ON THE NONLINEAR VISCOELASTIC BEHAVIOR SOFT BIOLOGICAL TISSUES*
OF
P. H. DEHOFF?: Dtvision of Interdisciplinary
Studies. Clemson University. Clemson. SC 29651. U.S.A.
Abstract-Tuo
approxtmate constitutive equations which have proved useful for characterizing the nonlinear viscoelastic behavior of polymers are proposed as candidate theories to characterize soft biological tissues. The equivalence of these theories with one form of the Fung equation for stress relaxation is demonstrated. It is shown that different results are predicted for constant strain rate tests for data presented by Haut and Little (1972) for collagen fibers.
proved useful for characterizing the nonlinear viscoelastic behavior of polymers and these theories could prove equally useful for biological tissues. There are two such theories of interest to the present study. One is an approximation suggested by Lianis (1963~ based on the finite linear theory of viscoelasticity originally proposed by Coleman and No11 (1961). The second is an incompressible elastic fluid theory developed by Bernstein, Kearsley, and Zapas (1963) and commonly called the BKZ theory. The purpose of the present study is to show that the equation suggest:d by H & L for the relaxation behavior of collagen can be derived from both the Lianis theory and the BKZ theory. It is also shown that differences esist among the three theories for constant strain rate tests at small strain levels.
ISTRODCCTIOS
In a recent paper. Haut and Little (1971) (hereafter referred to as H & L) proposed a constitutive equation to characterize the time dependent behavior of cohagen fiber bundles extracted from the tails of mature rats. The starting point for the development of their equation was the quasilinear viscoelastic integral equation originally proposed by Fung (1967) to describe the uniaxial viscoelastic behavior of rabbit mesentery. The authors were able to use material constants determined experimentally from relaxation tests to predict constant strain rate tests, hysteresis loops. and peak stress decays in sinusoidal tests. According to curves presented by the authors agreement between experiment and theory for the strain rate tests and hysteresis loops was remarkably good while prediction of sinusoidal response was not as good. Although the H & L study represents an initial effort to test the usefulness of a theory for tissue response by taking the data developed under one set of conditions to predict the results of an independent test, it nevertheless suffers from at least two limitations which makes its general applicability to tissues other than tendons seem unlikely. The maximum strain level of 1.S; is well below the physiological levels to which many of the soft tissues are subjected and the theory is limited to one dimensional strain fields. For a constitutive equation to have general applicability in characterizing biological tissues, it must be capable of containing time dependence. anisotropy, nonlinearity. and multi-axial situations (Kenedi, 1975). While attempts to include all of these factors in a single theory would most likely lead to intractable mathematical and experimental complexities. there are several continuum-based approximate constitutive theories containing these components which have
HALT
ASD
LITTLE
The Fung equation in the form a(t) =
!a)
G(r
EQU.4TIOSS
is given by Haut and Little
_
1)
d$[j.(r)]d2
d:.
11)
0
where D is the nominal stress and G’ is the elastic stress generated instantly. when the tissue is subjected to a step extension ratio (i.). C(f) is a normalized relaxation function defined by Fung such that G(o) = 1. For small strains equation (1) can be rewritten as t G(t - T)g [E(S)] y d:. a(r) = (2) i “0 where E is the uniaxial strain. Experimentally it is not possible to generate a step strain input and Fung (1972) has suggested that constant strain rate tests can be used if the material is strain rate insensitive at high rates. H & L apparently used this scheme to define ec as 6’ = C,$.
(3)
Based on relaxation data the authors took G(r) in the form
* Received 28 .Clarch 1977. f Visiting Associate Professor. : Permanent Address: Mechanical Engineering Department, Bucknell University, Lewisburg, PA 17837, U.S.A.
G(c) = A’ In I + B’. 33
(1)
P. H. DEHOFF
34
For a relaxation strain history given by e(r) = e,,H(r), substitution of equations (3) and (4) into equation (2) leads to c(t) =
4.09 ‘0
.Eei[jlln t + 11,
(3
where E = 2B’C’ and p = A’/B’. E and ,Kwere experimentally found equal to 23 x 101Odyn,cm” and -0.23 respectively. For a constant strain rate test for which E = fit equation (2) becomes ~(r)=~(‘[l+p(ln$-3~2)].
(6)
It will now be shown that both the Lianis and BKZ theories can be used to develop equation (5) but that different results are found for equation (6). For convenience, the Lianis equations and the BKZ equations will be developed under separate headings in the following sections. LMXUIS THEORY It has been shown by Lianis that the nonlinear uniaxial viscoelastic behavior of many polymers can be represented in the form
X “5 3.0Y w” 5o 2.0-
i
LO‘( b
Fig.
