Theocaris, Higher Spectra Approximations Derived/rein Multistep Loading crystallization a sufficient quantity of polyethylene is present. The ring-width increases with increasing crystallization temperature.

Further it is shown that in polyethylene spherulites grown from the melt the rotation of the indicatrix is connected with a non-crystallographic branching. We suppose, that also for the solution-grown spherulites the ring-system depends on branching.

Literatur

149

4) Bassett, D. C. und A. Keller, Phil. Mag. 6/63, 345 (1961). 5) Bassett, D. C. und A. Keller, P h i l Mag. 7/81, 1553 (1962). 6) Bryant, W. M. D., J. Polym. Sci. 2, 547 (1947). 7) Keller, A., J. Polym. Sci. 17, 351 (1955). S) Keller, A., J. Polym. Sci. 17, 291 (1955). 9) Hellwege, K.-H. und G. Hobq, Kolloid-Z. 170, 144 (1960).

1) Heber,I., Kolloid-Z. u. Z. Polymere 189, 112 (1963). 2) Kawai, T. und A. Keller, Phil. Mag. 11,116 (1965). 3) Bassett, D. C., F. C. Frank und A. Keller, Prec. Eur. Reg. Conf. on Electron Microscopy, Delft 1, 244 (1960).

A n s c h r i f t der V e r f a s s e r i n : F r a u D i p l . - P h y s . Irmgard Heber, Deutsches Kunststoff-Institut, (;100 D a r m s t a d t , S e h l o i ] g a r t e n s t r . 6 l~

From the National Technical University, Athens (Greece)

Higher Spectra Approximations Derived from Multistep Loading B y P. S. T h e o c a r i s With 10 figures in 11 details (Received A u g u s t 11, 1966)

Introduction I t follows clearly front the v e r y definition of r e t a r d a t i o n and r e l a x a t i o n spectra t h a t t h e y c a n n o t be calculated b y a n y straightf o r w a r d e x p e r i m e n t a l method, since none of the characteristic functions defining the mechanical b e h a v i o u r o f a p o l y m e r can be expressed a c c u r a t e l y b y an analytic function along the whole viscoelastic range of the material. Therefore, certain a p p r o x i m a t i o n s are necessary to derive the spectra from exp e r i m e n t a l d a t a available in numerical or graphical form. One t y p e of a p p r o x i m a t i o n entails the use of delta-functions. The first application of such m e t h o d s in viscoelasticity has been m a d e b y A1/rey and Doty (1) for the d e t e r m i n a t i o n of the well-known first approximations of spectra b y e m p l o y i n g time or f r e q u e n c y derivatives of the various compliances or moduli. This a p p r o x i m a t i o n has been discussed b y Ferry et al. (2), Andrews (3) and others. A r a t h e r complete list of the early papers dealing with the first a p p r o x i m a tion procedures is included in a p a p e r b y Williams and Ferry (4). Schwarzl (5-7), and Schwarzl, Staverman (8, 9) and Leaderman ( 1O) i n t r o d u c e d second a n d higher a p p r o x i m a t i o n procedures e m p l o y i n g various forms of delta-functions related to stress-relaxation curves given in t a b u l a r form. While the previous higher a p p r o x i m a t i o n s of spectra do not require the knowledge of the stress-relaxation curve in analytic f o r m the m e t h o d s introduced b y Ferry and Williams ( l l , 12) and

b y Andrews (13) assume an analytic form for the stress-relaxation curve, which m a y be a power or logarithmic law. The purpose of this p a p e r is to give a m e t h o d for the calculation of higher a p p r o x i m a t i o n s for the r e t a r d a t i o n or r e l a x a t i o n spectra b y using a multistep creep-creep or creep-recovery loading for the

retardation spectra and a multistep relaxation-relaxation loading for the relaxation spectra. Since transient composite curves of high precision exist actually for several polymers, they may be used to determine higher approximations of spectra.

