A New Formulation and Computation of the. Triphasic Model for Mechano-electrochemical. Mixtures

Transcription

1 A New Formulation and Computation of the Triphasic Model for Mechano-electrochemical Mixtures Y.C. Hon 1 M.W. Lu 2 W.M. ue 3. hou 4 1 Department of Mathematics, City University of Hong Kong, Hong Kong 2 Department of Engineering Mechanics, Tsinghua University, Beijing , PRC. 3 Department of Mathematics, Hong Kong Baptist University, Hong Kong 4 Beijing Special Engineering Design and Research Institute, Beijing , PRC. ABSTRACT From the generalized rst law of thermodynamics for an irreversible thermodynamical system, a new set of governing equations for the mixture theory is derived based on the triphasic model for mechano-electrochemical mixtures. It is shown that, in the case of electroneutral solution, a new biphasic mixture theory including the electrochemical eects can be derived from the new governing equations. The chemical-expansion stress representing both the inuences of deformation on the xed charge density and the electric potential of xed charge eld is given. For comparison and verication purposes, the numerical solution for a conned compression problem of a charged hydrated soft tissue is computed using the multiquadric method. INTRODUCTION Biological tissues are multiphasic materials consisting of various amounts of living cells, extracellular matrix, and interstitial uid. To a large extent, the biorheological properties of connective tissue are dependent upon the compositional and ultrastructural properties of the extracellular matrix. For soft connective tissues, the extracellular matrix is 1

2 composed of a collagenous brous network which is swollen by a highly hydrophilic glycosaminoglycan gel. All connective tissues exhibit some viscoelastic behaviors to various degrees. Some of these apparent viscoelasticity is ascribable to the intrinsic viscoelasticity of the matrix macromolecules, namely, collagen and glycosaminoglycan. Biological tissues consist of a large proportion of water. Much of this water is movable and could move as a result of deformation of the tissue matrix. The frictional drag as the interstitial uid moves relates to the tissue matrix has been regarded as one of the most important mechanisms of energy dissipation during tissue deformation. The physics of interstitial ows is important not only in tissue rheology, but also in other electromechanical phenomena as the electrolytic interstitial uid carrying mobile ions/charges moves relative to the tissue matrix with xed charges, as well as in other chemomechanical phenomena during the concomitant movement of the interstitial water as small mobile ions move down a concentration gradient. This understanding is important to our appreciation of the transport mechanisms in biological tissues as well as the transduction mechanisms of physical stimuli in the micro-environment within the tissue. In this paper, a new set of governing equations for the triphasic mixture theory based on the triphasic model proposed by Lai et al. [1] is derived from the generalized rst law of thermodynamics for an irreversible thermodynamical system. In the newly derived model, the ionic concentration is expressed by the chemical energy and the chemical expansion stress is represented by the electrostatic energy. The eect of the diusive resistance is then included in the energy dissipation. This greatly simplies the governing equations for the triphasic model given by Lai et al. [1]. For an electroneutral solution, i.e. in absence of electric current, the new governing equations can be simplied to a new biphasic mixture model which includes the electrochemical eects. For comparison and verication purposes, the numerical solution for a one-dimensional conned compression problem of a charged hydrated soft tissue is computed using the multiquadric (MQ) method which was developed by Hon et al. [12] for solving the biphasic mixture model. PRELIMINARY ASSUMPTIONS AND NOTATIONS IN TRIPHASIC MODEL Hydrated biological soft tissues such as articular cartilage have been idealized by a triphasic model [1, 2], which consists of three phases: (1) a solid phase including the collagen bres, proteoglycan aggregates (PGA) with its xed negative charges, superscribed by s, (2) an interstitial water phase, superscribed by w, and (3) an ion phase of dissolved salt 2

3 consisting of cations (Na + ) and anions (Cl? ), superscribed by i. Each phase is assumed to be intrinsically incompressible. The solution, which consists of solvent water and solute ions, is called uid phase, superscribed by f (f = w [ i). It is assumed that the solution is electroneutral and no electric current occurs. The electrostatic eect of xed charge groups along the glycosaminoglycan (GAG) chains is considered in this paper. Density and concentration. The volume fraction of each phase ( = s; w; i) is given by = ( = s; w; i) (1) in which is the true volume of phase, is the mixture volume, which is equal to the apparent volume of each phase, ( = s; w; i) satisfy the continuity condition of mixture: = 1: (2) The apparent mass density of phase ( = s; w; i) is = ( = s; w; i); (3) T where T tissue, is is the true mass density of phase. The total density of the mixture, i.e. the = = T : (4) In the study of the chemical eects, the molar concentration per unit solution (uid phase) volume, c = f M ( = s; w; i); (5) is used. Here, M is the molecular weight of species and i is assumed to be comparatively very small so that f = w + i w. Basically there are two mechanisms for tissue swelling eect resulted from the ionic phase. The rst is the well known Donnan osmotic pressure caused by the concentration dierence between the interstitial solution and the external bathing solution. The other is the so{called chemical{expansion stress eected by the electrostatic repulsive forces of the xed charges. It is assumed that the number of xed charges per unit mass of solid 3

