Correlation of Dynamic and Static Measurements on Rubberlike
Materials R. D. ANDREWS
The Dow Chemical Co., Midland, Mich. Dynamic and static measurements on rubberlike matewhere f is the stress, SO is the NE of the questions rials can be related in terms of a generalized Maxwell model which has been much fixed strain, and t is the time which is assumed to represent the mechanical behavior of discussed in the field of the elapsed since the sample was the system, the individual characteristics of the system m e c h a n i c a l properties of strained. being expressed by the nature of the relaxation time disrubberlike materials is the Dynamic modulus is orditribution of the model. relation between the elastic narily measured by applying Such properties as dynamic modulus and dynamic vismoduli measured by static sinusoidal vibrations t o the cosity measured in experimentsinvolving sinusoidal vibraand by dynamic experimental sample. T h e d y n a m i c tions and static modulus measured in experiments of rem e t h o d s . Some investigamodulus iR, like the static laxation of stress at constant strain can be expressed by tors have obtained results modulus, not a unique quanintegrals involving the distribution of relaxation times. which indicated that the tity, since it is a function of The distribution of relaxation times can therefore be calstatic and dynamic moduli frequency in general, even culated from experimental data of this sort by suitable were essentially equal, wherethough it may be fairly conmathematical methods. as other workers have obstant over certain ranges of Because of the complexity of such calculations when tained significant differences carried out rigorously, the use of approximate methods is between the two moduli. logarithmic frequency, just often desirable, A number of useful approximate relaTheoretical developments as the etatic modulus may be tions can be derived by investigation of the mathematical in this field in recent years fairly constant over certain structure of the integrals relating the distribution of rehave provided a fundamental ranges of the logarithmic laxation times to the observed properties and by examiviewpoint from which such time scale. The dynamic nation of the behavior of certain particular relaxation time questions can be examined. modulus, E d y n , as a funcdistributions. Attempts to correlate dynamic and static I n terms of this theory, the tion of circular frequency w data on the basis of this general theory, by use of the apm e c h a n i c a l b e h a v i o r of may be expressed in terms proximate relations discussed, have been reasonably rubberlike materials can be of the distribution of relaxasuccessful. regarded as arising from a tion times (sa, 28) combination of elastic and viscous behavior which can be represented by a so-called generalized Maxwell mode1,consisting of an infinite set of Maxwell units (spring and dashpot in series) connected in parallel. This type of model is characterized by a distribution function, E(T ) , which is The dynamic modulus of a material is fully expressed, therefore, called the distribution of relaxation times and which specifies the only by the complete curve of dynamic modulus versus frecontribution to the total “instantaneous” modulus (measured so quency. The question of comparing static and dynamic modulus quickly that no viscous flow can take place) that is associated with thus becomes in a more general sense the question of comparing a relaxation time, 7 . The term “distribution of relaxation times” the stress relaxation and E d y n versus frequency curves. is therefore not completely accurate, since E ( T ) is really a disSome rough conclusions can be drawn from the fact that curves tribution of partial modulus as a function of relaxation time. of static modulus versus time and dynamic modulus versus v i b r a Static elastic modulus can be measured either by applying a ‘ tion period (reciprocal of frequency in cycles per second) are t o a fixed strain or a fixed load to the rubber sample. The stress or crude approximation the same and that these curves must either elongation which is produced is more or less time-dependent, remain level or decrease with increasing time (or vibration period). however, and the measured value of stress or elongation at some It will generally be true, then, that the dynamic modulus will be arbitrary time value is ordinarily taken as defining the static approximately equal to or will be larger than the static modulus, modulus. Since the static modulufi measured a t fixed strain is since the vibration period in the dynamic measurement will orthe one most easily described in terms of the generalized Maxdinarily be significantly smaller than the time value a t which the well model, static modulus will be defined by that type of experistatic modulus is measured. The difference between the dyment in the present paper. Since the stress in a rubber sample namic and static modulus will depend on the magnitude of the held at k e d strain is time-dependent, it is obvious that there is relaxation time distribution between the Vibration period and no unique “static modulus” value for a rubberlike material but time values in question. only an apparent modulus which is LL function of time. The The dynamic modulus is not the only quantity of interest in static modulus of a material in the most general sense is therevibration experiments; the energy losses which occur during fore given by the complete curve of stress versus time a t constant vibration, or mechanical energy which is dissipated into heat, strain; this will be referred t o as the “stress relaxation” curve of is of great interest as well. This can also be interpreted on the the material. This may be expressed in terms of the distribution basis of the generalized Maxwell model, and if the vibration beof relaxation times (sa) havior is described in terms of a complex dynamic modulus, the real part of the complex modulus is the E h y n mentioned previfso ( t ) = E(T)e-t/Tdr (1) ously, while the imaginary part of the complex modulus, which April 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
707
-ELASTOMERS-Dynamic
Properties
1s related to the energy losses, can be written as
wTdyn where qdyn is an effective viscosity coefficient that i s a function of frequency. This latter quantity is related to the distribution of relaxation times by the expression (sa, 12)
terms of logarithmic time is simply related t o the distribution in terms of linear time E’(l0gio
7)
2.303 r E ( r )
=
(4)
which follows from the fact that by definition
(3) As a rough qualitative generalization in this case, energy losses will be greatest a t reciprocal frequency values where the distribution of relaxation times has the largest values; static and dynamic moduli will also show the greatest change with logarithmic time and frequency, respectively, in these regions.