Derivation of Higher Approximations The continuous retardation spectrum for the extension creep compliance D(t) is defined by the relation: co

D(t) : Dg 3- I Le(ln ~) (1 -- e-t/~) d(ln ~)

[1]

0

where the i n s t a n t a n e o u s compliance (Dg) is added to i n t r o d u c e a discrete c o n t r i b u t i o n with t = 0. Similarly the r e l a x a t i o n s p e c t r u m for the extension relaxation modulus is given as follows : oo

E (t) = E g 3- f He (ln r) (e-t/~) d (ln r)

[2]

0

where the i n s t a n t a n e o u s modulus (Eg) is i n t r o d u c e d to e n c o u n t e r the discrete contrib u t i o n with r = c~. Similar relations to eqs. [1] and [2] exist, which express the r e t a r d a t i o n and r e l a x a t i o n

150

Kolloid-Zeit~ehrlft und Zeitsehrift far Polymere, Band 215 9 Heft 2

spectra in shear (subscript s) in bulk deformation (subscript v), as well as the lateral contraction ratio in creep or relaxation (subscript ~). These equations relating a viscoelastic function with its spectrum, although accurate within the assumptions of linear viscoelasticity, fail to meet practical purposes as they stand, because an integration from 0 to + c~ is required for a function which is not known analytically all over the viscoelastic spectrum. Following the development b y S c h w a r z l and S t a v e r m a n (8, 9) we introduce the approximations for the retardation (L~) and relaxation (Hg) spectra co

Lk = ~/k(tlv) L(in T) d(ln 3) oo

[4]

where the approximations L~ and H~ are expressed as integrals over the exact spectra (L or H) multiplied b y the intensity functions ]~ or gk of order k. The intensity functions must be positive for T > 0, they are functions of (t/T) possessing only one maximum in the region ~ = t, they are normalized and they decrease to zero at both sides of the maximum. With the spectrum not varying in the region where /k or g~ are noticeably different from zero the following approximations hold, if we take into consideration the normalization properties of ]k and gk respectively: co

L~(In t) ~ L (In t) f fk (t/~) d (ha 7) = L (in t)

o

co

[5]

H~ (ln t) ~ H (ln t) f 9k (t/v) d (In 7) = H (ln t). 0

In order that relations [5] hold exactly the corresponding intensity functions ]~ and g~ must be delta functions. Then, the closer the resemblance of the intensity function to a delta function the higher the approximation. If we take

L(t) ~ Lk(t) = (-- 1)k-1/(k -- 1)! k k t~=-x ~p(k)(kt)

[7]

H(t) ~ Hk(t) = (-- 1 ) k - x / ( k - 1)!/c k t k-1 ~o(k)(k t) [8]

where k is the order of approximation seeked for the spectra and ~o(k)(kt) and q~(k)(kt) the ]~th order derivatives of the characteristic creep (y) or relaxation (~) functions, the spectra of which are investigated. From the above relations the following approximation spectra for the log time scale can be readily derived:

[9]

Ll(log t)= 0.434 [ ~ ]

[3]

0 Hk = ~ gk(t/v) H ( l n 7) d(ln 7) 0

functions. Following S c h w a r z l and S t a v e r m a n (8, 9) we can define the spectra:

L2(log t/2) = [0.434 dye(z) d (log v) d~o(v) Ls(log t/3) = [0.434 d (log 7) k

0 . 1 8 9 ~ )]

[10]

0.282 d2~(~) d (log7)2 + 0.041~1.

[ll]

Similar relations exist for the corresponding approximations of the relaxation spectra where the general function ~ is replaced b y the ~-function for relaxation and the plus sign before brackets is changed to minus sign. It may be concluded from the above relations that, in order to find the k th approximation of a spectrum at time log t, the experimental function must be known at time klog t. This remark indicates the range along which experimental data are needed to calculate certain approximations. It has been shown b y S c h w a r z l and S t a v e r m a n (9) that, in general, the higher the approximation the more details of the real spectrum are reproduced in right proportions. It has been also shown that the approximations of increasing order correspond to intensity functions of increasing sharpness.

