COHSTETUTWE EQUA'E‘IONS AND THE SOLUTBON OF SOME PROBLEMS OF iNYERACTlNG CONTINUOUS MEDIA Thesis for the Degree of Ph. D. MiCHéGAN STATE UNIVERSITY FARHAD TABADDOR 1968 '._ . .14L33maflm‘s‘w‘f . r) LiBH/lfiw '/ . E'“ :‘f. . - F p." .r'- . 1' 1- ' 9?"; bananas“ 1 .mn’n 3 This is to certify that the thesis entitled Constitutive Equations and the Solution of Some Problems of Interacting Continuous Media presented by Farhad Tabaddor J has been accepted towards fulfillment ‘ of the requirements for Pho D. degree in MEChaniCS fl $2)” 7 2 :‘ / {xi/(,tbumf (3' LCCCLLC/Zk Major professor Date JUIY 22, 1968 0-169 ff amome IY . "0A8 & 80W 3995.!lNQERUNC- ABSTRACT CONSTITUTIVE EQUATIONS AND THE SOLUTION OF SOME PROBLEMS OF INTERACTING CONTINUOUS MEDIA By Farhad Tabaddor A mathematical statement, describing the incompressi- it'uwe-*-rr-—e ere-rm, bility condition for a mixture of n incompressible Newtonian fluids and a linear elastic solid was obtained. Using a thermodynamic theory of interacting media proposed by Green and Naghdi, the constitutive equations were derived for a binary mixture of an incompressible Newtonian fluid and a linear elastic solid. The similarities between these governing equations and those for a binary mixture of a compressible fluid and a linear elastic solid were discussed. A system of field equations are formulated and the general solution for the displacements are presented for the steady-state case. A stress function solution for partial stresses of the solid was developed for steady—state plain— strain problems. These methods were applied to solve a two- dimensional problem. The reduction of the present theory to Biot's consolidation theory and Darcy's law of fluid flow through porous media is discussed. A diffusion law is given for a mixture of two ideal fluids flowing through a rigid porous material. Farhad Tabaddor Finally, following the main theory and employing the notion of hidden coordinates of irreversible thermodynamics, the constitutive equation for a binary mixture of a Newtonian fluid and a viscoelastic solid is derived. ficmt_+*-cre—.:—~»— .... my ;_ _ ‘ ’ ‘ CONSTITUTIVE EQUATIONS AND THE SOLUTION OF SOME PROBLEMS OF INTERACTING CONTINUOUS MEDIA BY Farhad Tabaddor A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1968 «at. \"- a“ «W til. i: .‘a ..', {‘- v.13...»- I 4‘ _ a.-...- J 4113.155, LU, ACKNOWLEDGMENTS The author would like to express his deep gratitude for the guidance and help of Dr. R. W. Little, associate professor of the Metallurgy, Mechanics and Material Science Department, under whose supervision the work was conducted. Thanks are also due to Dr. L. E. Malvern, professor and Dr. Summitt,assistant professor of the Metallurgy, Mechanics and Material Science Department, Dr. R. K. Wen, and Dr. 0. B. Andersland, professors of Civil Engineering Department for serving on his guidance committee. Thanks go to the Division of Engineering Research for the financial support throughout the entire work and my wife, Zahra, for her encouragement and devotion during this study. ii t-c fi-‘u—axar'smfio m 4. m' Ill! .3 -_' TABLE OF CONTENTS ACKNOWLEDGMENTS O O O O O O O O O O O O O O O O 0 LIST OF TABLES O O O O O O O O O O O O O O O O O 0 LIST OF SYMBOLS O O O O O O O O O O O O O O O O 0 CHAPTER I. II. III. INTRODUCTION 0 O O O O O O O O O O O O O O 1.1 Fick's Law of Diffusion . . . . . . 1.2 Truesdell's Hydrodynamic Approach 1.3 Thermodynamic Theories . . . . . . 1.4 Flow of Fluids Through Porous Media . . . . . . . . . . . . . 1 5 Further Theories . . . . . . . . . 1.6 Scope and Objectives . . . . . . . Y MIXTURE OF A NEWTONIAN FLUID AND AN ASTIC SOLID . . . . . . . . . . . . . 2.1 General Remarks . . . . . . . . . . 2.2 Basic Assumptions . . . . . . . . . 2.3 Incompressibility Condition . . . . 2.4 Constitutive Equations . . . . . . 2.5 Field Equations . . . . . . . 2.6 The Case of Compressible Fluids . 2.7 The Physical Interpretation of the Coefficients of Constitutive Equations for Partial Stresses . METHODS OF SOLUTION . . . . . . . . . . . 3.1 Displacement Equations . . . . . 3. 2 Solution of the Steady- -State Plain- Strain Problems by Means of Stress Function . . . . . . . 3. 3 One Dimensional Problem of Infinite Plate 0 I O O O O O O O O O O 3. 4 Semi- Infinite Strip Problem . . . . iii Page ii stun-'- 1’ vi \DwN 15 15 16 18 18 l9 19 22 28 29 30 33 34 38 41 43 Table of Contents (Continued) CHAPTER Page IV. DISCUSSIONS AND EVALUATIONS OF SOME THEORIES O O O O I O O O O I O O O O O O 51 4.1 Darcy's Law . . . . . . . . . . . . . 51 4.2 Brinkman's Theory . . . . . . . . . . 56 4O 3 Biot' 8 Theory 0 O O O O O O O O O O O 57 V. MIXTURE OF TWO IDEAL FLUIDS AND AN ELASTIC SOLID O O O O O O C C I C C O O O C O O I 60 5.1 Formulas and Notations . . . . . . . 60 5.2 Mixture of Two Ideal Compressible Fluids and an Elastic Solid under Isothermal Condition . . . . . . . 62 5.3 Case of Incompressible Fluids . . . . 64 VI. CONSTITUTIVE EQUATIONS FOR BINARY MIXTURE OF A NEWTONIAN FLUID AND A VISCOELASTIC SOLID O I O C I O C C O O O O O O C C O O 67 I s_ a l 1 I ! I r 6.1 General Remarks . . . . . . . . . . . 67 6.2 Hidden Coordinates . . . . . . . . . 68 6.3 Development of Constitutive Equations . . . . . . . . . . . . 69 REFERENCES 0 O O O O O O O O O O O O O I O O O O O O 77 iv LIST OF TABLES TABLE Page 4.1 Typical Values of permeability and porosity for various materials . . . . . . 54 (r) (r) (r) (r) (r) (r)a (r) O ik d.. 13 r.. 13 LIST OF SYMBOLS Component of the reference position vector (r)§ of a particle of the rth substance of the mixture. Component of the current position vector (r); of a particle of the rth substance of the mixture. (r)+ Component of the velocity vector V of a particle of the rth substance of the mixture. Component of the diffusive force vector of the rth S substance of the mixture. Density of the isolated rth substance. Initial density of the rth substance in the mixture. Current density of the rth substance in the mixture. Current density of the mixture. Component of the body force per unit mass of the rth component of the mixture. Component of the acceleration vector of the rth component of the mixture. Component of the partial stress tensor of the rth component of the mixture. Component of the velocity vector D of the solid in a fluid-solid mixture. Component of the acceleration vector of the solid in a fluid-solid mixture. Component of the rate of deformation tensor of the solid in a fluid-solid mixture. Component of the vorticity tensor of the solid in a fluid-solid mixture. Component of the displacement vector of the solid in a fluid-solid mixture. vi List of Symbols (Continued) ij A.. 1] Component of the body force per unit mass of solid in a fluid-solid mixture. Component of the velocity vector V of the fluid in a fluid-solid mixture. Component of the acceleration vector of the fluid in a fluid-solid mixture. Component of the rate of deformation tensor of the fluid in a fluid-solid mixture. Component of the vorticity tensor of the fluid in a fluid-solid mixture. Component of the body force per unit mass of the fluid in a fluid solid mixture. Component of outward unit normal. Internal energy per unit mass of the mixture. Temperature of the mixture. Entropy per unit mass of the mixture. Heat supply function per unit mass of the mixture. Flux of heat across area, per unit area and unit time. Mass supply function of the ith substance. Heat flux across xk - coordinates. Helmholtz free energy per unit mass of the mixture. Component of strain tensor of the solid in the mixture. Porosity factor. Ratio of pores compressibility to total compressi— bility. Difference between current and initial density of the dth substance in the mixture. KroneCker delta . Hydrostatic pressure. vii List of Symbols (Continued) (on) 13‘ (a)r.: 1] Permeability coefficient of the solid. .Component of the rate of deformation of the dth component of the mixture. Component of the vorticity tensor of the dth com- ponent of the mixture. Generalized coordinates. Generalized forces. Component of the stress tensor of the fluid in a fluid-solid mixture. Component of the stress tensor of the solid in a fluid-solid mixture. viii CHAPTER I INTRODUCTION :— The theory of mixtures, or heterogeneous media, has received a great deal of interest and attention in recent years. This branch of continuum physics is concerned with J that kind of medium which consist of two or more constitu- ents, where each constituent is a simple medium with its own physical and chemical properties in the absence of the others. When the components mix together, some changes will appear in both the physical and chemical properties of each constituent. The field of interacting media deals with the necessary modifications to these properties, and formulates the preper field and constitutive equations for the media under consideration. The problem is of great importance both from a the- oretical and practical viewpoint. The wide application of this branch is rather evident from the nature of the problem. As examples, consider the diffusion problem, seepage of water and other fluids through deformable or undeformable porous media, absorption of oils by plastics, water by fibers, and many others. In what follows the basic works in this area are briefly reviewed. 