4 phase is kept constant at any time [1]. The molar concentration of the negative xed charges based on the apparent volume of the solid phase is then dened by: c F = dnf d = cf 0 J; (6) where n F is the total molar number of xed charges, c F 0 = dn F =d 0 is the value of c F at the reference conguration. The volume ratio of apparent solid phase, J = d 0 =d, is related to the Green strain tensor E of the apparent solid phase as follows: 1 q J = 1 + 2J 1 (E) + 4J 2 (E) + 8J 3 (E) 1 + J 1 (E); (7) where J 1 (E) = tr(e), J 2 (E), and J 3 (E) are the rst, second, and third invariant of Green strain tensor E. We note here that the element of the triphasic mixture is referred to the porous solid matrix, i.e., the element of the apparent solid phase, so that J s = J, E s = E. For innitesimal deformation, we obtain from (7) that Substituting equation (8) into (6) we have J = 1? tr(e): (8) c F = c F 0 (1? tr(e)): (9) The xed negative charges on the GAG attract mobile cations Na + to maintain the macroscopic electroneutral state. By assuming that each negative xed charge attracts a cation Na +, the molar concentration of the attracted cations Na + per unit uid volume can be dened as and for innitesimal deformation, Electroneutrality. c a = dnf d f = cf f ; (10) f = 1? s 0J = f 0(1 + s 0 tr(e)): (11) f 0 Denote the concentration of free cations (Na + ) f and anions Cl? in a solution of electroneutral salt per unit uid volume by c + f = c? = c i. Due to the presence of additional attracted cations, the concentration of Na + is c + = c + f + ca = c i + c a : (12) 4

5 The electric current density, denoted by I, is calculated by I = F C w [c + f (v +? v s ) + c a (v a? v s )? c? (v?? v s )]; (13) where F C = 96485C/mol is the Faraday constant and v + ; v a ; v?, and v s are the velocities of the free cation, attracted cation, anion, and solid matrix respectively. Due to the absence of electric current and the fact that the velocity of attracted cations is macroscopically equal to v s, equation (13) becomes 0 = F C w [c i (v +? v s ) + c a (v s? v s )? c i (v?? v s )]: Hence, v + = v? v i : (14) In other words, if the electric current density in an electroneutral solution is zero, the ion phase can be treated as an uniform phase for the consideration of mechanical eects such as movement and deformation. Continuity of mixture consisting of incompressible components. The velocities v s ; v w and v i of phases s; w; and i respectively must satisfy the continuity condition. The continuity equation for each phase is D Dt =? 5 v ( = s; w; i); + 5 ( v ) = 0 ( = s; w; i): (16) Substituting (3) into (16) and using the assumption for intrinsic incompressible condition, T = constant, + 5 ( v ) = 0 ( = s; w; i): (17) The three equations in (17) and the continuity condition of mixture (Eq. combined to 5 ( Using the relations of tensor analysis: 2) can be v ) = 0: (18) 5 ( v ) = 5 v + v 5 ( = s; w; i); (19) 5

6 5 v = I : 5v = I : d ( = s; w; i); (20) we obtain from equation (18) the continuity equation of the mixture consisting of incompressible components: ( I : d + v 5 ) = 0; (21) where I is the identity tensor and d is the deformation rate tensor which is a symmetric component of the velocity gradient tensor 5v. ENERGY, WORK, HEAT AND DISSIPATION IN TRIPHASIC MITURES Consider a triphasic mechano-electrochemical mixture system in which the total energy, work, heat and dissipation of the mixture is expressed in a summation form of its value in each phase consisting of the mixture. The rate of kinetic energy _K of the triphasic mixture is given by _K = _K = ( v _v ) d; (22) where v and _v are the velocity and acceleration of the phase respectively. The notation ( _ ) represents a material derivative. The rate of internal energy _U of the triphasic mechano-electrochemical mixture consists of three portions: _U = ( _ U + _ C + _ E ) d; (23) where _ U, _ C and _ E are the rates of strain energy density, chemical energy density and electrochemical energy density of phase per unit mass respectively. We introduce a Helmholtz energy function F : F = U? T S; (24) where T and S are the absolute temperature and entropy of the considering system. The density of Helmholtz energy and entropy per unit mass of phase is denoted by F and respectively. By taking derivative of equation (24), we obtain _ F = _ F d = 6 _ F d