& INTENSITY FUNCTIONS
E‘(log1o 7 ) d loglo T = E(?-)d r
(5)
This is one case of the general relation for transforming distribution functions
N ( X )d x
= zk N ’ ( y ) d v
(6)
where N represents the distribution function and x and y are functions of some common parameter; the sign is chosen so that N is positive on both sides of the expression. Throughout the remainder of this paper, “log” will denote logarithms to the base 10 in all cases, and the subscript 10 will be dropped; natural logarithms will be denoted “In.” This approximation ckn be derived in the following way. By substitution of Equation 5, Equation 1 may be rewritten as
The negative slope of the static modulus curve plotted versus logarithmic time is then obtained by differentiation under the integral sign
o’2 0
tI
-2
-I
0
This expression is of the general form +I
+Z
Z
(log,oc -loslot)
Figure 1.
Intensity Functions in Integrals of Equations 7 and 8
Vibration behavior can be represented by a simple Voigt mechanical model (spring and dashpot in parallel) with an inertial mass in series, at any given frequency. However, for a material with a distribution of relaxation times, the modulus of the spring and the viscosity of the dashpot will both be functions of frequency and, in fact, are the Edyn and vdyn already given. Thus, both static and dynamic mechanical behavior can be regarded as resulting from the nature of the relaxation time distribution of the material and can be correlated on this basis. However, in deriving quantitative relationships the exact solution of the equations involved is often very difficult, and approximate methods are desirable for use in such capes. The present paper is concerned with an examination of approximate methods of this sort. Attempted correlations of static and dynamic mechanical data on the basis of these approximation8 are also discussed. STATIC MODULUS
The distribution of relaxation times E(7) can in principle be calculated from static modulus or relaxation of stress data, by inversion of the integral in Equation 1. This can be done by use of the Laplace transform, but the procedure involved is too complicated t o be conveniently usable in most cases. A simple approximate relation which has been proposed for use in this case is that the distribution of relaxation’times in terms of logarithmic time, E’(log1o r ) , a t any value of relaxation time 7, is equal to the negative slope of the relaxation curve plotted as f/so versus logarithmic time (loglo t ) a t the same valueof time t. A prime is added t o the symbol for the distribution function here to indicate that the logarithmic distribution will be a different function from the linear distribution E( T ) . The distribution of relaxation times in
708
=
+ E’(10g
T )
d log r
(9)
where $I is the function in square brackets in Equation 8. This function specifies the intensity of the contribution of various regions of the relaxation time distribution, E’(1og r ) , to the value of the integral, I , and therefore may be referred to as an intensity function. This function is plotted versus (log r log t ) in Figure 1; it has a fairly sharp peak a t 7 = t and drops off to zero on both sides of the peak. The negative slope of the relaxation curve is therefore determined by the nature of the relaxation time distribution in the immediate neighborhood of the time value, 1, at which the slope is measured. If the distribution function is a constant in this region, it can be taken from under the integral sign in Equation 8, and since the remaining integral has a value of unity
-
we obtain the desired relation
This relation may also be regarded as resulting from a replscement of the bracketed function in Equation 8 by a Dirae b-function, 8( T - t ) , having a numerical value unity, and no assumption is necessary regarding the form of E’(1og 7 ) ; Equation 10 indicates that the “normalization factor” of the &function is unity in this case. This general approach has been used by ter Haar (IO). This relation can be arrived a t by still another path, using a method adopted by Ferry (6),in which Equation 7 is written in approximate form as
f/so
(t) =
Lo;;
E’(1og
7)
d log
7
(12)
This approximation represents R replacement of the intensity function, e - t / r , in Equation 7 by a “cutoff function” which has a
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 4
-ELASTOMERS-Dynamic value of aero where log 7 is less than log t and a value of unity where log 7 is greater than log 1; the approximation that this involves can be seen by reference t o the correct intensity function, which is shown in Figure 1. Then, by differentiating both sides of Equation 12 with respect to log t, Equation 11 is immediately obtained. Actually, the approximation given in Equation 11 is capable of much greater generalization. Dividing the left-hand side of Equation 11 by 2.303 t and the right-hand side by 2.303 7, and utilizing Equation 4, one obtains after transformation
Properties-
This represents an improvement on Equation 11, in which a “second approximation” is obtained by addition of a correction term involving the second derivative of the relaxation curve. [An alternative second approximation expression can be derived by a method which is described in an accompanying article by Ferry et al. ( d ) . ] 1.0
2.303 t e
-t/+
!
