ v + I—s f M ÿ vr = s° f + I— s f M ÿ vr

which yields the material change of the total entropy in (93) as

(96)

DÅÅÅÅÅÅÅÅÅÅÅÅÅD t

=‡ Ir̀s s° s + r̀ f s° f + r̀ f I—s f M ÿ vr + g f I s f - ssMM d v = ‡ Is̀° + r̀ f I —s f M ÿ vr + g f I s f - ssMM d vwhere s̀

° is the saturated entropy change defined as

(97)s̀°

= r̀s s° s + r̀ f s° f

Legendre transformation between internal energy, free energy, entropy and temperature

We relate generically the internal energy, the free energy and the entropy to each other via theLegendre transformation

(98)e@A, sD = y@A, qD + s qwhere y is the Helmholtz free energy (or simply the free energy) representing the stored reversibleenergy as a function of the internal variables A . We emphasize that the set 8s, q, A< represents theindependent variables in eq. (98). As a result the linearized Legendre transformation reads

(99)∑e

ÅÅÅÅÅÅÅÅÅÅÅ∑ A

A°

+∑eÅÅÅÅÅÅÅÅÅ∑s

s° =∑ yÅÅÅÅÅÅÅÅÅÅÅ∑ A

A°

+∑ yÅÅÅÅÅÅÅÅÅÅ∑ q

q°

+ s q°

+ s° q ñ K ∑eÅÅÅÅÅÅÅÅÅÅÅ∑ A

-∑yÅÅÅÅÅÅÅÅÅÅÅ∑ A

O A° + K ∑eÅÅÅÅÅÅÅÅÅ∑s

- qO s° = K ∑ yÅÅÅÅÅÅÅÅÅÅ∑ q

+ sO q°corresponding to the conditions

∑eÅÅÅÅÅÅÅÅÅÅÅ∑ A

=∑ yÅÅÅÅÅÅÅÅÅÅÅ∑ A

, q =∑eÅÅÅÅÅÅÅÅÅ∑s

, s = -∑ yÅÅÅÅÅÅÅÅÅÅ∑ q

whereby a clear interpretation of the entropy is obtained, i.e. the “entropy” is the sensitivity of thefree energy with respect to the temperature.

In the following, let us apply the same transformation with respect to the individual phases. Hence,we introduce the relationships for the solid phase

(100)q ss = es - ys fl q s° s + ss q°

= e° s -∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ A

A°

-∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ q

q°

= e° s -∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ A

A°

+ ss q°

fl q s° s = e° s -∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ A

A°

where ys is the free energy of the solid phase. However, in the following we shall consider theisothermal condition leading to

(101)q ssêêêêê°

= q s° s = e° s - ys°

19

Likewise, for the fluid phase we introduce the Legendre transformation

(102)q s f = e f - y f fl q D f s fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t=

D f e fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t-

D f y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= e° f - y

° f+ I— s f M ÿ vr

where y f is the free energy of the fluid phase.

In addition, let us also consider the saturated entropy s̀° in (97). In view of (100) and (102) one

obtains

(103)q s̀°

= q r̀s s° s + q r̀ f s° f = r̀s e° s - r̀s y° s

+ r̀ f e° f - r̀ f y° f

= è°

- y°̀

The entropy inequality - Localization

In view of the relations (91),(92) and (96), the entropy inequality now reads

(104)DÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t

= ‡ Is̀° + r̀ f I —I s f MM ÿ vr + g f I s f - ssMM d v + ‡ 1ÅÅÅÅÅq Hq - h ÿ —qL d v ¥ 0It appears that the localized form of the entropy inequality may be written as

(105)Dmech + Dther ¥ 0

where we require that the thermal and mechanical portions of the inequality are assumed to thesatisfied independently as

(106)Dther = -h ÿ —q ¥ 0 fl h = -Kther ÿ — q fl Dther ¥ 0

(107)Dmech = q s̀°

+ r̀ f I — Iq s f M - s f — qM ÿ vr + g f I qs f - qssM + q ¥ 0In (107), it was used that q —s f = — Iq s f M - s f — q , and in (106) the thermal part of the entropyinequality is immediately satisfied via the introduction of Fourier’s law of heat condution, whereKther is the second order postive definite thermal conductivity tensor.

As to the mechanical portion, we develop Dmech in (107) using the the Legendre transformations inthe sequel (100-103). As a result we now obtain

(108)Dmech = è°

- y°̀

+ r̀ f I —Ie f - y f M - s f — qM ÿ vr + g f I e f - y f - Hes - ysLM + q ¥ 0Combination with the energy equation (89) yields:

(109)

Is - n f s f M : l - ns p e° vs - r̀s y° s +n f s f : l f +

-n f p e° vf - r̀ f y

° f+

r f vd ÿikjjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + 1ÅÅÅÅÅÅÅÅÅr f — ÿ s f - — y f - — ÿ ikjj pÅÅÅÅÅÅÅÅÅr f y{zz - s f — q - 1ÅÅÅÅÅ2 g fÅÅÅÅÅÅÅÅÅr̀ f vry{zzzz +

g f I-y f + ysM ¥ 0whereby the final result in fact may be interpreted in terms of a number of independent phenomeno-logical mechanisms of the mixture material. In view the appearance of the different terms in (109),this is formulated as

20

(110)D = Ds + Dvf + Di + De ¥ 0

where Ds ¥ 0 is the dissipation produced by the (homogenized) solid phase material considered asan independent process of the mixture. The Dvf ¥ 0 represents the dissipation developed by theviscous portion of the fluid stress s f . The Dnvf represents dissipation in the non-viscous stressresponse of the fluid. It is assumed that this dissipation can be neglected, i.e. Dnvf := 0. The Di ¥ 0represents dissipation induced by “drag"-interaction between the phases. Finally, the term De > 0represents dissipation motivated by mass transfer between the phases.