- I/i(r)]dr
+ 2
- l/i(r)]dT
+ 2
J’ &(t * Qdt - 5) J--r T)&[i2i2(T)
--P
-
l/n+)]dr
x
&[12(r)
+ (i.’ - l/i)
-k
2/i.(r)]dr,
(7)
where a, 6 and c are constants defining the long-time (equilibrium) behavior and the pi are time-dependent relaxation functions such that lim,,, &(r) = 0; i = d(t) is the uniaxial extension ratio at the present time t. It should be noted that equation (7) represents a reduction to the one dimensional case of a more general equation. The details of this development are omitted here but can be found in Lianis (1963). For a uniaxial stress relaxation test for which j.(r) = 1 r < 0 and i(r) = i. (constant) T > 0 it can be shown that equation (7) reduces to -
2’
a(t)
= [a + b + 2&(t) - 2M)l + 240(t) +
+ It
32(f)
+
+ C2b + c
‘44)l;
242(t) + d3(t)lC;.2-
11 h typical relaxation isochrone for polymers.
7
40 -
I/i.
=
[a + 24,(t) +
’ 90(t - 5) & [J.*(r)/j.’ s -cc
+2
1. A
11.
+ [b (8)
has been shown by Lianis that equation (8) is useful for materials which exhibit relaxation iso-
1 1.0
thrones (curves at constant time) similar to that depicted in Fig. I. Furthermore, if we consider only the range for which the curve is linear, equation (8) reduces further to
A’
G(f) = [a + b(i.* + 2/i - 3) + c/%](i.’ - l/j.)
I 0.5
0 . 0
wml
[c + W,(r) + 4Jsw;.
(9)
Ordinarily, the constants a and b are determined from a relaxation test in which the stress has decayed to its equilibrium value. In the present case we arbitrarily assume that biological tissues exhibit no equilibrium behavior so that n = c = 0. Additionaliy it is assumed that both 42(t) and &(t) are identically zero. Utilizing these approximations equation (9) takes the simple form
m p---= - l/i.
W,(t) +
?40w;.
(10)
The equivalence between equation (10) and the H & L equation (5) requires a transformation of equation (10) to account for small strains. Since 1. = 1 + er, we can expand i’ and retain terms to the O(e’) so that equation (10) becomes c(r) = 6%f_(~o(r)+ &(r)) - 4o(tkol.
(11)
Now, utilizing the relaxation results from H & L, several relaxation isochrones are plotted for the collagen bundles in Fig. 2. It is interesting to note that the straight line segment of a relaxation isochrone for polymers generally has a positive slope (Fig. 1) whereas the corresponding slope for biological tissues is generally negative. This behavior for tissues was also observed by Veronda and Westmann (1970) for the large strain elastic behavior of cat skin. While no explanation for this difference in behavior is attempted here, it can be noted that the negative slope leads to a stress-strain isochronic curve which is concave upward whereas the positive slope leads to a
37
Nonlinear viscoelastic behavior of soft bioloplcal tissues
and retain terms to the O(E)to obtain (15) Substitution of equation (13) into (15) and subsequent integration leads to
I/X Fig. 2.
Relaxation isochrones for tendon bundles calculated from the Haut and Little equation.
stress-strain isochronic curve which is concave downward. Returning now to the problem at hand, it can be observed that the relaxation functions 4i(t) and &,(T) in equation (10) can be evaluated from a series of such isochrones as the slope and the intercept respectively. For our purposes it is sufficient to observe that ~(t)‘(;.’ - 4) is zero for all isochrones at i = 1 so that 4i(t) = -4,,(t) for all time. Utilizing this information allows us to cast equation (11) in the form 4)
= 641(rk$.
(12)
Comparison of equations (12) and (5) leads us to conclude that these relaxation equations are equivalent if 4,(t) = -&(t)
=i@lnr
+ 11.
I’ -0
410 -
BKZ THEORY
The complete development of the BKZ incompressible elastic fluid theory can be found in Bernstein, Kearsley and Zapas (1963). This theory has been applied to the nonlinear viscoelastic behavior of polymers by Zapas and Craft (1965). Zapas (1966), Goldberg ec al. (1967). and more recently by Sadd and Morris (1976). For a uniaxial state of strain the BKZ theory can be written as u(t) =
(13)
Thus we have shown that the H & L form of the Fung equation for the relaxation behavior of collagen fibers can be obtained from continuum concepts. Although the comparison was developed for small strains, equation (1) is applicable over any strain range for which the relaxation isochrones are linear. There is some recent evidence (DeHoff and Bingham, 1977) to suggest that the relaxation behavior of the canine anterior cruciate ligament shows similar behavior for strains up to approx. 15%. In order to compare the constant strain rate behavior of the two theories we set a = b = c = qj>(t) = &(r) = 0 in equation (7) and rewrite it in the form u(t) = - 2
Equations (6) and (16) predict significantly different results for constant strain rate tests as can be seen in Fig. 3. Since H & L showed surprisingly close agreement between their theoretical predictions and experimental results for constant strain rate tests, one would be tempted to declare that the H & L equation is the more accurate equation. However. sufficient scatter seems to exist in the H & L data to allow a different interpretation. This will be discussed in more detail in a later section after the BKZ comparison has been developed.
