A solid-fluid mixture theory of porous media

Transcription

1 A solid-luid mixture theory o porous media I-Shih Liu Instituto de Matemática, Universidade Federal do Rio de Janeiro Abstract The theories o mixtures in the ramework o continuum mechanics have been developed throughout the sixties and seventies. In this paper, we shall ocus on one o the typical constitutive theories obtained by Bowen [1] or mixture o elastic materials. We employ a dierent procedure in the exploitation o the entropy principle, by the use o Lagrange multipliers or a solid-luid mixture. It is interesting to note that these two theories are completely equivalent concerning the constitutive restrictions. The general results are then applied to the studies o porous media by introducing the concept o porosity. The resulting theory is consistent with Darcy s law and the concepts o pore luid pressure and eective stress principle in soil mechanics. Keywords: Balance equations, entropy principle, Müller Liu procedure, volume raction, pore luid pressure, eective stress, Darcy s law. 1 Introduction The theories o mixtures in the ramework o rational continuum thermodynamics have been developed throughout the sixties and seventies, and subsequent development in various constitutive theories and thermodynamic analysis are too numerous to document or related reerences, see [10]. Unlike Irreversible Thermodynamics which postulates a generalized Gibbs relation and mandates linear constitutive relations in thermodynamic luxes/orces, Rational Thermodynamics relies on the entropy principle to exploit general restrictive conditions on constitutive unctions, linear or non-linear. There are basically two dierent approaches in exploiting the entropy principle. One is the so-called Coleman Noll procedure, by eliminating the entropy supply with the energy supply o the energy equation. While the other is the so-called Müller Liu procedure by employing the Lagrange multipliers. Generally speaking, the Coleman Noll procedure is more restrictive because it requires some speciic assumptions concerning the relation between the entropy lux/supply and energy lux/supply. However, it is more widely used owing to its relatively simpler mathematical calculations. For mixture theories, Müller was the irst to employ Lagrange multipliers or mixtures o luids [9] and later many similar theories have been presented mostly with linear constitutive equations to acilitate the tedious calculations and to avoid possible mathematical diiculties. One o the typical constitutive theory o elastic solid-luid mixture was presented by Bowen [1], employing Coleman Noll procedure or large deormation without linear constitutive assumptions. In this paper, we shall employ the Lagrange multipliers in analyzing the entropy principle or such a constitutive theory in order to compare with the general results 1

2 obtained by Bowen. In doing so, we rerain rom making isotropic and linear constitutive assumptions, but to acilitate mathematical diiculties in dealing with Lagrange multipliers, some simpliying assumptions are made, or one, the Lagrange multiplier or energy equation is identiied with the reciprocal o the absolute temperature. It is ound that these two theories are completely equivalent concerning the constitutive restrictions. Ater a brie review o essential eatures o the theories o mixtures and the exploitation o constitutive restriction or an elastic solid-luid mixture with Lagrange multipliers, the general results are applied to the ormulation o a theory o porous media by introducing the concept o porosity. The resulting theory is consistent with Darcy s law and the concepts o pore luid pressure and eective stress principle in soil mechanics. 2 Theories o mixtures We shall briely review some essential eatures or continuum ramework o mixture theories. We consider a mixture o N constituents, all o which are supposed to be able to occupy the same region o space simultaneously. Let B α denote the α th constituent and κ α be its reerence coniguration and denote B α = κ α B α. The motion o B α is a smooth mapping, χ α : B α IR IE, x = χ α X α, t, X α B α, or each constituent α = 1,, N. It states that or dierent motion o each constituent, at the instant t, there is a material point X α B α in each constituent, X α = κ α X α, that occupies the same spatial position x in the Euclidean space IE. The velocity and the deormation gradient o each constituent are deined as v α = t χ αx α, t, F α = Xα χ α X α, t. We introduce the ollowing quantities or the constituent α {1,, N}: ρ α mass density o constituent α. T α stress tensor o constituent α. b α external body orce on constituent α. ε α internal energy density o constituent α. q α energy lux o constituent α. r α external energy supply o constituent α. η α entropy density o constituent α. Φ α entropy lux o constituent α. s α external entropy supply o constituent α. τ α mass production o constituent α. m α interaction orce on constituent α. M α interaction moment o momentum on constituent α. l α energy production o constituent α. σ α entropy production o constituent α. 2.1 Balance laws o each constituent Following the pioneering work o Truesdell [13], the basic laws o a mixture are given by the ollowing balance equations or mass, linear momentum, moment o momentum, and energy 2