Therefore the distribution of relaxation times in terms of linear T is given approximately by the negative slope of the relaxation curve plotted versus linear t. Then, considering any general function of 7 or t, e, and applyingEquation 6 in the form
E(7)d r =
& E [ ~ ( T d) ]e(7)
=
(14)
E ~ ( ~e y)7 )I a7‘
we may divide the left-hand side of Equation 13 by O ‘ ( t ) and the right-hand side by e’( T ) , and this gives after transformation the completely general approximate relation
0.4
0.2 II
0 -2
I -I
0
(log,, z
‘
The absolute value is taken on the right-hand side because the distribution function must always be positive, whereas the first derivative may be either positive or negative, depending on whether e is a decreasing or increasing function of t. This relation can also be derived from Equation 11 even more directly by writing Equation 11in the form -d ( f / ~ o ) = E(1og t ) d log t where the prime on the distribution function has been dropped for convenience. Equation 6 may then be applied directly to the right-hand side of this expression, letting z = log t and y = O ( l ) , and on rearrangement one obtains Equation 15, Equation 11 is therefore simply one particular form of the general approximation expression (Equation 15) which represents an infinite family of similar approximation expressions. And since the various explicit forms of Equation 15, such as Equations 11 and 13, are all equivalent and mutually interconvertible, they can all be regarded a8 the “same” approximation and will all have the same intrinsic accuracy-that is, neglecting the possible variations in accuracy with which the different first derivatives could be measured in actual computations. This family of relations will be referred t o as “first approximation” expressions. I n some cases, it may be desirable t o use a more accurate expression than Equation 11. Such an expression can be obtained by considering the case where the distribution function, E‘( log T ) , is not a constant, but is a linear function in log 7 E’(log
7)
= a
+ p (log - log t ) 7
where a is the value of E’(1og distributidn into Equation 8
7)
:(,{ti +
--
=a
at
7
=1
1.
(17)
Substituting this
0.251 t9
By differentiation of Equation 18 with respect to log t again, remembering that a is a function of t, and by use of Equation 17
This may then be introduced for the value of j3 in Equation 18, and after rearrangement, the final result obtained is April 1952
Figure 2.
+I
+Z
- log,cJ t )
Graphical Representation of Derivation of Second Approximation, E q u a t i o n 20
The nature of the second approximation expression above is shown graphically in Figure 2. The intensity function in the integral of Equation 8, taken from Figure 1, is shown, with the distribution functions where E’(1og 7 ) is a constant and a linear function of log 7. The difference between the two distribution functions is shown as a shaded region, and this shaded region multiplied by the intensity function corresponds to the added correction term in Equation 18 and in turn to the added correction term in Equation 20. This ;haded region has opposite sign on opposite sides of log 7 = log t, and tends to partially compensate itself, but not completely. The graph showss case for which p is positive. Equation 20 is exact and not an approximation when the distribution function is of the form of Equation 17. The effect of the added correction term in Equation 20 w a ~ checked by calculating the distribution of relaxation times corresponding to an experimental stress relaxation curve of polyisobutylene, by use of the first and second approximation Equations 11 and 20, and the results are shown in Figure 3. The relaxation curve w e d here is a previously unpublished curve obtained by Andrews and Tobolsky for the standard polyisobutylene sample distributed by Marvin of the National Bureau of Standards (8). The relaxation curve corresponds to 25’ C., and the relaxation time distributions calculated from it therefore refer to 25” C. also. The first derivative was measured graphically from a magnified plot of the relaxation curve, and the second derivative was determined graphically from a magnified plot of the smoothed first derivative curve. The first approximation t o the distribution function is the Rame as the first derivative, and the second approximation was calculated from the smoothed first and second derivative curves by use of Equation 20. The difference between the first and second approximation distribution functions is small but nevertheless significant. It is also interesting in regard to the mechanical behavior of polyisobutylene that the distribution of relaxation times goes through a definite maximum in this region. A Romewhat different approximate method for calculating the distribution of relaxation times from the relaxation (static modulus) curve, which does not involve taking derivatives of the re-
INDUSTRIAL AND ENGINEERING CHEMISTRY
709
-
-ELASTOMERS-D
ynamic Properties
laxation curve, has been proposed by ter Haar (IO). He expresses his distribution function in term8 of “relaxation frequencies,” or reciprocal relaxation times, and in the symbolism of the present paper, the approximation which he proposes is [E(1/7)lr=t = t f- ( t )
(21 1
SO
y
eo
R E L A X A T I O N CURVE
I-
E(1/7)E N ( Y ) = const.