(17) *
6-
b ‘0
-
X
---LIANIS THEORY -HBL THEORY -----BKt THEORY A TABLE I Of H&L 60 FIG IO OF HBL
“s v \
9; 2 [.
---
i(t)
L(r)
i.‘(r) +
$+ dr, .
(14)
I
where we have also used the fact that &(t) = -4,(t). For a constant strain rate test in which j.(t) = 1 + c(t); e(t) = fit t > 0, we can expand equation (14)
0
0.5
1.0
1.5
STRAIN, % Fig. 3. Comparison of three predictions with constant strain rate data at 3.6%,‘min.
P. H.
38
where i. has the same meaning as before and
h($+)=2[g+gg].
(18)
Here U is a material function which depends on time and the two strain invariants I, and 12. For a uniaxial stress relaxation situation Equation (17) can be written as
DEHOFF
This equation is somewhat different from both the H & L squation and the Lianis. although it predicts very close to the Lianis case. A comparison of all three predictions is shown in Fig. 3 for an arbitrarily selected strain rate of 3.6yiJmin. For convenience, the experimental data scatter bands as presented in Fig. 6 of H & L (1973) are approximately represented in the figure.
m
= H(& f), A2 - 1i/i
DISCUsslON
where H(i, f) =
It
w h(k OX,
(20)
As previously mentioned it would appear that the H & L equation fits the experimental strain rate data better than both the Lianis and BKZ theories. However, H & L have also presented a table in which constant strain rate data has been fit with an empirical equation of the form
$$i.,t).
(21)
CJ= &“,
from which h(i., t) = -
It can be noted that equations (19) and (20) allow the determination of the form of h&t) from a simple relaxation experiment which then permits the use of equation (17) for other uniaxial strain histories. In the present case a comparison of equation (19) with (10) leads us to conclude that the BKZ and Lianis theories predict the same relaxation behavior if we take HG, 0 = 24,(t) + 2M);.
(22)
Hence all the arguments which were used for the relaxation behavior of the Lianis theory apply to the BKZ theory as well. Thus the equivalence between the H & L relaxation equation and the BKZ theory is indicated by equation (13). For the constant strain rate test, the situation is somewhat different. It can be shown that for a motion which starts at t = 0 following a rest history, equation (17) takes the special form:
- $)]h(&,
t - r)di.
Substitution of j.(t) = 1 + e(t) and linearization leads f fJ(t) = 12@ 4l(t - W I0 -
(23)
subsequent
128 ; 41(t - r)+)dr, I
(24)
where use has been made of equation (22) and e(t) = fit for a constant strain rate test. Substitution of (13) into (24) and performing the indicated integration yields t3(t)=_W{l
+p[Ini-il>.
(25)
(26)
where .-l and B were experimentally determined constants which were to some extent strain rate dependent. For the 3.6yJmin test A = 1.81 and B = 20.8 x lOLodyrn:/cm*.These constants, when substituted into equation (26). yield stress levels which are much higher than those predicted by the H & L equation. For convenience. several stresses calculated from Equation (26) have been plotted on Fig. 3 where it can be observed that these points lie above the Lianis prediction. A second factor which suggests that considerable scatter existed in the strain rate data is the fact that the loading portion of the 3.6y/dmin hysteresis loop (Fig. 10-H & L) exhibited stresses which were higher than the 3.6%/min rate curve. While it is somewhat difficult to pick data off published curves there nevertheless seems to be a significant difference. For comparative purposes, several points taken from Fig. 10 of H & L have been approximately plotted on Fig. 3 where it can be observed that these points fall above the H & L curve but below the Lianis and BKZ curves. While it is quite possible that the H & L data have been misinterpreted here, significant scatter in biological data is to be expected. This is especially true when very small strains and small loads are being measured. Thus it would seem that all three equations might predict constant strain rate response which falls within the expected scatter band. A final consideration in interpreting the accuracy of the constant strain rate predictions is the manner in which the relaxation constants were determined by H & L. Theoretically the relaxation test requires an instantaneous strain history (zero rise time) which, of course, can not be obtained experimentally. In the H & L study a screw type loading frame was used in which the maximum strain rate for the initial ramp of the relaxation strain history was apparently 4l%/min. Each specimen was strained at some finite rate to a predetermined strain level and held at this level for approx. 3 min. The constants E and p in
Nonlinear viscoelastic behavior of soft biologtcal tissues
39
equation (5) were then determined as an average of three tests at different strain rates for each level of strain. When relaxation tests are conducted in this manner the relaxation data are generally not considered valid until after perhaps ten times the rise time of the constant strain rate part of the history. Since the load response depends on the entire strain history the short time data will depend greatly on the way in which the “step” was generated. If a material possesses fading memory then the longer time relaxation response is assumed to depend only on the strain level and not the way in which this level was reached. Thus one could expect to predict constant strain rate response only at strain rates which are considerably lower than the lowest strain rate employed in the
In view of the success which the various continuum based theories have demonstrated for the nonlinear viscoelastic behavior of polymers it would seem reasonable that their potential for characterizing biological tissues should be thoroughly investigated Because of the complex structure of biological tissues it is highly unlikely that a single theory will prove capable of characterizing all tissues or in fact even a class of tissues in their various physiological strain environments. However, some additional flexibility in tissue characterization could possibly be obtained by
relaxation tests. This does not appear to be the case in the H & L study inasmuch as theoretical predictions were made for a 48Y$‘min rate which was probably as high as or higher than the strain rates used for the ramp part of the relaxation strain history.
