A Simplified Description of Viscoelastic Behavior of Polymers as Illustrated with Biaxially-Oriented

Po I y( E t hy I e ne Te re p h t ha I a te)

G. TITOMANLIO and G . RIZZO

Instituto cli lngegneria Chiiriica Unifiersit; di Palerino

Palerrno, I ta l y

A rheological model of solid polymers is proposed. Its mechanical analog is a parallel of a linear spring and a Maxwell element with variable viscosity. The viscosity of the dashpot is allowed to change with stress both directly, by an Eyring-type mechanism, and through free volume changes according to the Doolittle equation. Predictions of the model reproduce many of the features shown, expecially after yielding, by constant velocity stress-strain, stress-relaxation and creep data taken at room temperature on bi ax i a1 1 y -o r ie n t e d po 1 y ( e thy 1 e n e terephthalate) over a wide range of loading rates.

INTRODUCTION or sufficiently small values of strain (or stress), the F mechanical behavior of solid polymers is entirely

described (neglecting compressibility) by a single mate- rial function which enters the theory of linear viscoelas- ticity and can be determined by a single test. Unfortu- nately, below the glass transition temperature, the strain limit of linear viscoelasticity is always quite small

A large amount of experimental and theoretical work concerning the mechanical behavior beyond the linear region can be found in the literature. Most non-linear theories are given in the form of polynomial expansions made of integrals of the load history (7-9). The first term of the expansion is the well known Boltzman integral of linear viscoelasticity. The other terms are multiple in- tegrals which account for interaction effects among events occurring at different times in the past. It was noted (10) that only weak nonlinearities already require the use of the first three terms of the expansion and that going to higher stress levels or increasing the testing time involves further higher order integrals (11). As a consequence, the expansion approach has a limited pre- dicting power: as soon as one is interested in larger stress or strain values, more terms of the expansion are re- quired with unknown multidimensional kernel func- tions. Probably for this reason most workers have lim- ited their attention to stress values smaller than the yield stress and much less emphasis has been given to the viscoelastic behavior in the very large deformation re- gion (after yielding) of polymers which can undergo plastic deformations .

The constitutive equation proposed by Haward and Thackray (12) is of different form with respect to expan-

(1-6).

sion type equations. Its mechanical analog is a series ofa linear spring and of a particular Voigt element; the spring of the Voigt element is strain hardening and the viscosity, following Eyring (13), is assumed to decrease with stress. The model proposed in ref. (12) has the advantage of not being limited to relatively small defor- mations or stresses and it seems to describe some of the effects associated with the yielding phenomenon. How- ever other important effects are not described by this model, not even qualitatively. For instance, the model does not discriminate between tension and compression and predicts the same value for the yield stress in both cases, contrary to experimental observations (14). One further important drawback is found when the recovery of strain hardening materials is considered. This will be discussed thoroughly later in the paper. Finally the model suggested by Haward and Thackray is uni- dimensional and thus can not be applied in general.

The three-dimensional model recently considered in ref. (15) does a better job as far as stress-strain and recovery are concerned but it completely fails in de- scribing stress relaxation and creep tests.

Our effort here was that ofworking out a model which would describe, at least qualitatively, all known vis- coelastic effects in the high deformation non-linear range, possibly with a small number of parameters. As shown in the following, an appropriate modification of the model by Haward and Thackray is indeed adequate. It will be shown that a mechanism associated with free volume changes is to be included.

The model proposed here is properly three- dimensional so that it can be applied to arbitrarily com- plex strain histories such as those encountered in proc- essing and use of polymeric materials.

POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, NO. 70 767

G. T i t o m a n l i o a n d G. R i z z o

DESCRIPTION OF THE MODEL Classically the mechanical analog of a viscoelastic

solid is a series ofVoigt elements. In this paper the series is reduced to only two elements and the viscosity of one of them is taken zero. This mechanical model (Fig. l a ) , which up to now is the same as that proposed in ref. (12), is equivalent to a parallel of a Maxwell element and a spring, see Fig . 1 h. Beca~ise the three-dimensional form of the hlaxwell element has already been used in the literaturc., the schcme of Fig. I b will be used in the following.