(23)
between the limits of the distribution and t o zero outside those limits. The distributions used by ter Haar in checking the accuracy of his approximation were in many caqes of this form The results obtained here indicate that the accuracy would be best in those cases. The behavior observed in Figures 3, 4, and 5 shows that Equation 22 is much more restricted in its range of application than the approximation (Equation 11 or 15) which involves the first derivative. This latter approximation is referred to by ter Haar as the “Alfrey approximation,” and the form of this approximation Thich he uses to calculate N ( v ) or E( 1/7) can be readily derived from Equation 15 by setting 8(7) = 1/7 and e ( t ) = I / t :
100
90
the distribution function is arbitrarily cut off The distribution denoted here as E( I/T) iF symbolized by N ( Y) in ter Haar’s publications, and the distribution represented in Figure 6 when written in that form corresponds to
FIRST DERIVATIVE
The negative sign is retained in the final expression since the derivative will always be negative, and the use of this negative sign will therefore be equivalent to taking the absolute value.
I
SECOND DERIVATIVE
DYNAMIC MODULUS AND LOSS
The dynamic modulus is related to the distribution of relaxation times as given in Equation 2. This relation may be rewritten, by substitution of Equation 5, in the form DISTRIBUTION F U N C T l O h
B
IO
0 FI
W
0 -5
-4
-3
-2
0
-I
lOg,,t
I
2
3
(HOURS)
This integral is again of the form of Equation 9, and the “intensity function” in square brackets is shown plotted versus (log 7 - log l / w ) in Figure 7 . The distribution of relaxation times can be calculated approximately from the curve of dynamic modulus versus log l/w by use of the relation
Figure 3. Calculation of First and Second Approximation Dktribution Functions from Static Modulus (Stress Relaxation) Curve of Polyisobutylene at 25’ C.
In case there is a residual elastic stress which does not relax in the relaxation region under consideration, this must be subtracted from f/so in this equation. This expression can be written in terms of the distribution function, E’(1og 7)) by using Equation 6 to transform E( I/T), and the result obtained is [E’(log
~ ) ] r = t=
f (t) 2.303 so
(22)
According to this relation, the distribution function, E’(1og T), is a simple multiple (by a factor 2.303) of the relaxation curve, when log T and log t are plotted on the same scale. Comparison of the relaxation curve and approximate relaxation time distributions for polyisobutylene given in Figure 3 shows that the relation (Equation 22) does not hold in this particular case. Plots of the distribution function and corresponding relaxation curve (f/so) for three widely different forms of E‘(log T ) are given in Figures 4, 5, and 6. (These graphs are discussed in more detail in a later section.) Figures 4 and 5 show that Equat,ion 22 does not hold for relaxation time distributions of this form, either, even m a rough approximation. I n fact, for the distribution of Figure 5 , the distribution function increases sharply, while the relaxation curve decreases sharply. However, for the distribution shown in Figure 6, ter Haar’s approximation, in the form of Equation 22, is found to hold exactly for all time values well within the lower and upper limits, 71 and T,,,, at which
710
in which the distribution function is taken to be equal to the negative slope of the dynamic modulus curve. This approximation is the analog of Equation 11 for static modulus data, and as in that case is simply one member of an infinite family of equivalent expressions of the general form
which is the dynamic modulus analog of Equation 15, Equation 26 can be derived by the same general methods that were used in deriving Equation 11 in the static modulus case. The general expression for the negative slope of the curve of Ed, versus log (I/@)can be obtained by differentiating Equation 25 under the integral sign, giving the (exact) relation
The intensity function in square brackets is plotted in Figure 7 and has a fairly sharp peak, indicating that the slope of the dynamic modulus curve is determined by the nature of the relaxation time distribution in the immediate neighborhood of the value log l/w at which the slope is measured. If E’(log T ) can be assumed to be constant in the neighborhood of 7 = l/w, the distribution function may be brought out from under the integral sign, and since the remaining integral is equal t o unity
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 4
-ELASTOMERS-Dynamic
one immediately obtains the desired relation, given in Equation 26. As in the static modulus case, this relation can also be obtained by replacing the intensity function in Equation 28 by a Dirac &function, or by replacing the intensity function in Equation 25 by a cutoff function (6)
followed by differentiation of both sides of the equation with respect to log l/w.
Properties-
This is very similar to Equation 22 for static modulus data except that this expression involves a proportionality rather than an equality. According to this relation, the distribution function, E’(1og T), should be directly proportional to the dynamic modulus. As in Equation 22, Relation 33 clearly does not hold for relaxation time distributions of the forms shown in Figures 4 and 5; however, Equation, 33 does hold exactly for the distribution represented in Figure 6. Examination of the characteristics of the dynamic modulus function in Figure 6 indicates that the proportionality constant involved in Relation 33 is 2/7r, and when the right-hand side is multiplied by this factor we obtain the equality
(34) The range of applicability of this equation will be about the same as that of Equation 22, which will’be narrower than that of the relation involving the first derivative given as Equation 27.