7 where _ F = P = ( _ U + _ C + _ E? _ T? T _ ) d; (25) ( _ F )= represents the Helmholtz energy density of the mixture. The same terminology without superscript is also applied to _ U, _ C, and _ E. Both internal energy U and Helmholtz energy F are state functions depending on their state variables. For the triphasic mixture system, the following parameters are selected as the independent variables. These are the absolute temperature T as a thermal parameter, the Green strain tensor E as a mechanical parameter which is reduced to the Euler strain tensor in the case of innitesimal deformation, the apparent densities ( = s; w; i), which are related to the concentrations, as the chemical parameters, and the molar concentration of xed charges c F expressed as as a electrochemical parameter. The Helmholtz energy density can then be F = F (T; E; s ; w ; i ; c F ) ( = s; w; i): (26) Here, the Green strain tensor E of the apparent solid phase is selected as one of the state variables. The deformation eect of other phases will be considered by introducing the continuity equation (21) consisting of incompressible components. For the chemical energy density C in (25) we consider only the chemical eect of a mass transfer process, which is related to the concentration of species, but not the chemical reactions in which the mass of species are generated or exhausted. The chemical reaction is important for biological phenomena of the respiratory system and the digestive system, etc. However, it may be neglected for the deformation of passive biological tissues like cartilage [5]. Hence, the molar chemical potential ^ is given by where ~c, ~ ~ ( = s; w; i); (27) ~ F = F and ~ C = C are the molar concentration, Helmholtz energy and chemical energy per unit mixture volume respectively. n is the total molar number of species. The last equality of (27) due to the U; E and T in (25) are irrelevant to the chemical eect caused by the concentration dierence. The chemical potential density is dened to be: ~ @ = ^ M ( = s; w; i); (28) where M is the molecular weight of species. For a real solution at constant temperature 7

8 and pressure, the chemical potential density of species is equal to = 0 + RT M ln ( c ) ( = s; w; i); (29) where the constant 0 represents the chemical potential density of the reference state, R is the universal gas constant, is the activity coecient, which is equal to 1.0 for an ideal solution, and c is the molar concentration per uid volume. Chemical potential of the solid phase of articular cartilage comes mainly from the macromolecules of the proteoglycan aggregates (PGA), whose quantity is approximately 10% W/W or 7% / of the mixture. The inuence of collagen to chemical potential and osmotic pressure can be negligible [6]. A theoretical treatment on chemical potential and osmotic pressure of biological macromolecules is given by Tombs and Peacocke [7]. For the electro-chemical energy density E in (25) we consider only the electrostatic eect of the xed charges. In the absence of electric current, the electro{chemical eect related to the ow of mobile ions in an electroneutral solution may be neglected. The negative xed charge groups (SO? 3 and COO? ) along the GAG chains of proteoglycan aggregates (PGA) in the biological tissues are distributed very close to each other, spaced at 10 to 15 Angstroms apart [8] [9]. When the mixture deforms, the distance between the xed charges changes. This results in a variation of the eletrostatic potential, which is related to the chemical-expansion stress T C. The electrostatic potential at point P caused by a xed charge distribution q, which is considered to be composed of point charges, is ~ = q d; (30) 4"r where ~ E = E is the electro-chemical energy per unit mixture volume, " is the permittivity and r is the distance from point P to the xed charge distribution q. Although the exact distribution of xed charges is unknown, it is known that the electrostatic potential at P is mainly determined by its surrounding charges. We can assume that the distribution of the xed charges in the neighborhood of P keeps its pattern during the deformation. The distance r is then given by r = r 0 (1 + 1 tr(e)): (31) 3 Here tr(e) takes its value at P. Using q = F C c F 0 and substituting equation (31) into (30) and integrating over a neighborhood region of the point P, we obtain = tr(e) q d 0 (1? 1 tr(e)); (32) 4"r 0 3 8