3 or each constituent: ρ α t + divρ αv α = τ α, ρ α v α + divρ α v α v α T α ρ α b α = m α, t T α Tα T = M α, ρ α ε α + 1 t 2 ρ αv 2 α + div ρ α ε α ρ αv 2 α v α + q α Tα T v α ρ α r α ρ α b α v α = l α, 2.1 I the mixture reduces to a single constituent, the right hand side o the above equations are zero, and we recover the balance laws or a single body. The terms on the right hand side thus represent the physical transers among dierent constituents, such as chemical reactions, interaction orces and energy transer. In a mixture as a whole, we assume that such physical transers are solely due to exchanges among constituents. Thereore, we postulate that the mixture as a whole should behave like a single body. This is expressed by the ollowing relations: τ α = 0, m α = 0, M α = 0, l α = α 2.2 Balance laws o the mixture α The behavior o a mixture as a whole will be assumed to be governed by the same equations o balance as a single body. Indeed, by summing up the balance equations 2.1 over all constituents, we obtain the usual balance equations o mass, linear momentum, and energy or the mixture, ρ + divρv = 0, t ρv + divρv v T = ρb, t t ρε ρv2 + div ρε ρv2 v + q T T v = ρr + ρb v, α α 2.3 by the use o the relations 2.2 and the ollowing deinition o the corresponding quantities or the mixture, ρ = α ρ α, v = α ρ α ρ v α, T = α T α ρ α u α u α, b = α ρ α ρ b α, 2.4 and ρ α ρ ε α u2 α, ρ α ρ r α + b α u α. ε = α q = α r = α q α + ρ α ε α + 12 u2 αu α Tα T u α, 2.5 Note that in the relation 2.4 and 2.5, we have introduced the diusive velocity o constituent α relative to the mixture, u α = v α v. Also note that M α = 0 implies the symmetry o T α as well as the total stress T, while the partial stress T α is not symmetric in general. 3

4 2.3 The entropy inequality The balance o entropy o constituent α can be written as ρ α η α t + divρ α η α v α + Φ α ρ α s α = σ α. Since entropy is not a conservative quantity, we shall allow the partial production o entropy o a constituent be o any amount provided that the total production o entropy or the mixture be non-negative, σ α α By summing up the equations o all constituents α {1,, N} and introducing the corresponding quantities or the mixture, η = α Φ = α s = α Φ α + ρ α η α u α, 2.7 ρ α ρ η α, ρ α ρ s α, we obtain the usual entropy inequality or the mixture as a single body, ρη t + divρηv + Φ ρs Note that we do not postulate a similar entropy inequality or individual constituent. The partial entropy production could be negative provided that the total production o the mixture is non-negative, otherwise it will lead to very restrictive physical results. The use o the general entropy inequality 2.8 or the exploitation o its restrictions on constitutive models is oten quite sophisticate and could be rather tedious. In application, assumption or the speciic orm o the entropy lux and entropy supply or the constituents is usually adopted, namely, Φ α = 1 θ α q α, s α = 1 θ α r α, 2.9 where θ α is the temperature o the constituent α. Under this assumption, rom 2.7, we have Φ = qα + ρ α η α u α, s = ρ α r α. θ α α ρ θ α α For urther discussions, we shall assume that energy exchange among constituents is quick enough relative to characteristic observation time, so that all the constituents have the same common temperature. We can then rewrite the above relations as Φ = 1 q ρα K α + 1 θ 2 u2 αi u α, α s = 1 r ρ 2.10 α θ ρ u α b α, α where K α = ψ α I T T α ρ α, ψ α = ε α θη α, 2.11 will be called the chemical potential tensor and the ree energy o constituent α respectively. Even though we have adopted the Clausius Duhem assumptions 2.9 or each constituent, or the mixture as a single body, rom 2.10 such assumptions are not valid. In other words, the Clausius Duhem assumptions Φ = q/θ and s = r/θ may not be appropriate or thermodynamic considerations o material bodies in general. 4