Experimental Method

/ic= /x----[+ e-tIT]

[6]

we obtain the so-called first approximation for the retardation or relaxation spectra, which is rather poor yielding a maximum at the neighbourhood of the characteristic times at which retardation or relaxation processes occur but it only yields the overall shape of the spectra without details. To obtain better approximations it is necessary to construct expressions for eqs. [3] and [4] with more pronounced intensity

The k th approximation of a spectrum requires the knowledge of the corresponding experimental curve to a high degree of accuracy in order to allow a deduction of the derivatives up to the k th order at a time interval extending up to k log t. This reduces considerably the possibility of obtaining spectra with high accuracy from the values of the characteristic curves defined experimentally. An experimental composite curve yielding a characteristic function m a y allow with satisfactory accuracy to estimate the first

151

Theocarls, Higher Spectra Approximations Derived ]rom Multletep Loading

and second derivatives of the function. However, there are methods intended to measure higher derivatives directly and these methods yield these quantities with satisfactory accuracy. The first derivative of a creep function may be readily derived from a two-step loading programme with equal loads P applied at times log t = 0 and log t = 0. The same information m a y be derived from a creep test followed by a recovery test at time log t = 0 and lasting up to time log t = 2 0. The initial creep curve may be traced up to the time log t = 20 since the difference between corresponding points of the two subsequent creep curves or the creep and the recovery curve at times log t and (0 + log t) yields the value of the initial creep curve at time (0 + log t). Similarly, the first derivative of a relaxation curve may be derived from a two-step loading in relaxation with equal deformations e applied at times log t = 0 and log t = 0. Second derivatives may be derived by a similar reasoning by a three-step loading in creep or relaxation where the second derivatives are deduced from the differences of the corresponding points of the second and third creep or relaxation curves. Finally, third derivatives come out as differences between the mean values of corresponding points of second and fourth creep or relaxation curves and the corresponding value of the third creep or relaxation curve. Fig. 1 presents a creep extension compliance individual curve for a cold-setting pure epoxy polymer C-100-0-8 at a temperature T = 115 ~ traced on a log time scale, and the corresponding creep and recovery curves produced by reloading (for the creep curves) or releasing the loading (for the recovery curves) after sixteen, eight and four minutes from the initial loading time. The ordinates of these curves are directly related to ordinates of the initial individual creep curve at different times. The following simple relation may be derived from Boltzmann's superposition principle (~Pti -- ~Poi) = ~(t-o) + OPt

-

-

~o)

[12]

where ~Pt* and yJ0* express the corresponding ordinates of the ith subsequent-step creep deformations at times t and O, where 0 is the time of application of the ita loading, and ordinates without superscript correspond to the original creep curve extended all over the loading programme time. From this relation the first derivative creep or recovery curve

(i = l) comes out by starting the application of the above relation from the point considered, so t h a t the difference between the original and the first derivative curve at this point equals the difference in ordinates between the consecutive and the considered points of the original curve. Indeed, we have (E 0 E') = (F oF') and similarly (DOD1) = (P P'), (Co C1) = (NN'), (BOB1) = (M M') and (AoA1) = (L L'), where the points L, M, N, P correspond to loading times t = 17, 18, 20 and 24 minutes (fig. 1). Similarly, for the other points of the first-step curves it is valid t h a t (DoD~)= (EoE') and (COCa) = (DoD'). The same procedure may be followed to determine subsequent-loading step relaxation curves. i '

'

+El

i

i

SUBS.RECOVERYPOINTS/! 16L_~ +

l

/

SUBS9 CREEP - - - J/ I 7 e POINTS T

INITIAL CREEP I ! o INITIAL E POINTS ,' -

I

!9 N i

s/ '

12

. . . .

,

t

/i

1

~,

~,

o2

!

e ~6 LOG TIME (rain)

I 32

Fig. 1. Original individual extension creep compliance curve for a cold-setting pure epoxy polymer at T = 115 ~

as well as three creep and recoverycurves derived from subsequent loadings of equal amplitude and at times t = 4,8 and 16 minutesafter the applicationof the initial loading In this way we m a y deduce t h a t any information contained in a subsequent step creep or relaxation curve is condensed in the successive time intervals of the originM composite curve. Then, it is clear from the above argument t h a t the experimentally measured quantity must be also evaluated from the composite curve with a high degree of accuracy in order to obtain accurate spectra. The multistep creep or relaxation loading method is devised to yield the derivatives of the spectra directly and to increase therefore the resolving power of the method for their determination.