Motivated by the entropy inequality (109) the different components of the total dissipation aredefined as

(111)Ds = s : l - ns p e° vs - ns rs y

° s¥ 0

(112)Dvf = n f s f : l f ¥ 0

(113)Dnvf = -n f Jp e° vf + r f y° f N := 0(114)

Di = -hef ÿ vd ¥ 0 with he

f =

- r f ikjjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t + 1ÅÅÅÅÅÅÅÅÅr f — ÿ s f - —ikjj pÅÅÅÅÅÅÅÅÅr f y{zz - — y f - s f — q - 1ÅÅÅÅÅ2 g fÅÅÅÅÅÅÅÅÅr̀ f vry{zzzz

(115)De = g f Iys - y f M ¥ 0 fl g f ¥ERO 0 , ys - y f ¥ 0In (114), he

f is the effective drag force. Moreover, we introduced Ds ¥ 0 as the dissipation pro-duced by (homogenized) solid phase material, Dvf ¥ 0 is the dissipation developed by viscousportion of the fluid stress s f , Di ¥ 0 represents dissipation induced by “drag"-interaction betweenphases, and, finally, De > 0 is the dissipation motivated by mass transfer between the phases. Inaddition, from the assumption that the non-viscous portion of the fluid stress response is non-dissipative it immediately follows that the argument of y f is the compressibility strain ev

f definedby y f = y f Aevf E pertinent to a compressible fluid. It then follows from (113) that the fluid pressurecan be evaluated directly from the compressibility ev

f via the state y f Aevf E as(116)p e° v

f + r f ∑ y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ ev

fe° v

f = 0 fl p = - r f ∑ y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ ev

f

Alternatively, we may consider the constitutive relation in “compliance” form, in which case thefluid pressure p is the primary variable that produces the compressibility ev

f .

A note on the effective drag force

It is of significant interest to develop the effective drag force hef in terms of h f = -hs , which is the

actual one appearing in the momentum balance of the fluid phase in (66b). To this end, we first notethat the constitutive relation p = - r f ∑ y f ë ∑ evf yields the result

The entropy inequality - Localization 21

(117)

-—ikjj pÅÅÅÅÅÅÅÅÅr f y{zz - —y f = - 1ÅÅÅÅÅÅÅÅÅr f — p + ikjjjjp 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅHr f L2 - ∑y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ evf ∑evfÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ r f y{zzzz — r f =loomnooevf = -LogÄÇÅÅÅÅÅÅÅÅÅ r fÅÅÅÅÅÅÅÅÅr0f ÉÖÑÑÑÑÑÑÑÑÑ|oo}~oo = - 1ÅÅÅÅÅÅÅÅÅr f — p + ikjjjjp 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅHr f L2 + ∑ y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑evf 1ÅÅÅÅÅÅÅÅÅr f y{zzzz — r f = - 1ÅÅÅÅÅÅÅÅÅr f — pHence, one obtains the effective drag force he

f in (114) stated in the form

(118)-n f hef = n f — ÿ Is f - p 1M + r̀ f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz - n f s f —q - 1ÅÅÅÅÅ2 g fÅÅÅÅÅÅÅÅÅr f vr

Let us next assume, firstly, isothermal conditions corresponding to y f Aevf E fl s f = - ∑y fÅÅÅÅÅÅÅÅÅÅ∑q = 0, and,secondly, a non-erosive process g f := 0 which gives

(119)-n f hef = n f — ÿ Is f - p 1M + r̀ f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz

To arrive at the relation between hef and h f , let us reconsider the momentum balance relation (66b)

of the fluid phase written as

(120)

-h f = s f ÿ “ + r̀ f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz =

n f Is f - p 1M ÿ —+ r̀ f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzz + Is f - p 1M ÿ —n f - g f v f = -hef + Is f - p 1M ÿ —n fHence, the effective drag force is related to the actual one via the relation

(121)h f = n f hef - Is f - p 1M ÿ —n f

The momentum balance (66b) can thus be restated as

(122)r fD f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= — ÿ s f - — p + he

f + r f g

which is the momentum balance of the fluid phase represented in intrinsic quantities. We note thatin this relationship the effective drag force he

f plays the role of the “actual” drag force.

Theory questions

1. Establish the total entropy and the second law of thermodynamics in the context of a two-phase mixture material. On the basis of the involved material derivatives and principle of mass conser-vation, express the entropy inequality in terms of the solid reference configuration.

2. Define and discuss the suitable Legendre transformation in internal energy, Helmholtz free energy, temperature and entropy. Represent and localize the entropy inequality of the mixture material in terms of these quantities.

3. On the basis of the localized entropy for the mixture material, discuss the different components of the dissipation inequality and relate them to dissipative mechanics of the mixture material.

22 Entropy inequality

Constitutive relations

Guided by the dissipation inequality we outline, in this section, the constitutive relations of our two-phase continuum. We then consider in turn the different mechanisms outlined in the sequel (111)-(115). In the following, we shall assume that the mass production term g f = -gs := 0 pertinent to anon-erosive process.

