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A thermodynamically consistent constitutive equation for the elastic force-length relation of soft biological materials
.I. Biomechanics Printed
Vol. 22. No. 1 l/12. pp. 1203-1208,
m Great
1989.
co21~9290/89 nIxI+ .oo Pergamon Press plc
Brdam
A THERMODYNAMICALLY CONSISTENT CONSTITUTIVE EQUATION FOR THE ELASTIC FORCE-LENGTH RELATION OF SOFT BIOLOGICAL MATERIALS M. MAES, V. J. VANHUYSE, W. F. DECRAEMER and E. R. RAMAN Laboratory
for Experimental
Physics,
Rijksuniversitair Centrum Antwerpen, B-2020 Antwerpen, Belgium
Groenenborgerlaan
171,
Abstract-Starting from the laws of thermodynamics of reversible processes, a temperature-dependent constitutive equation is derived for the elastic force-length relation of soft biological tissues. These tissues are composed of a network of fibres (mainly collagen). The equation is based on a model which uses a simplified two-dimensional representation of the cc-helix of collagen.
NOMENCLATURE energy needed to break a cross-link force needed to hold the specimen at length I force needed to hold a fibre at length 1 internal energy of the specimen entropy of the specimen internal energy of a fibre entropy of a fibre number of configurations Boltzmann constant total number of zigzag elements in a fibre number of broken cross-links length of an element in the fibre model angle between two elements in the fibre model cos (b/2) length of fibre when no cross-links are broken fibre cross-section fibre Young’s modulus integration constant mean value of the normal distribution of fibre restlength at temperature T standard deviation of the normal distribution of fibre rest-lengths, at temperature T value of p at T=O value of s at T=O rest length of the specimen at temperature T rest-length of the specimen at T=O total number of fibres effective Young’s modulus (= NaK) temperature below which no cross-links are broken stress (= F/A,) relative length ( = l/l,) relative fibre length (= c/l,) specimen cross-section
1. INTRODUCTION
Many authors have proposed an analytical expression for the elastic force-length relation for soft biological materials in uniaxial tension, but none of them takes into account the influence of the temperature (Fung, 1972). Our constitutive equation has been derived from the laws of thermodynamics of reversible processes. It is an extension of our earlier developed equation (Decraemer et al., 1980), which is based on Received in jinal form 2 May 1989.
the fibrous structure of these materials: i.e. each fibre is assumed to behave in a linearly elastic manner and the initial lengths of the fibres are assumed to be normally distributed. Measurement of the force-length curve at different temperatures shows that the curve shifts to larger Ivalues as the temperature increases, from which it can be deduced that the entropy of the specimen increases with its length. Our approach is fundamentally different from the one used by Hooley and co-workers (1979, 1980): they use temperature changes as a tool for determining viscoelastic properties at one temperature, while we are interested in the viscoelastic properties at different temperatures.
2. MODEL The biological materials which have been studied (tendon, ligaments, tympanic membrane) are composed of a network of (mainly collagen) fibres, embedded in a gelatinous matrix. The model we use has been derived from the structure of the u-helix of collagen, which is shown in Fig. 1 (Kendrew, 1965): it consists of a large number of zigzag elements with cross-links (Fig. 2) and can be considered as a very simplified twodimensional representation of the cc-helix. To break a cross-link, an energy Z is needed, which can be provided by adding heat to the fibre, or by exerting a stretching force at its ends. Breaking of a cross-link results in a situation as shown in Fig. 3. ‘Breaking of a cross-link’ does not mean physical rupture of a fibre; it refers to some mechanism where a short-range internal force is active. When an external force is exerted, the distance between adjacent molecules gradually increases until, at a certain distance, the internal force is no longer active: the cross-link is broken. As soon as the external force decreases, and the molecules can get closer to each other, the cross-link can be rebuilt. We are aware of the fact that our model is a simplification of the real deformation mechanism. However, we used it first of all because it agrees very well with our experimental results, and also because
1203
M. MAES et al.
1204
a Fig. 3. Fibre with broken cross-link,
following relations, for the case of uniaxial tension:
Of=,-T(if),. (is>,= -(zJt>,
(1)
(2)
in which F, Z, U, S and T are respectively the force exerted on the specimen, its length, internal energy and entropy, and the (absolute) temperature. The same relations hold for a single fibre. From statistical physics, we know that there is a relation between the entropy of a system in an equilibrium state, and X, the number of substates in which the system can be found at that equilibrium: S=klnX,
(3)
k being the Boltzmann constant.
