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Cartilage is poroelastic, not viscoelastic (including and exact theorem about strain energy and viscous loss, and an order of magnitude relation for equilibration time)
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TECHNICAL
NOTES
CARTILAGE IS POROELASTIC, NOT VISCOELASTIC (INCLUDING AN EXACT THEOREM ABOUT STRAIN ENERGY AND VISCOUS LOSS, AND AN ORDER OF MAGNITUDE RELATION FOR EQUILIBRATION TIME)* Abstract-Cartilage is often called viscoelastic, yet when strain lags stress in cartilage it is not primarily becaus& of effects within the material of the cartilage skeleton itself. It is because the cartilage skeleton is bathed in fluid. Except in pure shear deformation, attaining equilibrium strain requires that pore fluid flow within the cartilage. Viscous forces retard this flow. This behavior is known as porcelastic. The equilibrium time is of the order L2/( Ya), where Y is the Young’s modulus, (I the permeability of the cartilage, and L is the length of the path along which liquid flows during equilibration. I show that this is true for any consolidation experiment, whatever the direction of consolidation and the direction of liquid flow. In thecourse of thisdemonsttation I prove that if load is applied abruptly to a Hookean material and is thereafter held constant, the strain energy at equilibrium equals the energy dissipated in the material during equilibration.
In two papers, Parsons and Black (1977, 1979) say that cartilage behaves as a viscoelastic solid, or, simply, that it is viscoelastic. Each paper gives a retardation time spectrum for the creep of cartilage. But cartilage is primarily poroelastic, not vi&elastic. In pure shear deformation, Hayes and Bodine (19781 have found that the elastic foras in the cartilagearc b@r than the damping forces by a factor of 5.5 at 1000 Hz, and a factor of 6.55 at 20 Hz When cartilage is compressed, however, damping forces may be many times the elastic forces, with the change in strain lagging the change in stresb by many minutes in a typical experiment (McCutchen, 1962). This is because attaining equilibrium compressive strain requires that pore fluid flow within (and out of) the cartilage. This flow is opposed by viscous forces in the fluid, and the order of the equilibration time r is given by t = L*/( Yo ),
(1)
where Y is the Young’s modulus, u the permeability of the material, and L is the length of the path along which liquid flows during eqDilibration. Note that I say equilibration time rather than relaxation time. It means the time required for most (which will not be more precisely defined)ofthe creep to occur. A poroelastic material does not have a single relaxation time, but characteristically creeps faster at the beginning, relative to its creep rate later on, than it would if thi creep followed an expo&tial curve (Biot, 1941. pp. l63-164:McCutchen. 1962.~. II :Kenvon. 1980.~. 1331.The retardation time spe&um fi;nd dy Pa&o& and ‘glack (1977, 1979) stretcheQl from 0.42s to II84 s. Because of the Lz dependencc, tht equilibration time depends sensitively on the size of the swimen, as well as on its material properties. It is well known that equation (I) holds in one-dimensional consolidation, where the liquid flow is parallel to the direction of consolidation (Biot, 1941; McCutchen, 1962; Kenyon. 1980). Likewis$ it is true for the consolidation of a poroelastic disk between itrtpervious plates that constrain the liquid to flow radially through the disk (McCutchen. 1962, p. 4). In l Rereeircl 25 Februcrry 1980: in rerisedform 1981.