Effective stress response

To arrive at the proper formulation of the effective stress response, let us formulate the dissipationproduced by the solid phase for the entire component as

(123)s = ‡ Ds d v = ‡0

J Ds d V ¥ 0

where the last expression was obtained after pull-back to the solid reference configuration.

It may be noted that the integrand J Ds in (111) may be rewritten in terms of the Kirchhoff stresst = J s (and the second Piola Kirchhoff stress S ) as

(124)J Ds = t : l - ns p e° vs - J ns rs y

° s=

1ÅÅÅÅÅ2

S : C°

- J ns p e° vs - r̀0

s y° s

where, in particular, the last equality was obtained from mass conservation due to the stationarity ofMs = M0

s = r̀0s .

Let us for simplicity assume hyper-elasticity “HE”, whereby we obtain the dependence ys@C, evsD inthe free energy for the solid phase. We then have J Ds := 0 leading to

(125)y° s@C, evsD = ∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C : C° + ∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ evs e° vs fl J Ds = 1ÅÅÅÅÅ2 KS - 2 r̀0s ∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C O : C° - J ns ikjjp + rs ∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ evs y{zz e° vs =HE 0

We thus obtain the constitutive state equations at hyper-elasticity:

(126)S = 2 r̀0s

∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C

(127)p = - rs ∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑ evs

where S = Sêê

- J C-1 p is the consequent effective second Piola Kirchhoff stress due to the Terzahgieffective stress principle, and p is the intrinsic fluid pressure. Like in the formulation of the fluidcompressibility in (116), we may alternatively consider the constitutive relation (127) in“compliance” form, whereby (again) the fluid pressure p drives the solid compressibility ev

s .

Solid-fluid interaction

Related to the constitutive relation for the densification of the gas, we consider the model for thedrag interaction motivated by the dissipation Di in (114) written as

(128)Di = -hef ÿ vd ¥ 0

23

where the effective drag force hef (or hydraulic gradient with negative sign) is chosen to ensure

positive dissipation, cf. the representation of Fourier’s law of heat condution in (106), according tothe linear Darcy law

(129)vd = -K ÿ hef

where K is the positive definite permeability tensor. To simplify the analysis let us restrict to isotro-pic permeability conditions using the scalar valued permeability parameter k , whereby K = k 1 andk > 0.

Experimental observations shows that the permeability k may be expressed in the effective gas-fluid viscosity m f and the intrinsic permeability Ks of the solid material. This is written as

(130)-hef =

m fÅÅÅÅÅÅÅÅÅÅKS

vd fl k =KsÅÅÅÅÅÅÅÅÅm f

Neglecting viscous part of the fluid stress s f in (122), we obtain

(131)vd = -kikjjj— p - r f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzzy{zzz

To conclude the developments in this subsection, we remark that the special situation of“impermeable material” (or closed cells in a cellular material), is named the “undrained case”. Inthis case we have vd := 0and Di := 0, corresponding to Ks Ø 0.

Viscous fluid stress response

It appears that the deviatoric fluid stress response s f generates the dissipative portionDvf = n f s f : l f ¥ 0. To ensure positive dissipation, the interpretation of s f is then that the corre-sponding stress response is purely viscous and thereby completely dissipative. This is formulated as

(132)s f = 2 m f Idev : l f fl Dvf = n f s f : l f = 2 m f n f l f : Idev : l f ¥ 0

where m f is the intrinsic fluid viscosity parameter and Idev is the forth order deviatoric projectionoperator.

Solid densification - incompressible solid phase

The intrinsic density of the solid phase is normally considered as a constant during the deformationprocess of the material. This corresponds to the usual assumption of incompressible solid phasematerial. Hence, we have ev

s := 0 and

(133)evs = -Log

ÄÇÅÅÅÅÅÅÅ rsÅÅÅÅÅÅÅÅr0s ÉÖÑÑÑÑÑÑÑ := 0 fl rs = r0sIn addition, for hyper-elasticity we obtain the dependence ys@C, evsD Ø ys@CD in the free energy forthe solid phase as was proposed in the specification of eq. (125).

Fluid densification - incompressible liquid fluid phase

The intrinsic density of the fluid phase considered as a liquid is also normally considered as aconstant during the deformation process of the mixture material. This corresponds to the usualassumption of incompressible fluid phase material. Hence, we have ev

f := 0 and

24 Constitutive relations

(134)evf = -Log

ÄÇÅÅÅÅÅÅÅÅÅ r fÅÅÅÅÅÅÅÅÅr0f ÉÖÑÑÑÑÑÑÑÑÑ := 0 fl r f = r0fIn addition, we obtain the dependence y f Aevf E Ø 0 in the free energy for the fluid phase.Fluid phase considered as gas phase

In some applications it is of interest to model the fluid phase considered as a compressible gasconcerning the non-viscous pressure response, e.g. gas-filled foam materials subjected to large rapiddeformations. To this end,the gas-pressure response is modelled using the ideal gas law (or theBoyle–Mariotte’s law) written as

(135)r f =mgÅÅÅÅÅÅÅÅÅÅR q

p

where R is the universal gas constant, q is the constant (absolute) temperature and mg is the molecu-lar mass of the gas. From the basic definition of the compaction of the gas in (43) we obtain

(136)r f = r0f e-ev

ffl p = p0 e-ev

fwith p0 =

R qÅÅÅÅÅÅÅÅÅÅÅÅmg

r0f

where p0 is the initial gas-pressure (normally considered as the atmospheric pressure). We remarkthat the linearized response of the gas phase may be represented as

(137)p° = -K f e° vf with K f = p0 e-ev

f

where K f is the compression modulus of the gas. Indeed, K f increases when the gas is densified,e.g. in the extreme case we obtain K f Ø ¶ when ev

f Ø -¶ .

