J. BIOMED. MATER. RES.
VOL. 9, PP. 661-674 (1975)
Viscoelastic Properties of Human Dentin EDWARD KOROSTOFF, Restorative Dentistry, School of Dental Medicine, and Orthopaedic Surgery, School of Medicine, and Bioengineering, College of Engineering and Applied Science, University of Pennsylvania, Philadelphia, SOLOMON R. POLLACK, Metallurgy and Materials Science, and Bioengineering, College of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania, and MANVILLE G . DUNCANSON, JR., Dental Materials, College of Dentistry, University of Oklahoma, Norman, Oklahoma
SummarJ Stress relaxation measurements were performed on thick-walled cylinders of radicular human dentin. The experimentally measured relaxation moduli were used t o obtain an approximation to the logarithmic distribution function of relaxation times. This distribution function was used to predict the behavior of other viscoelastic properties. I n particular, the prediction of the strain and strain rate dependence of the modulus was determined and compared with experimental results.
INTRODUCTION Since the initial work by Black' in 1895, the major effort in investigating the mechanical behavior of human dentin has been directed toward measurement of the elastic modulus, proportional limit, and ultimate compressive strength.2-6 The properties of these materials have traditionally been measured in association with metals whose mechanical behavior more closely allows for the determination of a time-independent modulus and a strain-rate independent proportional limit and ultimate compressive strength. Both human and bovine dentin have been shown or implied to be v i s c o e l a ~ t i c ~ - ~ like other connective tissue such as skin and mesentery. Therefore, in order to specify dentin's mechanical response, its time-dependent mechanical behavior must be investigated. 661 @ 1975 by John Wiley & Sons, Inc.
662
KOROSTOFF, POLLACK, AND DUNCANSON
This paper presents the results of compressive stress relaxation measurements on human dentin along with stress-strain measurements, both of which are used t o specify its viscoelastic properties. The stress relaxation method is one of several listed and mathematically developed b y Alfrey and Doty’O using linear viscoelastic theory. The use of the stress relaxation method allows one t o obtain an approximation t o the continuous distribution of relaxation times, H ( T ) ,by use of eq. (1).
E,(t)
Im
H(r)exp( -t/r)d In 7
= --m
(1)
where E,(t) is the time dependent relaxation modulus, and T is the relaxation time. Obtaining H ( T ) from eq. (1) then permits one method of determining other frequency (or time) dependent properties such as the loss modulus, E’(w), the storage modulus, E”(o), and the tensile flow viscosity, qt,ll as shown in eqs. ( 2 ) , (3), and (4).
vt
=
\
m
H(r)dr
(4)
Furthermore, the distribution function, H ( T ) ,can be used t o calculate a distribution function of retardation times, L(T),which is obtainable experimentally from creep measurements.* If one assumes linearity, then the superposition principle enables the determination of the stress (u) - strain (e) viscoelastic behavior from eq. ( 5 ) :
where E,(t) is the analytical expression for the relaxation modulus, X is the current time during the specific strain program, ~ ( t is ) the *Care must be exercised here to avoid complications arising from anisotropic effects in determiningE ( w ) and L ( r )from stress relaxation and creep experiments.
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
663
time dependent stress, and E(X) is the strain. Therefore, in principle, if one obtains an analytical expression for the relaxation modulus, the stress-strain behavior is completely determined. I n these experiments, the properties of dentin were explored in the high strain, high stress (approximately 0.6% strain and 50% of the compressive strength) region where permanent changes in the dentin can occur. Therefore, eq. (5) must be used with extreme caution in this analysis. That is, if E7(t) is determined from stress relaxation experiments during which permanent changes in the dentin occur, then eq. (5) may not predict the stress response in a “virgin” specimen. This will be developed later.