1.1 Fick's Law of Diffusion In analogy to heat conduction, and on no other grounds, Fick assumed that in a binary mixture, the rate of transfer of the diffusing substance through a unit area of the section is proportional to the concentration gradient measured normal to the section _ _ 3C .. F - 13-3-5; (1.1 l) where F = rate of transfer per unit area, 0 II concentration of diffusing substance, D diffusion coefficient- The fundamental diffusion equation is derived by substituting the above constitutive law into the continuity equation of diffusing substance. ac _ 2 _ Although the above relation was proposed for a binary mixture, Onsager (Ann. N.Y. Acd. Sci. 46,251, 1946) suggested * - the direct generalization of Fick's law. The above concept is not supported by principles of conti- nuum mechanics in general, and the many limitations and conceptual difficulties arising from Fick's hypothesis sug- gests an entirely different approach to the problem. A full account of linear theories of diffusion has been given by Truesdell [42,43] and the various theories of diffusion are analyzed. The assumption under which Fick's law is usually applied is that there is no mass exchange, that the pressure and total density are constant, and that there is no mean motion. The derivation of Fick's law of diffusion has also been considered by Adkins [1], Green and Adkins [24], and Mills [30] from different approaches, and the basic assump— tions of the classical theory are justified under special conditions. 1.2 Truesdell's Hydrodynamics Approach A general theory of mixtures, or alternatively het- erogeneous media was constructed by Truesdell [42]. The basic assumptions of the theory are as follows: A. Each space point x may be occupied simultaneously by several different particles. This assumption was first suggested by Fick and Stefan. The validity of this assump-1 tion is as good as the assumption of continuity of the matter. B. We can assign distinct kinematic quantities such as velocity and acceleration and mechanical quantities such as forces and stresses to each substance at a space point x. C. Since the diffusion involves relative motion of' different particles, a transfer of momentum between the com- ponents is involved, while the total momentum of media is conserved as a whole. D. To account for diffusion phenomena, the body force acting on each constituent can be subdivided into an extraneous body force, which is the same as for a single com- ponent, and a diffusive force. The theory calls for a consti- tutive equation for the diffusive force which satisfies all the necessary invariance requirements. E. "Each component is considered as being subject to a partial stress whose action upon any closed diaphragm is equipollent to the action of all constituents exterior to the diaphragm upon the material of the constituent under consid- eration within the diaphragm" [42]. Consequently the total stress is the sum of the partial stresses. Truesdell developed a general framework for hetero- geneous media,and,based on his general theory, he gave a comprehensive analysis of four different approaches to the diffusion problem. Some aspects of the linear theory were examined. Using Truesdell's approach, Adkins [1,2,3,4] has developed a non linear theory. In the following, the general framework of the problem is outlined. We refer the motion to a fixed system of Carte- sian coordinates Xi, and denote the initial position of each (r) substance as Xi where r denotes the rth substance. (r) The position of a particle Xi at time t is denoted (r) by xi where (r)x. = (r)x. ((r)X.' 1 1 3 We can express (1.2-1) in alternative form (r) t) r = 1,2....n (1.2-l) xi = (r)xi(‘r)xj, t) r = 1,2....n (1.2-2) For the above deformations to be possible in real materials we must have (r) (r) T3 xi 71—13 X. (1 2 2 ) > O > 0 . — a The summation convention does not apply to r. (r) (r) The substance Xi has a velocity vi at time t where r (r)V — E:_::i (1 2-3) 1 ‘ Dt ' (r) where —DE- denotes differentiation with respect to time hold- ing (r)Xi fixed. If the density of substance Sr is or at (r)xi then the density of the mixture is p where (1.2—4) and the mean velocity 5h of the mixture is defined to be n 055 _ z pr (r)vfi (1.2 5) r=l If w = w( xi,t) is any scalar or tensor function we observe that _ aw (r) am 5'1? + Vmfi; UIU (+6 I where B/Bt denotes partial differentiation with respect to time holding the spatial coordinates constant. If we define the operator D/Dt by D -3 _ 3 - Dt — t + vmggg (1.2 6) then n (r) n n n DW _ 3W (r) aw _ 32 (r) aw rél pr 5— _r£lpr(3t + Vmfiig) - rzlprat +r£lpr m3x; (r) 8w — 3w DU n = p + pv = p—— Dw _ Dy (1.2-7) ‘5? m5}; D r210]: '51? - th (r)— The diffusive velocity um of the substance Sr is defined to be: (r)- _ (r) _ - _ um _ Vm Vm (1.2 8) thus n n p(r)--m _ p(r)Vm _ 035 = zlps(r)vm _ les(S)vm s= s= = § 0 ((r)v _ (s)v ) (1 2-9) 5:1 5 m m ° so n n n n (r)— _ (r) _ _ (r) _ - rglpr m r£1(pr Vm prvm) _ rglpr Vm rElprvm = 95m - 03m = 0 (1.2-10) If we exclude the possibility of mass generation or mass dissipation by chemical reactions, adsorption or similar processes, the continuity equations can be written for each component of the mixture in the form, apr 8 (r) _ Dpr 3(r)vi .3_E_ + 53?;(01' Vi) : b-E— + (Dr—3i:— - 0 (1.2-11) Do 1 = - (If there exists a mass supply, equations (1.2-11) should be modified; see [42].) (r) We define Hi, the supply of momentum of rth sub- stance, to be ( ) < > 3 (r) r r Fi is the body force per unit density of rth com- (r) _ Oik 0 1T. - pr( 1 (1.2—13) (r) where ponent and (r)D The vort1c1ty tensors are defined to be 1 . . 1 .. = . . - . .. = '. . - . . 1.3-6 1] ‘2(u1,j uj,1) A1] 2(V1,j Vj,l) ( ) = + V’ — + E— = i3-—- + 6' a p D1 D2 p i plu1 p2V1 Dt 3t m5xm (1)D (2) th - pl Dt + p2 Dt as before. If A is an arbitrary fixed closed surface enclosing a volume V and nk is the outward unit normal, the following energy equation is postulated. §_.J[ [(p + p )U + ip u.u. + lp v.v.]dv +Jrln (p u + p v )U at V 1 2 2 l 1 1 2 2 1 1 H k 1 k 2 k + fplnkukui ui + :p p2 Hkvkv v l]dA = jr(pr + plFi ui + szi V. l)dV + j;(tiui + pivi)dA - j;h dA (1.3-7) where U is the internal energy of the mixture per unit mass. Fi’Gi are externally applied body forces per unit masses of S1 and 82. ti,pi are surface force vectors per unit area of A, such that tiui and pivi arethe ratesof work per unit area of A. r is the heat supply function per unit mass of the mixture due to external sources. h is the flux of heat across A per unit area and unit time. If we define m1 and m2 to be 1 ( )Dpl (2)Dp2 m1 ‘ "BE" + pluk,k m2 = “BE" + pZVk,k (1.3-8) and apply a uniform rigid body translation to (1.3-7), assum- ing that pl, p2, U, (Fi - ai), (Gi - gi), ti, pi, h, r, not 13 being altered by this, we obtain: jv[91(Fi ' a1) + p2(G1 ‘ 91) ' mlui ‘ mzvildv +f so (a) (a) 3 n_ 8 p _ _ —8t——Tt’for O. "' 1,20000n (203 12) If we substitute (2.3-12) and (2.3-10) into the.partial time derivative of (2.3-9) we obtain P § (“)c(“)f + (R - P )aemm = o (2 3-13) 0 d=1 kk 0 5t ° In the case of a binary mixture of a fluid and an elastic solid “'= l and (1)C = 1, therefore the equation (2.3—13) reduces to P — R 8e = 0 mm (2.3-14) fkk pO at — 22 0 < R g 1 0 < Po s l where fij is the rate of deformation tensor of fluid. Biot [9] stated the incompressibility condition for a mixture of a solid modeled by rigid spheres connected by helical springs, and an incompressible fluid. Such a relation can be obtained from relation (2.3-14) by setting R = l and integrating the equation. The incompressibility condition (2.3-14) reduces to fkk = 0 (2.3-15) in the following cases: aemm a) steady-state case where _FE— = 0 b) the sol1d is rigid so emm = 0 2.4 Constitutive Equations For a binary mixture of a linear Newtonian fluid and a non-linear elastic solid, Green and Steel [26] postulated the following constitutive equations. m1 = 0 m2 = O (2.4-1) Bxi A = A W,p2,T (2.4-2) 3 8x. 1 3 where A = U - TS (2.4-4) is Helmholtz free energy 1 _ _ _ 2(Oki + Oik) _ Aik + Aikrsfrs + Aikj(uj Vj) (2'4 5) 1 The + Tik) Bik + Bikrsfrs + Bikjhfi ‘ Vj) (2'4’6) 23 l — — = -— T - = 2(Cki 01k) 2‘ ki T1k) Dki + Dkirsfrs + D .. u. - V. (2.4-7) k13( J J) n = a . + a..