A suitable model of plastic polymers should obviously allow for large plastic deformations which, however, should also be largely recoverable by an appropriate teinperaturc rise. Other classical features of the experi- mental behavior of these materials are the existence of a yield point (or zone) and a weak dependence of the stress-strain curves on deformation rate.

Recently, an experimental study (16-18) of the vis- coelastic behavior of different plastic polymers has shown that: (i) The influence of the creep force is much larger below the yield point than above it where the creep behavior seeins essentially independent of the stress level. (ii) Both creep and stress-relaxation be- havior are largely influenced by the velocity with which the load was applied. In particular, after yielding, the creep and stress-relaxation data can be collected into single curves by means of a time shift factor proportional to the deformation rate previous to the (stress-relaxation or creep) test. As a consequence, the viscosity of the viscous elements in the model should be made to de- crease along a stress-strain curve up to the yield point and to stay essentially constant thereafter. This constant value should be smaller the larger the deformation rate and in particular it should be roughly proportional to the inverse of the deformation rate.

A qualitative discussion on the scheme o f F i g . Ib will show that appropriate non-linearities of the viscous ele- ment are sufficient to take account of these features. In particular it is immediately apparent that the viscosity 7 of the dashpot must be a strong decreasing function of crl (see Fig. 1 b). If this is the case during a stress strain' experiment, the value of 71 before yielding is very large and the deformation is absorbed essentially by the elas- tic part of the Maxwell element. As a consequence, crl increases and "/I decreases by effect of increasing crup to the yield point. Yielding will occur when the viscosity approaches the value

71 = u,/r. (1)

where is the imposed deformation rate. Most of the deformation is thereafter absorbed by the dashpot and both crl and 7 remain determined by the value of r,

(a> ( b ) Fig. 1 . Mechunicul analog of the model.

because E q I , in which r ) = q(crl), must be satisfied. A constant value of r assures constant values for the viscos- ity along the plastic zone of the stress-strain curve, as suggested by the data (16-18). Furthermore, if 71 is a strong function of crl, the weak dependence of the stress-strain curve from the deformation rate is satisfied and, as far as ul/q can be considered proportional to 1/71, the viscosit! changes proportionally to the inverse of the deformation rate.

The expression = q(crl) needs further qualifications. To begin with, it has of course the meaning that the viscosity is a function of the invariants of the tensor g T . The particular function used in this work has been suggested by both the activated energy model proposed by Eyring (13) and the free volume model.

The activated energy model relates the viscosity to the tangential stress in a shear field. A simple three- dimensional extrapolation of this model is suggested here and consists of substituting the tangential stress with the magnitude of the deviatoric part of gl. The functional dependence suggested in ref. (19) has been used for the free volume effect. In conclusion the follow- ing expression is proposed for the viscosity

where D , A arid B are constants,fis free volume,J is the identity tensor, tr indicates the trace and I I the mag- nitude of a tensor.

The insertion in Ey 2 of a free volume term, ignored by the model of Haward and Thackray (12), is dictated by the following consideration on the recovery phenome- non after a tensile test. If a tensile test is let to last until very large stresses (say about two times the yield stress) are reached and then the specimen is suddenly un- loaded, the Maxwell element goes abruptly from a large tensile to a large compressive stress. Without a inechanisin which discriminates according to the alge- braic sign (the free volume inechanisin in E q 2 ) the viscos- ity would remain small and an exceedingly large part of the deformation would be recovered. Furthermore, as mentioned in the introduction, ignoring the effect of free volume would not allow for different yield stresses in tension and in compression.

Free volume f can be split into two parts: one f o dependent upon temperature (and essentially constant below T,) and the otherf, dependent upon stress. The latter is taken to be proportional to the first invariant (or trace) of gl:

(3) where k is a proportionality constant. This is only a first order approximation and more complex relationships have been proposed (20-22). To be rigorous, one should also account for time effects (23-24).