what follows, I show, by an energy method, that these facts are not happenstances, that the equilibration time is of the order L’/(Yu) in any consolidation experiment, whatever the orientation of the deformation and flow path. 1 prove first that in any Hookean system at equilibrium the stored potential energy E, under i particular loading, is exactly half the integral W of the scalar uroduct ofdefonninn force and deformat& taken over the object. Hookean is h& defined to mean that e@libriwn strain is proportional to equilibrium stress. Because of the Hooktan requirclaent, the treatment is exact only t&r infinitesimal deformations ofreal materials. Including both surface and body forces this integral becomes
13 Junuar)
where f,, is the force applii to a unit area of the surface a, a0 is the force applied to one unit of volume u and do is the change in position of a point in the body consequent upon the application of the forces, Note that b0 is not in general parallel to f, or to ug. If the system that applies the forces is massless and lossless, so that the loadingcan be applied abruptly at the final value and then held constant however the body moves, W is the work done on the body. Body forces (like gravity or acceleration) act on the liquid as well as the solid of a poroelastic body. Loss or gain ofliquid at any place in the body during deformation will change the body fora distribution, which reinforces the limitation of the theorem to infinitesimal deformations. Now suppose that instead ofbeingapplied abiiptly at its final value the loading is increased slowly from zero. In the limit as the loading is done slower and slower there is no viscous loss. Provided that there is no speed-independent damping (Coulomb friction) the energy input will then equal the strain energy at the end of the ddormation. As the loading is increased, every point on the surface of the body and every point within it follows its own path through space. The work done per unit area of surfact at any point is the line integral of the applied stress along the path taken by the point. The work done per unit of volume at any point is the line integral of the 325
326
Technical Notes
applied body force per unit volume along the path taken by the point. Integrating these quantities over the surface and volume of the body respectively gives the work done on the whole body. Thus the strain energy E is given by a=*, 6=& u .dbdr. f.ddda+ E= (3) J.iD d=O SIV a=0 If we assume that the loading retains the same pattern as it increases from zero, so that only its magnitude varies, then in a Hookean material the deformation will also have a constant pattern and increasing magnitude. Hence f and u are proportional to 8, though not in general parallel to it. Thus f = c, lalt and II = C#(i, where C, and C2 are constants, and r and it are unit vectors. Letting d = 1818, where b is a unit vector, the expression for strain energy becomes
evaluated at the end of the immediate deformation. The strain energy EC at the end of the creep deformation is half W,., the value of equation (2) at the end of the creep deformation. Though W, is not necessarily the work done on the specimen in the immediate deformation, because some of the damping may occur outside the specimen, W,. - W, is quite accurately the work done on the specimen during creep, because little damping then occurs outside it. So the increase E,- - E, in strain energy during creep is (W,. - W,)/Z, or half the work done on the specimen during creep, and an identical quantity of energy goes into losses in the specimen. The loss due to flow during creep thus equals half the energy input during creep In a typical compression experiment this is half the load F times a,,, the distance moved by the loading plunger. The load in turn equals the equilibrium compressive strain SO/z,where Tis the thickness of the sample in the direction of compression, times the area A of the plunger, times the stiffness modulus of cartilage skeleton, which may be the Young’s modulus, the confined compression modulus, or something between them, depending on the experimental arrangement: the differena between these moduli is not large. The strain energy is thus given, exactly or approximately, by E = YA6;/2z. This equals the energy loss due to flow, which is the time integral of the volume integral of the product of the flow resistance by the square of the flow velocity. This integral can be evaluated very roughly by assuming that all the flow traverses the same route, and that it flows at a constant rate for a time t. The resistance of this flow path is, approximately, approximately, L/oB
(4)
which is half W as defined in equation (2). In deriving this relation we assumed that the stress pattern increased in a particular way. But strain energy depends on strain, not on strain history, so equation (4) holds however the system arrived at its final state. As the work done on the specimen is ideally W and the strain energy is E = W/2, the remaining energy, a quantity equal to the strain energy, must be absorbed by the losses. In a lightly damped system the loss component may be stored temporarily in kinetic and potential energy at different times and plaas in the specimen while the latter vibrates as the energy is being absorbed. In a practical experiment some of the loss may occur in the loading system. In this casz the work done on the specimen is less than 2E. In a poroelastic material with very small pores like cartilage, the strain following abrupt application of load occurs in two distinct stages, first a rapid deformation that ends before there has been significant flow of fluid, then a much slower creep that occurs only because fluid moves within the cartilage. Pore flow contributes littk to the losses that damp the quick &formation. These may be genuine viscoelastic losses in the specimen, or losses in the loading apparatus. The ditTerena between the rates of deformation is so large that their geometric mean is effectively infinitely slow relative to the immediate deformation, and instantaneous relative to the creep deformation. The preceding argument can therefore be applied in good approximation to the immediate deformation (but not directly to the creep deformation. Whereas the immediate deformation occurs in isolation, the creep deformation is always added to the immediate deformation.) The strain energy E, at the end of the immediate deformation is half W,, the value given by equation (2)
z L*/(aV),
where u is the permeability of the material, L is the length of the path, B its cross sectional area, and Y the volume of the specimen (or of that part of it that creeps, if parts of it do not). The rate of flow is given by the volume diilaczd Ab, divided by the time t. Here 6, is the total displaament, not that due to creep alone, and y lies between 1 and 1 - 2v depending on the particular experiment, where v is the Poisson’s ratio of cartilage skekton. There is negligibk flow during the immediate deformation. The energy loss during creep is the square of the flow rate, times the flow resistana, times the duration of the flow, or tL2A25;y2/(aVt2)
= (LA&y)‘/(&).