(u. - v.) (2.4-8) 1 1 1rs rs 13 j 3 8x 3x _ 8T r 8T r _ qk Bk T,§§;:02:§§—' + Bkj(Tp§§:102§§; (uj Vj) 'F. =‘3. + EL f + Efl.(u. - v.) (2.4-10) 1 1 1rs rs 13 j 3 8x . . r — _ where all the coeff1c1ents depend on gig, p2, T and pr“ The dependence of the constitutive equations on the acceleration is excluded simply because we then would have to include non linear terms in f , A and (u. - v.) to satisfy the invari- rs rs 1 1 ance conditions. The above coefficients should also satisfy suitable symmetry conditions such as B = - Biksr ikrs Bkirs (2°4-11) By a usual invariance argument, they deduced that the 3x r . dependency on FY— should appear as a funct1on of qu where Bxi axi P q If the coefficients Aik are assumed to depend on dis- placement gradients only through pl, then they become iso- tropic functions of pl, and T thus: 02 Bikrs = A6ikars + “(siréks + Gisékr) (2°4-13) where Gij is the KroneCkerdelta and A, U are scalar functions of 01, 02 and T. By use of (2.4-10 and (1.3-l9) into (1.3-14) they obtained 24 ai(ui " Vi) + airsfrs (ui - Vi) + aij (ui - Vi) (uj " Vj) = 0 (2.4-14) It is easily deduced that where a(ij) 15 the symmetric part of aij' Thermodynamic Consideration.--Using (1.3-19) in (1.3-21) and then applying to an arbitrary volume it was obtained that DS qkT'k 2 4 1 pTEE-pr+qk'k- T 20 (0-6) With the help of (1.3-8,20) and(2.4-lr9), this equation becomes 3 3 X. X — pts 131- {Aik - m; 31: (airs—22:101. + (Bik + pp2 %%;51k)fik + [a1 ' p1 3%3’;;§'+ % 92(aiis + §%§;';;f§] (ui - Vi) + (Brsi + airs)frs(ui - Vi) + Aikrsdikfrs + Aikjdik(uj ’ Vj) + aij(“i ' ViHuj ' Vj) + Bikrsfikfrs + Dki(I‘ik - Aik) + DkijU‘ik - Aik)(uj - vj) + Dkirsfrs(rik ’ Aik) ' E3: k ' Bk T kiuj - Vj) 3 0 (2'4‘17) The inequality (2.4-17) should hold for arbitrary values of dik' fik' ui - vi and Pik - Aik' incompressibility condition (2.3-l4) is satisfied. We re- provided that the write the incompressibility condition in the following form: f + f + f + al(dll + d + d = 0 (2.4-18) ll 22 33 22 33) 25 R-P where a1_= 0 Po and (2.4-l7) as Nikfik + Ndkdik + rest of the terms 2 0 (2.4-17a) where N = B + pp 3A 6. (2.4-19) ik ik 2 5oz 1k 3x. 3x _ _ l 1 k 3A 3A _ Mik ‘ Aik §“’§§"§§' 3e + 3e (2'4 20) r 3 rs sr Eliminating f from (2.4-l7) by use of (2.4-18) gives 11 _ .. M .— fzz‘sz N11) + f33(N33 N11) + f12N12 + 511‘ 11 a1N11) (2.4-21) To satisfy the inequality we should have 8A s — ~35 (2.4-22) Nik = p Gik (2.4-23) M = " - ik alp Oik (2.4 24) a = 3A 302 _ 1 3A + 3A i 01 53; 3;? ‘792 36“ 55" (2.4-25) 1 IS sr Dkl = o Dklj = o Dkirs = 0 (2.4-26) Aikj = o Aikrs = 0 (2.4-27) where 5': N11 is a scalar function. Now using (2.4-22-27) in (2.4-17) yields a(ij)(ui ' Vi) + (Brsi + airs)frs(ui ' Vi) + Bikrsfikfrs 3 0 (2.4-28) The above equation imposes some inequality conditions on + a. ) and B a(ij)' (Brsi 1rs ikrs’ Substituting the last results 26 into constitutive equgiions and assuming that Bikrs’ Bikj’ aij are functions of gii-through pl only, we obtain the 3' following: B. . = a. = 0 a.. = a 6.. (2.4-29) 1k] 1rs 13 13 3x. 3x 1 1 k 8A 3A — o. = o . = —-p + + a p 6. (2.4-30) 1k k1 2 BXr 3XS(3ers Besr 1 1k T = T = -pp 8A 6 + - 6 + lf 6 + 2p f. ki ik 2 53; ik P ik rr ik 1k 3p 3e _ 8A 2 1 3A 3A rs _ "i ' p1 §B;'§§T' 592‘FE;; + aesr’ Bxi + a‘ui Vi) 1 (2.4~32) where we have u 2 o A + §u a o a 2 0 (2.4-33) Further Reduction for Linear Small Theory.--In order 'bo obtain the main constitutive equations subject to the laasic assumptions of this chapter, we adopt an expression siJnilar to the one used by Green and Steel [26] for the Ikelmholtz free energy, in the form — {_ 1 DA ’ A0 + OLlemm + 0‘2“ + 204emmenn + OLSemnemn l 2 + 2&6” + a8emmn (2.4-34) Where 3 = 31 + 32' the constants A0 , 0. .... depend on initial 1 dennsities of each substance. If we substitute (2.4-34) into (22.4—30-31) and retain the linear terms only we obtain: O‘131 01k - aléik + a4 - 5' emmaik + 2(al + 0L5)eik a1 _ + a8 + 7- n 6 k + alp 6ik (2.4-35) and _ a p a _ _. - —. — 2 — .1 2 T1k ‘ ' p2°‘2 ' P + [pzas + (92 + p) 3‘]” + p2(0‘8 '5 )emm 5ik + Afrréik + 2n fik (2.4-36) From (2.3-9) it is easily seen that: _ R - P0 Substituting for n in constitutive equations and applying the assumption of zero initial stresses, yields: A - PO Gik = “4 + 02 "§3“ “8 emmaik + 20‘seik + a1p 51k (2.4-38) T. =-b-OL+—2aae +—+Af 6 +2uf (24-39) 1k 2 8 fi .1 6 mm P rr ik ik ' “i = a(ui - Vi) (2.4-40) Now if we define p to be 2 -1 — — - p - a: [alp a1(pza8 + p2 ala6)emm] (2.4-41) the constitutive equations can be written in the form: . = - (2.4-42) 01k a1p 6ik + 2azeik + aBemm 51k rik = -p 51k + *frréik + 2p fik (2.4-43) where a2 = a5 a = a + 3 a a + a 23 2 a + a E a (2 4-44) 3 4 2 l 8 l 2 6 l 2 8 ° 28 and where p is equivalent to thermodynamic pressure defined for simple media. 2.5 Field Equations In this section we summarize the results obtained so far and state the field equations. Combining (1.3-12) and (1.3-17) and considering (2.4—1,40), we obtain the equations of motion: aoki _ _ §§;‘ + plFi ' a(ui ‘ Vi) = plai 3T . k1 — _ — Bxk + p261 + a(ui ‘ V ’ ‘ 0291 where 2 3 w 3v. 3w a = l g = g—i u = ‘—£ 1 at: ' 1 t ’ 1 at The constitutive equations are Gik = Oki = ‘a19 51k + 2a2eik + a3emm 51k Tik = Tki = ‘P 51k + Afrr 51k + zufik Hi = a(ui - Vi) and the strain and rate of deformation tensors are f.. 13 1 Bwi 8w. e..=—-—+—l 13 2 8x. 3x. 3 1 f mm kk = a1 "FE" (2.5-l) (2.5—2) (2.5-3) (2.5-4) (2.5—5) (2.5-6) (2.5-7) (2.5-8) (2.5—9) Equations (2.5-1-9) are 43 equations for 43 unknowns, namely Oki’ Tki, Ni, ai, gi, p, u., w., f.. and e. .. The 1 1 13 1] 29 system is mathematically complete provided that the appro- priate boundary conditions are given. 2.6 The Case of Compressible Fluid The constitutive equations for a binary mixture of a compressible Newtonian fluid and a linear elastic solid were obtained by Green and Steel [26]. Under the assumptions of this chapter those constitutive equations become: a. = 6. + 2a e. + den 6i (2.6-l) 1k OL4emm 1k 5 1k k T. = {- E a 2 6n - p2a8emm rr ik If we define p to be p = 32a6n + E2a8emm (2.6-3) then the constitutive equations can be written as Tik = -p6ik + Afrr éik + 2n fik (2.6-4) Oik - alp 61k + 2a2eik + a3emm 1k (2.6-5) where 2 a a — 8 —' — 8 a = _ a = a a = a - -——- (2.6-6) 1 p206 2 5 3 4 a6 Here again p is equivalent to a thermodynamic pressure. In analogy to the theory of simple Newtonian fluids, it is seen that in the case of a compressible fluid the hydrostatic pressure p has an equation of state of the form (2.6-3), while in the case of incompressibility,p introduces a new un- known to the system of equations together with an additional equation namely the incompressibility condition. It is seen that the constitutive equation of the solid is coupled with that of the fluid through the thermodynamic 30 pressure, while the partial stresses of the fluid are coupled with those of the solid by the solid dilatation through equa- tion of state or through the incompressibility condition. We keep in mind that the coefficients in these equations are not numerically the same as those of the corresponding single media, however we notice that all the equations must reduce to those for a single elastic solid or fluid when 92 or pl vanishes respectively. It is perhaps worth noting that the velocity gradients of the solidwere not included in the constitutive equations; however, if the partial stresses are assumed to depend on those variables as in [5] and [22] then the suitable modifi- cations should be made. 2.7 The Physical Interpretation oflthe Coefficients of Constitutive Equations for Partial Stresses The equations (2.4-38,39) show that the seven coef- ficients a4, a5, a6, a8. al, A and u are to be determined. However in the incompressible case, the knowledge of five coefficients a1, a2, a3, A and p is sufficient. This reduc- tion of number of independent constants is due to the fact that n is related to emm through the equation (2.4-37) and hence the constants a and a in the equation (2.4-34) can 6 8 4 and as. In order to relate