Substitution o f E y 3 into E q 2 gives for the viscosity:

f = f o + fg = f o + K trnl -

* Tensors are indicated by a double underline

768 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, ko. 10

A Simplified Description of Viscvelastic Behavior v f Polymers

g z = Gz

where R = K i f o and F = A v o . The value F = 40 as found in the literature, e.g. , in ref. (25), has been used in the comparison with experiments.

Eyuation 4 and the following set of equations coin- pletely specify the model

- - 0- = g 1 + g2 (5)

g1 = s = J 6 (6)

a; p1” - 1 0 0

0 a; pz” - 1 0

0 0 (Y: p: - 1

where g is thc total stress, g2 the stress contribution of the spring indicated by the index 2 in Fig. I h , pI is the stress of the XIaxwell element of which one part 3 is described by E q 7 and the other 6 * 1 is an isotropic tensor. 6 is determined by the bouadary conditions and when the material is unloaded, it equals the external pressure. G I and G2 are constant moduli, Q - is the stretching tensor, i .e. , the symmetric part o f the veloc- ity gradient _L, and the expression in brackets in E y 7 is the Oldrovdcolltravariant derivative (26). &(u) - is the Finger strain tensor of the virgin unoriented material configuration (indicated by “u” in E y 8) with respect to the present configuration: in a rectangular Cartesian coordinate system the components of E((u) are

(9)

where X j is the coordinate position of a material element in the unoriented configuration, x j is the coordinate position of the same element at present time t and aij is the kroneker delta.

Obviously g2 - may be different from zero also if the material is unloaded. In this case g, would exactly cotn- pensate g2. This will be the case for “frozen in” stresses.

As most of the data available in the literature refer to elongational ,kinematics, in the following section the model will be specified for this type of deformation.

SPECIFICATION OF THE MODEL FOR ELONGATIONAL DEFORMATION

With referrnce to a rectangular Cartesian coordinate system, the kinematics of elongational flow is defined by

vi = ri(t) xi (10)

where ui are the velocity components and x i are the coordinates of a material point.

The elongational deformation defined by E q 10 is realized in several experimental situations. In particular the kinematics of usual stress-strain, stress-relaxation and creep tests on samples with rectangular cross sec- tional area are described by Eq 10. In this case indicated by 1 the loading direction the stresses acting on the material are

U1’ = Uil + T l 1 + 6

4 2 = up + $2 + 79 = 0

0-33 = u 2 3 + 733 + 6 = 0

(11)

(12)

(13)

(17) While E q 17 describes the change in time of uz, E q 7

describes how the stress of the Maxwell element, 2, changes with time. Use ofEy 7 requires a knowledge of the velocity gradient & and stretching tensor Q. These are given by

POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, NO. 10 769

G. Titomanlio und G. Rizzo

In stress-strain or stress-relaxation tests, the machine imposes rl(t) and the calculation task is to determine how d' changes with deformation or time. Ifconversely the testing machine imposes a load, as in a creep test, o m needs to calculate r, = r,(t). In both cases because of the non-linearities involved the equations have to be solved numerically.

In the case of stress-strain and stress-relaxation tests (rl(t) known) if the material is unoriented or if a2 = a3 the computational procedure is trivial. This is due to the fact that because of the symmetry around the load axis, directions 2 and 3 behave alike and, neglecting com- pressibility, continuity gives

(22)

which completely specifies the kinematics. If az and a3 are not equal, directions 2 and 3 behave differently and continuity is no more sufficient to determine completely the kinematics. In the Appendix a computational scheme for this case is illutrated.

Ifthe test piece is elongated with aconstant velocityv, r,(t) is given by

1 2 rz(t) = r3(t) = - r,(t)

0 vll, 1 1 + V t l l ,

rl(t) = - =

where 1, is the initial length of the test piece. For stress- relaxation tests after constant velocity strain ramps r,(t) is given by E q 23 until the stress relaxation starts and thereafter by r,(t) = 0.