’ I
tt u
ARBITRARY UNITS
E (r)=C E’(log,()2)=2.303
Static Modulus, Dynamic Modulus, and Dynamic Loss €unctions Corresponding to the Distribution Function in Equation 42a
Figure 4.
zc
I
ARBITRARY UNITS
In order to obtain a second-order approximation of improved accuracy, we may substitute a distribution of the form E’(l0g
.
7)
= a
+ p (log
7
- log l/w)
(31)
into Equation 28. However, because the intensity function in Equation 28 is symmetrical around its peak a t log 7 = log l/w, it is evident that the contribution of the second term in Equation 31 to the value of the slope will be zero. Referring to Figure 2, the positive and negative contributions to the slope which result when the shaded areas aze multiplied by the intensity function of Equation 28 will in this case exactly compensate each other. Thus Equation 26 represents a second-order approximation as well as a first-order approximation. Computation of E’(1og 7 ) from the slope of the dynamic modulus curve by use of Equation 26 is therefore a more accurate method than computation from the slope of the static modulus curve from the analogous Equation 11. It is evident, in addition, that if E’(1og 7 ) is expanded in a power series in (log 7 - log l / ~ )the , contribution of all odd power terms to the slope will be zero, as a result of the symmetry of the intensity function in Equation 28. An approximate relation by which the distribution of relaxation times can be computed without taking derivatives has been proposed by ter Haar (IO)for dynamic modulus data also. He presents this relation in terms of a distribution function N ( w ) = [N(v)IVIW, but his expression may be written in our symbolism a8
This represents a proportionality rather than a n equality. If the ) transformed to E’(1og 7 ) by use of distribution function E( 1 / ~ is Equation 6, this relation takes the form
April 1952
Figure 5. Static Modulus, Dynamic Modulus, and Dynamic Loss €unctions Corresponding to the Distribution Function in Equation 45a
The energy loss term for sinusoidal vibrations has been given in Equation 3. This may be written in terms of the logarithmic distribution of relaxation times by substitution of Equation 5, giving
A plot of the intensity function in this integral, in the form of a dashed curve, is shown in Figure 7. It is a symmetrical curve around its peak at log-7 = log l / w , as is the intensity function for the slope of the dynamic modulus curve in Equation 28. It is therefore not necessary to differentiate this expression in order to obtain a peaked intensity curve, as was true for the static and dynamic modulus in Equations 7 and 25. The peak in this curve
INDUSTRIAL AND ENGINEERING CHEMISTRY
71 1
-ELASTOMERS-Dynamic
Properties-
is not quite so sharp as that of the intensity curve in Equation 28, but the value of Uvdyn is still determined by the nature of E’(1og 7 ) in the immediate neighborhood of log 7 = log l/w. This curve is essentially the square root of the intensity curve in Equation 28.
I
From the symmetry of the intensity function in Equation 35, it can be concluded that Equation 37 also represents the secondorder approximation of this type, since the contribution to qd,.,, from the second term in Equation 31 will be aero. Similarly, the contribution from all odd power terms in a generalized expansion of the form of Equation 31 vi11 be zero, as in the case of the slope of the dynamic modulus curre. To relate dynamic loss and relaxation time distribution without taking derivatives, ter Haar ( 1 0 ) has proposed the following relation, which vhen written in the author’s symbolism is W?dyn(W)
ZI
J
cz
W;E(l’T)ll/~=w
(40)
where a proportionality rather than an equality is specified. If E ( ~ / Tis)transformed to E’(log T ) , this expression amumes the form
ARBITRARY UNITS
This is essentially the same relation as in Equation 37 and becomes identical with 37 if the right-hand side is multiplied by a proportionality conetant of ~ / 2and the proportionality is changed to an equality. This relation therefore requires no separate discussion. It is of interest to note, however, that unlike the other two relations proposed by ter Haar, this relation holds essentially for a relaxation time distribution of the form shown in Figure 4,and less satisfactorily for the distributions in Figures 5 and 6. PROTOTYPE DISTRIBUTION FUNCTIONS
Figure 6. S t a t i c Modulus, Dynamic Modulus, a n d Dynamic Loss Functions Corresponding t o the Distribution Function i n Equation 47a
As an alternative to mat,hematical analysis of the structure of t8heintegrals involved, the problem of deriving approximate relations may be approached by studying the behavior of certain particular forms of relaxation time distribution. One type of distribution which has been found particularly useful in this connection (1, 3, 11) is a “logarithmic box” distribution, which is a constant in terms of log 7 between a lower and upper limit 71 a n d T~ and zero outside those limits
7 m )
E’(1og 7 ) = 2.303 C (71 Approximate expressions for q d y n may be derived in the same manner as previously. If E’(1og 7 ) = const. is substituted into Equation 35, and brought out from under the integral sign, the
This corresponds to a distribution in linear
E(7) =
n.