5 2.4 Jump conditions at semi-permeable surace Let S be a surace within the mixture and [ A] be the jump o a quantity A across S, deined by [ A] = A + A. We assume that the one-side limits A + and A exist and the jump is smooth on S. Such a surace is called a singular surace. Let u and n denote the velocity and the unit normal o S respectively. In addition to the balance laws 2.3 and 2.8 at a regular point where relevant ields are continuous, at a singular point, i.e., a point on S, the balance laws or the mixture take the orm o the ollowing jump conditions o mass, momentum, energy and entropy: [ ρv u ] n = 0, [ ρv v u T ] n = 0, [ q T v + ρε v2 v u ] n = 0, [ Φ + ρηv u ] n We do not postulate the similar jump conditions or each constituent, because it is generally too restrictive and may even lead to unrealistic results 1. A singular surace will be called ideal, i across it the temperature is continuous, [ θ ] = 0. We assume urther that across an ideal singular surace, the entropy jump is null, [ Φ + ρηv u ] n = 0, 2.13 and the entropy lux Φ is given by the relation The diusive velocity o constituent α relative to the surace S is deined as V α = v α u. I V α = 0, we say that S is not permeable to the constituent α, i.e., the constituent does not go across the surace. A singular surace which is not permeable to all constituents except the constituent ν is called a semi-permeable surace with respect to the constituent ν, or simply a ν permeable surace. We have the ollowing result see [4]: Theorem. Let S be an ideal ν permeable surace, then by the use o the jump conditions ,2 and 2.13, the energy jump condition becomes [ Kν V 2 ν Iρ ν V ν ] n = 0, 2.14 where K ν is the chemical potential tensor o the constituent ν. 3 An elastic solid-luid mixture We consider a non-reacting binary elastic solid-luid mixture by the constitutive equation o the orm: = Fθ, F s, F, grad θ, grad F s, grad F, v s, v, It is shown in the end remark o [4], that the jump conditions or each constituent cannot be valid in the case o porous media 5

6 where θ is the temperature, F s and F are the deormation gradients o the solid and luid constituents respectively, with the ollowing balance equations rom 2.1 1,2 and 2.3: `ρ s + ρ s div v s = 0, `ρ + ρ div v = 0, ρ s`v s div T s + m = ρ s b s, ρ `v div T m = ρ b, ρ ε + div q trt grad v = ρr, 3.2 where the material derivatives with respect to the constituent and the mixture are used, ỳ α = y α t + grad y αv α, y = y + grad yv. t Moreover, or given density ρ s in the reerence coniguration, the partial balance equation or the solid constituent can be integrated to ρ s = ρ s det F s. 3.3 Note that since only a common temperature is considered or all constituents, only the energy equation o the mixture, instead o the partial energy equations o all constituents, is needed in the system o governing equations 3.2. Moreover, since the partial stresses are not assumed to be symmetric, the equation o partial moment o momentum is not explicitly needed in 3.2. In other words, the interaction moment M α can simply be determined rom the skew-symmetric part o the partial stress T α. To establish ield equations o the basic ield variables, {ρ s, ρ, χ s, χ, θ}, constitutive equations or the quantities in the balance equations, = {T s, T, ε, q, m } 3.4 o the material model must be speciied. The main objective o constitutive theory is to derive restrictions on constitutive unctions based on some physical principles. First o all, according to the principle o material rame-indierence, since the above constitutive quantities are objective tensor, scalar and vector quantities, an immediate consequence ollows, namely, constitutive unctions can not depend on partial velocities independently, rather must depend on the relative velocity, say, V = v v s. Secondly, the material symmetry condition or luid constituent requires the dependence on the deormation gradient be reduced to the dependence o the determinant o the deormation gradient only. Thereore, we may redeine the constitutive unctions 3.1 in terms o the ollowing state variables, = Fθ, ρ, F s, grad θ, grad ρ, grad F s, V. 3.5 For given ields o external supplies {b s x, t, b x, t, rx, t}, any solution o the system o ield equations, {ρ x, t, v x, t, v s x, t, F s x, t, θx, t}, will be called a thermodynamic process. Some useul identities For convenience, we deine the material time derivative y or any unction y, and the relative velocity V with respect to the solid constituent, y = y t + grad yv s, V = v v s. 6