Hence, in view of (116) and (136) we find that the stored mechanical energy in the gas phase maybe formulated in terms of the compaction of the gas-phase, i.e. y f = y f Aevf E , so that

(138)∑ y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ evf

= -p

ÅÅÅÅÅÅÅÅÅr f

= -R qÅÅÅÅÅÅÅÅÅÅmg

A remark on the intrinsic fluid flow

We note that in view of the constitutive relations (132), (130) and (135) that

(139)s f = 2 m f Idev : l f , hef = -

m fÅÅÅÅÅÅÅÅÅÅKS

vd , p = - r f ∑y fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑ evf

= ;r f = mgÅÅÅÅÅÅÅÅÅÅR q

p? = R qÅÅÅÅÅÅÅÅÅÅÅÅmg

r f

Upon inserting these relations into (122) we obtain corresponding to the compressible Navier-Stokes equation representing the fluid flow in terms of the intrinsic properties. This is formulated as

(140)r f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= 2 m f “ ÿ IIdev : l f M - R qÅÅÅÅÅÅÅÅÅÅÅÅ

mg “ r f -

m fÅÅÅÅÅÅÅÅÅÅKS

vd + r f g

Erosion modelling

Fluid densification - incompressible liquid fluid phase 25

Theory questions

1. Derive the state equation for the effective stress in the context of an hyper-elastic material. Formulate also the transformation between our different stress measures. Carry out the deriva-tion with the restriction to a non-erosive mixture material. What would happen if the material would have been erosive?

2. Motivated by the dissipation Di in eq. (114), formulate Darcy’s law as a constitutive representa-tion of the drag interaction between the phases. Discuss under which circumstances we can write: vd = -k I— p - r f Ig - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅD t MM where k is the (isotropic) permeability constant.

3. Formulate the representation of the fluid phase as a gas phase. In this development, derive the fluid compressibility!

Balance relations for different types of porous media

In the following subsections we consider a couple of specific porous media specified as specializa-tions from the developed framework of balance relations of porous media. In addition, the govern-ing coupled solutions for the different porous media are formulated in weak form.

Classical incompressible solid-liquid porous medium

Let us consider as a first prototype the classical incompressible solid-liquid porous medium. To thisend, we assume firstly (which was in fact already assumed in the development of the constitutiverelations) that the medium is non-erosive corresponding to g f = -gs := 0. Secondly, as alluded toin the title, we assume that the phases are incompressible corresponding to the compressibilitystrains

(141)evs := 0 , ev

f := 0

for the solid and the fluid phases, respectively, cf. eqs, (42-43). Hence, the intrinsic densities of thephases are stationary, i.e. we have r° s = 0 and r° f = 0 in this case. Thirdly it is assumed that thedissipative fluid stress is small enough to be neglected, i.e. we set s f := 0.

Balance of mass

In view of the aforementioned restrictions, we conclude in view of the balance of mass in (48) ofthe mixture material that we have for incompressible phases the relationship:

(142)“ ÿ v = -1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vd MIn addition we, of course have the mass balance conditions in (38) for the individual phases asusual:

(143)M° s

= 0, M° f

+ J — ÿ Ir f vdM = 0which may be elaborated on for the present case of incompressible phases as

(144)M

° s= 0 fl ns rs J =

n0s r0

s fl ns = 8rs = r0s < = n0sÅÅÅÅÅÅÅÅJ fl n° s J + ns J° = 0 fl n° f = ns J°ÅÅÅÅÅÅJ = I1 - n f M “ ÿ v

26 Constitutive relations

Hence, we may develop the balance of mass for the fluid phase in (143b) as

(145)M

° f+ J — ÿ Ir f vdM = J r̀° f + J° r̀ f + J — ÿ Ir f vd M = J n° f r f + J “ ÿ v r̀ f + J — ÿ Ir f vdM =

J I1 - n f M “ ÿ v r f + J n f “ ÿ v r f + J — ÿ Ir f vd M = J “ ÿ v r f + J — ÿ Ir f vdM = 0In view of sequence of developments in (145), we thus conclude that the balance of mass in (142) isindeed embedded in balance of mass for the fluid phase. In addition, from the developments in (144), we conclude that the conditions for the volume fractions based on the assumption of incompress-ible solid phase can be formulated as:

(146)ns =n0

s

ÅÅÅÅÅÅÅÅJ

, n f =J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

Jfl n° f = ns

J°

ÅÅÅÅÅÅJ

= I1 - n f M “ ÿ vBalance of momentum

As to the balance of momentum, we simply specialize the momentum balance of the mixture mate-rial in (60) from the assumption of a non-erosive material to the equilibrium relation

(147)sêêê ÿ “+ r̀ g = r̀s v° s + r̀ f Iv° f + l f ÿ vrM " x œwhere it may be noted the sêêê is the total stress which in turn is related to the effective (constitutive)stress s and the fluid pressure p via the Terszaghi effective stress principle in (87), i.e.