We suppose that, when an energy Z is added to a fibre, each cross-link has the same probability to be broken. The number of possible configurations in which n, cross-links, of which mi are broken, is then given by the number of combinations of mi elements out of ni. This number equals 2A where f is the probability for a binomial distribution with p=q =OS. For large values of n, (as will be found in the fibres), the binomial distribution can be replaced by a normal distribution whose mean and standard deviation are respectively given by (da)
(4b) Fig. 1. The a-helix of collagen.
The agreement between the binomial and the normal distribution will be best for m,-values close to the mean value, and will be worse for m,-values far away from the mean. We then find, for the number of possible configurFig. 2. Model for the fibres, with its parameters d and b.
ations: Xi==%
other, more elaborate models did ‘not allow a closed mathematical description.
3.
DERIVATION
OF A CONSTITUTIVE
exp[ -t(;-mi)‘]
and for the entropy of the fibre, according to (3):
(5)
.
EQUATION
The speed at which an elastic force-length relation is recorded is low enough for the process to be considered as a sequence of equilibrium states, which implies that the laws of thermodynamics of reversible processes are valid. From these laws, we derive the
(6)
Since in the right side of (6), only mi is a function of 1, we have: ami -.
ar
(7)
Elastic force-length relation of soft biological materials The relation between m, and I is given by (I is measured along the fibre axis, d is the length of a straight segment as seen in Fig. 2): /==2[mid+(ni-m,)dcosb/2] =/P+2m,d(l-cosP/2)
(8)
The integration ering that:
1205
constant l,=lo
So, the fibre rest-length written as:
I can be determined T=T,.
at
Ii at temperature
with 10= 2n,d cos /Ii2 the length of the fibre when no cross-links Equation (7) can now be written as: k
l?S I=
i-l
(11)
Supposing that each fibre’s behaviour is linearly elastic-we showed previously that this leads to a good agreement between experimental results and the theoretical prediction (Decraemer et al., 1980kthe force needed to hold a fibre with rest-length Ii, at length I ( > ii) is given by: aK(T).
(12)
The rest-length of the fibre and K, its Young’s modulus (which is assumed-without experimental evidence-to be the same for all fibres) are temperaturedependent; its cross-section a is considered as a constant and also as being the same for all fibres. From equation (12), we find: adK -_-2._ ( li aT
aK li
al,
-a$.
Since equation (2) must be valid for every value of 1,the coefficient of 1 and the constant term in equations (10) and (13) must be equal; this leads to: aK 31,
2ak
li3T
lf c;T
dI;(l-a)’
=l;f(T)
(184
k(T-TO)(1+cz)+Cd(l-cr)2
(W
2ak(T-T,)+Cd(l-cc)’
In a previous paper (Decraemer and Maes, 1982), we showed that the choice of a normal distribution for the rest-lengths of the fibres is justified. The mean value p and standard deviation s of this distribution will follow the same temperature dependence as Ii: AT)=pOf(T)
(194
s(T)=s’f(T).
(1W
The total force needed to hold a specimen at a given length is obtained by summing the forces necessary to stretch the individual fibres with rest length I,
’ l-l,(T)
F(1, T)=
-
s
o
[i(T)
- (ii(T)-P(T))’ 2?(T)
(13)
aT >
aaK
f(T)=
(10)
!x= cos 812.
aFi ==l
1
where
by putting
l--I(T) F,(l)=---UT)
-a)’
T may be
are broken.