7 From these deinitions, we have the ollowing identities: u = v v = ρ s ρ V, u s = v s v = ρ ρ V, since ρv = ρ s v s + ρ v, and ỳ s = y s, ỳ = y +grad y V, and y = y + ρ ρ grad yv, grad v = grad v s + ρ ρ grad V + V grad ρ. ρ These identities will be requently used later without urther reminder. From 2.4 and 2.5, we also have T = T I ρ sρ ρ V V, ε = ε I ρ s ρ ρ 2 V 2, 3.6 where T I and ε I stand or the internal parts without the explicit dependence on the relative velocity, T I = T s + T, ε I = ρ s ρ ε s + ρ ρ ε. Another well-known identity, grad v = F F 1, or the solid constituent gives grad v s = F s F 1 s. 3.7 Note that rom the material time derivative o the relation 3.3, ρ s = ρ s tr F s Fs 1, the trace o the relation 3.7 reduces to the mass balance o the solid constituent In the evaluation o entropy principle, this tensor equation 3.7 will be taken into account replacing the scalar equation Entropy principle The essential task o constitutive theory lies in the evaluation o the entropy principle, which requires that the entropy inequality rom 2.8, ρ η + div Φ ρs 0, 3.8 be satisied or every thermodynamic process. The entropy density and the entropy lux will also be given by constitutive equation o the orm 3.5. Since material properties should not depend on external supplies which characterizes the environment the material bodies encountered, thereore, one can consider supply-ree bodies or the evaluation o constitutive restrictions rom the entropy principle, by assuming that there is no external entropy supply when there are no any external body orce and energy supplies on the body. In this case, according to Liu [3], the requirement that the entropy inequality 3.8 must hold or any thermodynamic process can be analyzed with the use o Lagrange multipliers. 7

8 Method o Lagrange multipliers. There exist Lagrange multipliers Λ Ls, Λ ρ, Λ vs, Λ v, and Λ ε, depending on the state variables, such that the inequality ρ η + ρ grad η V + div Φ Λ Ls ρ s F s F 1 s + grad v s Λ ρ ρ + grad ρ V + ρ div v Λ vs ρ s vs div T s + m Λ v ρ v +ρ grad v V div T m Λ ε ρ ε +ρ grad ε V + div q trt grad v 0, 3.9 must hold or any analytic ields {ρ x, t, v x, t, v s x, t, F s x, t, θx, t}. 2 Note that ater introducing the constitutive relations into 3.9, the inequality assumes the ollowing orm: S a X a + σ 0, 3.10 a where X a = F s, ρ, V, v s, θ, θ, ρ, F s ; V, v s, θ, ρ, F s, while S a and σ are unctions o state variables. We have written the grad operator as. Note that the inequality 3.10 is linear in X a, and the values o X a can be given independently o the values o S a and σ. This implies that S a respecting the part involved with the symmetry o the second gradients in the corresponding X a must vanish, otherwise, it is possible to choose some values o X a such that the inequality is violated. By introducing the notations, H X = η ε Λε X X, M X = Φ q Λε X X + Λvs T s X + Λv T X + V ρ H X, we have the ollowing entropy density conditions: 3.11 and ρ H Fs + ρ s Λ Ls Fs T = 0, ρ H ρ Λ ρ = 0, ρ H V ρ Λ v = 0, ρ s Λ vs ρ Λ v = 0, 3.12 H θ = 0, H θ = 0, H ρ = 0, H Fs = Moreover, we have the entropy lux conditions: M V + Λ ε ρ ρ T Λρ ρ I V ρ Λ v = 0, Λ ε T ρ s Λ Ls ρ Λ ρ I V ρ Λ v = 0, 3.14 and M θ sym = 0, M ρ sym = 0, M Fs sym = 0, For general inormation on the applicability o the method o Lagrange multipliers, please reer to [3] or Chap. 7 o [6]. 8

9 where sym stand or the symmetric part o the expression resulting rom the symmetry o the second gradient in X a in There is a remaining entropy production inequality, σ = M θ θ + M ρ ρ + M Fs F s + Λ ε T V ρ Λ ρ V ρ ρ Λ vs Λ v m The relations concerning the entropy density and the entropy lux rom 3.12 through 3.16 are the complete set o consequences rom the entropy principle. It remains to analyze the restrictions on the constitutive unctions rom these relations and the physical signiicance o the Lagrange multipliers. 3.2 Further evaluations We shall remark that, even in the case o a single elastic solid body, a rigorous evaluation o the entropy lux relation with the Lagrange multiplier remains generally impossible without some urther assumptions, such as material isotropy or some explicit representations o constitutive unctions see [2, 7, 8, 9]. To acilitate urther evaluations and avoid technical mathematical diiculties, we shall also make some simpliying assumptions consistent with the general description o the classical mixture theory given in the previous section, but rerain rom making linear and isotropic constitutive assumptions. Entropy density relation The entropy relations 3.12 and 3.13 can be summarized in the ollowing dierential orm, dη = Λ ε dε + ρ ρ Λv dv + 1 ρ Λρ dρ ρ s ρ ΛLs F s T df s, which by the use o becomes dη = Λ ε dε I + Λ ε ρ sρ + ρ 2 ρ s V + ρ ρ Λv dv Λ ε 1 2 ρ 3 ρ s ρ V ρ Λρ dρ Λ ε 1 ρ 2 ρ 3 ρ ρ sv 2 I + 1 ρ ΛLs T ρ s F s df s This resembles a generalized Gibbs relation, which suggests that or simplicity, one could identiy the Lagrange multiplier Λ ε as the reciprocal o the absolute temperature θ, and take see [9] and [2] Sec. 7.6 Λ ε = 1 θ, 3.18 Λ v = Λ ε ρ s ρ V = 1 θ u, Λ vs = Λ ε ρ ρ V = 1 θ u s Note that the second relation ollows rom the irst relation and and is consistent with the identity ρ s u s + ρ u = 0. With the assumptions 3.18 and , the relation 3.17 reduces to dη = 1 θ dε I + 1 ρ Λρ I dρ ρ s 9 ρ ΛLs I F s T df s, 3.20