(148)sêêê = s - p 1

Entropy inequality of mixture material

Based on the fact that our mixture material can be characterized with respect to the isothermalcondition, incompressible phases and non-erosive material, we make the following restrictions fromthe general entropy inequality established in the sequel (110-115) as:

(149)ys@A, qD Ø ys@AD, y f Aevf , qE Ø 0, s f Ø 0, g f Ø 0We emphasize that this restriction could not have been done prior to the establishment of theentropy inequality. We rather make the restriction when the expression of the entropy inequalityhave been fully developed.

The specialized entropy inequality is now reduced to two independent mechanisms, namely, thedissipation produced by the incompressible solid phase considered as a porous skeleton and theinteraction between the phases. This is written as

(150)Ds ¥ 0, Di ¥ 0

where

(151)Ds = s : l - ns rs y° s

¥ 0 , Di = -hef ÿ vd ¥ 0

As to the effective drag force, we restrict from the final expression in (122) with s f Ø 0 to therelation

(152)hef = — p - r f g + r f

D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t

Classical incompressible solid-liquid porous medium 27

As to the constitutive relations in the present context, we are now guided by the relation (150) topropose for hyperelastic response and isotropic linear Darcy interaction between the phase theconstitutive response:

(153)J Ds =1ÅÅÅÅÅ2

S : C°

- r̀0s

∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C

: C°

:= 0 fl S = 2 r̀0s

∑ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C

(154)Di = -hef ÿ vd ¥ 0 fl vd = -k he

f fl vd = -kikjjj— p - r f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzzy{zzz

where k is the isotropic solid-liquid permeability parameter, cf. eq. (130).

Weak form of governing equations

In order to resolve e.g. the effective stress S and the Darcy flow vd pertinent to a structure problem,we consider the constitutive relations, the balance of mass, the balance of momentum formulated asa coupled set of relations. In the present case, these coupled relations may be established withrespect to the current configuration , shown in Figure 5, and we formulate the momentum balanceof the mixture from (147), the balance of mass for the mixture in (142), and the intrinsic fluid Darcyflow in (154) as

(155)

Hs - p 1L ÿ “ + r̀ g = Ir̀s + r̀ f M v° + r̀ f v° r + r̀ f Hl + lrL ÿ vr“ ÿ v +

1ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir̀ f vrM = 0he

f = — p - r f g + r f D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

D t= — p - r f g + r f Iv° + v° rM + r f Il + lrM ÿ vr =Darcy - n fÅÅÅÅÅÅÅÅÅ

kvr

Figure 5. Current domain along with the mechancal solid-fluid actions

We identify the primary variables for the coupled problem as the placement x = j@XD fl v = j° forthe solid particles, the fluid pressure p , and the relative velocity vr , i.e. the primary solution isidentified as = 8v, p, vr< .

28 Balance relations for different types of porous media

The idea is then to establish the solution œ V in weak form where V is the function space associ-ated with = 8v, p, vr< so thatV @w, h, wrD = 8w œ P, h œ S, wr œ H<where in turn P is the function space containing the virtual displacement field w@xD , S is the func-tion space containing the virtual fluid pressure field h @xD , and H is the function space containingthe virtual relative (or seepage) velocity field wr@xD .The coupled set of equations in (155) are recast into weak form following the standard steps: 1)Multiply eqs. (155) with the proper virtual field and 2) integrate over the current domain as

(156)

‡ sêêê ÿ l@wD d v + ‡ Ir̀s + r̀ f M w ÿ v° d v + ‡ r̀ f w ÿ v° r d v + ‡ r̀ f w ÿ Hl + lrL ÿ vr d v =‡G

w ÿ sêêê ÿ n d G + ‡ w ÿ r̀ g d v‡ h r f “ ÿ v d v + ‡ h “ ÿ Ir f vdM d v =‡ h “ ÿ v d v - ‡ H“ hL ÿ vd d v + ‡G

h r f vd ÿ n d G = 0‡ wr ÿ — p d v - ‡ r f wr ÿ g d v + ‡ r f wr ÿ Hv° + v° rL d v +‡ r f wr ÿ Hl + lrL ÿ vr d v + ‡ n fÅÅÅÅÅÅÅÅÅk wr ÿ vr d v = 0where the last equalities were obtained using the divergence theorem. Introduce the notation:

tê

= sêêê ÿ n , Q = r f vd ÿ n

which in view of (148) and (146) is used to reestablish eq. (156) as

(157)‡ Hs - p 1L ÿ l@wD d v + ‡ Ir̀s + r̀ f M w ÿ v° d v + ‡ r̀ f w ÿ v° r d v + ‡ r̀ f w ÿ Hl + lrL ÿ vr d v =‡

G w ÿ tê d G + ‡ r̀ w ÿ g d v " w œ P

(158)‡ h r f “ ÿ v d v - ‡ J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅJ H“ hL ÿ vr d v = -‡Gh Q d G " h œ S(159)

‡ wr ÿ — p d v + ‡ r f wr ÿ Hv° + v° rL d v + ‡ r f wr ÿ Hl + lrL ÿ vr d v +‡ 1ÅÅÅÅÅk J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅJ wr ÿ vr d v = ‡ r f wr ÿ g d v " wr œ H

We emphasize that these relation defines the solution œ V .