9 where F 0 = C c F 0 d (33) 4"r 0 is the electrostatic potential of xed charge eld at the reference state. When the ion concentration of solution c increases, the permittivity " also increases due to the charge shielding eect caused by the ion cloud moving into the space between two repulsing xed charges. This results in an exponential decrease of electrostatic potential 0 : 0 = a 0 F C exp(?k c ); (34) where F C is the Faraday constant for the charge value of 1 mol charges, a 0 and k are two experimental material constants. From equations (30), (32) and (34) we obtain the molar eletrostatic potential of the xed charge eld F as follows: The rate of works E = F F C = a 0 exp(?k c )(1? 1 tr(e)): (35) 3 _W e done by the external forces is given by _W e = f f v d + S t v ds g; (36) where f is the body force per unit mass, t = is the drag force applied on the surface element, is the Cauchy stress tensor, and is the external normal on the surface element. From the Gaussian gradient formula and the symmetry of stress tensor, the surface integration in (36) can be transformed into a volume integral as follows: For = s; w; i, S t v ds = ( v ) ds = S Substituting into (36) we obtain _W e = 5 ( v )d = The rate of work done by the pressure p is?p _. [(5 ) v + : 5v ]d: [(5 + f ) v + : 5v ]d: (37) However, in the incompressible case _ = 0, it may be computed by the right hand side of the continuity equation (21) consisting of incompressible components: _W p =? p ( I : d + v 5 ) d: (38) 9

10 The total rate of work is then equal to _W = W _ e + W _ p : (39) The rate of heat transfer _Q = _ Q into the system is dened as f d? S q ds g; (40) where is the rate of heat generation per unit mass of phase, which may include the heat generation of chemical reaction, and q is the heat ux vector. Again, by using the Gaussian gradient formula we obtain _Q =? (5 q? ) d: (41) The energy dissipation can also be considered in the energy formulation of irreversible thermodynamics [10]. In the triphasic mixture theory for articular cartilage and biological tissues, the dissipation is mainly caused by the diusive resistance due to the relative ow between two dierent phases and. We assume that the diusive drag is proportional to their relative velocity: = K (v? v ); (42) where K = K is called diusive drag coecient and represents other phases which is dierent from in the mixture. It follows from (42) that The rate of energy dissipation _D = _ D is dened as = 0: (43) v d: (44) For a biphasic mixture consisting of solid phase and uid phase, equation (44) leads to _D = _ D s + _ D f =? K sf (v f? v s ) (v f? v s )d: (45) It follows that the rate of energy dissipation _D is always a negative function. GOERNING EQUATIONS OF TRIPHASIC MODEL 10

11 The governing equations of triphasic model can now be derived from a generalization of the rst law of thermodynamics. The balance relation on the rate of energy including the dissipation for an irreversible thermodynamical system is given by _K + _ U? _D = _W + _Q: (46) Substituting equations (22), (23), (25), (39), (41) and (44) into (46), we derive that = [ _v v + ( _ F + _ T + T _ )? v? p 5 v ]d f[(5 + f ) v + : 5v ]? p I : d? (5 q? )gd: (47) The material derivative of the Helmholtz function (26) is: For = s; w; i, F_ (T; E; s ; w ; i ; c F T _E + From the continuity equations (15) and (20) F _cf : (48) _ D Dt =? I : d ( = s; w; i): (49) From equation (6) and the fact that _ J =?J 5 v s we obtain _c F =?c F 0 J I : d s : (50) Notice that _E = (F s ) T d s F s ; (51) where F s and (F s ) T Using equation (51) are the deformation gradient tensor and its transpose respectively. : _ : ( (Fs ) T d s F s ) = (Fs ) T ) : d s : (52) Substituting equations (49), (50) and (52) into (48) we obtain the formula for the material derivatives of the Helmholtz function: For = s; w; i, F_ _ + (Fs ) T? c F F I) I : d : (53) Using : 5v = : d; (54) 11

12 and substituting equation (53) into (47) we have:? [5 + f? _v + + p 5 ] v d + f[f F ( ) (Fs ) T? c F J ( [?? p I? F ~ )I ] ~ )I ] : d gd [5 q + T _? ]d ~ + ] _ T d = 0; (55) where d s is a component of d when = s. Since the rates v ; d ; and _ T change independently, all of the terms contained in each square bracket of equation (55) should be equal to zero. A set of governing equations for the triphasic mixture model can now be derived. Before listing down all the equations, the physical meaning of some terms in equation (55) is discussed in the following. Using equation (28) the chemical term ~ ~F ) = ( = s; w; i): (56) The electrostatic term in equation (55) is related to the chemical-expansion stress T C which appeared in the triphasic model developed by Lai et al. [1] but was not fully understood. Since the other components of F are irrelevant to the electrostatic eect, we F = ~ F From equations (8),(57), and (35) we derive that T C (c F 0 ; c ; E) c F 0 J = T C0 (1? 4 3 tr(e)); ~ F : F = T F C 0 (1? tr(e))(1? 1 tr(e)); (58) 3 and T C0 (c F 0 ; c ) = a 0 c F 0 exp(?kc ): (59) 12