MATERIALS AND METHODS Freshly extracted teeth were stored in a simulated extracellular fluid (Abbott Laboratories, Normosol R, pH 7.4, No. 4474) and refrigerated a t approximately 5°C until preparation for mechanical testing. The teeth utilized were of approximate cylindroconical root morphology, i.e., maxillary incisors and cuspids, and mandibular cuspids and premolars. This selective process was required in order t o obtain maximum specimen volume and uniform radial orientation of dentinal tubules, within each specimen and from specimen to specimen. Proper alignment of the pulp canal of the appropriate teeth allowed for the machining of thick-walled right circular cylinders of dentin with the following dimensions: height, h, = 0.250 in.; o.d., do = 0.150 in.; and i.d., d , = 0.045 in. Figure 1 shows the configuration and orientation of the specimen with respect t o the entire tooth. A mechanical testing instrument (Intsron Corp., Model TTCL-128) was used to apply a constant compressive strain, e,, to the specimen while the changes in load with time were indicated on the chart recorder of the instrument. The compression apparatus, a modification of that used by Tomalin,13 contained two linear variable displacement transducers (Daytronic Corp., Model DS200) which responded t o changes in height of the specimen; and the signal produced was transmitted through a transducer-amplifier indicator (Daytronic Corp., Model 300D) and recorded on a X-Y recorder (Hewlett-packard, Model 7005B) equipped with chart drive. Thus continuous recordings of load and displacement could be accom-
664
KOROSTOFF, POLLACK, AND DUNCANSON
Fig. 1. Orientation and configuration of dentin specimens.
plished. The engineering stress and strain, u(t),and c,, respectively, are given by:
and Ahc
€c
= -
ho
where
L(t)
experimentally observed load
=
do = outer diameter of specimen
di Ahc h,
= = =
inner diameter of specimen experimentally induced height change original specimen height
The relaxation modulus, Er(t),is therefore
E,(t)
=
4 )= ~
e,
4hoL(t) AhCr(doz- di2)
(7)
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
665
An environmental chamber, enclosing the compression apparatus, simulated in vivo temperature and humidity: 37°C ;and 100% relative humidity (by direct application to the specimen of the above mentioned physiologic solution). After calibration of the measurement circuitry and temperature equilibration of the environmental chamber, the specimen was strained a t a rate of do = 1.6 X sec-' until a total strain of d, was achieved. The strain was then held constant, and the stress relaxation behavior observed. See Figure 2 for the circuit diagram of the experimental set-up. The day after each stress relaxation run, stress-strain curves were obtained from the same samples a t 4 different strain rates, d1 = 1.6 X i, = 4.1 X da = 8.2 X d4 = 1.6 X I n all cases the maximum strain was approximately 0.6%. Approximately 3 hr was allowed to lapse between subsequent stress-strain runs.
RESULTS On imposition and maintenance of a constant compressive strain, smooth monotonically decreasing load curves of an exponential character were observed. After a period of 6 hr no experimentally measurable decrease in load was observed, i.e., a residual load level e,,
lnstron
X - Y Recorder
Fig. 2. Block diagram of experimental set-up.
KOROSTOFF, POLLACK, AND DUNCANSON
666
was asymptotically approached. Utilizing eq. (S), the relaxation modulus, E,(t), was tabulated. Although eq. (1) can theoretically be used t o obtain H ( T ) , it is perhaps best looked upon as a formal definition of H ( T ) rather than a direct mathematical method for obtaining H ( T ) . Gross14J5has worked out exact integral inversion formulae which, although cumbersome and complicated, do permit the distribution functions of relaxation and retardation times to be obtained as a n analytical function from zero to infinity. However, these integral inversion formulae do not lend t,hemselves to convenient numerical computation. One is therefore led t o the use of approximations as a n alternative t o the analytical approach, since the rigorous Laplace inversion approach is of such limited practical use. For slowly varying E,(t) , the first approximation to the logarithmic distribution of relaxation times is :16-17
Experimentally, E,(t,) is observed the ratio of the total initial stress to the total constant strain as seen in Figure 3. Representative plots of the normalized relaxation modulus, Bl(t)/E,(tO) versus loglo t are shown in Figure 4. As seen from theoretical eq. (9), H1(7) = constant X logarithmic derivative of E,t). I n order t o minimize I
I
STRESS
(0-1
I
I
I
&I
START OF STRESS RELAXATION DATA
CT (1,)
Ec
STRAIN E
I 0
to
TIME ( 1 )
-
Fig. 3. Stress and strain during stress relaxation experiment.