#J(l+cc)-2ct1]
lod(l-a)’
2ak(T-T,)+Cd(l
(9)
consid-
Equation
(20a) reduces to (20b)
F(l,
’ l-l,(T) T)=dE:T, o
s ri(T)
1’ . dl,(T)
(20a)
eVFYi)d’i(T~20bl
(14) by putting
and dK
b(T)=NaK(T) (15)
“z=d(l-o1)2. Integrating
yi’-
(15), we find immediately: K(T)=
k(l +cO (T-T,)+; ad(l -#
(16)
where TO and C are both part of the integration constant. TOstands for a ‘threshold temperature’ and its meaning will be clarified later. Substitution of this expression for K(T) in (14) leads to the temperature dependence of Ii: I(T)=10
(21)
k(l+cr)
k(l +cc)(T2crk(T-
T,)+Cd(l T,)+l-/;
-a)’
1
(17)
(ii(T)-P(T))’ 2$(T)
(l”-~o)2 =y-pr.
(22)
N being the total number of fibres. This constitutive equation contains three parameters, b, p and s, whose values were obtained from a least squares fit based on the computer program Minuit (James and Roos, 1975). The temperature dependence for p and s is given by equation (19): for b it may readily be derived using (16) and (21): b(T)=N
k(l+a)(T-TO)+Cd(l-a)2 d(l -ci)’
1.
(23)
In order not to overload the present paper we have limited the amount of experimental data necessary to
1206
M. MAESet al.
Fig. 4. Force-length relations at different temperatures. Dots represent experimental observation, the solid line the theoretical
fit.
temperatures (measured on a strip of human tympanic membrane) together with the theoretical adaptations, is shown in Fig. 4. The corresponding values for b, p and s are shown in Fig. 5. The temperature dependence of li can be obtained in a different way. We consider a pair of segments of a tibre as a system with two energy levels: a ground state in which a cross-link is present, and an ‘excited’ state, in which the cross-link is broken. The energy difference between both states equals Z. The probabilities for the system to be found in the ground state and the excited state are respectively given by: (Boltzmann distribution, see Landau and Lifshitz, 1963) 1 (24)
“=l+exp(-Z/kT)
(25) Equation
/
(8) can now be written as: li=lo+2d(l
I
experimental range
Fig. 5. Comparison
Z/p of the functionsf,,fand
g.
the goodness of fit of the model to a minimum. In a subsequent paper with the emphasis on experimental results we shall show that the present model may be successfully used to describe the behaviour of various biological materials. A typical set of experimental force-length relations at different illustrate
-cc)&
exp( -Z/kT)
1+exp(-Z/kT)
a+exp(-Z/kT) a(1 +exp(-Z/kT))
=19&z-).
(26)
For large T-values equation (26) shows that Ii reaches the value lo (1 + tx)/2a. According to equations (8) and (9), this is the total length I of the fibre when mi = nJ2. This result is in agreement with the law of quantum mechanics, which states that as T approaches infinity,
Elastic force-length relation of soft biological materials
1207
M. MAES et al.
1208
all energy levels have the same probability to be filled, and also with the law of thermodynamics, which states that as T approaches infinity, the entropy of a system reaches its maximum value. In principle, it is possible to obtain the best values for tl and 2 from a least squares fit of expression (26) to the experimental p-values. Although we could obtain an excellent agreement between experimental and theoretical values, we did not use this method, since the couple of values (tl, Z), was not uniquely defined: several completely different couples (a, 2) gave a very good agreement, and it was impossible to decide which of these were the exact values for a and Z. Figure 6 shows that in the experimental temperature range, we obtain a good agreement between the functionsf(T) and g(T). The disagreement between g andffor lower temperatures might have been caused by the fact that for small values of m,, there is a difference between the binomial distribution and the normal distribution by which it was approximated. For this reason, and in order not to obtain negative pvalues, we will use the function&(T) instead off(T). = 1
N a K(T) F(‘y T)=JGs(T) x exp
-
T< To.
equation
(27)
can finally be
’ I-pi so
u= F/A,
(A, is the cross-section of the strip)
(31a)
(3lb) s(T) s’=_=_
so
lo(T)
18
I=----
1
lo(T) A,
=
’
(3W
(314
h(T) C’ -_ I,(T)-c’
The experimental results for c(, C and To for specific biological tissues will be treated in a separate paper. (Typical values are CI= 0.2, To = 275 K, C = lo-l2 N. The way these values are calculated from the experimental results is not straightforward. This is another reason why we will treat this separately.)