these coefficients to those of the be absorbed by a solid and fluid components, let us first consider a medium under some external forces exerted by a highly permeable 31 agent such that the fluid pressure remains zero on the boundary at all times. After the transient part of the motion, the system arrives at an equilibrium state where the fluid pressure as well as the fluid velocity are zero. The constitutive equations (2.5-4) or (2.6-5) hold during the motion as well as in equilibrium state. We conclude that a = a2 = Ge a3 = 33 = Ae where Ge and Ae are the shear modulus and Lame's constant of the saturated porous elastic material. As it is pointed out in [13] a dry porous medium might not exhibit the same elastic properties as that of the saturated one. As an example they cited the case where elastic prOperties result from surface forces of a capillary nature at the interfaces of the fluid and solid. Whenever such differences are negligible the above coefficients are the ordinary prOperties of porous elastic solid and indepen- dent of the fluid components. The constant al is also a property of solid where a knowledge of initial porosity P0 and compressibility coefficients R of non-porous part is re- quired. Again it is conceivable to assume that these two constants may also depend on the fluid properties and the porosity may depend on the penetrability of the fluid. The physical interpretation of A and u is the same as of the single fluid; however, the numerical values of A and 32 u might differ because of the presence of the solid. Finally, in the compressible case the constant 31 remains to be determined which in turn requires the know— ledge of a and a8. In order to determine these two con- 6 stants use should be made of the equation of state (2.6-3). We may apply a constant fluid pressure p to a medium con— F1 fined in a rigid boundary such that emm = O and a6 would be found directly. Then a pressure p may be applied to the medium keeping the solid boundary tractions zero. The above two tests will supply the values of a6 and a8. L CHAPTER III METHODS OF SOLUTION IV The purpose of this chapter is to furnish the gen- eral solution of the system of equations for the steady-state case by means of displacement and stress functions. First the equations of Chapter II will be reduced to a system of .‘ ".m ”Qumran-nun— ‘1. differential equations in terms of displacements of the solid and the velocity vector of the fluid and the hydrostatic pressure only. A general method of solution is presented by means of scalar and vector functions satisfying harmonic, biharmonic and Helmholtz's equations. A stress function is presented to determine the partial stresses of the solid for steady-state plain—strain problems. The choice between Ege two different methods of finding the partial stresses of/soiid depends very much on problem at hand and especially on the prescribed boundary conditions. For the sake of illustration, the theory is applied to one and two dimensional steady-state problems. Atkin gt4al. [5] has presented a uniqueness theorem under a set of boundary conditions. Although the present theoryn together with the sets of boundary conditions used in the problems of this chapter is different from that used by Atkin, a similar 33 34 uniqueness theorem may be constructed; however we do not attempt to do so. 3.1 Displacement Equations If we substitute the rate of deformation and strain tensors in terms of displacements and velocities into the constitutive equations for partial stresses and then write the results in tensorial form, we obtain 3 = ~alp 3 + a2($$ + $6) + a3($:$)3 (3.1-l) : = -p 3 + u($$ + $6) + 1(v.v)3 (3.1-2) where symbols with z overhead denote second order tensors and those with arrows on the top denote vectors. Substitution of the eqUations (3.1-1,2) into the equations of motions (2.5-1,2) and using'Q.5-6) for diffusive force yields: 2+ + + + — 3 w -al$p + a2§°(§w + w$) + a3§($°w) - a(fi - V) = 1 SE7 and (3.1-3a) + -$p + WW-VI) + 1&ch + x7?) + ad? - V) = 32 g—‘EI- (3.1-3b) where the body forces are neglected. Observing the fact that: if?) vz‘fl <74 )‘v‘ = KW?) The equations (3.1-3,4) together with the incompressibility and condition can be written as the following: $ ( + 6 $~+ v23 (5 v — ‘ al p + a2 a3) ( w) + a2 - a - ) — pl Ft— 2 -$p + (A + u)$($-V) + uv v + a(U - V) = 62 33 (3.1-5) 35 + a ' O + - V v — al 5? 6 w (3.1 6) In steady-state case the time derivatives of dependent variables as well as the velocity of the solid vanish, hence we have: -al$p + (a2 + a3)$($-$) + aZVZE + a6 = 0 (3.1-7) -6....v2$ - .6 = o (3.1—8) 6-6 = o (3.1-9) where use has been made of the equation (3.1-9) in (3.1-8). The equations (3.1-7-9) constitute seven differential equa- tions for seven unknowns, 6, 5 and p. By application of 3 operator to the equation (3.1-8) and making use of (3.1—9) we obtain Vzp = 0 (3.1-10) It is seen that the hydrostatic pressure satisfies the Laplace equation, therefore any harmonic scalar function which satisfies the required boundary conditions would be a prOper expression for p. The general solution of 6 consists of the general solu- tion of the reduced equation uvzg - a? = 0 (3.1-11) plus any particular integral of the equation (3.1-8). If we (r) denote the general solution of (3.1-ll) by 5 and notice that a particular solution for the equation (3.1-8) is: § particular = -%§p (3.1-12) then the complete solution for (3.1-8) is obtained to be: 17 = ‘7‘“ + l .. aVP (3.1-l3) 36 (r) of the re- We observe that the general solution 6 duced equation must satisfy the condition (3.1-9); this im- poses some restrictions on the solution. In two dimensional case equation (3.1-9) reduces to (r) 3:§——— + 321....- O (3 1 14) 3x 3y - ' - It then follows that the velocity vector $(r) can be derived from a scalar function w SUCh that: Vx(r) = %% vy(r) = _ g¥ (3.1-15) Substituting for vx(r) and vy(r) into the equation (3.1-ll) yields: %_[uv2¢ - aw] = 0 (3.1-16) Y (3.1-l7) II o gituvzw - aw] The above pair of equations simply imply that the expression inside the bracket is a constant. This constant can be as- sumed to be zero without any loss of generality in the velocity solution. Therefore the problem reduces to finding a function w satisfying the Helmholtz equation. quw - aw = 0 (3.1-18) It is seen that the general solution of the velocity field consists of the linear combination of two scalar func- tions which satisfy the Laplace and Helmholtz equations. In order to obtain a general solution for the displacements of solid, we add equation (3.1—7) and (3.1-8) 37 2? = 0 (3.1-19) -(a + l)$p + (a2 + a3)§($-w) + a v23 + uV l 2 Equation(3.l-l9) shows that the vector function a2V2$ + uVZV is irrotational(referring to Helmholtz's rep- resentation) and hence can be expressed as the divergence of a scalar function $, a V25 + uV 2+ 2 V = $$ (3.1-20) where without loss of generality and under sufficient smooth- ness and integrability conditions we can find another scalar function ¢ such that 2 $ = V ¢ (3.1-21) Observing that $v2( ) = v2$( ) (3.1-22) we obtain, v2[a2$ + “a - $¢1 = 0 (3.1-23) Let aZE + .6 - $¢ = I (3.1-24) where w is a vector function which satisfies 2- V w = 0 (3.1-25) The general solution of E is a = L (fl; + $¢) .. Li; (3.1-26) a2 a2 Now if we apply the operator 6 to the equation (3.1-7) and make‘v use of (3.1—9) and (3.1-10) we obtain v2($-J) = 0 (3.1-27) (Rxmbining (3.1-l9), (3.1-20) and (3.1-21) yields: -(a + l)$p + (a2 + a3)$($-$) + 6V2¢ = 0 (3.1-28) l 38 By applying 6 operator to the equation (3.1-28) and using the relations (3.1-10) and (3.1-27) we obtain v4¢ = 0 (3.1-29) Hence ¢ is a biharmonic scalar function, and we again observe that the general solution of the system of equations (3.1-7-9) reduces to a linear combinations of the general solutions of some classical equations whose properties are rather well established. The solution to a particular prob- lem would be obtained by choice of these functions such that they satisfy the prescribed boundary conditions. Note that ¢ and W are not independent and they must satisfy 2 (a1 + l)’a2 a2 v ¢ = 2a2 + a p - ———————— v-T' (3.1-30) 3 3.2 Solution of the Steady-State_Plain Strain Problems by Means of Stress Function In the steady-state plain strain case the equations of motion (2.5-l), in the absence of external body force, reduce to the following: {aoxx 80X —§§—~+ .5?! + avx = O (3.2-1) < 30 30x i——XXI + ———X + av = O (3.2-2) 3y 3x y Substituting for vx and vy from (3.1-15) and rearranging the terms, we obtain 3 3x 3 (OXX - p) + 3§'(%Ky + aW) = O (3.2-3) 3 8 _ _ .37 (oyy - p) + -3—x- (Oxy - axp) — 0 (3.2 4) In analogy to the theory of linear elasticity [23] it is readily seen that there exist. two different scalar func- tions ¢ and x such that _ a¢ _ _3¢ _ OXX "' p - W O‘Xy + at! - '3—5'6 (3.2 5) 8x 3 - =-—— o - a = -—K 3.2-6 0yy 9 3x xy w 3y ( ) where ¢ and x are functions of X and y only. By adding the equations (3.2-5)2 and (3.2-6)2 and taking into account that o = 0 we obtain XY YX a _ 3¢ ' _ 5%.