Bernstein, B., Kearsley, E. A. and Zapas, L. S. (1963) A study of stress relaxation with finite strain. Trans. Sot. Rheol. 7, 391110. Coleman, B. D. and Nell, W. (1961) Foundations of linear viscoelasticity. Rec. Mod. Phys. 33. 239-249. DeHoff. P. H. and Bingham, D. (1977) Nonlinear viscoelastic characterization of the canine anterior cruciate ligament. 6th Canadian Congress of Applied Mechanics. Vancouver, B.C. Fung. Y. C. (1967) Elasticity of soft tissues in simple elongation. Am. J. Phpsio[. 213. 1532-1544. Goldberg. W., Bernstein, B. and Lianis. G. (1969) The exponential extension ratio history, comparison of theory with experiment. far. jI. Xon-linear :tlech. 4. 277-300. Haut, R. C. and Little, R. W. (1972) A constitutive equation for collagen fibers. J. Biomechnnics 5, 423-430. Jenkins. R. B. and Little, R. W. (1974) A constitutive equation for parallel-fibered elastic tissue. J. Biomrchanics 7, 397-402. Kenedi, R. Xl., Gibson, T.. Evans, S. H. and Barbenel, J. C. (1975) Tissue mechanics. Phys. Med. Biol. 20, 699-l 17. Lianis, G. (1963) Constitutive equations of viscoelastic solids under large deformation. A & ES Report No. 63-j, Purdue University. Sadd, M. H. and Xlorris, D. H. (1976) Rate-dependent stress-strain behavior of polymeric materials. J. Appl. Po/ym. Sci. 20. 421-433. Veronda, D. R. and Westmann, R. A. (1970) Mechanical characterization of skin-finite deformations. J. Biomechnnics 3, 111-11-l. Zapas. L. J. and Craft. T. (1965) Correlation of large longitudinal deformations with different strain histories. J. Res. Kar. Bur. Srond. 69A. Zapas, L. J. (1966) Viscoelastic behavior under large deformations. J. Res. .Vat. Bur. Srand. 7OA, 525-532.
C04CLL’SION It has been shown that the relaxation equation of H & L proposed to characterize the behavior of collagen fiber bundles can be obtained from existing continuum-based theories as a special case. The advantage which such theories have over essentially empirically developed theories, such as the Fung equation, lies in the fact that constitutive equations derived on the basis of continuum concepts are generalized theories. That is, they are capable of handling nonlinear, time dependent, and multi-dimensional stress and strain histories for both isotropic and anisotropic materials. Many of the situations of interest in biological tissues can be treated as special cases of the continuum theories. There have been several studies which have been based on continuum formulations but Kenedi et a/. (1975). while recognizing them as sound and elegant, have stated that features such as time dependence and anisotropy have been disregarded in many of these.
In addition. for any theory to have real significance it must be capable of taking material coefficients found from one set of experiments and using them to predict the behavior in an independent situation. This was generally not done in early studies but the H & L study suggests that the Fung equation can be used successfully in this way for collagen fibers over a small range of strains. More recently, Jenkins and Little (1974) have generalized the Fung equation further to introduce strain dependence in the normalized relaxation function and have reported success in characterizing the nonlinear uniaxial response of bovine ligamentum nuchae samples up to 60% strain. Again constants found from one set of data were reportedly successfully used to predict constant strain rate response.
the application
of these theories.
REFERESCES
NOMENCLATURE a
A b” B B’ C
C B E G h H
Elastic constant (dyn/cm’) Empirical strain rate exponent Non-dimensional relaxation constant Elastic constant (dyn,cm’) Empirical strain rate constant (dyn/cmz) Non-dimensional relaxation constant Elastic constant (dyn,‘cm’) Strain rate constant (dyn/cm”) Empirical strain rate constant (dyn;cm’) Equivalent elastic modulus (dynjcm’) Normalized relaxation function Material function Stress relaxation material function
10
1; f L’ /I E
P. H. Strain invariants Present time (min) Material function Strain rate (/min) Unit strain
DEHOFF
& 2. p G T
Relaxation function (dyn,‘cm’) Uniaxiai extension ratio Equivalent normalized time constant Normal stress (dkn.‘cm’) Time history (min).