T> To
.W)=.W) For T> TO the constitutive written as:
with
l,(T)
(Ii(T)-I(T))’
2sz(T)
14(T)
4. CONCLUSION
We derived a constitutive equation for the elastic force-length relation of soft biological tissues at different temperatures. It takes into account the temperature dependence of the Young’s modulus and of the rest-length of the fibres embedded in the tissues and governing their elastic properties. The fibre model which was used is a simplified representation of the structure of collagen.
REFERENCES
Decraemer, W. F. and Maes, M. (1982)Letter to the editor: reply to the comments of Dr. Y. Lanir. J. L&mechanics 15, x exp[
-w]lp.
(28)
From the definition of the rest-length 1, (Decraemer et 1980) and the temperature dependence of p. and s, we find:
al.,
~o(T)=P(T)-4s(T) =(P’-4s”)fo(T) =l;f(T).
The force-length
(29)
relation (28) can also be written as:
(30)
409411. Decraemer, W. F., Maes, M. and Vanhuyse, V. J. (1980) An elastic stress-strain relation for soft biological tissues based on a structural model. J. &mechanics 13.463468. Fung, Y. C. (1972) Stress-strain history relations of soft tissues in simple elongation. Biomechanics, its Foundations and Objectives pp. 181-208. Prentice-Hall, Englewood Cliffs, NJ. Hooley, C. J. and Cohen, R. E. (1979) A model for the creep behaviour of tendon. Znt. J. Biol. Macromol. 1, 123-132. Hooley, C. J., McCrum, N. G. and Cohen, R. E. (1980) The viscoelastic deformation of tendon. J. Biomechanics 13, 521-528. James, F. and Roos, M. (1975) A system for function minimization and analysis of the parameter errors and correlations. Comp. Phys. Commun. 10, 343. Kendrew, J. C. (1965) The three-dimensional structure of a protein molecule. The Living Cell, pp. 170-184. Freeman, New York. Landau, L. D. and Lifshitz, E. M. (1963) Statistical Physics. Pergamon Press, London.
Vol. 22. No. 1 l/12. pp. 1203-1208,
m Great
1989.
co21~9290/89 nIxI+ .oo Pergamon Press plc
Brdam
A THERMODYNAMICALLY CONSISTENT CONSTITUTIVE EQUATION FOR THE ELASTIC FORCE-LENGTH RELATION OF SOFT BIOLOGICAL MATERIALS M. MAES, V. J. VANHUYSE, W. F. DECRAEMER and E. R. RAMAN Laboratory
for Experimental
Physics,
Rijksuniversitair Centrum Antwerpen, B-2020 Antwerpen, Belgium
Groenenborgerlaan
171,
Abstract-Starting from the laws of thermodynamics of reversible processes, a temperature-dependent constitutive equation is derived for the elastic force-length relation of soft biological tissues. These tissues are composed of a network of fibres (mainly collagen). The equation is based on a model which uses a simplified two-dimensional representation of the cc-helix of collagen.
NOMENCLATURE energy needed to break a cross-link force needed to hold the specimen at length I force needed to hold a fibre at length 1 internal energy of the specimen entropy of the specimen internal energy of a fibre entropy of a fibre number of configurations Boltzmann constant total number of zigzag elements in a fibre number of broken cross-links length of an element in the fibre model angle between two elements in the fibre model cos (b/2) length of fibre when no cross-links are broken fibre cross-section fibre Young’s modulus integration constant mean value of the normal distribution of fibre restlength at temperature T standard deviation of the normal distribution of fibre rest-lengths, at temperature T value of p at T=O value of s at T=O rest length of the specimen at temperature T rest-length of the specimen at T=O total number of fibres effective Young’s modulus (= NaK) temperature below which no cross-links are broken stress (= F/A,) relative length ( = l/l,) relative fibre length (= c/l,) specimen cross-section
1. INTRODUCTION
Many authors have proposed an analytical expression for the elastic force-length relation for soft biological materials in uniaxial tension, but none of them takes into account the influence of the temperature (Fung, 1972). Our constitutive equation has been derived from the laws of thermodynamics of reversible processes. It is an extension of our earlier developed equation (Decraemer et al., 1980), which is based on Received in jinal form 2 May 1989.
