A finite deformation theory for cartilage and other soft hydrated connective tissues—I. Equilibrium results

1. Btihnmm Vol 2.3.No. L pp. 145-155. 1990. Pnntcd 111 Gral Bntua

m-9_%

90 SJ.W* 00

S I990 Rrpmon

Pressplc

A FINITE DEFORMATION THEORY FOR CARTILAGE AND OTHER SOFT HYDRATED CONNECTIVE TISSUES-I. EQUILIBRIUM RESULTS MICHAEL K. KWAN,* W. MICHAEL

and VAN C. Mow+

Llrrt

&ision Of Orthopaedic Surgery. University of California at San Diego, La lolla, CA 92093, U.S.A.; and tOrthopaedic Research Laboratory. Department of Orthopaedic Surgery. College of Physicians and Surgeons and Dept. of Mechanical Engineering College of Engineering and Applied Science, Columbia University. New York. NY 10032, U.S.A. l

Rrcei~ired in final

jorm19 Oecemher 1986.

Abstract-The determination of valid stress-strain relations for articular cartilage under finite deformation conditions is a prerequisite for constructing models for synovial joint lubrication. Under physiological conditions of high strain rates and/or high stressesin the joint. large strains occur in cartilage. A finite deformation theory valid for describing cartilage. as well as other soft hydrated connective tissues under large loads, has been developed. This theory is based on the choice of a specific Helmholtr energy function which satisfies the generalized Coleman-No11 (GCN,) condition and the Baker-Ericksen (B-E) inequalities established in finite elasticity theory. In addition, the finite deformation biphasic theory includes the effects of strain-dependent porosity and permeability. These nonlinear efTectsare essential for properly describing the biomechanical behavior of articular cartilage. even when strain rates are low and strains are infinitesimal. The finite deformation theory describes the large strain behavior of cartilage observed in onedimensional confined compression experiments at equilibrium. and it reduces to the linear biphasic theory under infinitesimal strain and slow strain rate conditions. Using this theory. WC have determined the material coefficients of both human and bovine articular cartilages under large strain conditions at equilibrium. The theory compares very well with expcrimcntal results.

J,.J,.J,

NOMENCLATURE

principal invariants of e material derivatives following the motions of solid and fluid ditlusivc drag cocllicicnt material constants Lam& constants for the solid matrix material constants material constants stress in axial dircctiun stretch ratio for the solid matrix in axial direction solid displaccmcnt coordinate in axial direction

D( )‘.‘/Dr apparent densities for the solid and fluid phases mixture density dclined as p = p’ + p’ initial mixture density solid-to-fluid volumetric ratio velocities for the solid and fluid phases ditlusion velocities for the solid and fluid phases baricentric velocity for the mixture accelerations for the solid and fluid phases body forces for the solid and fluid phases rate of deformation tensurs fur the solid and fluid phases apparent stressesfor the solid and fluid phases apparent extra stressesfor the solid and tluid phases pressure ditTusive drag forces (or momentum supplies! for the solid and fluid phases normal vector internal energy functions for the solid and fluid phases mixture internal energy function energy supplies to the solid and fluid phases entropy functions for the solid and fluid phases mixture entropy. pq=p’q’+p’q’ lempcraturc Helmholtz free energy functions for the solid and fluid phases mixture Hclmholtz free energy function defined as pA =p’A’+p’A’ left Cauchy-Green strain tensor for the solid matrix

K KU. bf A.. I’. I.. u,. a,. u 4,. d,

U’ Z

To understand

how normal

and pathological

diarth-

rodial joints function. it is essential to understand stress-strain articular

behavior

cartilage

of normal

and

under physiological

loading condi-

tions. In general. the salient biomechanical of articular provides

cartilage a

miniscule, (Malcom.

near

in diarthrodial

frictionless

under

normal

1976; Lipshitz

the

pathological functions

joints are: (I)

bearing conditions,

and Glimchcr,

it

surface

with

wear

rates

1979; Mow

and Lai. 1980, Dowson er al.. 1981); and (2) it spreads the loads resulting

from joint

function

(Ahmed

and and

1983; Brown

Hayes,

1984; and others). Most recently Holmes

co-workers

and Shaw.

1983; Huberti

Burke,

(1984a. b) have characterized

with which articular

cartilage

and

the manner

can also act to absorb

energy during cyclical compressive

deformation.