13 It is now clear that the chemical-expansion stress T C is related to the concentration of xed charges, the ion concentration of solution and also the inuences of deformation. The factors (1? tr(e)) and (1? 1 tr(e)) represent the inuences of deformation on the 3 xed charge density and the electric potential of xed charge eld respectively. The mechanical term in equation (55) is corresponding to the stress tensor of solid phase, in which is the second Piola-Kirchho stress tensor, ~ ~ = s E F s s E (Fs ) T = s E (60) is the Cauchy stress tensor. Finally, by substituting equations (54), (56), (58), and (60) into (55), and equating the terms contained in each square bracket in equation (55) to zero, we obtain the following governing equations for triphasic mixtures: Momentum equations 5 + f? _v + + p 5 = 0 ( = s; w; i); (61) Constitutive equations s =? s pi + s E? s s I? T C I ( for solid phase ); (62) =? pi? I ( for = w; i ); (63) For isotropic elastic material, s E is given by s E = str(e)i + 2 s E; (64) where s and s are the Lame elasticity constants. Heat transfer equation 5 q + T _? = 0 ( = s; w; i): (65) A denition for the entropy density can also be obtained from (55) : =? F ( = s; w; i): (66) 13

14 From equations (2) and (43), the three equations for the three phases s; w; i in (61) can be summarized as a total momentum equation for the triphasic mixture model: 5 + f? _v = 0; (67) where = P, f = P f =, and the density average velocity v is dened by v = ( v )=: (68) Equations (62){(64) governs the mechano-electrochemical coupled constitutive relations. From the derivation of equation (55), it can be observed that the coupling eect, including the chemical term and electrostatic term T C, comes from the volume, and therefore the density and concentration variation. For the case of electroneutral solution, the following derives a new biphasic formulation of the triphasic mixtures by treating the uid phase as the water phase and the ion phase as an uniform electrolytic solution phase. A NEW BIPHASIC FORMULATION OF THE TRIPHASIC MITURES Based on the conditions of electroneutrality and an absence of electric current, the ion phase including the cations and the anions can be considered as an uniform electroneutral salt phase, the velocity of which is denoted by v i (see equation (14) ). Furthermore, due to the absence of external electric eld, there exists no driving force pushing the ions to move macroscopically dierent from the water. It may then be assumed that v i = v w v f : (69) This means that the electroneutral salt and the water can be considered as an uniform electrolytic solution phase from the view point of the motion and deformation. The triphasic model derived in the previous section can then be simplied to a mechanoelectrochemical biphasic mixture model, which consists of a porous solid phase and an uniform electrolytic uid phase. The basic equations of this new biphasic mixture model can be derived directly from the above governing equations of the triphasic mixture model. By composing the equations for the water and ion phases together, equations (61){(63) and (16) are reformulated into the following governing equations for mechano-electrochemical biphasic mixture model: 14

15 Momentum equations r s + s f s? s _v s + s + pr' s = 0 ( = s); (70) r f + f f f? f _v f + f + pr' f = 0 ( = f = w [ i); (71) From equation (42) and the assumption v w = v i v f we have f = K sf (v f? v s ) =? s ; (72) where K sf = K sw + K si : Constitutive equations s =?' s pi + s E? (s s + T c )I ( = s); (73) f = f I =?' f pi? f f I ( = f); (74) where s E = str(e)i + 2 s E: (75) Continuity + r ('s v s ) = 0 ( = s); + r ('f v f ) = 0 ( = f); (77) where ' f = ' w + ' i ' w ; f = w + i w ; (78) f = w + i ; f f = ( w f w + i f i )= f ; f = w + i ; (79) v f = ( w v w + i v i )= f = v w = v i ; (80) f = ( w w + i i )= f ; (81) with the approximation sign \" denoted for dilute solution. For an osmotic process at constant temperature, the chemical potential and pressure change with respect to the concentration variation due to osmosis. Hence, they satisfy the Gibbs-Duhem equation: s d s + f d f = dp: (82) 15