the fibrous structure of these materials: i.e. each fibre is assumed to behave in a linearly elastic manner and the initial lengths of the fibres are assumed to be normally distributed. Measurement of the force-length curve at different temperatures shows that the curve shifts to larger Ivalues as the temperature increases, from which it can be deduced that the entropy of the specimen increases with its length. Our approach is fundamentally different from the one used by Hooley and co-workers (1979, 1980): they use temperature changes as a tool for determining viscoelastic properties at one temperature, while we are interested in the viscoelastic properties at different temperatures.
2. MODEL The biological materials which have been studied (tendon, ligaments, tympanic membrane) are composed of a network of (mainly collagen) fibres, embedded in a gelatinous matrix. The model we use has been derived from the structure of the u-helix of collagen, which is shown in Fig. 1 (Kendrew, 1965): it consists of a large number of zigzag elements with cross-links (Fig. 2) and can be considered as a very simplified twodimensional representation of the cc-helix. To break a cross-link, an energy Z is needed, which can be provided by adding heat to the fibre, or by exerting a stretching force at its ends. Breaking of a cross-link results in a situation as shown in Fig. 3. ‘Breaking of a cross-link’ does not mean physical rupture of a fibre; it refers to some mechanism where a short-range internal force is active. When an external force is exerted, the distance between adjacent molecules gradually increases until, at a certain distance, the internal force is no longer active: the cross-link is broken. As soon as the external force decreases, and the molecules can get closer to each other, the cross-link can be rebuilt. We are aware of the fact that our model is a simplification of the real deformation mechanism. However, we used it first of all because it agrees very well with our experimental results, and also because
1203
M. MAES et al.
1204
a Fig. 3. Fibre with broken cross-link,
following relations, for the case of uniaxial tension:
Of=,-T(if),. (is>,= -(zJt>,
(1)
(2)
in which F, Z, U, S and T are respectively the force exerted on the specimen, its length, internal energy and entropy, and the (absolute) temperature. The same relations hold for a single fibre. From statistical physics, we know that there is a relation between the entropy of a system in an equilibrium state, and X, the number of substates in which the system can be found at that equilibrium: S=klnX,
(3)
k being the Boltzmann constant.
We suppose that, when an energy Z is added to a fibre, each cross-link has the same probability to be broken. The number of possible configurations in which n, cross-links, of which mi are broken, is then given by the number of combinations of mi elements out of ni. This number equals 2A where f is the probability for a binomial distribution with p=q =OS. For large values of n, (as will be found in the fibres), the binomial distribution can be replaced by a normal distribution whose mean and standard deviation are respectively given by (da)
(4b) Fig. 1. The a-helix of collagen.
The agreement between the binomial and the normal distribution will be best for m,-values close to the mean value, and will be worse for m,-values far away from the mean. We then find, for the number of possible configurFig. 2. Model for the fibres, with its parameters d and b.
ations: Xi==%
other, more elaborate models did ‘not allow a closed mathematical description.
3.
DERIVATION
OF A CONSTITUTIVE
exp[ -t(;-mi)‘]
and for the entropy of the fibre, according to (3):
(5)
.
EQUATION
The speed at which an elastic force-length relation is recorded is low enough for the process to be considered as a sequence of equilibrium states, which implies that the laws of thermodynamics of reversible processes are valid. From these laws, we derive the
(6)
Since in the right side of (6), only mi is a function of 1, we have: ami -.
ar
(7)
Elastic force-length relation of soft biological materials The relation between m, and I is given by (I is measured along the fibre axis, d is the length of a straight segment as seen in Fig. 2): /==2[mid+(ni-m,)dcosb/2] =/P+2m,d(l-cosP/2)
(8)
The integration ering that:
1205
constant l,=lo
So, the fibre rest-length written as:
I can be determined T=T,.
at
Ii at temperature
with 10= 2n,d cos /Ii2 the length of the fibre when no cross-links Equation (7) can now be written as: k
l?S I=
i-l
(11)
Supposing that each fibre’s behaviour is linearly elastic-we showed previously that this leads to a good agreement between experimental results and the theoretical prediction (Decraemer et al., 1980kthe force needed to hold a fibre with rest-length Ii, at length I ( > ii) is given by: aK(T).