It is

generally believed that the biphasic nature of cartilage Received

in jinoljorm

I9 December

1986.

is responsible

for providing

all these important

func-

146

M. K. KWAN. W. M. Lu and V. C. MOW

tional characteristics in the joint (Mow et al., 1980, 1984, Maroudas, 1979; McCutchen. 1962). It has been known for a long time that early in the ostcoarthritic process, cartilage water content increases (Bollct and Nance. 1966; Mankin and Thrasher, 1975; McDevitt and Muir, 1976; Maroudas. 1976, 1981; Maroudas and Vcnn. 1977). This compositional disturbance has been shown to have profound biomechanical implications. An increase of water content greatly diminishes the intrinsic elastic moduli and increases the permeability of the collagen-proteoglycans solid matrix of cartilage (Mow and Lai, 1980; Armstrong and Mow, 1982; Hoch et al., 1983). Mow and Lai (1980) and Kwan et al. (1983) have shown that these material property changes have profound effects on the way cartilage supports the loads applied onto its surface and the manner in which the interstitial fluid flows and, thus, joint lubrication. Theoretical prediction of the functional responsesof cartilage associated with increased hydration show that cartilage may no longer function as the near-frictionless and wearresistant bearing material of the joint. Thus a vicious cycle is set up to cause further breakdown of the tissue, possibly by increases in both the friction and wear rates of the articular surfaces; eventually osteoarthritis develops. It is generally belicvcd that abnormal biomechanics oTcartilage plays an important part in the development of osteoarthritis. Thus, since the early 1920s. many attempts have been made to determine the mechanical propcrtics of articular cartilage. These attempts includcd the use of different theoretical models to describe the cartilage stress-strain behavior and geometry. Numerous experimental procedures were also used to determine the deformational behaviors ol cartilage (Benninghoff, 1924. 1925; Bar, 1926; Gocke, 1927; McCutchen, 1962; Sokololl’, 1966; Kempson et 01.. 1971; Colletti et ul., 1972; Hayes et al.. 1972; Parson and Black, 1977, 1979). Most recently, Mow and co-workers (1980) have developed a biphasic throry which takes into account the heterogeneity of cartilage constituents by considering the tissue as a multi-component mixture. Under strictly controlled laboratory conditions, these investigators showed that the infinitesimal strain biphasic theory for cartilage can accurately describe the compressive behavior of the tissue at low load leveis and low strain rates (Mow et al.. 1980. 1984; Armstrong and Mow, 1982; Holmes et al.. 1985). All theoretical models proposed for cartilage SOfar, however, have limited ranges of validity due to the use of the infinitesimal strain theory. They are not applicable to problems where loads are large or strain rates are high, such as those observed in a joint under physiological conditions where the stressesacting on articular cartilage may be as high as several megapascals (Ahmed and Burke. 1983; Brown and Shaw, 1983; Huberti and Hayes. 1984). Since the Lame constants or equilibrium modulus of the solid cartilage

matrix are only on the order of 1 MPa (Mow et al., 1980, Armstrong and Mow, 1982 Myers and Mow, 1983), such high applied stresseswould generate finite deformations within the tissue(Armstrong et al.. 1979; Takei et aI., 1985). In addition, physiological loading rates on the order of one cycle per second would augment the tendency for generating large deformations (Lai et al.. 1981; Holmes et al.. 1985). Thus, the objective of this investigation was to develop a finite deformation biphasic theory capable ol describing the behavior of cartilage and other hydrated connective tissues such as meniscus, nasal septum and intervertebral disc under large strain conditions. A further objective was to determine the intrinsic material coefficients associated with this finite deformation biphasic theory for human and bovine cartilages. A FINITE