- §§.+ 2am (3.2 7) The condition (3.2-7) imposes a restriction on functions ¢ and x. We observe that this condition would be fulfilled if a function 6 is defined such that Y X x=§+afwdy gg-afwdx (3.2-8) J We now observe that if 6 is any arbitrary function of x and y, the ¢ and x functions derived from (3.2-8), would satiSfy (3.2-7) and hence the equations of motion. Similar to the case of two dimensional elasticity, we call 9 a stress function. Substituting for ¢ and X in relations (3.2—5) and (3.2-6) we obtain: 2 ‘X 3 6 3w OXX " P + g-y-i- - a '5? dx (3.2-9) 2 y _ a e aw Oyy ' P + ‘flx + a] 'a'x dy (3.2-10) 2 _ _a e _ Oxy — W (3.2 1].) Again, as in the theory of elasticity, the stress function 6 is the only unknown function, but it is necessary to use the compatibility condition which puts a condition on the otherwise arbitrary stress function. We observe that the compatibility conditions of the present theory are identi- cal to those of linear elasticity. For the compatibility conditions are nothing more than the mathematical conditions which insure the integrability of the strain-displacement equations. Since the strain-displacement relations for the solid are identical in form to those of classical elasticity, the compatibility conditions must be of the same mathemati- cal form. We, therefore, have aze 32a 32e __’2‘£ + __)51 -_- 2 15.3% (3.2-12) 3y 8X The non-vanishing stresses in this case are {Oxx =-a1p + 2a2 e + a3(exx + eyy) (3.2-13) < ny = -a1p + 2a2 eyy + a3(exx + eyy) (3.2-l4) \ny = 2a2 exy (3.2-15) Finding strains in terms of stresses yields: _ 1 eyy _ 4a2<32 + a3) [(2.12 + a3) (Oyy + alp) 513(0)“ + alp)] 1 exx = 4a2(a2 + a3? Bzaz + a3HPXX + alp) ' a3(Oyy + alp)] (3.2-l7) exy = 75; UXY (3.2-'18) Substituting for exx’ eyy and exy into compatibility equation and making use of relation (3.1-10) yields: 2 2 2 2 2 a Oxx 3 c a Gxx 3 o 3 Ox (2a2+a3)—T+——§1-a3——2—+—%Y—=4(a2+a3)xy 3y 3x 3x 3y (3.2-l9) . . o o o . . Substituting for xx’ yy and xy in terms of stress function in the above equation and simplifying the resulted equation, yields X y 3 3 1 4 f a [a 11) — v e = —3- dx - —3. dy (3.2-20) a ay ax The general solution for 6 is obtained by addition of the particular solution of (3.2-20) to a biharmonic scalar func- tion. Again the arbitrary constants should be chosen to satisfy the prescribed boundary conditions. 3.3 Qne Dimensional Problem of Infinite Plate As an illustration, we will solve the following simple problem. Let us consider an infinite plate resting on a rigid highly permeable medflnnand bounded in Cartesian system by the faces x1 = 0 and x1 = h. A constant fluid pressure pO is applied to the face x1 = 0. We assume that the lateral dis— placements can be neglected. Under the above conditions, the only non-vanishing components of the velocity and displacement vectors are wl = w1(xl) v1 = vl(xl) (3.3-l) IiI-dL-ml‘l -—.— u.,-s... n...‘ u_-_.-.un- 42 and the boundary conditions are: at x1 = 0 o (3.3-2) at x1 = h ml = 0 p = 0 (3.3-3) The equation (3.1-10)reduces to: a - .— 3;; — o (3.3 4) Integrating the above equation and using the boundary condi- tion (3.4-3)2 yields to p = c0(x — h) (3.3-5) where C0 is an arbitrary constant. From the relations (3.1-9) and (3.1-8) we conclude that (3.3-6) Using the results (3.3-5) and (3.3-6) into (3.1-7) and inte- grating, gives (a1 + l)C (a1 + l)C u’1 = 2(2a2 + a ' l 2(2a2 + a O 2 X + C X 0 2 3) 1 3) h - Clh (3.3-7) l where use has been made of the boundary condition (3.3-3)l and C1 is a new arbitrary constant. The fluid stresses are obtained from (3.1-2) as follows: The solid stresses are awl o _ - 11 (a3 + 2a2)§gI alp (3.3 9) Bwl 022 = 033 = -a1p + a3 3x; (3.3-10) 43 Oij = 0 if 1 # 3 (3.3-11) Substituting for ml and p yields to all = C0 (x + alh) + C1(2a2 + a3) (3.3-12) (a1 + 1)a3C0 022 = 033 = -alC0(x - h) + (2a2 + a3) + Cla3 (3.3-l3) The only remaining boundary condition (3.3-2) gives the con- stant C in term of C l 0 -p0 + C0h(l + al) C = C l (2a2 + a3) 0 (3.3-14) It may be seen that the solution is indeterminate within a constant C0. However, this indeterminacy may be re- moved by specifying the surface porosity at the face x = 0, and therefore prescribing the separate values of all and p at that face. 4.3 Semi-Infinite Strip Problem As another illustration, we consider an infinitely deep and long strip of an elastic solid with width n. We take the Cartesian system (x, y, 2) as shown in the figure below. 2 \f X see— 44 A fluid pressure p is applied to the surface y = 0. Under the above conditions it is conceivable to assume all vari- ables to be functions of x and y only. We assume the following boundary condition. at y = 0 p = A1 cos x vx = 0 Uyy = -A2 cos x ny = 0 (3.4-l) at x = i;- p = o vy = f(y) on = 0 0xy = o (3.4-2) at y = w p = 0 vx = vy = O (3.4-3) Let us observe that the solution for a general fluid pressure is similar to the present problem, because any applied fluid pressure f(x) for + n/2 2 x 2-w/2 can be expanded in Fourier series _ m nnx m . nnx _ f(x) — a0 + 2 an cos "c— + 2 bn 811’] _c (3.4 4) n=1 n=l N|l-' where 2C is the interval of Fourier expansion and not neces- sarily 2n, provided that f(x) satisfies the necessary re- quirements of Fourier expansion. We also notice that the problem of a strip with width K can be converted to the above problem by a simple change of variable. Let us further com- ment that the constants Al and A2 depend on fluid pressure and porosity factor P0. The equation (3.1-10) reduces to 2 2 L§+,§—§= o (3.4-5) 3x By The general solution for p can be written as: 45 p = §ay(Aa cos ax + Ba sin ax) + eay(Ca cos ax + Da sin ax) + ng(EB cos By + F8 sin By) + eBx(GB cos BY + H8 sin BY) For all values of a and B, A......H are constants. Due to the linearity of the equation, the above solutions can be summed or integrated over the values of a and 3. Considering that p should be even in x and decaying in y, it is readily seen that the solution has the following form p = Ae-By cos Bx Applying the boundary conditions (3.4-l)l and (3.4-2)1 gives the particular solution for p to be: p = Ale.y cos x (3.4-8) The equation (3.1-18) reduces to 2 2 u i—‘é’fifl + 23—2351)— - aw(x,y) = o (3.4-9) where the general solution of the above equation may be written as follows: 1 a w(x,y) = e-(a + U) Y(Aa cos ax + Ba sin ax) l l 2 a - _ 2 a ' + e(a + fi)2Y(Ca cos ax + Dd sin ax) + e (B + u)§y(EB cos By (82 + 3);" + F8 sin By) + e u y(GB cos By + H8 sin By) (3.4—10) Observing that the velocity components vx and vy are decaying in.y and are odd and even in x respectively, we assume the following form for w: 46 2 '1' °° % a w = B e-(a + fig y sin ax +I/r C cos yy sh (y2 + 3) X dy a 0 Y H where the second term is just a different form obtained from the combination of some of the terms in the above general solution. This term is retained in anticipation of its neces- sity to satisfy the boundary condition (3.4-2)2. The velocity components of the fluid are obtained from (3.1-13), where use has been made of (3.4-11) and (3.4-9L and are as follows 1 1 2 a 7 v = £.A e y sin x - B (a2 + E)2 e-(a + U) y sin ax x a l a n -J/- CY Y sin yy shy/Y2 + g-x dy (3.4-12) 0 .1. A 2 a 2 _ l -y _ -(a + -) y vy — ar-e cos x Bade n cos ax w l -J/- CY(Y2 + %)2 cos Yy chVY2 + % x dY (3.4-13) 0 The application of boundary condition (3.4-l)2 implies that l A _ a§'__l _ a — l and Ba(1 + E) — a (3.4 14) The only condition on the fluid velocity remaining to be satis- fied is (3.4-2)2. The constant CY should be chosen such that w l f(y) = -Jr CY(Y2 + %)7 ch/y2 + %.g dy (3.4—15) O In order to avoid mathematical complications, we take the case where f(y) = 0 and therefore CY = O. The velocity components become .1 2 A a v = —l[e-Y - e-(1 + Hay] Sin x (3.4-l6) x a 3 A l a _ l -y _ a -— -(1 + —) Y] cos x (3.4-l7) Vy '- a—[e (l + F) 2 e U The fluid stresses are avx Txx = -p + 2n §§— (3.4-18) 8v T = - + 2 3.4-19 W p u—lay < ) avx 3v TXY = 1.1 -3—y—- + 43X (3.4“20) Substituting for vx and vy and p from (3.4-8), (3.4-16) and (3.4-17), yields 1 a E ' — EH _ -y _ 2n -(l + -) y _ TXX - A1 ‘a l)e 3— e u 1 cos x (3.4 21) 2n 2 (1 + aFY = - __ “Y _ _E ' " _ Tyy A1 (a + l,e a e U cos x (3.4 22) 3 )JA _ l _'1 _ 3 Txy = —al - 2e y + (+ %)2 + (1 + %) 2 e (l + u) y sin x Since the boundary conditions of the solid part are such that the surface tractions are all known, then the prob- lem can be solved with the help of a stress function. The displacement solution may be ignored unless it is required. Obtaining w from (3.5-11) NIH A 1 a _ l a -— -(l + -9 Y - the equation (3.2-20)becomes 48 1 a 2 V46 = Alhe_'(l + E) y cos x (3.4-25) where h= (1+3) - (1+??