Were this expression exactly the energy loss it would equal the strain energy E, so we have, E = YAS;/(2r)
a (LA6,y)*/(uVr),
so,asA,=V, t f 2b+y2L2/(s; Yu),
(5)
or, sina the whole calculation is a very crude approximation anyway,
I z L2/( Vu).
(6)
which is equation (1 k Because, in particular cases, equation (1) can easily be derived by calculating the flow rate directly from the pmsure gradient, the energy method used here may seem roundabout. Its virtue is that, in effect, it automatically works out the pressure gradient and saves us from having to calculate it for each case of interest. It works for any direction offlow, which might be parallel to the compression. as when a thin disk is squeezed between porous plates, or at right angles to it, as happens when the plates are impervious. In the latter case the equilibration time is much longer than in the former because the flow path is longer (McCutchen, 1962, p. 3). lfwe take L to be the radius of the disk when the Rats are impervious, and half the thickness of the disk when the flats are porous, then the ratio between the two values is 16. so the ratio of
327
Technical Notes equilibration times should be 256. The data give a ratio nearer 100, which is not surprising, &cause taking Las the radius of the disk should overestimate its effective length. Because of the radial symmetry of the disk, the inner portion of the path conducts very littk fluid. Furthermore, any flow channels left by imperfect mating of the cartilage surfaces to the impervious glass plates will speed the equilibration. This pair of between-flats experiments illustrates what happens in a working joint. Under load, escaping fluid has to take the long path parallel to the rubbing surfaces. The equilibration time is long. When the cartilage is unloaded the fluid flows into it through its surface, and the equilibration time is much shorter. When an experiment is done on geometrically similar objects of difTerent sizes, the equilibration time is precisely proportional to the square of the size of the specimen. Even without similarity, the L2 dependence is so strong that it swamps any effect ofimperfect similarity unless Lisnearly the same in both cases. Thedisk-squeezing experiment shows this clearly, and a kitchen-table experiment is equally convincing. Try to dent cartilage on a bone end by pressing on it with finger or thumb. No obvious dent is produced, but the cartilage surface will bear a sharply-incised impression of one’s fingerprim. Papillary ridges are about 0.5 mm apart. The fingertip is over 1 cm across. Fluid must flow at least 20 times farther to escape from under the whole fingertip than to get out from under a papillary ridge. The equilibration time is
at least 400 times longer, so it is not surprising that no fingersized dent is formed. Laboratory of Pathology NIAMDD National Institute of Health Bethesda, MD20205 U.S.A.
C. W. MCC~JTCHEN
REFERENCES Biot. M. A. (1941) General theory of three-dimensional consolidation. J. appl. Phys. 12, 155-164. Hayes, W. C. and Bodine, A. J. (1978) Flow independent viscoelastic properties of articular cartilage matrix. J. Biomechanics 11, 407-419. Kenyon, D. E. (1980) A model for surface flow in cartilage. J. Biomechanics 13, 129-134. McCutchen, C. W. (1962) The fractional properties of animal ioints. Wear 5. 1-17. The auantitv “a” should be deleted Born all equations on page; 10 and 11 of this paper. Parsons, J. R. and Black, J. (1977) The viscoelastic shear behavior of normal rabbit articular cartilage. J. Biomechanics 10, 21-29. Parsons, J. R. and Black, J. (1979) Mechanical behavior of articular cartilage: Quantitative changes with alteration of ionic environment. J. Biomechanics 12, 765-773.