(12)
The rest-length of the fibre and K, its Young’s modulus (which is assumed-without experimental evidence-to be the same for all fibres) are temperaturedependent; its cross-section a is considered as a constant and also as being the same for all fibres. From equation (12), we find: adK -_-2._ ( li aT
aK li
al,
-a$.
Since equation (2) must be valid for every value of 1,the coefficient of 1 and the constant term in equations (10) and (13) must be equal; this leads to: aK 31,
2ak
li3T
lf c;T
dI;(l-a)’
=l;f(T)
(184
k(T-TO)(1+cz)+Cd(l-cr)2
(W
2ak(T-T,)+Cd(l-cc)’
In a previous paper (Decraemer and Maes, 1982), we showed that the choice of a normal distribution for the rest-lengths of the fibres is justified. The mean value p and standard deviation s of this distribution will follow the same temperature dependence as Ii: AT)=pOf(T)
(194
s(T)=s’f(T).
(1W
The total force needed to hold a specimen at a given length is obtained by summing the forces necessary to stretch the individual fibres with rest length I,
’ l-l,(T)
F(1, T)=
-
s
o
[i(T)
- (ii(T)-P(T))’ 2?(T)
(13)
aT >
aaK
f(T)=
(10)
!x= cos 812.
aFi ==l
1
where
by putting
l--I(T) F,(l)=---UT)
-a)’
T may be
are broken.
#J(l+cc)-2ct1]
lod(l-a)’
2ak(T-T,)+Cd(l
(9)
consid-
Equation
(20a) reduces to (20b)
F(l,
’ l-l,(T) T)=dE:T, o
s ri(T)
1’ . dl,(T)
(20a)
eVFYi)d’i(T~20bl
(14) by putting
and dK
b(T)=NaK(T) (15)
“z=d(l-o1)2. Integrating
yi’-
(15), we find immediately: K(T)=
k(l +cO (T-T,)+; ad(l -#
(16)
where TO and C are both part of the integration constant. TOstands for a ‘threshold temperature’ and its meaning will be clarified later. Substitution of this expression for K(T) in (14) leads to the temperature dependence of Ii: I(T)=10
(21)
k(l+cr)
k(l +cc)(T2crk(T-
T,)+Cd(l T,)+l-/;
-a)’
1
(17)
(ii(T)-P(T))’ 2$(T)
(l”-~o)2 =y-pr.
(22)
N being the total number of fibres. This constitutive equation contains three parameters, b, p and s, whose values were obtained from a least squares fit based on the computer program Minuit (James and Roos, 1975). The temperature dependence for p and s is given by equation (19): for b it may readily be derived using (16) and (21): b(T)=N
k(l+a)(T-TO)+Cd(l-a)2 d(l -ci)’
1.
(23)
In order not to overload the present paper we have limited the amount of experimental data necessary to
1206
M. MAESet al.
Fig. 4. Force-length relations at different temperatures. Dots represent experimental observation, the solid line the theoretical
fit.
temperatures (measured on a strip of human tympanic membrane) together with the theoretical adaptations, is shown in Fig. 4. The corresponding values for b, p and s are shown in Fig. 5. The temperature dependence of li can be obtained in a different way. We consider a pair of segments of a tibre as a system with two energy levels: a ground state in which a cross-link is present, and an ‘excited’ state, in which the cross-link is broken. The energy difference between both states equals Z. The probabilities for the system to be found in the ground state and the excited state are respectively given by: (Boltzmann distribution, see Landau and Lifshitz, 1963) 1 (24)
“=l+exp(-Z/kT)
(25) Equation
/
(8) can now be written as: li=lo+2d(l
I
experimental range
Fig. 5. Comparison
Z/p of the functionsf,,fand
g.
the goodness of fit of the model to a minimum. In a subsequent paper with the emphasis on experimental results we shall show that the present model may be successfully used to describe the behaviour of various biological materials. A typical set of experimental force-length relations at different illustrate
-cc)&
exp( -Z/kT)
1+exp(-Z/kT)
a+exp(-Z/kT) a(1 +exp(-Z/kT))
=19&z-).