152

Kolloid-Zeitschri/t und Zeitschri/t /fir Polymere, Band 215 9 He/t 2

Testing P r o e e , l u r e In order to show the potentialities of the multistep loading method it has been applied to determine higher approximations (up to the third) of retardation and relaxation spectra for two polymeric substances, that is a high molecular weight crosslinked polymer (cold-setting pure epoxy polymer C-1000-8) and a polyurethane elastomer (Hysol Elastomer 8705). The first substance has been tested in tensile creep, while the second in tensile relaxation. For the specification of the materials and the mode of preparation of the samples see references 13 and 14. Single tensile strip specimens, cut from the plates prepared from the substances, were used for the tests. The length of the free section of the specimens was taken long enough to assure a uniform tensile field along the gauge area of each specimen. Tensile tests in creep and relaxation were carried out at various steps of temperature from the glassy to the rubbery region of each material in order to obtain the original extension compliance or relaxation modulus composite curve for each substance. The test equipment contained a simple lever loading frame with dead weights for the creep tests. An Instron tester has been used for the relaxation tests. Measurements of strains for the creep tests and stresses for the relaxation tests were taken at equal time intervals in a log scale from 1/4 to 16 minutes for each step of temperature. The stresses measured at the Instron tester were accurate to better than ~= 0.25 percent and the strains measured in the simple lever loading frame, with a mechanical-optical extensometer using a Tuclcerman gage as optical element, were accurate to ~ 10-~ over a range of 2.5 • -2 strain. Since the polymers did not exhibit any kind of flow a repeated loading technique has been applied in consecutive steps of higher temperature after complete recovery from any previous loading history. The specimens were continuously immersed in a tank containing a silicone-fluid matching in refractive index with the specimens. The oiltank has been used as a temperature controlling bath and it was equipped with electric heaters, stirrers, mercury thermometers and thermocouples and an automatic temperature controller with a sensitivity of :~ 0.1 ~ Each temperature step was readily reached and maintained for long time periods to an accuracy of 4- 0.1 ~ because of the great thermal capacity of the oil.

Each test yielding the original individual creep curve was followed by a multistep creep loading programme consisting of jumpings of load at times t = 2, 4, 8, 16 minutes of constant load amplitude and equal to the initially applied load. Care was taken not to overpass the elastic limit of the material during the last loading step by properly ehosing the initially applied load. This multistep creep curve together with the original individual curve yielded all the necessary information for tracing the whole family of creep curves for each temperature step. The experiments wei-e followed by a multistep | 3 W ~: e,,

o

I

2

I !

j

L~ o. UJ

I

....

I

o=1

"I

I| L

. . . .

_ _ CREEP STRESS CURVE - _ _ RECOV. STRESS CURVE (=)

51 4

~3

I

r Z

o

2

.

_ _ I

f

.

.

i .

t--

RELAXATION STRAIN CURVE

0

i e

: (b) : 2e 3e

j 4e

50

TIME t (rnin.) Fig. 2. Multisbep loa~Jng p r o g r a m m e ~ w i t h t i m e (a) for creep or recovery, (b) for r e l a x a t i o n

recovery programme, where the initial creep loading has been released in four steps of equal amplitude at times t = 2, 4, 8 and 16 minutes (fig. 2a). Similarly, for the relaxation experiment the initial relaxation test has been followed by four successive relaxation steps of equal amplitude of strain applied at times t = 2, 4, 8 and 16 minutes (fig. 2b). Fig. 1 shows the original individual creep curve for the pure epoxy polymer at temperature T = 115 ~ as well as the 4, 8, and 16 minutes creep and recovery curves derived from the multistep loading method. The values of the first approximation creep or recovery test may be used to extend the