DEFORMATION

BIPHASIC THEORY

Articular cartilage is a white denseconnective tissue which covers the bone ends within a diarthrodial joint. It is a soft, highly hydrated and permeable tissue. About 80% of articular cartilage is water (Linn and Sokoloff, 1965; Venn and Maroudas. 1977; Muir, 1980: Armstrong and Mow. 1982; Torzilli et aI., 1982). The solid component of the tissue is formed by an extracellular matrix which is predominantly composed of Type II collagen fibrils (Stockwell. 1979) and proteoglycans (Muir, 1980). The collagen fibrils arc arranged in a specific librillar network (Clark, 1971; Ghadially, 1978) whose interfibrillar space is ftllcd with protcoglycans, water, and some nonspecific glycoproteins and chondrocytcs. The composition of cartilage varies with age, site in the joint, and depth from the articular surface. For various cartilaginous tissues, collagen content ranges from 50% to 70% by dry weight and proteoglycan content ranges from 20% to 40% by dry weight (Muir, 1980). The fibrillar network and protcoglycan gel somehow interact with each other to form a porous solid matrix filled with water. The precise nature ol the interaction is not clearly understood. A significant body of data is available which indicates that there is coulombic interaction (Serafini-Fracassini and Smith, 1966), as well as physical entanglements between the proteoglycans and collagen (Smith et nl., 1967; Hamerman et a!., 1970) and among the collagen fibrils themselves (Weiss et al., 1968; Clark, 1971). In etTecect. these structural macromolecules interact to form a cohesive fiber-reinforced material which possessesall the essential mechanical characteristics of a porous solid matrix swollen with water (Mow et a!., 1984). Most of the water in cartilage moves freely through the interstitium of the extracellular matrix. This fluid movement can be caused by diffusion (Maroudas, 1975; Tonilli et al.. 1982), by compression of the tissue (Linn and Sokoloff, 1965; Mow er al., 1980). or by application of a pressure gradient (McCutchen, 1962; Mansour and Mow, 1976). The flow of this water

Finite deformation theory for eartilagt-I through the tissue is determined by the molecular and microscopic organization. compressibility, and permeability of the tissue. The relative motion and the subsequent frictional interaction between the interstitial fluid and solid matrix appears to be the predominant factor governing the observed viscoelastic behaviors of the tissue (Mow et al., 1980, 1984; Kwan et al.. 1984). A binary mixture theory, based on the multiphasic mixture theory of Craine et a!. (1970) and Bowen (1976). was developed by Mow et al. (1980) to study the fluid flow and deformation behavior of articular cartilage. In this biphasic theory. cartilage was depicted as a soft. porous and permeable elastic solid filled with water, with both solid and fluid phases assumed to be intrinsically incompressible. For such a binary mixture, the equation of continuity is given by

(1) where x is the solid-to-fluid volumetric ratio called the solid content. 8 and ir’ are solid and fluid velocity, respectively. Equation (I) describes the principal of mass conservation for the mixture and has a simple interpretation. Since l/(1 +a) is the porosity and g/(1 +z) is the solidity, div[B’/(l +a)] is the net loss of fluid volume per unit mixture volume, and div [aP/( I + a)] is the net loss of solid volume per unit mixture volume. Obviously, the total loss of volume must bc zero since both phases arc incompressible. For Anitc deformation. Kwan (1985) showed that the solid content a is related to the deformation by a0

a=

(23)

(I +a,)JZij-a0 where x0 is the initial solid content and B is the left Cauchy-Green strain tensor of the solid matrix. For completeness, we have provided the derivation of this relationship in the Appendix. We note that, under finite deformation, this simple kinematic relationship shows that the change of volumetric ratio z( = V’/V’) is related to the deformation of the solid matrix through the determinant of the deformation tcnson B (seeAppendix). However, under conditions of infinitesimal deformation, the left Cauchy-Green tensor becomes B = I + 25 and det( B) = I + 2e. Here E is the infinitesimal strain tensor, and e is the dilatation given by tr(e_).Thus, for small deformation. equation (2a) reduces to the relationship r=z,[1

-(I

+r,)eJ.

(2b)

This linear relation gives a simple way to calculate the change of the volumetric ratio a of the solid matrix during small deformation. For example, volumetric compression of the solid matrix, where e is negative, will increase a linearly. The constitutive equation for the binary mixture can be derived using an approach which reduces some of the complexities of the general mixture theory by

147

limiting the theoretical consideration, to the extent possible, to only mechanical effects.In this derivation, temperature variation, heat flux vectors and heat supplies are absent. It is assumed that no chemical reaction is taking place between the constituents. External body forces are also neglected. A full derivation, where these assumptions were not made, can be found in several sources, including Kwan (1985). For a control volume V, bounded by surface S,, the balance of energy for the solid constituent is described by the following equation:

=

(a’tidS).b’+

-

,W( I ss

PdV

(;‘dV).b’+ I y.

I s.