OF
P. H. DEHOFF?: Dtvision of Interdisciplinary
Studies. Clemson University. Clemson. SC 29651. U.S.A.
Abstract-Tuo
approxtmate constitutive equations which have proved useful for characterizing the nonlinear viscoelastic behavior of polymers are proposed as candidate theories to characterize soft biological tissues. The equivalence of these theories with one form of the Fung equation for stress relaxation is demonstrated. It is shown that different results are predicted for constant strain rate tests for data presented by Haut and Little (1972) for collagen fibers.
proved useful for characterizing the nonlinear viscoelastic behavior of polymers and these theories could prove equally useful for biological tissues. There are two such theories of interest to the present study. One is an approximation suggested by Lianis (1963~ based on the finite linear theory of viscoelasticity originally proposed by Coleman and No11 (1961). The second is an incompressible elastic fluid theory developed by Bernstein, Kearsley, and Zapas (1963) and commonly called the BKZ theory. The purpose of the present study is to show that the equation suggest:d by H & L for the relaxation behavior of collagen can be derived from both the Lianis theory and the BKZ theory. It is also shown that differences esist among the three theories for constant strain rate tests at small strain levels.
ISTRODCCTIOS
In a recent paper. Haut and Little (1971) (hereafter referred to as H & L) proposed a constitutive equation to characterize the time dependent behavior of cohagen fiber bundles extracted from the tails of mature rats. The starting point for the development of their equation was the quasilinear viscoelastic integral equation originally proposed by Fung (1967) to describe the uniaxial viscoelastic behavior of rabbit mesentery. The authors were able to use material constants determined experimentally from relaxation tests to predict constant strain rate tests, hysteresis loops. and peak stress decays in sinusoidal tests. According to curves presented by the authors agreement between experiment and theory for the strain rate tests and hysteresis loops was remarkably good while prediction of sinusoidal response was not as good. Although the H & L study represents an initial effort to test the usefulness of a theory for tissue response by taking the data developed under one set of conditions to predict the results of an independent test, it nevertheless suffers from at least two limitations which makes its general applicability to tissues other than tendons seem unlikely. The maximum strain level of 1.S; is well below the physiological levels to which many of the soft tissues are subjected and the theory is limited to one dimensional strain fields. For a constitutive equation to have general applicability in characterizing biological tissues, it must be capable of containing time dependence. anisotropy, nonlinearity. and multi-axial situations (Kenedi, 1975). While attempts to include all of these factors in a single theory would most likely lead to intractable mathematical and experimental complexities. there are several continuum-based approximate constitutive theories containing these components which have
HALT
ASD
LITTLE
The Fung equation in the form a(t) =
!a)
G(r
EQU.4TIOSS
is given by Haut and Little
_
1)
d$[j.(r)]d2
d:.
11)
0
where D is the nominal stress and G’ is the elastic stress generated instantly. when the tissue is subjected to a step extension ratio (i.). C(f) is a normalized relaxation function defined by Fung such that G(o) = 1. For small strains equation (1) can be rewritten as t G(t - T)g [E(S)] y d:. a(r) = (2) i “0 where E is the uniaxial strain. Experimentally it is not possible to generate a step strain input and Fung (1972) has suggested that constant strain rate tests can be used if the material is strain rate insensitive at high rates. H & L apparently used this scheme to define ec as 6’ = C,$.
(3)
Based on relaxation data the authors took G(r) in the form
* Received 28 .Clarch 1977. f Visiting Associate Professor. : Permanent Address: Mechanical Engineering Department, Bucknell University, Lewisburg, PA 17837, U.S.A.
G(c) = A’ In I + B’. 33
(1)
P. H. DEHOFF
34
For a relaxation strain history given by e(r) = e,,H(r), substitution of equations (3) and (4) into equation (2) leads to c(t) =
4.09 ‘0
.Eei[jlln t + 11,
(3
where E = 2B’C’ and p = A’/B’. E and ,Kwere experimentally found equal to 23 x 101Odyn,cm” and -0.23 respectively. For a constant strain rate test for which E = fit equation (2) becomes ~(r)=~(‘[l+p(ln$-3~2)].
(6)
It will now be shown that both the Lianis and BKZ theories can be used to develop equation (5) but that different results are found for equation (6). For convenience, the Lianis equations and the BKZ equations will be developed under separate headings in the following sections. LMXUIS THEORY It has been shown by Lianis that the nonlinear uniaxial viscoelastic behavior of many polymers can be represented in the form
X “5 3.0Y w” 5o 2.0-
i
LO‘( b
Fig.