In the case the testing machine imposes a load (creep tests) ob\iously the initial values o f g z , ~ - - and r, depend upon the deformation history previous to the creep it- self, i.e., upon how the material is loaded. Their calcula- tions belong to the previous case. The computational procedure must be changed at the instant of time when the constant load has been reached because thereafter r,(t) is no more an input but an output of the calcula- tions. Also for this case a computational scheme is given in the Appendix.

PARAMETER EVALUATION In Figs. 2-6 the predictions of the model are compared

with tensile constant velocity, stress-relaxation and con- stant force creep data taken on biaxially-oriented po- ly(ethy1ene terephthalate) (PET) (Mylar manufactured by du Pont) and reported in references (16-17).

In order to determine the intial strains, at, of the material, a few shrinkage tests were performed on strip samples cut in directions 1 and 2 out of the same Mylar sheet, 9 mils thick, which was used in references (16-17). The shrinkage was very small up to temperatures very close to the melting point, where it was about 15 percent in both directions. On the other hand, close to the melting point, during the shrinkage there is a simul- taneous partial melting of the material and probably a part of the orientation does not show up into shrinkage. Although this point might deserve a deeper investiga- tion, we have tentatively assumed.

The fact that the directions 1 and 2 have the same initial strain was independently checked by comparing the mechanical behavior along the two directions.

"E 20 E m 1

- m o d e l p r e d *

0 0 4 h '

A 16

0 160

b

10

w

0 0.5 1

Fig. 2 . Constant velocity stress-strain curves. The parametw of the c u r w s i s the initial deformation rate vll,,. The model predic- tions are calculated f o r G, = 164 kglmm2 - GP = 5 25 kglmm' - R = 3.5 - B = 340. The prestrains are as in Ey 24.

./ I NE . 201 m Y

If

N

E 2 0 - . m Y

0

10 . - model pred. experimental

I I I I I

0 0.2 0.4 0.6

Fig. 3. Constant velocity stress-strain cumes of prestrained samples. Both the strain and deformation rate are referred to the length of samples previous to any elongation. For the model parameters see caption of Fig. 2.

10 .c---_

Fig. 4 . Stress-relaxation curves at strains smaller than the yield strain. Full lines represent model predictions; f o r the model parameters see caption of Fig. 2.

770 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, No. 10

A Simplified Description of Viscoelastic Behavior of Polymers

20,

m 1

4

5 15 -

b

10 -

..A v l l o = 16 h '

0.0.~. v l I s = 0.4 h '

6 lo-' 1 10 10' 10 lo4 10 1 0 6

Fig. 5 . Stress-relaxation curves at strains larger than the yield strain. Full lines represent model predictions; f o r the model parameters see caption of Fig. 2.

t . sec

." 1 10 lo2 l o 3 1 o4 lo5

Fig. 6. Constant creep curves at different values of loud and initial deformation rate; f o r the model parameters see cuption of Fig. 2.

All the other model parameters may be determined by fitting the tensile constant-velocity data. G l and G2 are easily calculated from the stress-strain curves by using the initial slope which gives 3(G1 + G,) 500 kg/mm2 and the slope soon beyond the yield point where, as discussed previously, gl is essentially con- stant and, once the initial strains have been specified, the stress growth is related only to the value of G2.

The parameters entering the viscosity equation need a longer discussion. It is convenient to rewrite the vis- cosity equation in the form

where

andplois the value ofg l when the material is unloaded.

The quantity q, is the viscosity of the material at rest. The model predictions proved to be quite insensitive to the value of q,. To be realistic, a value of 20 years was chosen for the "rest" relaxation time qo/G1 of the mate- rial, but very small variations of either R or B in Ey 25 would compensate for large variations of -qo without significantly altering the quality of the fitting. Only if qo is changed by several orders of magnitude do the corre- sponding changes of R or B become significant. In par- ticular, by increasing the vo value, R or B would de- crease and consequently the dependence of the stress- strain curve upon the velocity would be weakened.