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
667
subjective error in graphically obtaining the logarithmic slope, a computer analysis was performed for each specimen obtaining both the slope and the correlation coefficient, K . (For K = f 1, the correlation is perfect, while K = 0 implies no correlation.18-20) Of principal interest here is the degree of variation from linearity of plots of E7(t)/E7(0)versus loglot. This is expressed quantitatively by the square of the coefficient of correlation, called the coefficient of determination and this is the ratio of the explained variation to the total variation. The unexplained variation can be considered a combination of experimental error and the contribution of the higher logarithmic derivatives to the approximation of H ( T ) . Table I shows values for H ~ ( T K ) ,2 ,as well as patient data obtained from clinical records. Table I1 shows the mean value of the relaxed modulus obtained for all samples. The stress-strain curves, obtained for each sample on the following day, were analyzed and the slope (dalde) determined a t a strain value of 0.5y0for each strain rate. Figure 5 shows a representative sec-'. stress-strain curve obtained at a strain rate i = 1.6 X
DISCUSSION The experimental protocols described above give the following experimental parameters for each sample tested: J%( ), the relaxed modulus from stress relaxation; H1(7), the distribution of relaxation
.900
h
,800
c, L
Y
W
,700 -
0
19 20 0 21 A
1
2
3
Log t(sec.) Fig. 4. Representative plots of E,(t)/E,(O) vs. log,, t .
4
668
KOROSTOFF, POLLACK, AND DUNCANSON
HI
(T),
TABLE I K2, and Patient Data for All Specimens
Specimen Number
Clinical Data*
x 109 dynes/cm2
K2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
M-40-11-C F-86-21-C F-29-11-C F-29-21-C F-32-21-C F-50-21-C F-50-11-C M-28-23-C M-70-33-C M-36-454 M-57-11-P F-41-13-C F-50-23-P M-83-11-P F-32-43-P F-43-13-P M-50-33-P F-19-21-P F-40-45-P M-41-44-P M-60-21-C M-41-23-C M-41-11-C M-28-11-P M-70-43-C
3.47 3.04 1.92 2.61 5.78 2.96 3.87 1.98 4.61 5.56 6.56 3.78 4.93 3.45 3.10 4.82 5.48 2.08 4.78 5.87 2.07 3.14 2.92 2.96 3.30
0.958 0.929 0.976 0.990 0.863 0.958 0.949 0.941 0.980 0.898 0.992 0.914 0.996 0.996 0.992 0.990 0.884 0.962 0.956 0.990 0.990 0.949 0.958 0.976 0.982
H ~ ( T )
*Gender, age, tooth (FDI Notation), primary extraction pathology (C - caries;
P - periodontitis). TABLE I1 Mean and Standard Deviation of Values Age (year)
H ~ ( TX) l O Q dynes/cmZ K2 E,( m )
- the relaxed modulus
45.8 3.80 0.959 12.0 X
f 17.2 f 1.36 f 0.037 10'0 dynes/cm2 ( = 1.7 X
lo6psi
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
669
15.0 14.0
13.0 12.0 11.0
10.0
n
-
lo
7.0
x
6.0
5.0 4.0 3.0 2.0 I .0
-
0 0
1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 x
~
~
3
Fig. 5. Representative stress-strain curve.
times from the slope of E,(t) versus log t; &(to), the relaxation modulus at the start of the stress relaxation measurements; and the stress-strain curves at 4 strain rates taken on the day following the stress relaxation measurements. I n order to utilize the integrals in eqs. (1) through (5) it is necessary to determine two more parameters rmaxand rmin,the upper and lower limits on the distribution of relaxa~-~ developed ~ a procedure for obtion times, H ( r ) . T o b o l ~ k y ~has taining T~~~ and rminfrom stress relaxation data. Implicit in his approach is that the stress relaxation data has been taken starting with values of to (see Fig. 3) sufficiently small that a constant high strain rate modulus has been observed. I n this case, H ( T ) in eq. (1)
670
KOROSTOFF, POLLACK, AND DUNCANSON
is replaced by the constant experimental value of H1, from eq. (9), and eq. (1) is integrated giving
Er(t)
HI(T) [Ei(-t/Tmin)-
=
Ei(-t/rmax)]
(10)
where Ei(x) is the exponential integral function which has been numerically tabulated,25and rmaxand rminare graphically determined by extrapolating the stress relaxation data to the observed constant high strain-rate modulus (t-rmin) and the observed constant low strain rate modulus (t-rmax). The logarithmic derivative of eq. (10) then yields
from which it is clear that for rmin
VOL. 9, PP. 661-674 (1975)
Viscoelastic Properties of Human Dentin EDWARD KOROSTOFF, Restorative Dentistry, School of Dental Medicine, and Orthopaedic Surgery, School of Medicine, and Bioengineering, College of Engineering and Applied Science, University of Pennsylvania, Philadelphia, SOLOMON R. POLLACK, Metallurgy and Materials Science, and Bioengineering, College of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania, and MANVILLE G . DUNCANSON, JR., Dental Materials, College of Dentistry, University of Oklahoma, Norman, Oklahoma
SummarJ Stress relaxation measurements were performed on thick-walled cylinders of radicular human dentin. The experimentally measured relaxation moduli were used t o obtain an approximation to the logarithmic distribution function of relaxation times. This distribution function was used to predict the behavior of other viscoelastic properties. I n particular, the prediction of the strain and strain rate dependence of the modulus was determined and compared with experimental results.