16 For dilute solution, integrating equation (82) we have s ( s? s 0) + f ( f? f 0) = p osm : (83) where s 0 and f 0 are the chemical potentials at the reference state, p osm is the osmotic pressure. The gradient of (83) is r( s s ) + r( f f ) = rp osm : (84) By summarizing the equations of the solid phase and the uid phase, and using equations (2), (72), (83) and (84), we obtain the following basic equations for the electrochemical biphasic mixtures: Momentum equation r + f? _v = 0; (85) in which rp osm is included in the gradient of the stress tensor. Constitutive equation = s E? (p + T c)i; (86) in which p osm is included in the pressure p. Continuity equation r (' s v s + ' f v f ) = 0: (87) Substituting the constitutive equations (73) and (74) into (70) and (71) we obtain:?' s rp + r s E + s? r( s s + T c ) + f s? s _v s = 0 for solid; (88)?' f rp + f? r( f f ) + f f? f _v f = 0 for uid: (89) Using equation (84) and the relation of partial pressure and substituting (86) into (85), we obtain: Mechano-electrochemical coupled momentum equations:?' s rp + r s E + s? rt c + f s? s _v s = 0 for solid; (90)?' f rp + f + f f? f _v f = 0 for uid; (91) r s E? r(p + T c) + f? _v = 0 for mixture: (92) in which p osm is included in the pressure p. To further understand the eect of the ion phase in the triphasic model, the above mechano-electrochemical biphasic model is compared with the classical one developed by Mow and Lai [3]{[5]. The result of comparison is listed as follows: 16

17 (I) the momentum equations (70), (71), and (85) and the continuity equations (76), (77), and (87) are the same as the classical biphasic model. Two dierences in [5] are: (i) the terms pr' ( = s; f) in the continuity equations are merged into the diusive drag ; and (ii) the body forces f and the inertial forces? _v are neglected. (II) The electrochemical eects are embodied only in the constitutive equations. If the chemical term and the electrostatic term T c are neglected, equations (73), (74), and (86) can be reduced to the classical biphasic constitutive equations. (III) Equations (73) and (74) indicate that from the view point of mechanics, the chemical potential multiplied by the mass density and the electrostatic potential multiplied by the charge concentration can be considered as a generalized stress, which is an isotropic stress tensor similar to the pressure. (I) The true stress tensors are obtained from equations (73) and (74) by dividing a volume fraction ' : s T = s E ' s? (p + s T s + T c )I; (93) ' s f T = f T I =?(p + f T f )I: (94) Dividing the scalar coecient of equation (94) by? f T, we obtain f? f T f T = p f T + f ; (95) where f is the net chemical potential caused by the concentration variation of species and f is a generalized chemical potential including the eect of pressure. From the viewpoint of chemists, the true stress tensor s E ='s in (93) divided by the solid density s T considered as a non-isotropic chemical potential tensor of deformed solid phase. can be () Neglecting the body force and the inertial force, we reformulate equation (89) into f r f + f r f + ' f rp = K sf (v f? v s ): (96) It can be seen that the gradients of chemical potential, pressure and concentration are the driving forces to push the uid to ow and to overcome the diusion resistance. NUMERICAL COMPUTATION To verify and compare the newly developed triphasic mixture model with the biphasic model, the numerical solution for a one-dimensional conned compression problem of 17

18 charged hydrated soft tissue with similar settings given by Hon et al. [12] is given. The proposed multiquadric (MQ) method in [12] is used so that the original computer program for solving the biphasic model can easily be modied to obtain the numerical solutions. Consider a specimen of charged hydrated soft tissue placed snugly inside a circular conning chamber (Fig. 1). The sample is equilibrated in a NaCl solution. Before the applying of the ramp displacement, the tissue is in a swollen state, which is taken as the reference conguration. The history of the ramp displacement load is illustrated in Fig. 2. Governing Equations. In one-dimensional case, the continuity equation (87) can be written as Substituting s = 1? f into equation (97), (s v s + f v f ) = 0: @z [f (v s? v f )]: (98) Since the velocities v s and v f are zeroes at z = 0 for all time, the integration of equation (98) with respect to z = f (v s? v f ); (99) where u s is the displacement of the solid phase. If the body force and the inertial force are neglected, the momentum equations (90) and (91) can be written + K sf(v s? v f ) = 0; + ( s + 2 s c + K sf(v f? v s ) = 0: (101) From equations (99) (101), we obtain the governing = (f ) 2 K sf [( s + 2 s u c ]: This equation, together with its initial and boundary conditions, is solved by using the MQ method [12]. Results and Discussions. 18