Classical incompressible solid-liquid porous medium 29

Compressible solid-gas medium

Let us consider, as the second prototype, the compressible solid-gas porous medium. To this end,like in the previous case, we assume that the medium is non-erosive corresponding tog f = -gs := 0. In addition, it is assumed that the solid phase is incompressible whereas the fluidphase is compressible, i.e. in view of eqs, (42-43) we have now

(160)evs := 0 , ev

f ∫ 0

It is still assumed that the dissipative fluid stress is small enough to be neglected, i.e. s f := 0.

In order to model the “fluid” compressibility, the fluid (or gas) phase is here considered as a com-pressible gas concerning the non-viscous pressure response, e.g. we consider gas-filled foam materi-als subjected to large rapid deformations, whereby the gas-pressure response is modelled using theideal gas law (already specified in (135)) written as

(161)r f =mgÅÅÅÅÅÅÅÅÅÅR q

p = r0f e-ev

ffl p = p0 e-ev

fwith p0 =

R qÅÅÅÅÅÅÅÅÅÅÅÅmg

r0f

where the last equality was obtained using the definition of the fluid phase compressibility in (42).We also find that from (42) and (161) that

(162)r° fÅÅÅÅÅÅÅÅÅr f

= -e° vf =

mgÅÅÅÅÅÅÅÅÅÅR q

p°

ÅÅÅÅÅÅÅÅÅr f

=p°

ÅÅÅÅÅÅÅp

Balance of mass

In view of the aforementioned restrictions, we conclude in view of the balance of mass in (48) ofthe mixture material that we have for incompressible phases the relationship:

(163)“ ÿ v - n f e° vf = -

1ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdMLet us also in this case elaborate on the mass balance conditions in (38) for the individual phaseswritten as

(164)M° s

= 0, M° f

+ J — ÿ Ir f vdM = 0where it is noted that M

° s= 0 yields (just as in the case incompressible phases) the conditions

(165)M

° s= 0 fl ns rs J =

n0s r0

s fl ns = 8rs = r0s < = n0sÅÅÅÅÅÅÅÅJ fl n° s J + ns J° = 0 fl n° f = ns J°ÅÅÅÅÅÅJ = I1 - n f M “ ÿ vHence, the balance of mass for the fluid phase in (164b) may be expanded as

30 Balance relations for different types of porous media

(166)

M° f

+ J — ÿ Ir f vdM =J r̀

° f+ J

°r̀ f + J — ÿ Ir f vd M = J n° f r f + J n f r° f + J “ ÿ v r̀ f + J — ÿ Ir f vd M =

J I1 - n f M “ ÿ v r f + J n f “ ÿ v r f + J n f r° f + J — ÿ Ir f vdM =J “ ÿ v r f + J n f r° f + J — ÿ Ir f vdM = 0 fl “ ÿ v + n f r° fÅÅÅÅÅÅÅÅÅ

r f+

1ÅÅÅÅÅÅÅÅÅr f

— ÿ Ir f vdM = 0which indeed is the balance of mass specified in (163). In addition, from the developments in (144)and (165) we conclude the conditions for the volume fractions from the assumption of incompress-ible solid phase as:

(167)ns =n0

s

ÅÅÅÅÅÅÅÅJ

, n f =J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

Jfl n° f = ns

J°

ÅÅÅÅÅÅJ

= I1 - n f M “ ÿ vBalance of momentum

Concerning the balance of momentum for the mixture material in (60) we obtain, from the assump-tion of a non-erosive material and the fact that the gas density r̀ f

(172)J Ds =1ÅÅÅÅÅ2

S : C°

- r̀0s

∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C

: C°

:= 0 fl S = 2 r̀0s

∑ysÅÅÅÅÅÅÅÅÅÅÅÅ∑C

(173)Di = -hef ÿ vd ¥ 0 fl vd = -k he

f fl vd = -kikjjj— p - r f ikjjjg - D f v fÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD t y{zzzy{zzz

where k is the isotropic permeability parameter of the solid-gas interaction.

Weak form of governing equations

In order to formulate the coupled solid-gas interaction problem we combine the balance of mass,the balance of momentum and the intrinsic fluid flow as a coupled set of equations. This is formu-lated with respect to the current configuration as

(174)

Hs - p 1L ÿ “ + r̀s g = r̀s v°“ ÿ v - n f e° v

f +1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdM = “ ÿ v + n f p°ÅÅÅÅÅÅÅp

+1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdM = 0he

f = — p = -n fÅÅÅÅÅÅÅÅÅk

vr

where the constitutive relation for the gas densification (162) was embedded into the balance ofmass. Like in the previous fully incompressible problem, the problem is formulated in the solidphase placement x = j@XD fl v = j° , the fluid pressure p , and the relative velocity vr . Hence, weformally identify the primary solution as = 8v, p, vr< . However, in the present case the we reducethe problem via simple elimination between the two last equations in (174), i.e. we may eliminatethe seepage velocity vr so that Ø = 8v, p< and (174) is rewritten as

(175)

Hs - p 1L ÿ “ + r̀s g = r̀s v°“ ÿ v - n f e° v

f +1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdM = “ ÿ v + J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅJ

p°

ÅÅÅÅÅÅÅp

-1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ik r f — pM = 0where the expression for the fluid phase volume fraction n f in (167) was inserted in the lastequality.