The parameters R and B play the same role in tensile tests. More precisely, the predictions of the model for a pair of R and B values can be reproduced within a few percent by compensating a smaller value of one of them with alarger value of the other. This is true, however, as long as the ratio RIB is not too small. In fact, as already observed, if the free volume effect is neglected, i.e., if R/B vanishes, most of the plastic deformation becomes recoverable by simply removing the stress, which is contrary to experimental indications. Recovery compu- tations after a hundred percent sample elongation showed that taking R = 0 would imply about eighty percent strain recovery after a free stress time interval of only one hour. Small values of R decrease remarkably the strain recovery. This only establishes a lower limit to the ratio RIB. In order to separate completely the two parameters, the compressive behavior also should be considered. From tensile data only, any acceptable pair of R - B values can be determined by the value of the yield-stress.

In conclusion, apart from a better definition of the individual values ofR and B , the model parameters are determined by the usual stress-strain data and some information on frozen-in stresses. The model is then predictive for all other kinds of stress or strain histories.

COMPARISON WITH DATA Some of the qualitative predictions of the model have

already been discussed above. Actually, the main new features of the proposed model were suggested by an effort to meet the requirements of a number of seini- quantitative experimental indications.

In the following, we shall attempt a quantitative coin- parison with the data reported in references (16-17). The comparison is shown in Figs. 2-6. Figure 2 shows the stress-strain constant velocity curves whose fitting de- termined all the model parameters which are reported in the figure caption. We recall that F has a value already reported in the literature (25).

In Fig. 3 is shown the comparison of the tensile stress with strain behavior of Mylar samples elongated in two steps. More precisely samples of Mylar were elongated with initial deformation rate 16 h-' up to deformations larger than the yield deformation; they were then un- loaded and, after a free stress recovery period of about half an hour, they were tested again by elongation at the same velocity. By reporting on the abscissa the strain AUlo as referred to the initial sample length, both ex- perimental stress-strain curves and model predictions,

77 1 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, No. 10

G. Titomanlio und G. Rizzo

after an initial elastic stress-growth, approach the stress of samples elongated con t i nu0 ti s lv (wit ho ti t inter - mediate unloading). The model predictions have been calculated accounting for both the Mylar initial strain, see Ey 24, and the strain with respect to hIylar as measured aftor the unloadings. This figure shows that the model accounts properly for the strains frozen in the material,

Figures 4 and S show generally fair agreement be- tween the inodel predictions and the experimental stress relaxation curves both below and above the yield strain. At strains smaller than the yield strain, (see Fig. 4) the experimental behavior is quite well reproduced especially at longer times where all the curves, both experimental and theoretical, approach each other. At larger strains ( s e ~ Fig. .5), the effect of the deformation rate which existed prior to the relaxation (the parameter of the curves) is much larger. In particular both the experimental and theoretical curves show faster stress relaxation at larger values of the deformation rate and the experimental time shift factor coincides with that predicted by the model. On the other hand, it should be mentioned that while the data show that the amount of stress relaxation slightly increases with the strain (notice that the data cover a wide stress range), the model predicts that the stress relaxation is essentially indepen- dent of the strain.

At stresses larger than the yield stress also for the creep behavior there is a good qualitative agreement (see F i g . 6). In fact, along a stress-strain curve, i.e., at a given velocity previous to the creep, both experimental and predicted creep curves are essentially independent of stress. Also i n this case the effect on the creep curves of the deformation rate previous to the creep is quite well reproduced by the inodel.

The theorctical predictions for creep curves at stresses smaller than the yield-stress do not fit the data sufficientlv well and sho~7 a dependence on creep load much larger than that shown by the experimental re- sults.

We note finally that the model predicts that during elongation there is a much smaller percentage reduction in the sample thickness than in the sample width, this being due to the effect of the prestrains (see Ey 24). This feature was qualitatively observed on all Mylar samples used in the experiments reported in references (16-17).