10 where we have deined the internal parts without explicit V -dependent Lagrange multipliers as Λ ρ I = Λ ρ + Λ ε 1 ρ s 2 ρ 2 ρ s ρ V V, Λ Ls = Λ Ls + Λ ε ρ I 2 ρ 2 ρ ρ sv V I. The relation 3.20 can also be rewritten as dψ I = η dθ θ Λ ρ I dρ ρ ρ s Λ Ls F T I s df s, 3.22 or dρψ I = ρη dθ + ψ I θλ ρ I dρ ψ I I θλ Ls I ρ s F T s df s, where ψ I = ε I θη = ρ s ρ ψ s + ρ ρ ψ, is the internal part o the ree energy density. It ollows rom this Gibbs relation that ψ I = ψ I θ, ρ, F s, Λ ρ I Λ Ls I = 1 θ ρ ψ I ρ = 1 θ = 1 θ ρ ρ s ψ I F s F T s η = ψ I θ, ψ I ρψ I, ρ = 1 ψ θ I I + 1 ρψ I Fs T ρ s F s Entropy lux relation The second part o evaluation concerns the entropy lux relations 3.14 and This is in act the most challenging and rewarding part o the evaluation procedure with Lagrange multipliers see [5]. For example, in [7] or isotropic viscoelastic bodies, the mathematically complicated procedure leads to the entropy lux heat lux relation and the proo that the Lagrange multiplier Λ ε is a unction o the temperature only which or simplicity, we have taken as an assumption in In the present case, a similar analysis remains a very challenging mathematical problem in general. Meanwhile, we shall cast the relation , Φ V 1 q θ V = Λ vs T s V + Λv T V ρ η V 1 θ ε V V + 1 θ + Λ ρ ρ I. ρ ρ T V ρ Λv into a more convenient orm. By the use o 3.19, , , and 3.6, with the ollowing intermediate results: and Λ vs T s V + Λv T = 1 θ V ρ η V 1 θ = 1 θ ρ s ρ ρ ρ V + 1 θ ρ T V ρ Λv { V 1 T s ρ s V 1 T ρ V T T + s ρ s T T + ρ } s ρ ρ V V + 1 θ T T. ε = 1 ψ V ρ I V θ V + ρ sρ 2 ρ 2 V V { ψ ρ ψ ψ I I + V ρ V V ρ ψ ψ I V + ρ sρ 2 } ρ 2 V V, 10

11 we arrive at the relation, Φ 1 V θ q + k = 1 T T θ ρ ψ I + ρ ρψ I ρ In this relation, we have introduced the deinition o an extra lux, I ρ V ψ V k = ρ ψ ψ I V ρ sρ T T s ρ ρ s T T V 1 ρ s ρ ρ 2 ρ 2 ρ s ρ V V V, 3.25 which by the use o 3.19, can be written as k = α=s, ρ α ψ α I Tα T ρ αu α u α I u α. On the other hand, with the extra lux k, the entropy lux relation , given in the previous section, becomes Φ = 1 q + k θ To proceed with our evaluation, without attempting a rigorous proo, we shall make another simpliying assumption that this entropy lux relation 3.26 holds. An immediate consequence ollows rom 3.24 that the right hand side should vanish, which gives T = ρ ψ I ρψ I ψ ρ ρ I + ρ V V, T s = ρ s ψ s I + ρψ I F s F T s + ρ s ψ s V V The second relation ollows rom the condition by the use o and Furthermore, with 3.25, 3.26 and , the deinition 3.11 gives rise to M X = 1 θ V ρ ψ X or X = { θ, ρ, F s }, and the lux relations 3.15 imply that the symmetric part must vanish, V ψ X + ψ X V = 0, which in turns implies ψ / X = 0. To see this, consider A B + B A = 0, then Thereore, we have proved that A A B + B A B = 0, A AB B + A B 2 = 0, A = 0 and/or B = 0. ψ = ψ θ, ρ, F s, V, ψ s = ψ s θ, ρ, F s, V