- I/i(r)]dr
+ 2
- l/i(r)]dT
+ 2
J’ &(t * Qdt - 5) J--r T)&[i2i2(T)
--P
-
l/n+)]dr
x
&[12(r)
+ (i.’ - l/i)
-k
2/i.(r)]dr,
(7)
where a, 6 and c are constants defining the long-time (equilibrium) behavior and the pi are time-dependent relaxation functions such that lim,,, &(r) = 0; i = d(t) is the uniaxial extension ratio at the present time t. It should be noted that equation (7) represents a reduction to the one dimensional case of a more general equation. The details of this development are omitted here but can be found in Lianis (1963). For a uniaxial stress relaxation test for which j.(r) = 1 r < 0 and i(r) = i. (constant) T > 0 it can be shown that equation (7) reduces to -
2’
a(t)
= [a + b + 2&(t) - 2M)l + 240(t) +
+ It
32(f)
+
+ C2b + c
‘44)l;
242(t) + d3(t)lC;.2-
11 h typical relaxation isochrone for polymers.
7
40 -
I/i.
=
[a + 24,(t) +
’ 90(t - 5) & [J.*(r)/j.’ s -cc
+2
1. A
11.
+ [b (8)
has been shown by Lianis that equation (8) is useful for materials which exhibit relaxation iso-
1 1.0
thrones (curves at constant time) similar to that depicted in Fig. I. Furthermore, if we consider only the range for which the curve is linear, equation (8) reduces further to
A’
G(f) = [a + b(i.* + 2/i - 3) + c/%](i.’ - l/j.)
I 0.5
0 . 0
wml
[c + W,(r) + 4Jsw;.
(9)
Ordinarily, the constants a and b are determined from a relaxation test in which the stress has decayed to its equilibrium value. In the present case we arbitrarily assume that biological tissues exhibit no equilibrium behavior so that n = c = 0. Additionaliy it is assumed that both 42(t) and &(t) are identically zero. Utilizing these approximations equation (9) takes the simple form
m p---= - l/i.
W,(t) +
?40w;.
(10)
The equivalence between equation (10) and the H & L equation (5) requires a transformation of equation (10) to account for small strains. Since 1. = 1 + er, we can expand i’ and retain terms to the O(e’) so that equation (10) becomes c(r) = 6%f_(~o(r)+ &(r)) - 4o(tkol.
(11)
Now, utilizing the relaxation results from H & L, several relaxation isochrones are plotted for the collagen bundles in Fig. 2. It is interesting to note that the straight line segment of a relaxation isochrone for polymers generally has a positive slope (Fig. 1) whereas the corresponding slope for biological tissues is generally negative. This behavior for tissues was also observed by Veronda and Westmann (1970) for the large strain elastic behavior of cat skin. While no explanation for this difference in behavior is attempted here, it can be noted that the negative slope leads to a stress-strain isochronic curve which is concave upward whereas the positive slope leads to a
37
Nonlinear viscoelastic behavior of soft bioloplcal tissues
and retain terms to the O(E)to obtain (15) Substitution of equation (13) into (15) and subsequent integration leads to
I/X Fig. 2.
Relaxation isochrones for tendon bundles calculated from the Haut and Little equation.
stress-strain isochronic curve which is concave downward. Returning now to the problem at hand, it can be observed that the relaxation functions 4i(t) and &,(T) in equation (10) can be evaluated from a series of such isochrones as the slope and the intercept respectively. For our purposes it is sufficient to observe that ~(t)‘(;.’ - 4) is zero for all isochrones at i = 1 so that 4i(t) = -4,,(t) for all time. Utilizing this information allows us to cast equation (11) in the form 4)
= 641(rk$.
(12)
Comparison of equations (12) and (5) leads us to conclude that these relaxation equations are equivalent if 4,(t) = -&(t)
=i@lnr
+ 11.
I’ -0
410 -
BKZ THEORY
The complete development of the BKZ incompressible elastic fluid theory can be found in Bernstein, Kearsley and Zapas (1963). This theory has been applied to the nonlinear viscoelastic behavior of polymers by Zapas and Craft (1965). Zapas (1966), Goldberg ec al. (1967). and more recently by Sadd and Morris (1976). For a uniaxial state of strain the BKZ theory can be written as u(t) =
(13)
Thus we have shown that the H & L form of the Fung equation for the relaxation behavior of collagen fibers can be obtained from continuum concepts. Although the comparison was developed for small strains, equation (1) is applicable over any strain range for which the relaxation isochrones are linear. There is some recent evidence (DeHoff and Bingham, 1977) to suggest that the relaxation behavior of the canine anterior cruciate ligament shows similar behavior for strains up to approx. 15%. In order to compare the constant strain rate behavior of the two theories we set a = b = c = qj>(t) = &(r) = 0 in equation (7) and rewrite it in the form u(t) = - 2
Equations (6) and (16) predict significantly different results for constant strain rate tests as can be seen in Fig. 3. Since H & L showed surprisingly close agreement between their theoretical predictions and experimental results for constant strain rate tests, one would be tempted to declare that the H & L equation is the more accurate equation. However. sufficient scatter seems to exist in the H & L data to allow a different interpretation. This will be discussed in more detail in a later section after the BKZ comparison has been developed.