I y,

5”. H dS) -

+PP)(PfidS)

(3)

I s,

where p’, E’, 5’ and 2’ are apparent density, internal energy, interaction momentum supply (or diffusive resistance) and the interaction energy supply for the solid phase. Equation (3) states the rate of change of energy for the solid phase inside the control volume V, is due to: (I) the rate of work done by the stress vector b’ acting on the solid portion of the surface S, bounding the volume V,; (2) the rate of work done on the solid by the fluid by force fi’ acting at every point inside V, due, for example, to the relative motion of the fluid with respect to the solid; (3) the rate of mechanical energy input to the solid, E’. through solid-fluid interactions. This energy supply is related to the work done by the momentum supplies i’s as described by equation (7) below; and (4) the influx of internal and kinetic energies due to convection through the surface S, of the control volume. Making use of the divergence theorem, the identity div(~‘.8’)=tr[a’D’]+t’.div~‘, where D’ is the solid rate of deformation tensor and ~‘=[q’]~, and the relation div c’ + ir’ = p’( D’t’/Dt), equation (3) can be reduced to D’E’ ‘-dV= ’ Dr

tr(a’Q%f V+

t’dV.

(4)

In a local form, equation (4) becomes DIG’ p’Dr=tr($D’)+P’. A similar equation can be obtained for the fluid constituent:

, D’s’ ’

-

Dr

= tr($ D’ + 2.

Combining the energy equations for the solid and fluid constituents. i.e. equations (5), we obtain (6)

148

M. K. KWAX W. M. LAI and V. C. Mow

The total interaction energy supply (P+g’) in equation (6) is related to the interaction momentum sup plies 5’ and 5’ by ~‘+~f+~‘.~s+~f.~f=O

(7)

where 0’. f = p’. r - 5, where fi is the barycentric velocity defined by p5 = p’6’+ p’fi’. Equation (7) states that the rate of work done by the internal interaction forces is at the expense of the total interaction energy supply. This equation can be derived from the requirement that equation (6) be consistent with the energy equation postulated by Truesdell (1957) for the mixture, namely

a’= -rp!+g:

g’= -@Pl+g;

or

(12a)*

and _br= -p!+g!

or

gr= -#Pi+,!

(IZb)*

where the isotropic terms arise from the incompressibility condition with p denoting the apparent hydrostatic pressure and P denoting the pore pressure; 4 and 4’ denote the solidity and porosity, respectively; $ and gi represent the extra stress tensors. From equations (12) we can find

(13)

DE PDt=tr(2D)-divij

where we note tr Drer =div it’*‘. Substituting equation (13) into equation (IO) and making use of the continuity equation, i.e., equation (I), we then have

where g=a*+af_p”~‘6’-pr~f9r

written as

(9a)

P&=P’&‘+Pr&r+~(P’iY”~‘+pf~r.~r)

(9b)

D = b[grad D+ (grad i?)T]

(9c)

) D’rS

--PDc

r Dfsr -+tr(g:p+2tD’) PDc -[iCpE].(j’-GO.

(14)

4s - (+Y + 2’ \if) + P’(& + i 6’. \t’)c+

+#(d + ) kt’ ’ biJ)bv.

(94

The postulate for the mixtures of continua has its origin on the kinetic theory of gases (Truesdell and Toupin. 1960; Bowcn, 1976). In equations (9). the terms involving the 3’s (the velocity of each phase rclativc to the center of mass of the two phases), are analogous to the k&tic cncrgy terms in the kin&c theory of gases. Substituting equation (7) into equation (6). WC obtain

I’

D‘E’ ‘-+pf= Dt

D’e’

=tr(n’p’+e’o’)-ir.(d’-8’)

(IO)

where fi = irr= - y. Since both constituents are incompressible, the mixture can sustain an equilibrium state of spherical total stress, -Pi, without any deformation and flow of the solid and tluid. Under such a condition, the true stresson either the solid or fluid is exactly -P!. and the apparent stressesare

(I la) and a f E--

P I +2

(1IbJ

In arriving at the above equations, the solid-to-fluid area ratio on any cross-section of the tissue has been assumed to be the same as the volumetric ratio z. In general, the apparent stressescan thcrcforc be

Since there is dissipation due to the frictional drag between the fluid and solid, it is necessaryto take into account the constraint imposed by the second law of thermodynamics. Under the assumption that heat flux vectors and heat sources are absent, and the temperature of the mixture is constant, the second law of thermodynamics can be written: D’q’ D’q’ ‘-+prTjrZO ’ Dt

where +’ are entropies per unit mass for the two phases. The inequality (15) is equivalent to the inequality postulated by Truesdell and Noll (1965) under the assumptions mentioned above, namely DII p-ij;-+div(drl’~‘+p’q’iv’)~O

(16)

where tt is the mixture entropy defined by pq=p’q’ +P’$. The internal energy and entropy are related by A’=e’-0~’

(t7a)

A’=&‘-(Iv’

(l7b)

and

where 0 is the constant temperature. Substituting equation (17) into inequality (IS). we then find lw D’c’ ~~+PrE-P~-_Pf I’

Dr

D’A”

D’A’ -30.