Theocaris, Higher Spectra Approximations Derived/rom Multlstep Loading

153

lation of the first, second and third derivatives, of the original composite curve. In practice log t = 0.2 to 0.4 is a reasonable choice. We have decided to t a k e log t = 0.2. Smaller values of log t m a y cause large errors and fluctuations due to the limited time

original creep curve to the n e x t time interval (from 16 to 32 minutes). Fig. 3 presents the composite extension creep-compliance curve for the cold-setting pure epoxy polymer as it has been derived by shifting the individual curves of the mate32s

90

PURE EPOXY POLYMER C-100-0-8 24.0

~0

D(t)- CREEP CURVE FIRST APPR. SPECTRUM (k=l) SEC. APPR. SPECTRUM (k=2) THIRD APPR. SPECTRUM (k=3)

.

"u16.0

x

!

~- B.o

0q

0

0

4

,

i

30

o

10 8

12

LOG TIME

(rain)

~.o

16

24

Fig. 3. Composite extension creep compliance curve for a cold-setting pure epoxy polymer C-100-0-8, as well as first, second a n d t h i r d approximations of its retardation spectrum

(a)

i

t24

[

"c

(+) PURE EPOXY POLYMER

II

C-tO0-0-e

I

,, +

"~'(O)l B

.

'

.

.

" +

"]'I /'~

S E e . DERIVATIVE CURVE THIRD DERIVATIVE CURVE '

6

II I I

'~ ~

+

.

0

.

=r ~ 0

2

-0

6

8

lZ

LOG TIME (rnin)

16

20

24

Fig. 4. First, second and third derivative composite curves for a cold-setting pure epoxy polymer (C-100-0-8) rial according to the principle of reduced variables. Simultaneously the subsequent individual creep or recovery curves derived from loading or unloading jumpings at times log t = a, log t = 2 a, log t = 3 a etc. have been shifted according to the same principle and three additional composite curves have been derived, related to the original composite curve by the fact t h a t t h e y allow the calcu-

period of loadit)g during the nmltistep creep programme or unloading during the multistep recovery programme, while larger values of log t yield coarser sequences of points to define the various derivatives and therefore poor accuracy. Fig. 4 presents the first, second and third derivative curves for the epoxy polymer C-100-0-8 as t h e y have been derived from

154

Kolloid-Zeitschri/t und Zeitschri/t fi~r Polymere, Band 215 9 Heft 2

I t may be concluded from figs. 3 and 5 t h a t the accuracy of an approximation increases with increasing order of approximation. Indeed, higher approximations of spectra become sharper at the area of the unique maximum existing at the vicinity of the in-

the multistep loading procedure. I t may be derived from the rather reduced scatter of the points in higher derivatives curves t h a t this procedure yields consistent results. The values of these derivatives have been introduced into relations [9], [10] and [11] .

.

.

.

.

!

T

I ..

I 8 U

i

o E (t)-RELAX. CURVE ,*, FIRST APPROX. SPECTRUM + SEC. APPROX. SPECTRUM

.~

i

POLYURETHANE ELASTOM~:~ (HYSOL 8705 )

r

M

2 (:9 O v

,., 4

2

-~

-16

-12

-8

LOG

0

-4

TIME

(rnin)

Fig. 5. Composite extension relaxation modulus curve for a polyurethane elastomer (Hysol 8705), as well as first, second and third approximations of its relaxation spectrum

] 9

+

(+)

9

~

y

HI ~

3

o FIRST DERIV. CURVe SEC. DERIV. CURVE '+ THIRD DERIV. CURVE

J F i g . 6. F i r s t , s e c o n d , a n d t h i r d

-15

derivative

-li

LOG

0.1

~ O-O--- 0

t.