12 Entropy production inequality With previous results, ater a long and straightorward calculation, the remaining entropy production inequality 3.16 reduces to { σ θ, ρ, F s, V = 1 θ 2 q + k 1 θ ρ ψ } θ + η V θ + 1 θ ρ s ψ s ρ V ρ 1 θ ρ ψ F s V F s 1 θ V m + o V 3 0, 3.29 where o V 3 stands or the terms containing V V V rom the velocity-dependent parts o the internal energy ε and the stress tensor T, which are irrelevant or urther evaluation. Note that σ0, ρ, F s, 0 = 0, thereore, we can deine an equilibrium process, namely without entropy production, as a process with θ = 0 and V = 0. From 3.29, it states that the entropy production density σ θ, ρ, F s, V attains its minimum at an equilibrium process, which is necessary that the ollowing conditions be satisied: σ X A E = 0, In particular, or σ V E = 0 and m 0 = ρ sψ 0 s ρ 2 σ E 0, or X A = { θ, V }. X A X B σ = 0, we obtain θ E where the superscript 0 indicates the value in equilibrium. Summary o results grad ρ ρ ψ0 F s grad F s, q 0 = Thermodynamic considerations o such a mixture theory has been considered by Bowen [1], see also [13] Appendix 5A in which consequences o the entropy principle have been analyzed with Coleman-Noll procedure based on the assumptions ,2. It is quite interesting and perhaps even expected that the present analysis with Müller-Liu procedure, as sometimes reerred to would give rise to exactly the same results, which can be summarized in the ollowing constitutive equations 3.4 and 3.5: ψ I = ψ I θ, ρ, F s, ε I = ψ I θ ψ I θ, ψ = ψ θ, ρ, F s, V, ψ s = ψ s θ, ρ, F s, V, T = ρ ψ I ρψ I ψ ρ ρ I + ρ V V, 3.31 and in equilibrium, T s = ρ s ψ s I + ρψ I F s F T s + ρ s ψ s V V, m 0 = ρ sψ 0 s ρ grad ρ ρ ψ0 F s grad F s, q 0 = Chemical potential at luid-permeable surace I we deine the equilibrium chemical potential o the luid and the equilibrium partial luid pressure as µ = ρψ I, p ρ = ρ µ ψ 0,

13 then the equilibrium luid stress reduces to a pressure, and rom 2.11, the chemical potential tensor o the luid becomes T 0 = p I, 3.34 K = µ I V ψ V At a luid-permeable surace considered in the previous section, rom the relation 2.14, we have the ollowing energy jump condition, [ ] K V 2 Iρ V n = 0, At this surace, v s = u, so that V = v u = V. Hence, by the use o 3.35, the condition becomes [ µ V 2 V ψ ] ρ V V n = 0, where we have noted that rom the mass jump condition at the luid-permeable surace ρ V n is continuous, and since it is not zero in general, it ollows that [ µ V 2 V ψ ] = V In particular, we have [ µ ] 0 when V 0. Thus, we conclude that in equilibrium v = u, the chemical potential o the luid constituent is continuous across a luid-permeable surace. This conirms the well-known result o the classical theory see [4]. 4 Saturated porous media The solid-luid mixture considered in the previous section can be regarded as a model or saturated porous media provided that the concept o porosity is introduced. For mixture theory o porous media, a material point is regarded as a representative volume element dv which contains pores through them luid constituent can low. Physically, it is assumed that a representative volume element is large enough compare to solid grains connected or not, yet at the same time small enough compare to the characteristic length o the material body. Let the volume raction o pores be denoted by φ, then the volume ractions o the luid and the solid are dv = φ dv, dv s = 1 φdv, i the porous medium is saturated. Remember that in the mixture theory, the mass densities are deined relative to the mixture volume, so that the luid and solid mass in the representative volume element are given by dm = ρ dv = d dv, and hence, dm s = ρ s dv = d s dv s, ρ = φ d, ρ s = 1 φd s, 4.1 where d and d s are the true mass densities o luid and solid constituents respectively. 13