Dt

Dt

(18)

Thus, from equation (l4), we have

, PA’ ‘EqurGons (12) can be derived from the second law of thermodynamics where p is the Lagrange multiplier multiplying the constraint condition given by the incomprcssibility continuity equation (see Mow er id.. 1980).

(1%

-p%--P

r D’A’ -+tr(@‘+2Q) Dr -[

k+Ps].(,‘-,.),O.

(19)

Finite deformationtheory for cartilage-1

149

To derive the constitutive equations for the binary mixture, we postulate that A’ = A’( f$ p ‘)

(2Oa)

Af=A’(13.pf)

Wb)

fi = II( @.p’, grad & grad p’, P - i’) where g is the left Cauchy-Green

(2Oc)

g‘= -pl-p’ i= -p-

deformation

grad1 I+z

tensor.*

D~+@~D~_p’%D’

8p’ ,$(fi’-P).gradp’

(2fa)

and

D’A’

[_I

2A’ 8 x

tr

Dr

i

T

JA’

[grad!!I

(2lb)

where (dA’/dfb) [grad 81 is a vector defined as $[grad where i,(i=

1.2.3)

Substituting

fI]-[$3]$,

(22)

are the Cartesian

equations

unit base vectors.

(2la. b) into equation

(191, we

find

se-P’!!

dA' T

[

x

_I1

_I

dA' T

dA'

c:-p*!!

-P'B

apf

where II,.@’ -it’)bO.

(25)

In view of inequality (25). we further assume that fi,=K(ii’-fi’)

(26)

where K is a non-negative coefficient called the diffusive drag coefficient. In general, K is a function of the porosity or volumetric ratio z of the tissue. Here, based upon our strain-dependent permeability data (Mansour and Mow, 1976; Mow et al.. 1980; Lai and Mow, 1980). we assume that

[

z

dAr

-P’!!JB

Q’

(27)

where K, and M are two material properties of the tissue. It has been shown (Kwan et al., 198% under infinitesimal strain conditions, that equation (27) reduces to the expression which has been used to account for the observed strain dependence of the permeability of the tissue (Lai and Mow, 1980). This strain-dependent permeability effect is necessary in order for the biphasic theory to describe the nonlinear compressive creep, stress-relaxation, and energy dissipation elects for articular cartilage under the conditions of imposed infinitesimal strains (Holmes et al.. 1984a. b; Hou ct ul.. 1986). For the binary mixture, each phase must satisfy an equation of motion which, under quasi-static conditions, is given by diva’~‘+~‘*‘=~.

+p’g[gradB]

.(I?-P)>O. 1

(23)

Since Q’, Q’, and ir’- P are independent quantities, thus to satisfy the inequality (23) and to be consistent with the constitutive assumptions (20). we must have

a’=

Wb)

dA’ dA’ +p’-gradp’-prX[gradf33+ir,

h’=h’,exp[-M(Jz-&-I)]

+(e’ -6’) . x

tr

!

(244

Using the chain rule of differentiation, and after a considerable amount of manipulations where the conservation of mass equation and several identity equations for deformation tensors were used, we obtain

-=

(24a)

b

-ap!+Ei{pfg+[~p]

‘It may be remarked that other forms OC finite deronnation tensors such as the right Cauchy-Grccn and the Lagrange deformation tensor can alsobe used.The tensorg has the advantage and simplicity that it is an objective tensor. It is noted that the use of B also impfics the auumption of isotropy for the solid phase [see discussion rollowing equation (ZS)].

(28)

A recent publication (Mizrahi et al., 1986) demonstrated possible anisotropic deformation of cartilage under compressive loading. These material property anisotropies are similar to those known for cartilage subject to tension (Kempson et al., 1968; Woo et al., 1976; Roth and Mow, 1980). Nevertheless, under the confined compression condition where no tensile stressor strain are expected, it is reasonable to assume that articular cartilage behaves in an isotropic manner (Mow et al., 1980; Lai et al.. 1981; Armstrong and Mow, 1982). Thus, in the present paper, only an isotropic constitutivc law is considered. Indeed. it can be shown that, without any additional difficulties, anisotropic cases can be developed. In this case, the dependence of the Helmholtz free energy function on @ reduces to a dependence on the three invariants of II. namely J,. J, and J,, where J,=trfI,