TECHNICAL
NOTES
CARTILAGE IS POROELASTIC, NOT VISCOELASTIC (INCLUDING AN EXACT THEOREM ABOUT STRAIN ENERGY AND VISCOUS LOSS, AND AN ORDER OF MAGNITUDE RELATION FOR EQUILIBRATION TIME)* Abstract-Cartilage is often called viscoelastic, yet when strain lags stress in cartilage it is not primarily becaus& of effects within the material of the cartilage skeleton itself. It is because the cartilage skeleton is bathed in fluid. Except in pure shear deformation, attaining equilibrium strain requires that pore fluid flow within the cartilage. Viscous forces retard this flow. This behavior is known as porcelastic. The equilibrium time is of the order L2/( Ya), where Y is the Young’s modulus, (I the permeability of the cartilage, and L is the length of the path along which liquid flows during equilibration. I show that this is true for any consolidation experiment, whatever the direction of consolidation and the direction of liquid flow. In thecourse of thisdemonsttation I prove that if load is applied abruptly to a Hookean material and is thereafter held constant, the strain energy at equilibrium equals the energy dissipated in the material during equilibration.
In two papers, Parsons and Black (1977, 1979) say that cartilage behaves as a viscoelastic solid, or, simply, that it is viscoelastic. Each paper gives a retardation time spectrum for the creep of cartilage. But cartilage is primarily poroelastic, not vi&elastic. In pure shear deformation, Hayes and Bodine (19781 have found that the elastic foras in the cartilagearc b@r than the damping forces by a factor of 5.5 at 1000 Hz, and a factor of 6.55 at 20 Hz When cartilage is compressed, however, damping forces may be many times the elastic forces, with the change in strain lagging the change in stresb by many minutes in a typical experiment (McCutchen, 1962). This is because attaining equilibrium compressive strain requires that pore fluid flow within (and out of) the cartilage. This flow is opposed by viscous forces in the fluid, and the order of the equilibration time r is given by t = L*/( Yo ),
(1)
where Y is the Young’s modulus, u the permeability of the material, and L is the length of the path along which liquid flows during eqDilibration. Note that I say equilibration time rather than relaxation time. It means the time required for most (which will not be more precisely defined)ofthe creep to occur. A poroelastic material does not have a single relaxation time, but characteristically creeps faster at the beginning, relative to its creep rate later on, than it would if thi creep followed an expo&tial curve (Biot, 1941. pp. l63-164:McCutchen. 1962.~. II :Kenvon. 1980.~. 1331.The retardation time spe&um fi;nd dy Pa&o& and ‘glack (1977, 1979) stretcheQl from 0.42s to II84 s. Because of the Lz dependencc, tht equilibration time depends sensitively on the size of the swimen, as well as on its material properties. It is well known that equation (I) holds in one-dimensional consolidation, where the liquid flow is parallel to the direction of consolidation (Biot, 1941; McCutchen, 1962; Kenyon. 1980). Likewis$ it is true for the consolidation of a poroelastic disk between itrtpervious plates that constrain the liquid to flow radially through the disk (McCutchen. 1962, p. 4). In l Rereeircl 25 Februcrry 1980: in rerisedform 1981.