CONCLUSIONS We have seen that a simple model made of two springs

and one dashpot with a variable viscosity is sufficient to describe many features of the viscoelastic behavior of a biaxially-oriented poly(ethy1ene terephthalate). The pa- rameters of the inodel are fully determined by the con- stant velocity stress-strain behavior and some informa- tion on the prestrain.

The fact that some discrepancies remain, i.e., that the model is not yet fully quantitative, is probably due to the oversimplification inherent in a model with a single relaxation time. It is physically plausible that a set of relaxation times gives a better description but of course the simplicity of the model is correspondingly reduced.

The main discrepancies which were observed are (i) the creep below yielding, as predicted by the model, is too strongly dependent upon the applied load, and (ii) the model predicts aconstant amount of stress relaxation after yielding while the material shows some increase of the stress relaxed by increasing the previous strain. Both discrepancies can be removed by parallel Maxwell ele- ments. In fact, along a stress-strain curve different ele- ments would yield at different strains and all the phe- nomena would be more gradual. This graduality would apply also to the transition between the linear viscoelas- ticity region and the plastic region and thus, before yielding, the predicted creep behavior would be less stress dependent. Also for large deformations, because strain increments involve progressively new elements (those with longer relaxation times) in the yielding proc- ess, the relaxable stress would increase with the strain.

APPENDIX Computational Procedures

(a) I?, = I?&) is known Subtracting Ey 21 from Ey 20 gives

where $3 = 72' - 733 and k is defined by the. equations

rz(t) = - k r,(t) r3(t) = - (1 - k ) r,(t) (27)

which identically satisfy continuity. From Eys 12, 13 and 17 one can easily derive

p;) $3 = - (u;' - uj3) = - Gz(& pz" - (28)

= - 2 ~ , ( ~ ; p; r2 - a; p; r,)

- - a(w;z - ug3) -- at d t

Substituting Eqs 27 and 28 into Eq 26 gives K =

(29)

In conclusion, the following computational scheme is suggested

c (b) Constant Load

Because in this case r,(t) is unknown, E q 29 is not sufficient to determine the kinematics. Another relation involving the quantities which appear in Ey 29 can be determined as follows. Because the test piece is sub- jected to a constant load, the total stress and its deriva- tive are given by

d' = (To lilac (30)

772 POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, NO. 10

A Simplijed Description of Viscoelustic Rehuvior of Polymers

where mo is the initial creep stress, 1 is the current length ofthc. test piece, 1, is the initial value, lo, is the value ofl at thc beginning of the creep and c , is the creep velocity.

Substituting Eys 12 , 13 and28 into Ey 14 and differ- entiating gives

p - - 1 a ( P - 722) rl at

1 - - Pi ~0 - 2G2(4 P: + k az” Pz”) (32)

Subtracting E q 20 from E q 21 gives another expres- sion for a(r” - ?)/at. Finally equating the latter to the right hand side of E q 32 gives for rl

Eyuutions 29 and33 are sufficient to calculate rl and k for each set of the variables Pi, ai, rii, a, and l,ll,c. A possible computational procedure is then

The inital conditions are obviously determined by the loading procedure which has a known kinematics and is thus calculated with the previous scheme.

ACKNOWLEDGMENT The work has been supported by C .N.R . Grant No.

75.00290.03.

NOMENCLATURE A = constant of the viscosity equation (see Ey 2 ) B = constant of the viscosity equation (see E q 2) D = constant of the viscosity equation (see Eq 2)

= the stretching tensor p = free volume fo = free volume at zero stress fv = free volume contribution due to stress F = Alf, GI = modulus of the Maxwell element G z = modulus of the elastic element &(u) - = finger strain tensor of the unoriented material

configuration with respect to the present one k = defined by E q 27 K = proportionality constant between f, and the

1 = length of the test piece 1, = length of the test piece before any elongation l o , = length of the test piece at beginning of a creep

test - - L = velocity gradient R = Klf, t = time u oi = velocity components xi

trace of gl

= relative velocity between the test piece ends

= coordinates of a material point at present time

X i

Xi

@i

Pi

r ri 7) 7)O

6 A P - 0-

n1

n z a0

-

7

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.