-l u The solution of the above equation is 1 2 a * _ u -(l + —) y (3.4-26) 6—6h+Alh;2-e Ll COSX where 6h is a biharmonic scalar function of x and y and the second term is the particular solution of (3.4-25). We see that the above problem is equivalent to the similar strip problem of classical elasticity under identical loading ex- cept for the extra term coming from the interaction in the form of a body force. We conclude that the whole machinery of classical elasticity is applicable to the solution of the steady-state problems of the present theory for solid part, provided that the effect of interaction is treated as a prescribed body force. Since the elasticity solution of the semi-infinite strip, under the most general loading on the finite face and stress free elsewhere, has been solved, [29] we convert the present problem into two parts. 1. The original problem except for an additional stress distribution f(x) at face y = 0, such that we have oyy =--A2 cos x + f(x) at face y = 0, where f(x) is totally arbitrary at this stage. 2. A semi-infinite elastic solid with no body force and under the traction Oyy = -f(x) at face y = O and stress free elsewhere. 49 To solve the first part let us assume a to be 1 2 a 2 9 = Alhigf e-(1 + U) y cos x.+ Cle y cos x + Czye y cos x a ” gish Y ; + E cos yy x sh yx - ——————?— ch yx dy (3.4- where C1 and C2 are constants and EY is the coefficient of Fourier integral. 0 = A h E— (1 + Czyemy cos x - 0 fl ch YX dY (3.4- Cth 1 2 -l a 2 —1+—)y o = —A h E— + l + 3 ( ' C ‘ 1 yy 1 a2 ( u) n cos x ( )e - Czye-y cos x + Jr E cos yy 2y ch yx + y x sh yx 0 n ' n 25‘th 2 - Y ch YX dY (3.4- Cth l u2 a i a'7 xy = -Alh _2 (l + —)2 e-(1 + Ho y sin x - C e-y sin x a l - Czye-y sin x + Cze-y sin x + me EY Y sin Yy sh yx O + yx ch yx - sh Yx dY (3.4- The stresses are found to be e-(l + EJEP a _ 'Y + H) Al n cos x + (Cl+l)e 2C2e-Y cos x -J[“ YZEY cos yy [x shyx - ch 27) COS X 28) 29) 30) 50 The boundary condition (3.5-2Ly is identically satisfied. We now choose f(x) to be 00 TT TT 2 ESPY? 2 f(x) = E 2Y ch Yx + Y x sh Yx - ——————?— Y ch Yx dY The application of the remaining boundary conditions yields the following relations u2 a l - Alh —7 (1 + E)2 - C1 + c2 = 0 a U2 a _ .__ _. _ - + = A2 A1 h 2 (1 + u) 1 cl 1 0 a .2 .1 .3- A1h ‘2 (l + “)7 e' (l + £9 Y + [c1 - c2 + c2y1e'Y ” n n n — sh Y 7 ch Y 7 + Y 7) . - E Y Sin Yy dY (3.4-35) Y n 0 (3th The values of EY' Cl and C2 are obtained from the above equation. The complete solution of the problem is obtained by superposition of the two parts, while the solution to the second part, under the prescribed traction, f(X), can be ob- tained by method of [29]. We do not present it here. The present problem could also have been solved by use of dis- placement function presented in the early part of this chap- ter. The later method is particularly useful when either the displacements are desired or the prescribed boundary condi- tions are partially or totally in terms of displacement. CHAPTER IV DISCUSSIONS AND EVALUATION OF SOME THEORIES In this chapter we will briefly review the equations of fluid. flow through undeformable porous media based on Darcy's law. We will further review the Biot theory for flow of fluids through deformable media, a generalization of the former theory, where a modified Darcy's law is adopted. The purpose of this chapter is to examine the above theories from the standpoint of the theory of the present work. Of interest is also Brinkman's drag theory [37], which happens to be a useful modified Darcy's law. 4.1 Darcy's Law The first assumption throughout the classical field of fluid' flow through porous media is that the solid is un- deformable. Hence the pores of the media are fixed and their boundary surfaces are geometrically describable. Formally speaking, the problem is but a special case of the general problem of viscous flow of fluids between impermeable bound- aries. It is quite apparent that a flow problem through such a tortuousiiregular channel. is mathematically so compli- cated that the pure hydrodynamic approach is out of the question. 51 H. \J a? Mifin‘JLx- .'I.‘fl. I” mama 13.11432.- a. 52 Because of the above mentioned difficulties, an em- pirical dynamic equation, the Darcy's law, was established. This law asserts that, macroscopically the velocity is pro— portional to the pressure gradient acting on the fluid + < __E _ _ U Up (4.1 l) where u is the viscosity of the fluid and k is the permeabil- ity of the solid. The permeability k in the above equation has dimension of square length and expresses the ease of the fluid flow through porous media. The monographs and the literature on the field give detailed discussions of the permeability and the various formulas expressing it in terms of porosity and other variables. In the presence of the body force F we have 6 = -§.($p + f) (4.1-2) and in the case that the body force is derivable from a potential function G, we have F = - VG (4.1-3) and hence + v = - $¢ (4.1—4) where k ¢ ='F(p - G) (4.1-5) The expression (4.1-4) is the generalized form of Darcy's law. This law together with the equation of state and continuity equation constitute a complete system of equations. These equations supplemented by initial and boundary conditions provide all the necessary information 53 for the solution of any particular problem. For incom- pressible fluids, the equations of state and continuity reduce to p = constant V‘V = 0 (4.1-6) and hence Vzp = v2¢ = o (4.1-7) According to the above equations, we observe that there is no distinction between steady—state and nonsteady- state problems for incompressible fluids. In order to compare the above fundamental equation to those of the present theory, we assume the solid to be undeformable and hence 3 = E E 0. The equations (3.1—5) and (3.1-6) reduce to _ 1 2+ + _ L1 - aVV+V‘ 51% (4.18) 6-6 = o Vzp = o (4.1-9) In order to reduce (4.1-8) to (4.1-4), we see that the dif- fusive coefficient a should be assumed as _ u (4.1-10) a ' E This immediately implies that the diffusive force vanishes for ideal fluids. Substituting (4.1-10) into (4.1-8) we obtain - kV v + v = -— VP (4.1-11) We see that the above equation can be reduced to (4.1-l) if the term kVZV would be negligible compared to V. Since we have assumed that the velocity as well as its space deriva- tives are small of the same order, it is concluded that k “In. J! mytwma .fi juwjh-ifl 54 has to be small. In the following table, the values of per- meability are given for some materials. Table 4.l--Typical values of permeability and porosity for various materials. [37] Porous Solid Permeability (Darcy) Porosity Fraction Sand 2 - 180 0.31 - 50 Sandstone 10"7 - 11 0.08 - 0.40 Brick 0.0048 - 0.22 0.12 - 0.34 Soil 0.29 - 14 0.43 - 0.54 The permeability coefficient, for the above typical materials, is small enough that the first term can be ne- glected. However on the other extreme if k tends to infinity the equation (4.1-ll) becomes + _. 1.1V V = Vp (4.1-12) which is the equation for slow viscous flow of bluids. It is seen that the reduced equation (4.1-8) with the special choice of a from (4.1-10) includes two different extremes, namely a pure slow viscous flow of fluids, and flow of fluid through highly impermeable materials. The limitations of the resulting formulas for either case can be stated rigorously from the construction of the theory. We see that, under cer- tain limitations and conditions, the present theory gives almost identical formulas for flow of fluids through porous media as the classical theory. These limitations can be m “manna-tn.- Pam—VAX-..” 