-18

0.z

POLYURETHANE ELASTOMER HYSOL 8705)

-7 TIME

I

i

-3 (min)

0

-0.t

3

c o m p o s i t e c u r v e s for a p o l y u r e t h a n e

and yielded the first, second and third approximations of the corresponding retardation spectra of the material, which are shown in fig. 2. Fig. 5 shows the extension relaxation modulus composite curve for the polyurethane elastomer Hysol 8705, as well as the three approximations of the relaxation spectra of the substance, while fig. 6 exhibits the curves of the three derivatives for this substance.

elastomer

( H y s o l 8705)

flection point of the composite curves. This shape is consistent with the behaviour of a cross-linked polymer for which the extension creep-compliance or relaxation modulus curve changes slowly with log time in both sides of this maximum.

ltlgher Approximations of Spectra from Composite Curves We have already deduced t h a t all information included in creep and recovery or

Theocaris,Higher Spectra Approximations Derived/rom Multistep Loading relaxation curves obtained by a multistep loading programme is condensed in the successive time intervals of the original composite curve. We will use this fact to derive higher approximations of relaxation spectra directly from extension relaxation composite curves for polymeric substances for which we

30

I

- - "

data published by Tobolsky and Catsi//(15), while the same composite curve for PMMA was taken from data by McLoughlin and Tobolslcy (16). Both groups of values constitute data of a high degree of accuracy for the transient extension relaxation modulus.

i

N~.201......

o z~ + •

/ Q

j:~

15

N.B.S. POLYISOBUTYLENE

I

155

7

E(t)-RELAX. CURVE FIRST APPROX. SPECTRUM SECOND APPROX. SPECTRUM THIRD APPROX. SPECTRUM

5

10

o= 5

-14

-13

-11

-9 -7 LOG TIME (hrs)

-5

-3

Fig. 7. Composite extension relaxation modulus curve for NBS Polyisobutylene, as well as first, second a n d t h i r d approximations of its relaxation spectrum (o) g") 120124

80 16

~~o

-,ok

'

I (+) ]6

i

4

'

.

~

.

!~ ~

.

.

~,

I

/

.

:

I~

'

o

t-,

......

-,,or,,--

_2oo[~_%/ -14

-u

-I

i -li

.

;

r

;

l'~

-? -s -3 LOG TIME (hrs.) Fig. 8. First, second and t h i r d derivative composite curves for NBS Polyisobutylene

have sufficiently accurate composite curves in tabular or graphical form. As such materials we can use the NBS Polyisobutylene (PIB) and the Polymethyl-methacrylate (PMMA). Both substances are uncross-linked polymers and the first is a low molecular weight, while the sample used for the second is from a high molecular weight batch. The extension relaxation modulus composite curve for P I B was taken from

-s

The composite curves of both materials have been plotted in a large semi-log time scale and smoothed from eventual errors. The values of the composite curves corresponding to 0.2 of the log time scale were evaluated. I t has been decided to calculate higher approximations of spectra at these equidistant points of each composite curve. The values of the first, second and third derivatives of each point of the composite

156

KoUoid-Zeitschri/t und Zeitschri/t /iir Polymere, Band 215 9 Heir 2

curves were calculated from the subsequent relaxation curves related to each point and corresponding to times (log t~ + 0.2), (log t~ + 3

0l

~

/k

i

~P

.

I J (

-'~

, "~ "~

0 -5

-3

~

~ ~ ,#z( ~

-,-

i

0

3

first, second and third derivatives curves for the same substances derived from relations [13] to [15]. Again, higher approximations ~-

,

P.M.M.A..

6

0 E(t)-RELAX. CURVE ... FIRST APPROX.SPECTRUM J + SEC. APPROX.SPECTRUM

,5

~; x

4 u,

1 :~

THIRD APPROX. SPECTRUM

6 w LOG TIME (hrs.)

12

"TE

- 15 - - 1~

Fig. 9. Composite extension relaxation modulus curve for Polymethylmethaerylate, as well as first, second and third approximations of its relaxation spectrum (o) (4) '7,

120

(*)

P.M.M.A.