(26)
For large T-values equation (26) shows that Ii reaches the value lo (1 + tx)/2a. According to equations (8) and (9), this is the total length I of the fibre when mi = nJ2. This result is in agreement with the law of quantum mechanics, which states that as T approaches infinity,
Elastic force-length relation of soft biological materials
1207
M. MAES et al.
1208
all energy levels have the same probability to be filled, and also with the law of thermodynamics, which states that as T approaches infinity, the entropy of a system reaches its maximum value. In principle, it is possible to obtain the best values for tl and 2 from a least squares fit of expression (26) to the experimental p-values. Although we could obtain an excellent agreement between experimental and theoretical values, we did not use this method, since the couple of values (tl, Z), was not uniquely defined: several completely different couples (a, 2) gave a very good agreement, and it was impossible to decide which of these were the exact values for a and Z. Figure 6 shows that in the experimental temperature range, we obtain a good agreement between the functionsf(T) and g(T). The disagreement between g andffor lower temperatures might have been caused by the fact that for small values of m,, there is a difference between the binomial distribution and the normal distribution by which it was approximated. For this reason, and in order not to obtain negative pvalues, we will use the function&(T) instead off(T). = 1
N a K(T) F(‘y T)=JGs(T) x exp
-
T< To.
equation
(27)
can finally be
’ I-pi so
u= F/A,
(A, is the cross-section of the strip)
(31a)
(3lb) s(T) s’=_=_
so
lo(T)
18
I=----
1
lo(T) A,
=
’
(3W
(314
h(T) C’ -_ I,(T)-c’
The experimental results for c(, C and To for specific biological tissues will be treated in a separate paper. (Typical values are CI= 0.2, To = 275 K, C = lo-l2 N. The way these values are calculated from the experimental results is not straightforward. This is another reason why we will treat this separately.)
T> To
.W)=.W) For T> TO the constitutive written as:
with
l,(T)
(Ii(T)-I(T))’
2sz(T)
14(T)
4. CONCLUSION
We derived a constitutive equation for the elastic force-length relation of soft biological tissues at different temperatures. It takes into account the temperature dependence of the Young’s modulus and of the rest-length of the fibres embedded in the tissues and governing their elastic properties. The fibre model which was used is a simplified representation of the structure of collagen.
REFERENCES
Decraemer, W. F. and Maes, M. (1982)Letter to the editor: reply to the comments of Dr. Y. Lanir. J. L&mechanics 15, x exp[
-w]lp.
(28)
From the definition of the rest-length 1, (Decraemer et 1980) and the temperature dependence of p. and s, we find:
al.,
~o(T)=P(T)-4s(T) =(P’-4s”)fo(T) =l;f(T).
The force-length
(29)
relation (28) can also be written as:
(30)
409411. Decraemer, W. F., Maes, M. and Vanhuyse, V. J. (1980) An elastic stress-strain relation for soft biological tissues based on a structural model. J. &mechanics 13.463468. Fung, Y. C. (1972) Stress-strain history relations of soft tissues in simple elongation. Biomechanics, its Foundations and Objectives pp. 181-208. Prentice-Hall, Englewood Cliffs, NJ. Hooley, C. J. and Cohen, R. E. (1979) A model for the creep behaviour of tendon. Znt. J. Biol. Macromol. 1, 123-132. Hooley, C. J., McCrum, N. G. and Cohen, R. E. (1980) The viscoelastic deformation of tendon. J. Biomechanics 13, 521-528. James, F. and Roos, M. (1975) A system for function minimization and analysis of the parameter errors and correlations. Comp. Phys. Commun. 10, 343. Kendrew, J. C. (1965) The three-dimensional structure of a protein molecule. The Living Cell, pp. 170-184. Freeman, New York. Landau, L. D. and Lifshitz, E. M. (1963) Statistical Physics. Pergamon Press, London.