what follows, I show, by an energy method, that these facts are not happenstances, that the equilibration time is of the order L’/(Yu) in any consolidation experiment, whatever the orientation of the deformation and flow path. 1 prove first that in any Hookean system at equilibrium the stored potential energy E, under i particular loading, is exactly half the integral W of the scalar uroduct ofdefonninn force and deformat& taken over the object. Hookean is h& defined to mean that e@libriwn strain is proportional to equilibrium stress. Because of the Hooktan requirclaent, the treatment is exact only t&r infinitesimal deformations ofreal materials. Including both surface and body forces this integral becomes
13 Junuar)
where f,, is the force applii to a unit area of the surface a, a0 is the force applied to one unit of volume u and do is the change in position of a point in the body consequent upon the application of the forces, Note that b0 is not in general parallel to f, or to ug. If the system that applies the forces is massless and lossless, so that the loadingcan be applied abruptly at the final value and then held constant however the body moves, W is the work done on the body. Body forces (like gravity or acceleration) act on the liquid as well as the solid of a poroelastic body. Loss or gain ofliquid at any place in the body during deformation will change the body fora distribution, which reinforces the limitation of the theorem to infinitesimal deformations. Now suppose that instead ofbeingapplied abiiptly at its final value the loading is increased slowly from zero. In the limit as the loading is done slower and slower there is no viscous loss. Provided that there is no speed-independent damping (Coulomb friction) the energy input will then equal the strain energy at the end of the ddormation. As the loading is increased, every point on the surface of the body and every point within it follows its own path through space. The work done per unit area of surfact at any point is the line integral of the applied stress along the path taken by the point. The work done per unit of volume at any point is the line integral of the 325
326
Technical Notes
applied body force per unit volume along the path taken by the point. Integrating these quantities over the surface and volume of the body respectively gives the work done on the whole body. Thus the strain energy E is given by a=*, 6=& u .dbdr. f.ddda+ E= (3) J.iD d=O SIV a=0 If we assume that the loading retains the same pattern as it increases from zero, so that only its magnitude varies, then in a Hookean material the deformation will also have a constant pattern and increasing magnitude. Hence f and u are proportional to 8, though not in general parallel to it. Thus f = c, lalt and II = C#(i, where C, and C2 are constants, and r and it are unit vectors. Letting d = 1818, where b is a unit vector, the expression for strain energy becomes
evaluated at the end of the immediate deformation. The strain energy EC at the end of the creep deformation is half W,., the value of equation (2) at the end of the creep deformation. Though W, is not necessarily the work done on the specimen in the immediate deformation, because some of the damping may occur outside the specimen, W,. - W, is quite accurately the work done on the specimen during creep, because little damping then occurs outside it. So the increase E,- - E, in strain energy during creep is (W,. - W,)/Z, or half the work done on the specimen during creep, and an identical quantity of energy goes into losses in the specimen. The loss due to flow during creep thus equals half the energy input during creep In a typical compression experiment this is half the load F times a,,, the distance moved by the loading plunger. The load in turn equals the equilibrium compressive strain SO/z,where Tis the thickness of the sample in the direction of compression, times the area A of the plunger, times the stiffness modulus of cartilage skeleton, which may be the Young’s modulus, the confined compression modulus, or something between them, depending on the experimental arrangement: the differena between these moduli is not large. The strain energy is thus given, exactly or approximately, by E = YA6;/2z. This equals the energy loss due to flow, which is the time integral of the volume integral of the product of the flow resistance by the square of the flow velocity. This integral can be evaluated very roughly by assuming that all the flow traverses the same route, and that it flows at a constant rate for a time t. The resistance of this flow path is, approximately, approximately, L/oB
(4)
which is half W as defined in equation (2). In deriving this relation we assumed that the stress pattern increased in a particular way. But strain energy depends on strain, not on strain history, so equation (4) holds however the system arrived at its final state. As the work done on the specimen is ideally W and the strain energy is E = W/2, the remaining energy, a quantity equal to the strain energy, must be absorbed by the losses. In a lightly damped system the loss component may be stored temporarily in kinetic and potential energy at different times and plaas in the specimen while the latter vibrates as the energy is being absorbed. In a practical experiment some of the loss may occur in the loading system. In this casz the work done on the specimen is less than 2E. In a poroelastic material with very small pores like cartilage, the strain following abrupt application of load occurs in two distinct stages, first a rapid deformation that ends before there has been significant flow of fluid, then a much slower creep that occurs only because fluid moves within the cartilage. Pore flow contributes littk to the losses that damp the quick &formation. These may be genuine viscoelastic losses in the specimen, or losses in the loading apparatus. The ditTerena between the rates of deformation is so large that their geometric mean is effectively infinitely slow relative to the immediate deformation, and instantaneous relative to the creep deformation. The preceding argument can therefore be applied in good approximation to the immediate deformation (but not directly to the creep deformation. Whereas the immediate deformation occurs in isolation, the creep deformation is always added to the immediate deformation.) The strain energy E, at the end of the immediate deformation is half W,, the value given by equation (2)
z L*/(aV),
where u is the permeability of the material, L is the length of the path, B its cross sectional area, and Y the volume of the specimen (or of that part of it that creeps, if parts of it do not). The rate of flow is given by the volume diilaczd Ab, divided by the time t. Here 6, is the total displaament, not that due to creep alone, and y lies between 1 and 1 - 2v depending on the particular experiment, where v is the Poisson’s ratio of cartilage skekton. There is negligibk flow during the immediate deformation. The energy loss during creep is the square of the flow rate, times the flow resistana, times the duration of the flow, or tL2A25;y2/(aVt2)
= (LA&y)‘/(&).