INTRODUCTION Since the initial work by Black' in 1895, the major effort in investigating the mechanical behavior of human dentin has been directed toward measurement of the elastic modulus, proportional limit, and ultimate compressive strength.2-6 The properties of these materials have traditionally been measured in association with metals whose mechanical behavior more closely allows for the determination of a time-independent modulus and a strain-rate independent proportional limit and ultimate compressive strength. Both human and bovine dentin have been shown or implied to be v i s c o e l a ~ t i c ~ - ~ like other connective tissue such as skin and mesentery. Therefore, in order to specify dentin's mechanical response, its time-dependent mechanical behavior must be investigated. 661 @ 1975 by John Wiley & Sons, Inc.
662
KOROSTOFF, POLLACK, AND DUNCANSON
This paper presents the results of compressive stress relaxation measurements on human dentin along with stress-strain measurements, both of which are used t o specify its viscoelastic properties. The stress relaxation method is one of several listed and mathematically developed b y Alfrey and Doty’O using linear viscoelastic theory. The use of the stress relaxation method allows one t o obtain an approximation t o the continuous distribution of relaxation times, H ( T ) ,by use of eq. (1).
E,(t)
Im
H(r)exp( -t/r)d In 7
= --m
(1)
where E,(t) is the time dependent relaxation modulus, and T is the relaxation time. Obtaining H ( T ) from eq. (1) then permits one method of determining other frequency (or time) dependent properties such as the loss modulus, E’(w), the storage modulus, E”(o), and the tensile flow viscosity, qt,ll as shown in eqs. ( 2 ) , (3), and (4).
vt
=
\
m
H(r)dr
(4)
Furthermore, the distribution function, H ( T ) ,can be used t o calculate a distribution function of retardation times, L(T),which is obtainable experimentally from creep measurements.* If one assumes linearity, then the superposition principle enables the determination of the stress (u) - strain (e) viscoelastic behavior from eq. ( 5 ) :
where E,(t) is the analytical expression for the relaxation modulus, X is the current time during the specific strain program, ~ ( t is ) the *Care must be exercised here to avoid complications arising from anisotropic effects in determiningE ( w ) and L ( r )from stress relaxation and creep experiments.
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
663
time dependent stress, and E(X) is the strain. Therefore, in principle, if one obtains an analytical expression for the relaxation modulus, the stress-strain behavior is completely determined. I n these experiments, the properties of dentin were explored in the high strain, high stress (approximately 0.6% strain and 50% of the compressive strength) region where permanent changes in the dentin can occur. Therefore, eq. (5) must be used with extreme caution in this analysis. That is, if E7(t) is determined from stress relaxation experiments during which permanent changes in the dentin occur, then eq. (5) may not predict the stress response in a “virgin” specimen. This will be developed later.