19 The parameters used in this numerical calculations are quoted from a private communication paper of Mow et al. as follows: T = 298K R = 8:314J=mol K c F 0 = 0:15mEq=ml c = 0:15M k = 7:5M?1 K sf = 5.25e-14 Ns/m 4 s + 2 s = 0:3MP a f = T 1000kg=m3 a 0 = 2:5MP a=(meq=ml) = = = 1:0 The strain history with respect to the time t at dierent normalized height z=h is given in Fig. 3. For comparison purpose, the strain at the surface z=h = 1:0 resulted from the biphasic theory is represented by a dashed curve. It can be observed from Fig. 3 that the curves representing the strain history of the triphasic mixtures are similar to that of biphasic theory with an expectation that these strain values are lower due to the resistance from the osmotic pressure and the chemical-expansion stress. Fig. 3 indicates that the maximum strain value always occurs at the surface. Also, after the ramp displacement (t > 200s), the strain at the upper part of the tissue relaxes but still increases at the lower part until the nal equilibrated value of 10 %. Fig. 4 shows the nonlinear strain distribution with normalized height at various times. The solid curves and the dashed curves are corresponding to the compressing and relaxing stages respectively. The nal equilibrated state tends to an uniform strain distribution indicated by the curve H. For osmotic pressure and chemical-expansion stress, their history and distribution are similar to Fig. 3 and Fig. 4. The solid curves in Fig. 5 represent the time history of the pressure at dierent normalized height and the dashed curve is the total stress history. The total stress distributes uniformly throughout the height. The signicant dierence between the pressure history and the strain history is that the pressure at all normalized height undergoes a relaxing stage, but the strain at the lower part increases to reach the equilibrated value. Fig. 6 gives the distribution of pressure with normalized height at various times. It can be seen that the pressure takes its maximum at the bottom and decreases toward surface. This pressure gradient overcomes the diusive drag and drives the uid to ow out from the surface. To further compare the dierences between the triphasic and the biphasic theories, Fig. 7 gives the history curves of the pressure and total stress near the bottom calculated using the triphasic theory (the solid curves) and the biphasic theory (the dashed curves) respectively. Curve D represents the diusive pressure p dif f, which tends to zero near the 19

20 equilibrated state, as calculated by where Ha = s + 2 s p dif f = (Ha T C j z=h) (103) are the volume modulus and the strain of the solid phase respectively. If the eect of the chemical-expansion stress on the volume modulus, (4=3)T C0, is neglected, curve E from the biphasic theory can be obtained. represents the pressure p, which is calculated by Curve C p = p dif f + p osm j z=h (104) where p osm j z=h is the Donnan osmotic pressure at the surface. The total stress of the mixture, represented by the curve A, is calculated by? = (p? s E + T C ) j z=h : (105) The curve B represents the chemical-expansion stress T C j z=h but not the elastic stress? s E j z=h of the solid phase. The total stress from the biphasic theory is indicated by the curve F. It can be observed that, according to the triphasic theory, a large part of loading in the equilibrated state is caused by the pressure, i.e. the osmotic pressure. This is dierent from the equilibrated pressure calculated from the biphasic theory, which always tends to zero. Fig. 8 shows the inunces of the salt concentration c to the strain (curves A; A 1 ), the pressure (curves B; B 1 ), and the total stress (curves C; C 1 ). The solid curves and the dashed curves are corresponding to the maximum loading (t = 200s) and the equilibrium (t = 600s) state respectively. CONCLUSION Based on the triphasic mixture model developed by Lai et al. [1], all the expressions on the rate of energy, work, heat, and dissipation are given, in which the eect of the ionic concentration is expressed by the chemical energy, the eect of the chemical-expansion stress is reected in the electrostatic energy, and the eect of the diusive resistance is included in the energy dissipation. From the generalized rst law of thermodynamics for irreversible thermodynamical system, a new set of governing equations for the triphasic mixture theory is derived in this paper. In the case of an absence of external electric eld and electric current (electroneutral solution), it has been assumed that the velocities of the free cations and of the anions 20

21 are equivalent, i.e. v + = v? v i. The ion phase can then be treated as an uniform phase for the consideration of the mechanical eects. By assuming that the ion phase ows macroscopically with respect to the water phase, i.e. v i = v w v f, a new mechano-electrochemical biphasic mixture model is derived, which consists of (1) a porous charged solid phase including the collagen ber and proteoglycan aggregates with its xed negative charges and (2) an electroneutrally electrolytic uid (solution) phase including the interstitial water, cations and anions. This new set of governing equations for the biphasic model is directly derived from simplifying the newly derived set of equations for the triphasic mixture theory. It is also shown that, when the electrochemical eects are neglected, the basic equations of the new model can be reduced to the classical biphasic theory [3]{[5]. If only the electrostatic eects is neglected, it becomes a mechano-chemical biphasic theory similar to the model given Fung [11]. The chemical-expansion stress comes from the electrostatic eect of the xed charges. A new expression of chemical-expansion stress consisting both the inuences of deformation on the xed charge density and the electric potential of xed charge eld is presented. It is shown that the chemical-expansion stress is composed of two terms: one is related to the local tr(e) which is represented by an equivalent volume modulus 4T 3 C 0 as given in (103) and the other, T C j z=h in (105), represents the surface chemical expansion. A one-dimensional conned-compression problem of a charged hydrated-soft tissue is solved by using the multiquadric (MQ) method. Comparison and discussion of the results from the triphasic and biphasic theories are given. The numerical formulation and computation for the two-dimensional cases are in progress and will be reported soon. ACKNOWLEDGMENTS This project is supported by the Research Grant Council of project No and the City University of Hong Kong of grant No The authors would also like to thank Prof..C. Mow and Prof. A.F.T. Mak for their invaluable comments on the rst draft of this paper. References [1] W.M.Lai, J.S.Hou and.c.mow, 'A triphasic theory for the swelling and deformation behaviors of articular cartilage', ASME J. Biomech. Eng., 113, , (1991). 21