55 removed without any essential difficulties from continuum approach, whereas the classical field is based strictly on empirical viewpoints for a certain range and does not provide a basis for all possible generalization. Let us recall that the theory on which this work is built covers a very wide range of heterogeneous media under quite general conditions. } Atkin 21121: [S] has given one possible set of boundary conditions for which the problem has unique solu- tion. He remarked on the necessity of specifying at each point of the boundary two vector boundary conditions and a scalar functionfor thermal consideration. It is easily seen from equations (4.1-8) and (4.1-9) and the remarks in [5], that in steady-state case, a vector boundary condition should be prescribed at each point of the boundary for the fluid part only in order to have a complete solution. That this is true is also apparent from purely physical considerations. Contrary to the above remarks the classical equations do not allow to us to specify a vector boundary condition but only a scalar function. From our analysis, it is seen that this occurred because of neglect of the term kVZV. We con- clude that, in the case of low permeability, the error arising from the above simplification is insignificant far from the boundary while the error might be quite serious near and at the boundaries. The two dimensional problem of the former chapter illustrates this effect, for instance, vx vanishes at boundary 56 y = 0 according to (3.4-l6) however the classical approach gives vX = sin x. However for y==0 the dominating term is A —§ e y sin x, the term obtained by classical theory. 4.2 Brinkman's Theory It may be of interest that the same equation (4.1-ll) has been proposed in a series of papers by Brinkman [37]. His theory is based on the assumption that the solid parti- cles are spheres of radius R and that they are kept in posi- tion by external forces as in a bed of closely packed parti- cles supporting each other by contact. In the absence of the particles the stresses give rise to a force F dV which is given by Navier-Stokes equation 2 F = - Vp + uV V (4.2-1) 1 The presence of the solid spheres causes a damping force deV on the fluid elements. It was assumed that the damping force is proportional to the mean velocity and viscosity of the fluid and to the reciprocal of permeability so ‘* — 1‘. '* _ Since Therefore —§7*p+pv§7-%\7=0 (4.2-4) For high particle densities the term uVZV is negligible com- pared to E'V. This implies that Darcy's law is the limiting form of equation above for low permeability. The boundary conditions are that the tangential and normal velocity at the surface of the spheres to be zero. firm “I r. :.W n‘Tv‘i‘i‘ifim “—- 57 Although he obtained the same equation as (4.1-ll), the derivation and assumptions are drastically restricted, and the proof is not rigorous. 4.3 Biot's Theory The next major extension of the classical theory of flow through porous media has been done by Biot [6,17] who considered the solid to be elastically deformable. In the following we briefly review his equations. The constitutive equations for the stresses are Oij = 2Neij + Mekkdij + 05 (4.3-1) 1.. = Qekk + Leéi. = 06.. (4.3-2) where e: is the dilata‘ticndf the fluid defined by e = 6-5 where Up is the fluid displacement vector and o = - Bp, where B is the fraction of fluid element per unit section and p is the fluid pressure. The equation of motion in the absence of body forces for the quasi-static theory is (Oij + osij),j = 0 (4.3—4) and the modified Darcy's law is V0 = a(V - U) (4.3—5) While for the dynamic theory [12], the equations become: Q I 0310) r1- (pllui + plzvi) - a(vi - ui) (4.3-6) Vi) + a(vi - ui) (4.3-7) Q ll QJI o; d. (912“1 + 022 58 where 011 + 012 = pl, 022 + 012 = 02, 012 is a mass coupling parameter. Here again it is seen that our theory can be reduced to Biot's theory if terms with viscosity coeffi- cients A and u are eliminated from the constitutive equation for fluid. This can be justified if the viscosity is low enough such that the only dominating term in the expression for fluid stress would be the hydrostatic pressure. In the dynamic theory,equations (4.3-6,7) can be written in the following form 301. Sui 8 Sigl = p1 FE‘ + 012 3? (Vi ‘ Pi) ' a(vi ‘ “1) (4°3'8) 39— — 111.- 3 ( - u ) + (v - u ) (4 3-9) Bxi ‘ p2 as 012 3E V1 1 a i 1 ° From the standpoint of Green and Naghdi's theory, it is deduced that the diffusive force according to Biot's for- mula above, is 3 (v - ui) - a(vi - ui) (4.3-10) "i = p12 3? i This tells us that in the dynamic theory the diffusive force is the same as of quasi-static case plus the term .8— (v 012 at i of frame indifference, but the second term does not, and hence - ui). The first term satisfies the requirement are ‘ not allowed to appear in constitutive equations. As it is remarked in [26] the effect of acceleration cannot be in- troduced in linear constitutive equations. The above analysis shows that Biot's dynamic theory is incorrect to this extent. Biot [9] has extended his theory for the most general 59 anisotrOpic solid and also viscoelastic case. He and co- workers have worked out some problems based on his theory; however because of mathematical complexity, very little progress has been made. It is seen that the above theories can be explained in the light of the continuum mechanics approach while this approach enables us to rigorously analyze the existing theories and find out the pitfalls as well as their range of validity. u)...“ CHAPTER V MIXTURE OF TWO IDEAL FLUIDS AND AN ELASTIC SOLID As it is pointed out in Chapter I, the theory of Green and Naghdi [25] for a binary mixture has been general— ized for n components mixture by Mills [31]. The former authors have also proposed a theory for n components mixture [27], removing some of the restriction of the former one. We, however, use Mills' result in formulating the problem of the mixture of two incompressible fluids and an elastic solid. 5.1 Formulas and Notations We first consider a mixture of n substances which are in relative motion to each other. The equations (1.2-1-10) are all valid and we will use them in this section. The rate of deformation and vorticity tensors are respectively defined to be 1,2..1'1 (501—1) (0) _ l (a) (a) fij - 2—( Vi'j + Vj,i) for 04 (a)? ij for G = 1,2..n (5.1-2) %._((0()V_ _ (01) 1,j j,i where a comma denotes partial differentiation with respect to space coordinates. 60 61 In view of the energy balance relation proposed by Green and Naghdi [25], the following relation is postulated by Mills as a generalization of the former one 35.];(0U + 1 % pa (“1(GM)dV + j;{UM 2 Pa W)Vk n +% .1: Po. O”(0.)Vk(m)vi(on)vi)nk M = L‘or + 2 Po. (a)Fi(a)vi)dV +f(a:11( ti(a)v. _ h 1 A where the symbols have the same meaning as those of Chapter L .-H dA (5.1-3) II. By essentially the same method as that used by Green and Naghdi, namely the invariance requirement under different rigid body motions, the following relations were obtained: (a) _ (a) _ _ Gki,k + pa Fi ai - Ni (5.1 4) (a) n 2 (“)ni = 0 (5.1-5) a=l If we denote the symmetric part of stress tensor by O' (ik) and antisymmetric part bycfiik], we have n (0') The energy equation becomes n-l DU (8) B (n) (a)O (a) “BB ‘ Pr + qk,k ‘ Bil "1 (Vi ‘ Vi) ‘ all O(k1) dik (B) (n) _ _ Zolki] rik - rik)_ 0 (5.1 7) 62 The entropy production inequality was postulated to be DS r h fvpfi-dV-fva-dv -f;i,-dA>O (5.1-8) 5.2 Mixture of Two Ideal Compressible FIuids and an Elastic Solid Under Isothermal Condition In view of the constitutive equations for a mixture of a Newtonian fluid and an elastic solid, the following con- stitutive equations were postulated as the generalization of the former ones. A = A (ers’ pl' 92) (5.2-1) 5 = s (ers, pl. 02) (5.2-2) (a)o(ki) = (“’Aik a = 1,2,3 (5.2-3) (a)o[ki] = (a)Dik “ = 1'2’3 (5-2-4> (a)Tri = (a)ai + (e)aij((1)vj _ (3)Vj) +«nbij((2)vj _ (3)Vj, B = 1,2 (5.2-5) where a = 1,2 is referred to fluids one and two and a = 3 corresponds to the elastic solid. Substitution of the above constitutive equations into the entropy production inequality and using the same argument as before yields the following (a) o[ik]= 0 for a = 1,2,3 (5.2-6) (1) _ _ 3A _ Oik ’ pp1 53; 51k (5'2 7) m- -’-'I 63 ——— 6. (5.2-8) (3) i k 3A _ Oik P axr ax aers (5'2 9) (B)a. = g 0 3A BOY _ 0 3A BOB + 0 3A Berg (5 2-10) 1 Y=l B 80Y 5xi 508 5xi B Sers 5xi For isotrOpic case, the Helmholtz free energy, A, can be expanded in Taylor series to be 6A _ l a 2 -'2 a4emmenn+ aSemnemn + a6emmnl + a7"mmn2 + a8 n1 2 + a9 n2 + alonln2 (5.2-ll) where terms less than the second are not included because of the zero initial stresses. Substituting for A in constitutive equations and re- taining the linear terms only, results in the following equations: (1)0 _ _ _ — _ _ ik — {pla6emm + 2a8010l + plalon2>61k (5.2 12) (2) _ _ - -‘ 5 _ Cik — <92a7emm + Zagfiznz + 02a100%} ik (5.2 13) (3) —- 0 0 a - Oik"a4emm ik + 2aseik + asni ik + a7”2 ik (5'2 14) (l) _ (l) _ (3) (2) _ (3) _ ti - Kll( vi vi) + K12( vi Vi) (5.2 15a) (2) _ (l) _ (3) (2) _ (3) _ "i - K21( vi vi) + K22( vi vi) (5.2 16b) The equations of motion are (l)O = (1)fl_ (5.2—l6) ki,k 1 64 (2) _ (2) _ Oki,k — Ni (5.2 17) (3)0 + (1)w. + (2)n. - 0 (5.2-18) ki,k i i In the case of rigid solid e. 0, therefore 13' (1)0 =-E'(2a n + a n ) 6 (5 2-19) ik l 8 l 10 2 ik ' (2) (5.2-20) Gik ="’2(2“'9"2 + a1on1) 51k Substituting for partial stresses into equations of motion and making use of continuity equations, yields an 1 — .