MATERIALS AND METHODS Freshly extracted teeth were stored in a simulated extracellular fluid (Abbott Laboratories, Normosol R, pH 7.4, No. 4474) and refrigerated a t approximately 5°C until preparation for mechanical testing. The teeth utilized were of approximate cylindroconical root morphology, i.e., maxillary incisors and cuspids, and mandibular cuspids and premolars. This selective process was required in order t o obtain maximum specimen volume and uniform radial orientation of dentinal tubules, within each specimen and from specimen to specimen. Proper alignment of the pulp canal of the appropriate teeth allowed for the machining of thick-walled right circular cylinders of dentin with the following dimensions: height, h, = 0.250 in.; o.d., do = 0.150 in.; and i.d., d , = 0.045 in. Figure 1 shows the configuration and orientation of the specimen with respect t o the entire tooth. A mechanical testing instrument (Intsron Corp., Model TTCL-128) was used to apply a constant compressive strain, e,, to the specimen while the changes in load with time were indicated on the chart recorder of the instrument. The compression apparatus, a modification of that used by Tomalin,13 contained two linear variable displacement transducers (Daytronic Corp., Model DS200) which responded t o changes in height of the specimen; and the signal produced was transmitted through a transducer-amplifier indicator (Daytronic Corp., Model 300D) and recorded on a X-Y recorder (Hewlett-packard, Model 7005B) equipped with chart drive. Thus continuous recordings of load and displacement could be accom-
664
KOROSTOFF, POLLACK, AND DUNCANSON
Fig. 1. Orientation and configuration of dentin specimens.
plished. The engineering stress and strain, u(t),and c,, respectively, are given by:
and Ahc
€c
= -
ho
where
L(t)
experimentally observed load
=
do = outer diameter of specimen
di Ahc h,
= = =
inner diameter of specimen experimentally induced height change original specimen height
The relaxation modulus, Er(t),is therefore
E,(t)
=
4 )= ~
e,
4hoL(t) AhCr(doz- di2)
(7)
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
665
An environmental chamber, enclosing the compression apparatus, simulated in vivo temperature and humidity: 37°C ;and 100% relative humidity (by direct application to the specimen of the above mentioned physiologic solution). After calibration of the measurement circuitry and temperature equilibration of the environmental chamber, the specimen was strained a t a rate of do = 1.6 X sec-' until a total strain of d, was achieved. The strain was then held constant, and the stress relaxation behavior observed. See Figure 2 for the circuit diagram of the experimental set-up. The day after each stress relaxation run, stress-strain curves were obtained from the same samples a t 4 different strain rates, d1 = 1.6 X i, = 4.1 X da = 8.2 X d4 = 1.6 X I n all cases the maximum strain was approximately 0.6%. Approximately 3 hr was allowed to lapse between subsequent stress-strain runs.
RESULTS On imposition and maintenance of a constant compressive strain, smooth monotonically decreasing load curves of an exponential character were observed. After a period of 6 hr no experimentally measurable decrease in load was observed, i.e., a residual load level e,,
lnstron
X - Y Recorder
Fig. 2. Block diagram of experimental set-up.
KOROSTOFF, POLLACK, AND DUNCANSON
666
was asymptotically approached. Utilizing eq. (S), the relaxation modulus, E,(t), was tabulated. Although eq. (1) can theoretically be used t o obtain H ( T ) , it is perhaps best looked upon as a formal definition of H ( T ) rather than a direct mathematical method for obtaining H ( T ) . Gross14J5has worked out exact integral inversion formulae which, although cumbersome and complicated, do permit the distribution functions of relaxation and retardation times to be obtained as a n analytical function from zero to infinity. However, these integral inversion formulae do not lend t,hemselves to convenient numerical computation. One is therefore led t o the use of approximations as a n alternative t o the analytical approach, since the rigorous Laplace inversion approach is of such limited practical use. For slowly varying E,(t) , the first approximation to the logarithmic distribution of relaxation times is :16-17
Experimentally, E,(t,) is observed the ratio of the total initial stress to the total constant strain as seen in Figure 3. Representative plots of the normalized relaxation modulus, Bl(t)/E,(tO) versus loglo t are shown in Figure 4. As seen from theoretical eq. (9), H1(7) = constant X logarithmic derivative of E,t). I n order t o minimize I
I
STRESS
(0-1
I
I
I
&I
START OF STRESS RELAXATION DATA
CT (1,)
Ec
STRAIN E
I 0
to
TIME ( 1 )
-
Fig. 3. Stress and strain during stress relaxation experiment.