22 [2] W.Y.Gu, W.M.Lai and.c.mow, 'Transport of uid and ions through a porouspermeable charged-hydrated tissue and streaming potential data on normal bovine articular cartilage', J. Biomech., 26, , (1993). [3].C.Mow, S.C.Kuei, W.M.Lai and C.G.Armstrong, 'Biphasic creep and stress relation of articular cartilage in comparison: theory and experiments', ASME J. Biomech. Eng., 102, 73-83, (1980). [4] R.L.Spilker and J.K.Suh, 'Formulation and evaluation of a nite element model for the biphasic model of hydrated soft tissues', Comp. Struct., 35, , (1990). [5] J.K.Suh, R.L.Spilker and M.H.Holmes, 'A penalty nite element analysis for nonlinear mechanics of biphasic hydrated soft tissue under large deformation', Int. J. Num. Meth. Eng., 32, , (1991). [6] A.Maroudas, Physicochemical properties of articular cartilage, in M.A.R.Freeman (ed.), Adult Articular Cartilage, Pitman Medical, , [7] M.P.Tombs and A.R.Peacocke, The Osmotic Pressure of Biological Macromolecules, Clarendon Press, [8].C.Mow, A.Ratclie and A.R.Poole, 'Cartilage and diarthrodial joints as paradigms for hierarchical materials and structure', Biomaterials, 13, 67-97, (1992). [9] H.Huir, 'Proteoglycans as organisers of the intercellular matrix', Soc. Tran., 9, , (1983). [10] M.A.Biot, 'New variational irreversible thermodynamics of open physical-chemical continua', Proc. of the IUTAM Symp. on ariational Methods in the Mechanics of Solid, USA, 29-39, [11] Y.C.Fung, Biomechanics: Motion, Flow, Stress and Growth, Springer-erlag, [12] Y.C.Hon, M.W.Lu, W.M.ue and Y.M.hu, 'Multiquadric method for the numerical solution of a biphasic mixture model', Appl. Math. Comp., 88, , (1997). 22

23 Loading Piston o z h c* Tissue Sample Porous Platen Confining Chamber Fig. 1 Schematic of the 1-D conned compression of a charged hydrated soft tissue equilibrated with NaCl solution 23

24 u(mm) t0=200s t Fig. 2 Prescribed ramp displacement applied at the surface 24

25 Compressive strain Bi G F E D C B Curve Normalized height(z/h) A 0.08 B 0.29 C 0.50 D 0.71 E 0.81 F 0.92 G 1.0 Bi 1.0(biphasic) 0.05 A Time(second) Fig. 3 Strain history at dierent normalized height z=h 25

26 Compressive Strain Curve Time(s) A 20 B 60 C 120 D 200 E 208 F 220 G 260 H 600 C G D E F H 0.06 B A Normalized Height(z/h) Fig. 4 Strain distribution with normalized height at various times 26

27 Pressure and Total Stress(MPa) T A C E G Curve Normalized height A 0.08 C 0.50 E 0.81 G Time(second) Fig. 5 Pressure and total stress history 27

28 C D E Curve Time(s) A 20 B 60 C 120 D 200 E 208 F 220 G 260 H 600 Pressure(MPa) 0.08 F B 0.06 G A 0.04 H Normalized Height(z/h) Fig. 6 Pressure distribution with normalized height at various times 28

29 0.16 Normalized height= Pressure and Stress(MPa) D F C A B 0.02 E Time(second) Fig. 7 Pressure and total stress history near the bottom computed by using the triphasic and biphasic theories respectively 29

30 A Strain, Pressure(MPa), Total Stress(MPa) B B1 C C1 A c* (M) Fig. 8 Inunces of salt concentration c to strain, pressure, and total stress 30