II=f

[(tr f3)’ - tr(gz)],

J, =det I$

(29)

154

M. K. KWAN. W. M. LAI and V. C. Mow

By further assuming that A*=Af=A(I,.JZ.J,).* obtain from equation (24) aA J’jj-+J,%

dA

I[ l++, gg-1I

us= _ -api+2p

2

1

we

I

3

(304

1

a I =-pi

OW

grad a ii= -p --K(B’-6’). l+a

(30#

Substituting equations (30) into (28) and combining the resulting equations, we obtain one governing equation for an isotropic biphasic material:

form of the biphasic stress-strain relation developed by Mow et al. (1980) under infinitesimal conditions. (ii) Equations (32a) and (32~) contain only two material constants each. This is the same minimum number needed for the isotropic infinitesimal strain case. (iii) Equation (32b) is the generalization of the twodimensional form of the energy function proposed by Fung (1981) to describe the deformational behavior of the skin. This appears to be a reasonable choice. Indeed, we have tried many other forms for the Helmholtz free energy. To demonstrate what one can achieve with a particular choice, and what one needs to be concerned with in a particular choice, only these three forms are presented.

FINITE

DEFORMATION

EQUILIBRIUM

BEHAVIOR

OF

ARTICULAR CARTILAGE

+(l +a)K(ir’-P)=G. A BIPHASC FINITE DEFORMATION

(31) THEORY FOR

CARTILAGE AND OTHER SOFT HYDRATED TI.S!XIES

The general stress-strain law for a dissipative binary mixture has been shown to be derivable from its constituent Helmholtz free energy functions, equations(24). or under assumption WC made, from a single mixture Helmholtz free energy function [equations (30a. b, c)]. In this section we will examine three possible choices for the mixture Hclmholtz free energy function:

Nonlinear equilibrium stress-strain behavior of cartilage in compression has been reported (Sokoloff. 1966; Eisenfeld et al., 1978; Mow et ol., 1980) and observed in many of our laboratory results at high loading rates and applied stress. In general, with increasing compressive strain under uniaxial confined compression conditions, the solid matrix of articular cartilage. and possibly other hydrated tissue as well, stiffens for compressive strain beyond 20%. Under these conditions, the equilibrium stress-strain relations corresponding to equations (32a). (32b) and (32~). can be obtained. respectively, as follows: a,=f(rl,+Z~J[I 0,=2[1

+d,&-

+d,(l,-

l)]&(i:-

I)

~(~~_

1)]1,

(33a)

,)

I

3& + 2P, J +&+2P.J2 Pr J PoA=-q I yj-1-z 2

-I,@, +6u,+Za,)‘(i.;-

1)

+I,{-a,J,-a,l:-a,J, +f(at+6a2+2a,),(6J,-J:) (33b)

+Cxp[a,(J,-3)+n,(J;-9)+n,(J,-3)]} (32b)

and a,=a(&+2P,)Cl

(32~) Our choices of these three forms of the Hclmholtz free energy function were motivated by the following considerations. (i) For infinitesimal strains, all three forms of the Hclmholtz free energy function yield exactly the same *This assumplion is made solely for the purpose of simpliwithout loss olgcncrality. In a formulation where this assumption is not imposed, we must fying the theory and analysis

deal with many more material coefficients. We ankipalc that. in the future. ic will be necessary 10 relax this assumption if the present theory is found to be too restrictive. tT’o simplify calculation, the higher order term involving [grad 81 in i is neglected.

+d,(&-

1 1 [ 1

1)1x

“j-2

(33c)

where I, is the stretch in the direction of the compressive load and 0=(1:-l)[a,(l:-l)+u, +6u2+ 24,J. d, = d,(a, + l)/(a, + do). and a0 and do are initial solid content and fluid-to-solid true density ratio. The average value of a0 is around 0.20 (Mow ef al., 1980). We note that the value of d, is estimated to be between 0.70 and 0.90, based on tissue composition, densities of Type II collagen. proteoglycan aggregates and monomers, chondrocytcs and other glycoproteins. Due to the variability of cartilage cotiposition under various physiological conditions, the exact value of d, for each specimen can only be determined after complete biochemical analysis. By using a nonlinear regression procedure, equations (33) may be

151

Finite deformation theory for cartilage-l

used to curve-fit our experimental data. Figure la shows that equation (33a) does not provide the stiffening effect observed in the experiments (o represents a typical set of experimental data), while equations (33b) and (33~) give good fits to the data within the range of stresses and strains tested (Figs 2a and 3a). For equations (33b) and (33~). the coefficients for the finite deformation theory are listed in Figs ta and 3a. The theoretical stress-strain behaviors for 0
(a) -u, (MPal + 0.6 -A, I,.