(17) *
6-
b ‘0
-
X
---LIANIS THEORY -HBL THEORY -----BKt THEORY A TABLE I Of H&L 60 FIG IO OF HBL
“s v \
9; 2 [.
---
i(t)
L(r)
i.‘(r) +
$+ dr, .
(14)
I
where we have also used the fact that &(t) = -4,(t). For a constant strain rate test in which j.(t) = 1 + c(t); e(t) = fit t > 0, we can expand equation (14)
0
0.5
1.0
1.5
STRAIN, % Fig. 3. Comparison of three predictions with constant strain rate data at 3.6%,‘min.
P. H.
38
where i. has the same meaning as before and
h($+)=2[g+gg].
(18)
Here U is a material function which depends on time and the two strain invariants I, and 12. For a uniaxial stress relaxation situation Equation (17) can be written as
DEHOFF
This equation is somewhat different from both the H & L squation and the Lianis. although it predicts very close to the Lianis case. A comparison of all three predictions is shown in Fig. 3 for an arbitrarily selected strain rate of 3.6yiJmin. For convenience, the experimental data scatter bands as presented in Fig. 6 of H & L (1973) are approximately represented in the figure.
m
= H(& f), A2 - 1i/i
DISCUsslON
where H(i, f) =
It
w h(k OX,
(20)
As previously mentioned it would appear that the H & L equation fits the experimental strain rate data better than both the Lianis and BKZ theories. However, H & L have also presented a table in which constant strain rate data has been fit with an empirical equation of the form
$$i.,t).
(21)
CJ= &“,
from which h(i., t) = -
It can be noted that equations (19) and (20) allow the determination of the form of h&t) from a simple relaxation experiment which then permits the use of equation (17) for other uniaxial strain histories. In the present case a comparison of equation (19) with (10) leads us to conclude that the BKZ and Lianis theories predict the same relaxation behavior if we take HG, 0 = 24,(t) + 2M);.
(22)
Hence all the arguments which were used for the relaxation behavior of the Lianis theory apply to the BKZ theory as well. Thus the equivalence between the H & L relaxation equation and the BKZ theory is indicated by equation (13). For the constant strain rate test, the situation is somewhat different. It can be shown that for a motion which starts at t = 0 following a rest history, equation (17) takes the special form:
- $)]h(&,
t - r)di.
Substitution of j.(t) = 1 + e(t) and linearization leads f fJ(t) = 12@ 4l(t - W I0 -
(23)
subsequent
128 ; 41(t - r)+)dr, I
(24)
where use has been made of equation (22) and e(t) = fit for a constant strain rate test. Substitution of (13) into (24) and performing the indicated integration yields t3(t)=_W{l
+p[Ini-il>.
(25)
(26)
where .-l and B were experimentally determined constants which were to some extent strain rate dependent. For the 3.6yJmin test A = 1.81 and B = 20.8 x lOLodyrn:/cm*.These constants, when substituted into equation (26). yield stress levels which are much higher than those predicted by the H & L equation. For convenience. several stresses calculated from Equation (26) have been plotted on Fig. 3 where it can be observed that these points lie above the Lianis prediction. A second factor which suggests that considerable scatter existed in the strain rate data is the fact that the loading portion of the 3.6y/dmin hysteresis loop (Fig. 10-H & L) exhibited stresses which were higher than the 3.6%/min rate curve. While it is somewhat difficult to pick data off published curves there nevertheless seems to be a significant difference. For comparative purposes, several points taken from Fig. 10 of H & L have been approximately plotted on Fig. 3 where it can be observed that these points fall above the H & L curve but below the Lianis and BKZ curves. While it is quite possible that the H & L data have been misinterpreted here, significant scatter in biological data is to be expected. This is especially true when very small strains and small loads are being measured. Thus it would seem that all three equations might predict constant strain rate response which falls within the expected scatter band. A final consideration in interpreting the accuracy of the constant strain rate predictions is the manner in which the relaxation constants were determined by H & L. Theoretically the relaxation test requires an instantaneous strain history (zero rise time) which, of course, can not be obtained experimentally. In the H & L study a screw type loading frame was used in which the maximum strain rate for the initial ramp of the relaxation strain history was apparently 4l%/min. Each specimen was strained at some finite rate to a predetermined strain level and held at this level for approx. 3 min. The constants E and p in
Nonlinear viscoelastic behavior of soft biologtcal tissues
39
equation (5) were then determined as an average of three tests at different strain rates for each level of strain. When relaxation tests are conducted in this manner the relaxation data are generally not considered valid until after perhaps ten times the rise time of the constant strain rate part of the history. Since the load response depends on the entire strain history the short time data will depend greatly on the way in which the “step” was generated. If a material possesses fading memory then the longer time relaxation response is assumed to depend only on the strain level and not the way in which this level was reached. Thus one could expect to predict constant strain rate response only at strain rates which are considerably lower than the lowest strain rate employed in the
In view of the success which the various continuum based theories have demonstrated for the nonlinear viscoelastic behavior of polymers it would seem reasonable that their potential for characterizing biological tissues should be thoroughly investigated Because of the complex structure of biological tissues it is highly unlikely that a single theory will prove capable of characterizing all tissues or in fact even a class of tissues in their various physiological strain environments. However, some additional flexibility in tissue characterization could possibly be obtained by
relaxation tests. This does not appear to be the case in the H & L study inasmuch as theoretical predictions were made for a 48Y$‘min rate which was probably as high as or higher than the strain rates used for the ramp part of the relaxation strain history.