remaining integral has the value __
4.606
c/7 ( T i < < 7
= 0(T
(42%)
T , ~ )
7
of the form
Tin)
(421, )
< 71, 7 > T m )
The static modulus, dynamic modulus, and dynamic loss functions corresponding to this distribution can be obtained by ~ u b stitution of Equation 42b into Equations 1, 2, and 3 and one obtains the first-order approximation that
flso =
c [Ea ( -
f/7L)
- Ei ( - t / r m ) ]
(43x1 (43b)
7r
This expression can also be obtained by substitution of __ 4.606 6 ( 7 - l / w ) for the intensity function in Equation 35. According to Equation 37, the distribution function, E‘(log T ) , is proportional to the quantity wq~iY,. Dynamic viscosity can also be represented by use of a cutoff function ( 5 )
where a is a constant with a value in the neighborhood of unity, and by differentiation of both sides of this equation one obtains the additional approximation that [E’(log T ) ] . i = 1 / w = 712
-aw
&k!E d log w
(39)
Wqdyn
= C
[ t a n - l w ~-~ tan-lW711
(431.)
These three functions are shown plotted, together with the distribution function itself, for a distribution which is five log cycles in width, in Figure 4. For the time, or reciprocal frequency range which lies well v-ithin the limits of the distribution function, the static and dynamic modulus curves are straight lines with slope equal to the height of the distribution function, E’(1og T ) , and the height of the dynamic loss curve is approximately constant and equal to ~/4.606times the height of the distribution curve. This particular relaxation time distribution therefore leads to the first-order approximations, Equations 11, 26, and 37, which hold exactly for this distribution; it is essentially equivalent to the substitution of E’(log 7 ) = const. into the various integlals involved. The general Equations 15 and 27 also hold exactly for this distribution.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 44, No. 4
-ELASTOMERS-Dynamic Another useful relation, between static and dynamic modulus, can be obtained from this distribution. I n the intermediate range, the static and dynamic modulus curves are identical except for a slight lateral shift. The amount of this shift is given by the relation that the time value, t, in the static modulus curve corresponding to any arbitrary value of modulus is related to the reciprocal frequency 1/w a t which dynamic modulus has the same value by the expression t = - 0.561
Using this relationship, the static and dynamic modulus curves can be related directly, without going through the intermediate step of calculating the distribution of relaxation times. It is therefore of interest to see how generally this relationship holds, and this can be checked by examining the behavior of the two relaxation time distributions shown in Figures 5 and 6, which differ considerably from Equation 42a and in opposite directions. The distribution shown in Figure 5 is of the form = 2.303
T )
=0
(T
C
T (71
< 7 < Tm)
(458)
E ( T ) = C ( T Z< T =s
< Ti, 7 > ~
m )
This modified form again has the advantage that it is independent of position along the logarithmic T scale, for a given value of C, The corresponding linear distribution is obtained by multiplying Equation 47b by rm.
INTENSITY FUNCTIONS
12c I .o
0.8
0.6
< 72, T > T n ) 0.4
This corresponds t o a linear distribution of the form
’
(T
(44)
w
E’(l0g
= 0
Propertie-
T n )
(45b) 0.2
If one divides Equation 45a by the constant, T ~ representing , the upper limit of the distribution, one obtains the modified form
0 -2
0
-I
+I
t2
( l o g l o 7 -l O Q , O & ,
0
(7
< T Z , 7 > Tm)
Figure 1. Intensity Functions in Integrals of Equations 25, 28, and 35 .