(l) _ (5.2-21) E— + pr V "' 0 3r) 2 - .(2) _ _ This will give us K 3n K 3n 2 2 _ ll 1 12 2 2a8V 01 + alOV 02 — gr: FE‘ + 3152 SE— (5.2-23) l K 8n K an 2 2 _ 21 l 22 2 310V T11 + 239V [)1 — $1.6; T + E:- SE— (5.2 24) In steady-state case the variables are independent of time, SO = 0 (5.2-25) 5.3 Case of Incompressible Fluids If the fluids are incompressible, the incompressi- bility condition (2.3-13) reduces to (1)C(1)d + (2)C(2)d + R ' Po 8emm _ 0 (5.3-1, kk kk R at _ 65 where (1)C + c = 1 TIT; = C (5.3-2) Similar to the argument made in Chapter II, the use of (5.3-1) and the entropy production inequality introduces a new unknown parameter p into constitutive equation for par- tial stresses. The results are (1) Oik - {pla6emm + Zaapln1 + plalon2 + p} 6ik (5.3-3) (2) 0 = - 3 a e + 2a 3 n + 3 a n l (l)C ik 2 7 mm 9 2 2 2 10 1 ‘TIT““ P (503-4) (3) _ e. _' 01k — {afinl + a7n2 + a4em¥>5ik + 2a5 1k (5.3 b) In the case of undeformable solid eij = 0 and (5.3-3,4) becomes (1) _ _ - — - Oik — {2a801nl + alopl + p} Sik (5.3 6) 1(l)C Substituting equations (5.3—6,7) into equations of motion and making use of continuity equations yields K 3n K 8n 2 2 2 _ 11 1 12 2 _ V p + 238V T11 + alOV T12 - 3576—- + 6. 6. at (5.3 8) 01 l 2 1 - (l)C V2 + a v2 + 2a v2 = K21 an1 + K22 an2 “TITE" P 10 n1 9 n2 EIEZ‘FE' ‘2'? t Tc“ 66 The incompressibility relation (5.3-l) reduces to (1)C n 1 _ (1)C n pl 1 "‘33“" 2 Eliminating p between (5.3-8) and (5.3-9) and using relation (5.3-10), we obtain: 2 and BV na = St- for a = 1,2 (5.3-ll) where _ ElC(2a8C - alo) + 32(alOC - 2a9) B = F’1 2 cb'z(K11c + K12) Tfi'lmxu + K 22) The equation (5.3-ll) is the diffusion equation for a mixture of two incompressible fluids and a rigid solid. Green and Adkins [24] derived the diffusion equations for a binary mixture of compressible fluids. Later Mills [30] gave the derivation of diffusion law for a mixture of two incom— pressible fluids. Equation (5.2-23) and (5.2-24), manipulated from Mills' results, are the diffusion law for a mixture of two compressible fluids flowing through a rigid body. Finally equation (5.3-ll) represents the diffusion of two incom- pressible ideal fluids through a solid. The equations (5.2-23) and (5.2-24) or (5.3-ll) may be considered as the modified Darcy's law of flow of two miscible fluids through porous rigid media. CHAPTER VI CONSTITUTIVE EQUATIONS FOR BINARY MIXTURE OF A NEWTONIAN FLUID AND A VISCOELASTIC o.-T-‘ SOLID 6.1 General Remarks In order to derive the desired constitutive equa- “." tions we follow the same approach as for binary mixture of an elastic solid and a Newtonian fluid. These equations will be derived under the assumptions of Chapter II. Before proceeding any further, it would be pertinent to consider the different approaches to the theory of visco- ' elasticity. The problem has been considered by many research workers where two main lines of work are of interest from thermodynamical viewpoints. The first line of activity is more or less based on Biot's linear thermodynamic theory [16-17] and the non-linear counterpart of it, where the idea of hidden coordinates has: been introduced to take care of dissipation phenomena. The other line is due to Coleman- [19-20], who has introduced the idea of materials with fading memory asserting that "deformation that occurred in the dis- tant past should have less influence in determining the present stress than those occurred in the recent past." The thermodynamic aspects of the problem are discussed [19]. The 67 68 viscoelastic materials are a special class of materials with fading memory. In the present work the idea of hidden coordinates is employed, However the alternative of using the idea of fading memory is possible, but has not been attempted. 6.2 Hidden Coordinates The thermodynamic system is assumed to have n degrees of freedom defined by n state variables 51....gn. These in- dependent state variables are alternatively called generalized coordinates. These coordinates are divided into two groups of hidden and observed ones. The system is assumed to be under the action of n generalized forces Qj in such a manner that Qidgi represents an incremental amount of work done on the system. The hidden variables are those whose correspond- ing conjugate forces are zero and are of interest only to the extent of their influence upon our observed variables. As an example, in a body under external loading, strain components are considered as observed variables and stress components as their conjugate external force, while the effect of "molec- ular configuration," interstitial atoms, dislocation, grain boundaries, etc., on stress-strain law can be accounted by hidden variables. The plan is to introduce the hidden co- ordinates in the equations of state and eliminate them from our ultimate stress-strain relationships. At this stage we are not concerned about the explicit form of 51. However, 69 they are assumed to be functionals of observed variables history. 6.3 Development of the Constitutive Equations We postulate the following constitutive equations in accordance with the equipresence principle. p A = A(eijl p2! Efll fijl dijl ui - Vi) (6.3-1) 5 = S(eijl 02: €£I fijr dijl ui - Vi) (6'3-2) f l _ _ -' L 2(Oki + Oik) ‘ Aik + Aikj (“j Vj) + Aikrsfrs + Aikrsdrs E, 1(T + T ) = B + B (u - v ) + B f + E d 2 k ik ik ikj j j ikrs rs ikrs rs _ l _ _ 2(Oki ' Gik) ‘ 2(Tki Tkl) ‘ Dki + Dkij(uj Vj) + Dkirsfrs Dikrsdrs (6°3-5) Hi = ai + aij(uj - vj) + airsfrs + airsdrs (6.3-6) ' I where Aik-oudepend on eij' 02, €£° The dependence on fi3 dij' ui - vi can be omitted from (6.3-l) and (6.3-2) by thermodynamic consideration as shown by Crochet and Naghdi [22]. A A(ei. J. 02. 4,) (6.3-7) S = S(e (6.3—8) ij' p2, EZ) Using (1.3-20) and the entr0py production inequality (2.4-16) yields 70 21 = - 21. _ 1 1 Pvt + "i‘ui Vi) + f(oki + Oik)dik + 2(Tki + Tik'fik U l Differentiating (6.2-7) gives, DA = 3A D02 + 3A Ders + 3A D52 BE “66“6E' Pae Dt Pa: Dt 2 rs 2 (6.3-10) D In view of (1.3—8) and (2.4-l), equation (6.3-10) becomes 3x. 8x. 962': ‘992 3%— fkk + 2 'ii ”21 (32A + 32A dij 2 r 3 rs sr 30 + (u _ v ) 9 3A 2 _ l 0 3A + 8A 8ers k k l 502 5yk 2 2 Sers §esr 5xk D5 + 9%A_.__1 (6.3-11) K D With the help of (6.3-11) and (6.3-3-6), equation (6.3-9) becomes DY 1 3x1 axk 3A 3A 8A EE“ Aik ' 2“ 3x 8x 3e + as dik + Bik+ PPz 53’51k fik r 5 rs sr 2 80 3e 3A 2 1 8A 3A rs + a. - p ——— ——— + — D ———— + (U- - V-) 1 l 802 ayi 2 2 aers Sesr axi i 1 + (Brsi + arsi) frs(ui - Vi) + (Arsi + arsi.) drs (ui - Vi) + Aikrsdikfrs + aij(ui ‘ Vi)(uj ' Vj’ + Bikrsfikfrs + Dki(rik ‘ Aik) + Dkij(rik ‘ Aik)(ui ‘ Vi) + Dkirsfrs(rik - A ) - 25—'EE-£-.+ A d d + E d f + E (P -A ) ik Pagz Dt ikrs rs ik ikrs rs ik kirs ik ik > 0 (6.3-12) 71 For a given state of deformation this inequality has to be satisfied for all arbitrary values of d. , f. , (u. - v.) 1k 1k 1 1 rik - Aik' Applying the above argument, we obtain: 3x. 3x 1 1 k 3A 3A A. = + (6.3-l3) 1k 7 er 3X5 Bers aesr _ _ 3A _ fl? 3p 3e i 3A 2 1 8A 3A rs a. = p ——— ——— + p + (6.3-15) l 1 3oz ayi 7 2(5ers Besr Bxi Dki = 0 Dkij = 0 Dkirs = 0 Dkirs = 0 (6'3'16) L} axr ' In the case where the coefficients are functions of 5?— s only through pl we will have Bikj = o Aikj = o aikj = 0 aikj = 0 (6.3-17) and the following relations = 6 6 6 6 Aikrs Aldik rs + u1( ir ks + is kr) ’ = A 6 6 6 6 6 6 Aikrs 2 ik rs + “2( ir ks + is kr) (6.3-18a) = l 6 6 6 6 Bikrs 3 ik rs + u3( ir ks + is kr) ’ = A 6 6 6 6 6 6 Bikrs 4 ik rs + “4‘ ir ks + is kr) . . . . _ l , . Con31der1ng the identity frsfrs - j frrfss + f rsf rs where f'rs is the deviatoric component of the tensor, we obtain the following inequalities 2 u 2 o u4 2 0 x3 + 3 u3 > o A + 3 2 o 4 2 ( +' )2 4 3 “4 lJ3“4 “1 “2 4M3 + § u3)(?\4 + § 114) > ml + *2) + §