Bernstein, B., Kearsley, E. A. and Zapas, L. S. (1963) A study of stress relaxation with finite strain. Trans. Sot. Rheol. 7, 391110. Coleman, B. D. and Nell, W. (1961) Foundations of linear viscoelasticity. Rec. Mod. Phys. 33. 239-249. DeHoff. P. H. and Bingham, D. (1977) Nonlinear viscoelastic characterization of the canine anterior cruciate ligament. 6th Canadian Congress of Applied Mechanics. Vancouver, B.C. Fung. Y. C. (1967) Elasticity of soft tissues in simple elongation. Am. J. Phpsio[. 213. 1532-1544. Goldberg. W., Bernstein, B. and Lianis. G. (1969) The exponential extension ratio history, comparison of theory with experiment. far. jI. Xon-linear :tlech. 4. 277-300. Haut, R. C. and Little, R. W. (1972) A constitutive equation for collagen fibers. J. Biomechnnics 5, 423-430. Jenkins. R. B. and Little, R. W. (1974) A constitutive equation for parallel-fibered elastic tissue. J. Biomrchanics 7, 397-402. Kenedi, R. Xl., Gibson, T.. Evans, S. H. and Barbenel, J. C. (1975) Tissue mechanics. Phys. Med. Biol. 20, 699-l 17. Lianis, G. (1963) Constitutive equations of viscoelastic solids under large deformation. A & ES Report No. 63-j, Purdue University. Sadd, M. H. and Xlorris, D. H. (1976) Rate-dependent stress-strain behavior of polymeric materials. J. Appl. Po/ym. Sci. 20. 421-433. Veronda, D. R. and Westmann, R. A. (1970) Mechanical characterization of skin-finite deformations. J. Biomechnnics 3, 111-11-l. Zapas. L. J. and Craft. T. (1965) Correlation of large longitudinal deformations with different strain histories. J. Res. Kar. Bur. Srond. 69A. Zapas, L. J. (1966) Viscoelastic behavior under large deformations. J. Res. .Vat. Bur. Srand. 7OA, 525-532.
C04CLL’SION It has been shown that the relaxation equation of H & L proposed to characterize the behavior of collagen fiber bundles can be obtained from existing continuum-based theories as a special case. The advantage which such theories have over essentially empirically developed theories, such as the Fung equation, lies in the fact that constitutive equations derived on the basis of continuum concepts are generalized theories. That is, they are capable of handling nonlinear, time dependent, and multi-dimensional stress and strain histories for both isotropic and anisotropic materials. Many of the situations of interest in biological tissues can be treated as special cases of the continuum theories. There have been several studies which have been based on continuum formulations but Kenedi et a/. (1975). while recognizing them as sound and elegant, have stated that features such as time dependence and anisotropy have been disregarded in many of these.
In addition. for any theory to have real significance it must be capable of taking material coefficients found from one set of experiments and using them to predict the behavior in an independent situation. This was generally not done in early studies but the H & L study suggests that the Fung equation can be used successfully in this way for collagen fibers over a small range of strains. More recently, Jenkins and Little (1974) have generalized the Fung equation further to introduce strain dependence in the normalized relaxation function and have reported success in characterizing the nonlinear uniaxial response of bovine ligamentum nuchae samples up to 60% strain. Again constants found from one set of data were reportedly successfully used to predict constant strain rate response.
the application
of these theories.
REFERESCES
NOMENCLATURE a
A b” B B’ C
C B E G h H
Elastic constant (dyn/cm’) Empirical strain rate exponent Non-dimensional relaxation constant Elastic constant (dyn,cm’) Empirical strain rate constant (dyn/cmz) Non-dimensional relaxation constant Elastic constant (dyn,‘cm’) Strain rate constant (dyn/cm”) Empirical strain rate constant (dyn;cm’) Equivalent elastic modulus (dynjcm’) Normalized relaxation function Material function Stress relaxation material function
10
1; f L’ /I E
P. H. Strain invariants Present time (min) Material function Strain rate (/min) Unit strain
DEHOFF
& 2. p G T
Relaxation function (dyn,‘cm’) Uniaxiai extension ratio Equivalent normalized time constant Normal stress (dkn.‘cm’) Time history (min).