1.2 ~12 o B " +

FIRST DERIV. CURVE SEC. DERIV. CURVE THIRD DERIV. CURVE

4

; 0.8 z,

0.4

_

o

x

0

4-

<)A

.~ -40 9-

0.8

-80

-120 .12 -160

-S

-1.2

i -3

1

LOG

TIME

(hrs)

Fig. 10. First, second and third derivative composite curves for Polymethylmethaerylate

0.4) and (log t, + 0.6). These three vahles for the point i come out to be: ~(')(log t) = [E(i) -- E(i+0.,)] ~(2)(log2t)-----[(E(i) + E(i+0.,) -- 2E(i+o.,)]

[13] [14]

of the spectra present a steeper maximum at the vicinity of the inflection point of the original composite curve of the substance, which is in compliance with the behaviour of these polymeric substances.

1 [E (1) -- 3E(i+0.~) + 3E(i+o.~)--E(i+o.e)]. ~(a) (log 3/) = ~-

Conclusions

[15] Figs. 7 and 9 present the extension relaxation modulus composite curves for P I B and PMMA, as well as the three approximations of the relaxation spectra for these substances, as they have been derived from the above data. Figs. 8 and 10 exhibit the

I t has been shown t h a t a multistep loading programme consisting of a series of creep or recovery tests of equal amplitude and shifted along the log time scale to a constant quantit y yield enough data for the accurate determination of higher approximations of the retardation spectra of a polymeric substance.

Theocaris, Higher Spectra Approximations Derived ]rom Multistep Loading

For the case of relaxation experiments the series of creep tests must be replaced by a series of relaxation tests. Moreover, it has been demonstrated that a composite curve for a substance contains in a condensed form all the above information yielded by the series of multistep creep or relaxation curves related to each point of the composite curve. Then, in the case where a composite curve has been defined to a high degree of accuracy, it yields sufficient data for the determination of higher approximations of its spectra. Examples taken from published data for various materials confirm the validity of the method. Acknowledgement The work contained in this paper has been sponsored by the NATO grant No. 245. The author is indebted to Mr. G. Samaras for making all the calculations contained in the paper. Summary A multistep loading programme for a cross-linked polymer with equidistant discontinuities jumping with constant amplitudes of stress or strain may yield higher approximations of the retardation or relaxation spectra to any degree of accuracy provided that the Boltzmann superposition principle is not violated. In the case of retardation spectra the multistep loading programme can he replaced by an initial creep loading followed by steps of recovery of equal amplitudes and time. In the case when an accurate composite curve of the material is available by application of the principle of reduced variables, this curve yields sufficientinformation for the tracing of retardation or relaxation spectra to any degree of approximation. Examples are given for the tracing of retardation and relaxation spectra up to their third approximation for an epoxy polymer and a polyurethane elastomer by using the multistep method. Three approximations of relaxation spectra for N.B.S. Polyisobutylene and Polymethylmethacrylate are also derived from their composite curves. Zusammen[assung Ein Programm, bei dam ein vernetztes Polymeres mit diskontinuierlichen Sprtingen yon gleichem zeitlichem Abstand und konstanter Amplitude in bezug auf Dehnung oder Spannung belastet wird, kann h6here N/the-

157

rungen fiir Retardations- und gelaxationsspektra mit einem ausreiehenden Grad an Genauigkeit liefern, vorausgesetzt, dab das Boltzmannsehe Superpositionsprinzip nieht verletzt wird. Im Falle der Retardationsspektren ist die Mehrstufenbelastung ersetzbar dureh eine anfiingliehe Kriechbelastung, gefolgt dureh Stufen der Entlastung mit gleiehen Amplituden und gleiehen Zeitabstgnden. In dam Fall, daft eine exakte Master-Kurve fiir das Material dureh Anwendung des Prinzips der reduzierten Variablen erlangbar ist, gewghrt diese Kurve geniigend Information ffir die Ermittlung des Retardations- oder Relaxationsspektrum mit ausreiehender Nfiherung. Beispiele warden fiir die Gewinnung yon Retardations- und Relaxationsspektren bis zu ihrer dritten N~herung fiir ein Epoxidharz und ffir ein Polvurethan-Elastomeres unter Verwendung der Mehrstufenmethode gegeben. Drei N~herungen der Relaxationsspektren fiir N.B.S., Polyisobutylen und Polymethylmethacrylat warden aus den Master-Kurven abgeleitet.

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