14 We shall also regard the partial luid pressure p in the mixture theory as the outcome o a microscopic pressure acting over the area raction o surace actually occupied by the luid in the pore, i.e., p da = P da, hence, p = φ a P, where P will be called the pore luid pressure and φ a = da /da is the area raction o the pores. In general, the volume raction φ and the area raction φ a may be dierent, yet or practical applications, we shall adopt a reasonable assumption that they are the same or simplicity, so that the pore luid pressure is deined as P = p φ. 4.2 The pore pressure is an important concept in soil mechanics, we shall see that the deinition 4.2 leads to results consistent with the well-known results in soil mechanics. 4.1 Pore luid pressure In most applications, it is reasonable to assume that in equilibrium the ree energy o the luid constituent is the same as the ree energy o the pure luid, i.e., we shall assume that ψ 0 = ψ θ, d. 4.3 From 3.33, we have µ = P d + ψ 0. Since the true density does not change, [ ] d = 0, at an ideal luid-permeable surace which allows the luid to go through, it ollows that [ ] [ ] P [ ] µ = + ψ θ, d d = 1 [ P ]. d Thereore the condition 3.36 implies that [ P ] + d [ 1 2 v u 2 v u ψ V ] = 0. In particular, i v = u, then [ P ] = This result agrees with our physical intuition that i the luid does not low rom one side to the other side through the pores then the pressures in the pores on both sides must be equal. A typical luid pressure measurement is the use o manometric tube attached to the specimen. In the case o a porous body, the junction o the manometric tube and the porous medium can be treated as a luid-permeable ideal singular surace, where on one side is a solid-luid mixture and on the other is a pure luid at rest. Consequently by virtue o the jump condition 4.4 in equilibrium, the luid pressure measured rom the manometric tube is the pore luid pressure see [4]. 14

15 4.2 Equations o motion The equations o motion 3.2 3,4 or the luid and the solid constituents in porous media can be written as φd `v = div T + m + φd g, 1 φd s`v s = div T s m + 1 φd s g, where `v and `v s are the accelerations o the luid and the solid respectively, and the external body orce is the gravitational orce g. Let us write the stresses in the ollowing orm, T = φp I + T, T s = 1 φp I + T s. 4.5 We call T the extra luid stress and T s the eective solid stress, since it reduces to the eective stress widely used in soil mechanics as we shall see later. The equations o motion then become φd `v = φ grad P P grad φ + div T + m + φd g, 1 φd s`v s = 1 φ grad P + P grad φ + div T s m + 1 φd s g. 4.6 On the other hand, note that the interactive orce m in equilibrium rom 3.32, with the use o 3.33, can be written as which now becomes by the use o 4.2 and Linear theory and Darcy s law m 0 = p ρ grad ρ ρ grad ψ 0 0, 4.7 P m 0 = P grad φ + φ d ψ d grad d d, 4.8 Since equilibrium is characterized by the conditions, grad θ = 0 and V = 0, in a linear theory, we shall assume that grad θ and V are small quantities. We shall call the non-equilibrium part o the interactive orce r = m m the resistive orce, since it is the orce against the low o the luid through the medium. One may also call r the drag orce acting upon the solid constituent. Furthermore, since it vanishes in equilibrium, we can write the resistive orce as r = φr V φ G grad θ + o2, 4.10 where material parameters R and G are tensor-valued unctions in general. On the other hand, rom 3.31 and 4.5, the extra luid stress, T = ρ ψ ψ 0 I + ρ ψ V V o2, is a second order quantity because the ree energy o luid constituent must be a scalar-valued isotropic unction. 15