Were this expression exactly the energy loss it would equal the strain energy E, so we have, E = YAS;/(2r)
a (LA6,y)*/(uVr),
so,asA,=V, t f 2b+y2L2/(s; Yu),
(5)
or, sina the whole calculation is a very crude approximation anyway,
I z L2/( Vu).
(6)
which is equation (1 k Because, in particular cases, equation (1) can easily be derived by calculating the flow rate directly from the pmsure gradient, the energy method used here may seem roundabout. Its virtue is that, in effect, it automatically works out the pressure gradient and saves us from having to calculate it for each case of interest. It works for any direction offlow, which might be parallel to the compression. as when a thin disk is squeezed between porous plates, or at right angles to it, as happens when the plates are impervious. In the latter case the equilibration time is much longer than in the former because the flow path is longer (McCutchen, 1962, p. 3). lfwe take L to be the radius of the disk when the Rats are impervious, and half the thickness of the disk when the flats are porous, then the ratio between the two values is 16. so the ratio of
327
Technical Notes equilibration times should be 256. The data give a ratio nearer 100, which is not surprising, &cause taking Las the radius of the disk should overestimate its effective length. Because of the radial symmetry of the disk, the inner portion of the path conducts very littk fluid. Furthermore, any flow channels left by imperfect mating of the cartilage surfaces to the impervious glass plates will speed the equilibration. This pair of between-flats experiments illustrates what happens in a working joint. Under load, escaping fluid has to take the long path parallel to the rubbing surfaces. The equilibration time is long. When the cartilage is unloaded the fluid flows into it through its surface, and the equilibration time is much shorter. When an experiment is done on geometrically similar objects of difTerent sizes, the equilibration time is precisely proportional to the square of the size of the specimen. Even without similarity, the L2 dependence is so strong that it swamps any effect ofimperfect similarity unless Lisnearly the same in both cases. Thedisk-squeezing experiment shows this clearly, and a kitchen-table experiment is equally convincing. Try to dent cartilage on a bone end by pressing on it with finger or thumb. No obvious dent is produced, but the cartilage surface will bear a sharply-incised impression of one’s fingerprim. Papillary ridges are about 0.5 mm apart. The fingertip is over 1 cm across. Fluid must flow at least 20 times farther to escape from under the whole fingertip than to get out from under a papillary ridge. The equilibration time is
at least 400 times longer, so it is not surprising that no fingersized dent is formed. Laboratory of Pathology NIAMDD National Institute of Health Bethesda, MD20205 U.S.A.
C. W. MCC~JTCHEN
REFERENCES Biot. M. A. (1941) General theory of three-dimensional consolidation. J. appl. Phys. 12, 155-164. Hayes, W. C. and Bodine, A. J. (1978) Flow independent viscoelastic properties of articular cartilage matrix. J. Biomechanics 11, 407-419. Kenyon, D. E. (1980) A model for surface flow in cartilage. J. Biomechanics 13, 129-134. McCutchen, C. W. (1962) The fractional properties of animal ioints. Wear 5. 1-17. The auantitv “a” should be deleted Born all equations on page; 10 and 11 of this paper. Parsons, J. R. and Black, J. (1977) The viscoelastic shear behavior of normal rabbit articular cartilage. J. Biomechanics 10, 21-29. Parsons, J. R. and Black, J. (1979) Mechanical behavior of articular cartilage: Quantitative changes with alteration of ionic environment. J. Biomechanics 12, 765-773.