Solid curves refer to dynamic modulus: dashed curve to dynamic loss
which is more convenient in that it is independent of position along the logarithmic T scale, for a given value of C. The corresponding linear distribution is obtained by dividing Equation 45b by T ~ .The static modulus, dynamic modulus, and dynamic loss functions corresponding to this distribution in its modified form are
Edyn
*
=
c (1 -
72/7m)
- w7,(tan-’wTn
- tan-lw71)
(46b)
For distributiona where T Z < < T ~ the , second term in the first square brackets of Equation 46a and the second term in the first parentheses of Equation 46b are essentially zero and may be dropped. These functions are plotted, again for the case where T Z and T~ differ by five logarithmic cycles, in Figure 5. The distribution in Figure 6 is of the form E’(l0g
7)
= 2 = c = 0
(7
(71
< 7 < Tm)
< 71, 7 > T m )
which may also be writtenmas a linear distribution
E(7)
3
= 0 (T
7 m )
C/T2 (72
T~
(47b)
gives the more convenient form
The static and dynamic functions corresponding t o this modified distribution are
If TZ a
EXPERIMENTAL INVESTIGATIONS
8
In the previous sections of this paper, a number of approximate relations between static and dg nainic mechanical quantities and the distribution of relaxation times have been given, as well as relations between these quantities themselves. The experimental data available for checking these relationships are not very extensive. However, some work has been done in this direction, and will be mentioned here. In all these cases, only the firstorder approximations were used, and in particular the first-order approximations derived from the logarithmic box distribution (Equation 42a). Kuhn and Kunzle ( 6 ) in 1947 attemptid t o correlate creep dat,a and free torsional vibration data on three samples cut from B sheet of vulcanized natural rubber. They showed that for a material with a creep curve which is linear in logarithmic time
2 6 t
m
a 4
a I
$
> 5
2
I I
0
a
:-2 a
g -4 0
w
where 01 is fractional elongation, u is strcm, and a and b are constants, the dynamic loss is given approximately by the expression
-6
-10
V
c
I 0
I
I
0.2
I
I
0.4
I
I
0.6
I
0.8
Figure 8. Relation between Second Derivatives of S t a t i c and Dynamic Modulus Curves and Horizontal Displacem e n t of Curves for Distribution 45a Shown i n Figure 5 Arrows by curves indicate direction of increasing time and reciprocal
frequency:
dashed line indicates displacement corresponding t o diatribution (Equation 42a) shown in Figure 4
fairly linear section, this corresponds to such a small region of the modulus curves that this furnishes no useful relation. A similar behavior would probably be found for Distribution 47a. Thus, this approach does not seem to lead to a useful second approximation for Equation 44. However, Distributions 45a and 47a are both fairly extreme in shape, and for distributions which are obtained in practice, which are probably closer to the form of 42a, Equation 44 should be a fairly satisfactory approximation. It was noted previously that the first approximation expressions (Equations 11, 26, and 37) hold exactly for a distribution of the form of Equation 42a which is illustrated in Figure 4 (provided that one remains well within the limits of the distribution function). However, Equations 11 and 26 also hold exactly for the Distribution 47a which is shown in Figure 6; and this is of course true also for these equations in their generalized forms (15 and 27). It was mentioned in addition that ter Haar's approximations (Equations 22 and 34) hold exactly for the Distribution 47a. The fact that E'(1og T ) can be exactly proportional to the modulus value and also to the slope of the modulus curve simultaneously, in both the static and dynamic cases, for this distribution, is explained by the fact that the distribution function, static modulus, and dynamic modulus curves are all decreasing exponen714
where Y is vibration frequency in cycles per second. Thoir calculations of dynamic loss are shown in Table I with the experimentally determined values. Good agreement is obtained. Dynamic modulus could also be calculated successfully from creep data, provided that creep was also measured in torsion.
Table I. Comparison of Observed Dynamic Loss w i t h Values Calculated f r o m Creep Curves for Vulcanized N a t u r a l Rubber (6) Sample I I1
I11
(Vibration period Type of Creep Curve Tension Tension Tension Torsion
=
20 seconds)
Y$dyn
x io-'
(Calcd.) 3.7 2.70 2.47 2.55
YHyn
x lo-'
(Obsd.) 3.9 2.75 2.66 2.55
Similar calculations of dynamic loss for various vulcanized rubbers from static modulus data have been carried out by Dunell and Tobolsky ( I I ) , essentially using Equations 11 and 37. Typical results are given in Table 11, where Eo is the slope of the relaxation curve plotted as static modulus versus log t. In general, experimental values are higher than calculated values by a factor of 2 to 10. However, the time range of the relaxation measurements did not overlap nTith the range of reciprocal frequency of the vibration measurements, and the relaxation data had to be extrapolated. This is probably the source of the discrepancies obtained here. Calculations of this sort for various textile materials gave generally better agreement. Values of dynamic loss calculated by these authors from creep data using the method of Kuhn and Kunzle also tended to be somewhat lower than the experimental values, but the same difficulties of extrapolation were present in these calculations.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 4
-ELASTOMERS-Dynamic
Propertie-
reduced modulus is identical with actual modulus at 25' C. A composite 1/3 f/8o, COMPOSITE STATIC static modulus curve for 25" C. extendM0DU L U S ing into the time range of the vibration 0 Gdyn, DYNAMIC SHEAR MODULUS experiments was constructed in this way and is shown as a solid curve in Figure 9. Static modulus, ?/SO, waa measured in tension and has been converted t o shear units by multiplying by ]/a. The position on the curve at which measurements could first be made at 25' C. is indicated by an arrow. The part of the c&ve in the region of the vibration data was measured at -50' C. Experimental value8 of dynamic shear modulus are shown -8 -7 -6 -5 -4 -3 -2 -I 0 I as ciroles and are plotted ver8us log 0.56l/w as abscissa, m indicated by log,, t, loglo (HOURS) Equation 44. T h e solid curve and experimental points show similar beFigure 9. Comparison of Static and Dynamic Shear Modulus Data on havior-a marked rise in the same Polyisobutylenq at 25" C. region, However, the agreement bePlotted as indicated by Equation 44 tween static and dynamic modulus values is not so good as one would expect from Figures 4, 5, and 6, and further work would be deTable 11. Comparison of Observed Dynamic Loss with sirable to investigate the reasons for this discrepancy. Perhaps Values Calculated from Stress Relaxation Data for Various the composite static modulus curve is not really identical with Rubber Stocks (21) the curve that one would obtain if the 'whole curve were measEO, ured directly a t 25' C. Dynes/Sq. qdyn X Wqdyn X Stock Temp., C . Cm. X 10-a (Calcd.) (Obad.) I n general, it can be said that the experimental work done so 0.24 Hevea g u m 40 0 11 0 026 far s e e m to confirm the correctness of the general theory de0.13 100 0.20 0.045 2.5 2.1 0.48 Hevea tread 40 scribed here and the usefulness of approximate relations derived 1.1 from it. Where discrepancies are observed, reasonable causes for 0.82 0.11 0.42 0.11 these discrepancies readily suggest themselves. Further experi0.85 Neoprene G N gum 40 1.5 0.33 0.68 100 1.5 0.33 mental work is needed to check various aspects of this theory and to extend the range of direct experimental knowledge in this field.