16 Thereore, by the use o 4.8 through 4.10, the equations o motion or the luid constituent in the linear theory becomes where Similarly, the equation becomes d `v = grad P κ + d g, 4.11 P κ = R V + G grad θ d ψ d grad d d φd s`v s = 1 φ grad P + div T s + φ κ + 1 φd s g, 4.13 which is the equation o motion o the solid constituent. We can obtain an interesting equation or the solid constituent i we multiply the equation 4.11 by 1 φ and subtract it rom the equation 4.13, 1 φd s`v s d `v = div T s + κ + 1 φd s d g In quasi-static case when the accelerations can be neglected, the two equations 4.11 and 4.14 or the luid and the solid constituents can be written as grad P + κ = d g, div T s κ = 1 φd s d g Note that the term 1 φd s d g, which represents the dierence between the gravitational orce o the solid and the luid in the volume raction o the solid, can be regarded as the buoyancy orce on the solid constituent. Furthermore, adding the two equations in 4.15 together, we obtain div T = ρg Clearly this is the equilibrium equation or the mixture as a whole, where rom 4.5 T = T + T s = P I + T s + o2, ρ = φd + 1 φd s, 4.17 are the total stress and the total mass density o the mixture respectively. Remark. The equation o motion or the luid constituent 4.11 is a generalization o Darcy s law. Indeed, it is shown in [4], that or the case o classical Darcy s experiment see [11], the equation reduces to the original orm o the Darcy s law. In particular, i only the essential contribution in 4.12 κ = RV, is taken into account and the solid matrix is stationary v s = 0, the equation become v = R 1 grad P d g Roughly speaking, it asserts that pore pressure gradient is the driving orce or the low o the luid through the medium. This is the Darcy s law in the usual classical orm, where the reciprocal o the resistivity constant R is call the permeability tensor. 5 Incompressible porous media We shall consider incompressible porous media, i.e., or which the true mass densities d and d s are constants. For a linear theory, we have T = φp I, T s = 1 φp I + T s, κ = RV + G grad θ. 5.1 where the pore pressure P is constitutively indeterminate and the eective solid stress is given by the constitutive relation, T s = T s φ, F s. 16

17 5.1 Equilibrium pore pressure From 4.15, the equations in equilibrium κ = 0 become grad P = d g, div T s = 1 φd s d g. 5.2 These two equations can be solved separately. Suppose that the x-coordinate is in the vertical downward direction, g = ge x. Then the irst equation can be integrated immediately to give P = d g x + P 0, P 0 = P This result asserts that the equilibrium pore pressure is the hydrostatic pressure. It agrees with the observation in soil mechanics rom experimental measurements that the manometric pressure in the soil is the pressure as i the medium were bulk luid, unaected by the presence o the solid constituent in the medium. We remark that this result is sometimes overlooked in the mixture theory o porous media. It is mainly due to the act that in the theory o simple mixture in the sense o Noll s simple materials, which omits the second gradients o deormations as independent constitutive variables, the equilibrium interactive orce m 0 is identically zero by constitutive hypothesis see However, rom the relation 4.8, m 0 is not a negligible quantity or a body with non-uniorm porosity, m 0 = P grad φ, and it is easy to see that in the absence o this term, the result 5.3 need not ollow. This remark, we shall regard as a strong evidence that porous media must be treated as non-simple mixtures even or a linear equilibrium theory. 5.2 Eective stress principle From 5.1, we have T = T + T s = P I + T s, where the pore pressure P is given by 5.3. Thereore, the eective stress o the solid constituent is the part o the total stress in excess o the pore pressure see also in more general case. This is known as the eective stress principle in soil mechanics irst introduced by Terzaghi see [12]. Both the concepts o pore pressure and eective stress are elementary in soil mechanics. However, regarding soil as porous media in terms o the theories o mixtures, such results are oten overlooked without giving due attention. Acknowledgement This work is supported by a research project rom Petrobras/Brasil and a research ellowship o CNPq, Brazil. Reerences [1] Bowen, R.M.: The theory o Mixtures in Continuum Mechanics, Edited by A.C. Eringen, Vol. III, Academic Press [2] Hutter, K.; Jöhnk, K.: Continuum Methods o Physical Modeling, Springer-Verlag Berlin Heidelberg [3] Liu, I-Shih: Method o Lagrange multipliers or exploitation o the entropy principle, Arch. Rational Mech. Anal. 46, [4] Liu, I-Shih: On chemical potential and incompressible porous media, Journal de Mécanique 19,

18 [5] Liu, I-Shih: On entropy lux-heat lux relation in thermodynamics with Lagrange multipliers, Continuum Mech. thermody. 8, [6] Liu, I-Shih: Continuum Mechanics, Springer-Verlag Berlin-Heidelberg [7] Liu, I-Shih: Entropy lux relation or viscoelastic bodies, Journal o Elasticity 90, [8] Liu, I-Shih: On entropy lux o transversely isotropic elastic bodies, Journal o Elasticity, 96, [9] Müller, I.; A new approach to thermodynamics o simple mixtures, Zeitschrit ür Naturorschung 28a, [10] Rajagopal, K.R.; Tao, L.: Mechanics o Mixtures, World Scientiic Singapore [11] Scheidegger, A.E.: Hydrodynamics in Porous Media, in Handbuch der Physik Vol. VIII/2, Flügge, Editor. Springer Verlag [12] Terzaghi, K.: Theoretical Soil Mechanics, Chapman & Hall, London [13] Truesdell, C.: Rational Thermodynamics, Second Edition, Springer-Verlag, New York Berlin