2#l,

l

0 002

-0

594

MPo

Mh

02.506

0.4

a, -6a**Za,.-003

--

.

0.2 _-

0.0

./

(1 -x,1

I /

I.6

.

I

IO.0 (b)

/

0.2

0.4

0.6

- ur

-u, (MPO) .

0.6 xs +

2rs 92.127 MPO .

0.4

--

.

Fig. 2. (a) A curve-fit of the observed large deformational behavior of articular cartilage in confined compression using equation (33b): l experimental data; - theory. (b) A representation of the stress-strain relation as predicted by equation (33b) for O-cL,-cco. (0.0

lb)

-

0.2

0.4

0.6

U3

I

I

I I

I

Fig. I. (a) A curve-fit of the observed large deformational behavior of articular catilage in confined compression using equation(33ak l experimental data; - theory. (b) A representation of the stress-strain relation as predicted by equation

describes the natural stilTening of the tissue under large compressive strains, and predicts a ‘large-strainmaintained-by-large-stress’ relationship. In a further study, equations (32~) and (33~) are found, under the one-dimensional confined compression conditions, to satisfy the generalized Coleman-No11 (GCN,) condition for the entire range of strain and the Baker-Ericksen (B-E) inequality for i., > ,/[v,/ (1 -II,)] (Kwan, 1985). Recent studies of normal and repaired cartilage indentation properties have reported that in situ v, is very small (~0.1) (Whipplc et 01.. 1985). Thus, for articular cartilage, equations (32~) and (33~) are physically feasible over a large range of compressive strains without violation of the constraint conditions known for finite deformation theory. Therefore, these equations appear to be a better choice if very large strains are anticipated. By inserting equation (32~) into equation (29). we nhtain the ocneral finite deformation biphasic consti-

M. K. KWAN. W. M. LN and V. C. Mow

152

variable K(g),

Porosity,

z(B). (2) nonlinear

and (3) finite deformation,

EXPERIMENTAL DEllWMlNATION DEFORMATION

diffusive drag,

B.

OF LARGE

CONFINED COMPRESSION BEHAVIOR OF CARTILAGE

Cylindrical imately

(b)

l/4

cartilage-bone

cores measuring approx-

of an inch in diameter were harvested

from the patellofemoral grooves of normal bovine and human knee joints. The osteochondral plugs were immediately placed in Ringer’s solution with PMSF enzyme inhibitors and stored at -20°C until the time of testing. At the time of testing, the specimen was thawed and allowed to equilibrate in Ringer’s solution at 4°C for I h. Immediately prior to the testing, the diameter and thickness of the articular cartilage were optically measured using a Bausch and Lomb stereomicroscope at x 10. Each of these values was obtained as an average of six measurements taken at 60” intervals around the circumference. The patellofemoral groove was chosen because cartilage thickness variation was found to be minimal. Confined compression test was then performed using an apparatus and method similar to those developed by Armstrong and Mow (1982). With minor modifications to this apparatus, the chosen increments of load may be easily applied. At each load the specimen was allowed to creep until equilibrium was attained. Typical load histories and compressive creep responses for the bovine and human cartilage are shown in Fig. 4. Human cartilage appears to be stitfer and to

-us

Fig. 3. (a) A curve-fit of the observed large deformational behavior olarticular cartilage in confined compression using equation (33~):l experimental data; -theory.(b) A reprrscntation of the stress-strain relation as predicted by equation (334 for 0<1, c a3.

INCREMENTAL LOAO HISTORY 4

“c

z

190s

tutive equations:

. TIME

+(%,+2~,)J:]~+[31,+2~,+2(i.,+211.)J,JB +(9J,+

lo&)J,Q-‘}

-pl

$f= *z-p

Three

different

these constitutive

INCREMENTAL FINITE CREEP

WW

grad a

T+Oc-K(G’-G’)

(344

where p/p0 are related to the deformation P -=I[1 PO Jzj

+

WI

+d,(Jz&

nonlinear

by

1)-J.

effects are embedded

equations,

(35)

in

i.e., effects due to (1)

Fig. 4. Typical load history and compressive creep response for bovine and human articular cartilage.