-
O
I
ACKNOWLEDGMENT
Table 111. Comparison of Observed Dynamic Viscosity in Torsion with Values Calculated from Creep Data for Various Plastic Materials (7) Material Polyethylene Polymethyl methacrylate Ebonite Polystyrene
b X 1013 220
9.8 2.25 2.5
U X 109 1.1
13.3 9.0 12.0
?dyn x 10' at 1 Cyole/Second Calad. Obsd. 2.9 2.0 18.7 15.0 2.0 1.8 3.9 2.5
The author would like to express his appreciation to R. F. Boyer and T. Alfrey, Jr., for their interest in the progress of this work, for useful suggestions, and for a critical reading of the manuscript prior to publication. The vibration data for the polyisobutylene sample were kindly supplied to the author by J. D. Ferry. Thanks are also extended to Grace S. Stockwell for preparing the drawings of the figures. LITERATURE CITED (1) Andrews, R. D., Hofman-Bang, N., and Tobolsky, A. V., J . Polymer Sci., 3 , 6 6 9 (1948). (2) Andrews, R. D., Holmes, F. H., and Tobolsky, A. V., unpub-
'
Lethersich (7) has recently attempted to calculate dynamic viacosity from creep data for various plastic materials, using these same methods (or approximations). His results are given in Table 111,where b is the slope of the creep curve versus log t, and G is dynamic modulus. Generally satisfactory agreement was obtained. The last two columns of this table actually represent VQdyn as well as V d m , since v = 1 here. As a final example, a comparison is possible between some shear vibration data on a polyisobutylene sample at 25" C. by Marvin, Fitzgerald, and Ferry (9) and static modulus data on a polymer of the same molecular weight by Andrews, Holmes, and Tobolsky (9). I n this case also, the time ranges of the static modulus measurements did not overlap with that of the vibration measurements. However, relaxation of stress curves were measured at different temperatures, and other studies (1, 8) have indicated that curves at different temperatures can be superposed on one another by horizontal shift along the log t scale when reduced modulus (modulus multiplied by 298/T) is plotted as ordinate;
April 1952
lished work. R. D., and Tobolsky. A. V., J. Polumer Sci.. 7 , 221
(3) Andrews, (1951).
(4) Ferry, J. D., Fitzgerald, E. R., Grandine, L. D., Jr., and Williams, M. L., IND.ENC).CHEM.,44, 703 (1952). (5) Ferry, J. D., Sawyer, W. M., Browning, G. V., and Groth, A. H., J. Applied Phys., 21, 513 (1950); see also Ivey, D. G . , Mrowca, B. A., and Guth, E., J. Applied Phys., 20, 486 (1949).
(6) Kuhn, W., and Ktinele, O., Helv. Chim. Acta, 30, 839 (1947). (6a)Kuhn, W., Ktinzle, O., and Preissmann, A., Helv. Chim. Acta, 30, 307, 464 (1947). (7) Lethersich, W., British J . Applied P h y s . , 1, 294 (1950). 44, 696 (1952). (8) Marvin, R. S., IND.%NG. CREM.,
(9) Marvin, R. S., Fitzgerald, E. R., and Ferry, J. D., J. Applied Phys., 21,197 (1950). (10) ter Haar, D., Physica, 16, 719, 738, 839 (1950); J . Polymer Sci., 6 , 2 4 7 (1951). (11) Tobolsky, A. V., Dunell, B. A., and Andrews, R. D., TextiZe Research J . , 21, 404 (1951). (12) Tobolsky, A., and Eyring, H., J. Chem. P h y s . , 11, 125 (1943). RECEIVED for review September 17, 1951.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED January 28, 1962.
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