The relations (175) are formulated in weak form so that the solution œ V where V is a functionspace defined by

V @w, hD = 8w œ P, h œ S<where in turn P is the function space containing the virtual displacement field w@xD , S is the func-tion space containing the virtual fluid pressure field h @xD .The coupled set of equations in (175) are recast into weak form following the steps: Multiply eqs(175) with the proper virtual field and integration over the current domain. This is carried out usingthe divergence theorem as

(176)‡ Hs - p 1L ÿ l@wD d v + ‡ r̀s w ÿ v° d v = ‡G

w ÿ sêêê ÿ n d G + ‡ w ÿ r̀ g d v

32 Balance relations for different types of porous media

(177)‡ h r f “ ÿ v d v + ‡ r f J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅJ h p°ÅÅÅÅÅÅÅp d v - ‡ h “ ÿ Ik r f — pM d v =‡ h r f “ ÿ v d v + ‡ r f J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅJ h p°ÅÅÅÅÅÅÅp d v + ‡ k r f H“ hL ÿ H— pL d v + ‡G h vd ÿ n d G = 0where the last equation (177) (balance of mass) was premultiplied by the density r f prior to theintegration in order to avoid the complicated application of the divergence theorem. Let us alsointroduce the standard notation for the traction and the normal gas flow on the external boundary,i.e.

tê

= sêêê ÿ n , Q = r f vd ÿ n

Hence, in view if (176-177) the solution œ V is established from the coupled set of relations:‡ Hs - p 1L ÿ l@wD d v + ‡ r̀s w ÿ v° d v = ‡G

w ÿ tê d G + ‡ r̀s w ÿ g d v " w œ P

‡ h r f “ ÿ v d v + ‡ r f J - I1 - n0f MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅJ h p°ÅÅÅÅÅÅÅp d v + ‡ k r f H“ hL ÿ H— pL d v = -‡Gh Q d G " h œ SRestriction to small solid deformations - Classical incompressible solid-liquid medium

In this subsection, we outline the formulation and consequences of the assumption of small soliddeformations pertinent to the classical incompressible solid-liquid medium. We emphasize that theassumption of small deformation concerns merely the solid phase response. As to the fluidresponse, the assumption of small deformation is of no relevance.

In the case of small solid deformations we make the simplifications: The current domain is thesame as the solid reference domain, i.e. we have that

(178)Ø 0 fl “X Ø “, d V Ø d v

The small deformation solid displacement u is the solid velocity

(179)v Ø u° fl l = u° ≈ “

whereby the solid spatial velocity gradient becomes equal to the rate of displacement gradient. Inaddition, the deformation gradient and its Jacobian tends to unity, i.e.

(180)F Ø 1, J Ø 1

and the engineering strain rate becomes equal to the linearized Lagrange strain tensor, i.e.

(181)e° Ø1ÅÅÅÅÅ2

C°

=1ÅÅÅÅÅ2

Hl + ltL = Hu° ≈ “Lsym, J° = “ ÿ v Ø “ ÿ u° = 1 : e° = e° vBalance of mass

From (142), corresponding to the case of incompressible phases, the mass balance reads

Compressible solid-gas medium 33

(182)e° v = -1

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir f vdMIn view of (146) and (180), we now have the conditions for the volume fractions

(183)ns = n0s , n f = n0

f fl n° f = ns J°

= I1 - n f M e° vConstitutive equations

On the basis of the solid entropy inequality (151), we now obtain in the small deformation case withys@C, qD Ø ys@eD the condition with respect to hyperelasticity

(184)Ds = s : e° - r̀0s

∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ

∑ e: e° := 0 fl s = r̀0

s ∑ ysÅÅÅÅÅÅÅÅÅÅÅÅ

∑e

where s@eD is now the small deformation effective stress tensor. As a prototype for small deforma-tions, we may choose the linear elastic response corresponding to

(185)r̀0s ys =

1ÅÅÅÅÅ2

e : E : e fl s = E : e

where E is the elastic (constant) stiffness modulus tensor.

Weak form of governing equations

From (155) we now reconsider the governing equations in the context of small deformations as

(186)

Hs - p 1L ÿ “ + r̀ g = Ir̀s + r̀ f M u° + r̀ f v° r + r̀ f Hl + lrL ÿ vre° v +

1ÅÅÅÅÅÅÅÅÅr f

“ ÿ In0f r f vrM = 0— p - r f g + r f Iv° + v° rM + r f Il + lrM ÿ vr = - n0fÅÅÅÅÅÅÅÅÅ

kvr

corresponding to the weak form with the solution œ V written as‡ Hs - p 1L ÿ l@wD d v + ‡ Ir̀s + r̀ f M w ÿ v° d v + ‡ r̀ f w ÿ vr d v + ‡ r̀ f w ÿ Hl + lrL ÿ vr d v =‡ w ÿ tê d v + ‡ w ÿ r̀ g d v " w œ P‡ h r f “ ÿ v d v - ‡ n0f r f H“ hL ÿ vr d v = -‡ h Q d v " h œ S‡ wr ÿ — p d v + ‡ r f wr ÿ Hv° + v° rL d v + ‡ r f wr ÿ Hl + lrL ÿ vr d v + ‡ n0f wr ÿ vr d v =‡ r f wr ÿ g d v " wr œ H

34 Balance relations for different types of porous media

Theory questions

1. Define and discuss the specialization to the classical incompressible solid-liquid porous medium. Which are the restrictions imposed? Elaborate, in particular, on the balance of mass in this context. Which are the conditions imposed on the volume fractions?