153

Finite deformation theory for cartilage-1

-z 3E

06 t

- 0

Human

15

30 ‘STRAIN’

+

45

/

/

Bovm

60

75

(- !$ )X

Fig. S. Equilibrium stress-strain behavior for human and bovine articular cartilage. A nonlinear stifiening effect can be seen. This mechanism limits the compresssive strain sustained by the tissue under large applied load.

creep at a slower rate than bovine cartilage. The equilibrium stress-strain data for the bovine and human specimens are shown in Fig. 5. Data analysis and rrsults

Equation (33~) expresses the equilibrium strcssstretch relationship for this theory. It was used to curve-tit the equilibrium experimental stress-strain data. In this one-dimensional problem, (I - 1,) is a convenient measure of strain for small. as well as for large. deformation. The aggregate modulus, II,, = i, + 2c(,, was found to be 0.563 f 0.272 M Pa (mean + standard deviation) for human cartilage and 0.399 f0.147 MPa for bovine cartilage. Statistically. the two mean aggregate moduli were found to be significantly different (p
A finite deformation biphasic theory, capable of describing the large deformational behaviors of cartilage and other soft hydrated biological materials, has been developed. The deformational behavior of these materials are dominated by three nonlinear effects:(1) the strain-dependent porosity effect, (2) the

strain-dependent permeability or diffusive drag effect, and (3) the finite deformational clfects. Creep, stress-relaxation and other viscoelastic effects exhibited by the material as a whole are thus controlled by the flow of the interstitial fluid through the deforming matrix; thus the viscoelasticproperties of these biphasic materials are nonlinear. It is noted that this theory does not consider the solid matrix to be viscoelastic. Such a theory. consideringthe solid matrix to be viscoelastic in a quasi-linear manner, has been developed by Mak (1986) under infinitesimal strain conditions. It was shown that, under the condition of confined compression, intrinsic matrix viscoelasticity contributes only in a minor manner to the behavior of the tissue. Finally. the present finite deformation theory has been used to successfullyanalyze the equilibrium confined compression behaviors of articular cartilage under large strain conditions. It is likely then that the mechanical properties determined in this way might provide for a better correlation with the biochemical composition and tissue micro-structure than the current results derived from the linear theory. It should also be remarked that, while the finite deformation stress-strain relations proposed in this paper satisfy the general principles of constitutive equations, whether or not they can actually describe the transient creep response awaits further analyses and experimentation. We have shown, however, that the theoretical prediction of the transient creep response of this finite deformation theory does provide good qualitative agreement with experimental data obtained under the confined compression condition (Mow et ul., 1986). At present we have not attempted to determine any of the material coefficients (i.e., the nonlinear permeability and solid matrix elastic coefficients) of this theory from the transient creep tests. In addition, further studies need to be undertaken to investigate the validity of our proposed theory to describe experimental conditions other than the very restrictive confined compression conditions reported in this paper. The main result of this study is, therefore, the development of a finite deformation biphasic theory which permits the study of articular cartilage and other soft hydrated tissue in a more comprehensive manner. Acknowlrdyummrs-This research was supported by the National Institute of Arthritis, Diabetes. and Digestive and Kidney Diseases grants AM 19094. AM 26440 and the National Science Foundation grant MSM 8211968. Any opinions, findings and conclusions or recommendations expressed in this paper arc those of the authors and do not necessarily represent the views of the National Science Foundation.

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155

APPENDIX Since we follow the solid component of the mixture during the deformational process. there is no change in solid mass. I.e.. dm’ = dm’, . (Al) (We note that quantities with a subscript 0 denote initial values at the undeformed state.) Also, since the solid component is incompressible. we have dV’=dV’,.

dV=,/~dV,.

(A3)

Note that dV’ dv’=dydV

=&dV

=&dV.

(A44

and dV’,=LdV,, I+a,

zc--

a0

I+zo

I _ d V. ,/det B

(AW

Substituting equations (A4) into equation (AZ). we can then find a

-t-P I+a

10, 205.

Torzilli. P. A.. Rose. P. E. and Dethcmcrs. D. A. (1982) Equilibrium water partition in articular cartilage. Biorheology 19, 51 S-537. Truesdell. C. (1957) Sulle basi delta thermomeccanica. Rend. LPncei (8) 22. 33-38, 158-166.

(A’)

Because of the fact that solid mass does not change the deformed volume of the mixture. d V. can be shown to rclatc IO the initial undeformed volume, d Fe, by

a0

I

l+hJGij’

or a=

a0 I(I +a,,)Jdet

(QED). B -a0

(AS)