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
667
subjective error in graphically obtaining the logarithmic slope, a computer analysis was performed for each specimen obtaining both the slope and the correlation coefficient, K . (For K = f 1, the correlation is perfect, while K = 0 implies no correlation.18-20) Of principal interest here is the degree of variation from linearity of plots of E7(t)/E7(0)versus loglot. This is expressed quantitatively by the square of the coefficient of correlation, called the coefficient of determination and this is the ratio of the explained variation to the total variation. The unexplained variation can be considered a combination of experimental error and the contribution of the higher logarithmic derivatives to the approximation of H ( T ) . Table I shows values for H ~ ( T K ) ,2 ,as well as patient data obtained from clinical records. Table I1 shows the mean value of the relaxed modulus obtained for all samples. The stress-strain curves, obtained for each sample on the following day, were analyzed and the slope (dalde) determined a t a strain value of 0.5y0for each strain rate. Figure 5 shows a representative sec-'. stress-strain curve obtained at a strain rate i = 1.6 X
DISCUSSION The experimental protocols described above give the following experimental parameters for each sample tested: J%( ), the relaxed modulus from stress relaxation; H1(7), the distribution of relaxation
.900
h
,800
c, L
Y
W
,700 -
0
19 20 0 21 A
1
2
3
Log t(sec.) Fig. 4. Representative plots of E,(t)/E,(O) vs. log,, t .
4
668
KOROSTOFF, POLLACK, AND DUNCANSON
HI
(T),
TABLE I K2, and Patient Data for All Specimens
Specimen Number
Clinical Data*
x 109 dynes/cm2
K2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
M-40-11-C F-86-21-C F-29-11-C F-29-21-C F-32-21-C F-50-21-C F-50-11-C M-28-23-C M-70-33-C M-36-454 M-57-11-P F-41-13-C F-50-23-P M-83-11-P F-32-43-P F-43-13-P M-50-33-P F-19-21-P F-40-45-P M-41-44-P M-60-21-C M-41-23-C M-41-11-C M-28-11-P M-70-43-C
3.47 3.04 1.92 2.61 5.78 2.96 3.87 1.98 4.61 5.56 6.56 3.78 4.93 3.45 3.10 4.82 5.48 2.08 4.78 5.87 2.07 3.14 2.92 2.96 3.30
0.958 0.929 0.976 0.990 0.863 0.958 0.949 0.941 0.980 0.898 0.992 0.914 0.996 0.996 0.992 0.990 0.884 0.962 0.956 0.990 0.990 0.949 0.958 0.976 0.982
H ~ ( T )
*Gender, age, tooth (FDI Notation), primary extraction pathology (C - caries;
P - periodontitis). TABLE I1 Mean and Standard Deviation of Values Age (year)
H ~ ( TX) l O Q dynes/cmZ K2 E,( m )
- the relaxed modulus
45.8 3.80 0.959 12.0 X
f 17.2 f 1.36 f 0.037 10'0 dynes/cm2 ( = 1.7 X
lo6psi
VISCOELASTIC PROPERTIES OF HUMAN DENTIN
669
15.0 14.0
13.0 12.0 11.0
10.0
n
-
lo
7.0
x
6.0
5.0 4.0 3.0 2.0 I .0
-
0 0
1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 x
~
~
3
Fig. 5. Representative stress-strain curve.
times from the slope of E,(t) versus log t; &(to), the relaxation modulus at the start of the stress relaxation measurements; and the stress-strain curves at 4 strain rates taken on the day following the stress relaxation measurements. I n order to utilize the integrals in eqs. (1) through (5) it is necessary to determine two more parameters rmaxand rmin,the upper and lower limits on the distribution of relaxa~-~ developed ~ a procedure for obtion times, H ( r ) . T o b o l ~ k y ~has taining T~~~ and rminfrom stress relaxation data. Implicit in his approach is that the stress relaxation data has been taken starting with values of to (see Fig. 3) sufficiently small that a constant high strain rate modulus has been observed. I n this case, H ( T ) in eq. (1)
670
KOROSTOFF, POLLACK, AND DUNCANSON
is replaced by the constant experimental value of H1, from eq. (9), and eq. (1) is integrated giving
Er(t)
HI(T) [Ei(-t/Tmin)-
=
Ei(-t/rmax)]
(10)
where Ei(x) is the exponential integral function which has been numerically tabulated,25and rmaxand rminare graphically determined by extrapolating the stress relaxation data to the observed constant high strain-rate modulus (t-rmin) and the observed constant low strain rate modulus (t-rmax). The logarithmic derivative of eq. (10) then yields
from which it is clear that for rmin