2. Formulate the entropy inequality in the context of the incompressible solid-liquid porous medium. Formulate and discuss the relevant constitutive relations in this context.

3. Set up the governing equations for the incompressible solid-liquid porous medium. Define the primary solution and establish the weak coupled equations corresponding to this solution.

4. Define and discuss the specialization to the compressible solid-gas medium. Which are the restrictions imposed in this case as compared to the previous one? Elaborate, in particular, on the balance of mass in this context. Which are the conditions imposed on the volume fractions?

5. Set up the governing equations for the compressible solid-gas medium. Define the primary solution and establish the weak coupled equations corresponding to this solution.

6. Formulate and discuss the consequences of the small solid deformation assumption from the kinematical and constitutive response point of view.

Numerical procedures - Classical incompressible solid-liquid porous medium

In this section we outline the numerical solution strategy to the classical incompressible solid-liquidporous medium. In particular, the temporal integration of the momentum balance relations is formu-lated as an explicit integration step in the spirit of the central difference scheme and the forwardEuler method. Likewise, the balance of mass is integrated using the backward Euler method. Thediscretization of the problem is proposed and the finite element out-of-balance force vectors areestablished. It is proposed to resolve the coupled problem using a staggered solution procedurewhere the fluid pressure is considered constant over the mechanical step whereas the configurationis considered fixed over the mass balance step. Finally, a 1D linear assignment problem is outlined.

Temporal integration

In order to integrate in time the coupled boundary value problem we shall consider the two prob-lems separated. We thereby consider the time interval I subdivided into a finite set of N time stepsso that I = 90 t, .. nt, t, .. Nt= . Typically, as a result of the time integration procedure we advancethe solution step-wise from one time-step nt with known values to the updated time t = n+1t . In thesubsequent representation of the integrated solution, we omit the supscript n+1è for quantities at theupdated time tn+1 . In this fashion, we may consider two different types of integration strategies: oneexplicit integration related to the momentum balance relation and another implicit step related tothe mass balance relation.

Theory questions 35

Momentum balance - Explicit integration step

As the point of departure, let us reconsider the governing balance relations (155) restated withrespect to the reference configuration, whereby these equations are pre-multiplied by the ”J ” due tothe change of integration domain. Hence, the governing relations are rewritten in terms of the“mechanical problems” and the “mass balance problem” as

(187)

J I tÅÅÅÅÅJ - p 1M ÿ “ + Ir̀0s + M f M g = IMs + M f M v° + M f v° r + M f Il + lrM ÿ vrJ — p - J r f g + J r f Hv° + v° rL + J r f Hl + lrL ÿ vr = - J n fÅÅÅÅÅÅÅÅÅÅk vr |oo}~ooMechanical problems HaL, HbL

J “ ÿ v +J

ÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir̀ f vrM = J° + JÅÅÅÅÅÅÅÅÅr f

“ ÿ Ir̀ f vrM = 0|oo}~oo Mass balance problem HcLConcerning the integration of the mechanical problems (187ab) we shall consider the well knownexplicit central difference scheme, where the solution is advanced in an explicit step applied to the(structural) or equilibrium equation. To this end, let us consider the central difference schemeapplied to the acceleration in the momentum balance relation (187a) whereas a standard forwardEuler integration step is proposed form the relative acceleration v° r in (187b). This yields the inte-grated placement j and the integrated relative velocity vr from the Taylor series expressions

(188)j = nj + n vD t +1ÅÅÅÅÅ2

n v° D t2

(189)vr = n vr + n v° r D t

where the “current” solid displacement velocity n v is defined with respect to the current and theprevious time steps as

(190)n v :=1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅn D t + D t

K D tÅÅÅÅÅÅÅÅÅÅÅÅnD t

In j - n-1 jM + nD tÅÅÅÅÅÅÅÅÅÅÅÅÅD t

Hj - n jLOMoreover, in eqs. (188) and (189) the accelerations n v° and n v° r are established from the “inertia”updated at t = nt written as

(191)

IM s + M f M n v° = BJ J tÅÅÅÅÅÅJ

- p 1N ÿ “+ Ir̀0s + M f M gFt=nt

- AIM f n v° r + M f n Hl + lrL ÿ n vrM Et=n*t

J r f n v° r =ÄÇÅÅÅÅÅÅÅÅ- J n fÅÅÅÅÅÅÅÅÅÅÅÅÅÅk n vr - J — p + J r f gÉÖÑÑÑÑÑÑÑÑ - AJ r f In v° + J r f n Hl + lrL ÿ n vrMEt=n*t

whereby due to the presence of the non-linear convective terms of the inertia the integrated solutionis characterized in an implicit fashion. In practice, the implicit integrated solution involving theconvective term n Hl + lrL ÿ n vr is established using a simple fix-point iteration procedure, as alludedto by the specification t = n*t in (191). However, concerning the establishment of the placement jin (188) we note the explicit character of the integrated solution, whereby in view of (190) it suf-fices to establish the acceleration at the central point for the advancement of one step. In thisadvancement, the velocity is nothing but the mean velocity over the last and the current time-steps(where nD t is time step-size of the last time-step). We also note that the displacement velocity n vimplicitly involves the current placement.

36 Temporal integration

Mass balance - Implicit integration step

In order to integrate the mass ba