26.

= coordinates of a material point in the initial configuration

= coordinates of a material point in the un- oriented configuration

= initial strains (see Ey 15) = strains with respect to the initial material

configuration (see E q 16) = elongational rate = elongational components of the velocity gra-

= viscosity = free load viscosity = see E y 6 = relaxation time, vlGl = see Ey 32 = total stress tensor = stress tensor of the Maxwell element = stress tensor of the elastic element = initial creep stress = stress tensor of the Maxaxwell element reduced

dient L -

of the isotropic i? . J

REFERENCES €3. Maxwell and A. C. Guimon, J . Appl. Polym. Sci., 6 , 83 (1962). I. V. Yannas and A. C. Lunn, J . Macromol. Sci.-Phys., B4, 603 (1970). I. V. Yannas, N. H. Sung, and A. C. Lunn, J . Macromol. Sci.-Phys., B5, 487 (1971). I. V. Yannas, J . Macrornol. Sci.-Phys., B6, 91 (1972). H. Bertilsson and J . F. Jansson,J.Appl. Polym. Sci., 19,1971 (1975). H. Bertilsson, “Proc. 7th Int. Cong. Rheology,” p. 362, Gothenburg, Sweden (August 23-27, 1976). A. E. Green and R. S. Rivlin,Arch. Ration. Mech. Anal., 1,1 (1957). A. E. Green, R. S. Rivlin, and A. J. M . Spencer,Arch. Ration. Mech. Anal., 3, 82 (1959). I. M. Ward and E. T. Onat, J . Mech. Phys. Solids, 11, 217 (1963). I. H. Hall,]. Polym. Sci., A-2, 5, 1119 (1967). I. V. Yannas, J . Polym. Sci.: Macromol. Rev., 9, 163(1974). R. N. Haward and G. Thackray, Proc. Roy. Soc., A, 302,453 (1968). H. Eyring, J . Chem. Phys., 14, 283 (1936). S. S. Sternstein and L. Ongchin, Am. Chem. Soc., Polym. Prepr., 10, 1117 (1969). G. Titomanlio, B. E. Anshus, G. Astarita, and A. B. Metzner, Truns. Soc. Rheol., 20, 527 (1976). G. Titomanlio and G. Rizzo, “Master Curves ofViscoelastic Behavior in the Plastic Region of a Solid Polymer,”J. Appl. Polyrn. Sci., in press. G. Titomanlio and G. Rizzo, “Proc. 7th Int. Cong. Rheol- ogy,” p. 552, Gothenburg, Sweden (August 23-27, 1976). G. Titomanlio, G. Rizzo, and D. Acierno, “Creep and Stress-Relaxation in the Plastic Region of Polymeric Mate- rials,’’ presented Joint Meeting of British, Netherlands and Italian Societies ofRheology, Pisa, Italy(April13-15, 1977). A. K. Doolittle, J . Appl. Phys., 22, 1471 (1951). A. J. Matheson, J . Chem. Phys., 44, 695 (1966). K. Hellwege, W. Knappe, and P. Lehmann, Kolloid-Z., 183, 110 (1963). L. A. Wood,]. Polym. Sci., B2, 705 (1964). S. Kastner, Kolloid-Z., 206, 143 (1966). A. J. Kovacs, Trans. Soc. Rheol., 5, 285 (1961). I. M. Ward, “Mechanical Properties of Solid Polymers,” p. 154, John Wiley, New York (1971). G. Astarita and G. Marrucci, “Principles of Non-Newtonian Fluid Mechanics,” p. 217, McGraw-Hill, New York (1974).

POLYMER ENGINEERING AND SCIENCE, MID-AUGUST, 1978, Vol. 18, No. 10 773