AFS

UTEC CE 72-145

ON THE GENERAL THEORYOF STEKLOV - AGING MATERIALS

By

J. Edmund Fitzgerald

August 1972

DDC\

For presentation at the SEP 27 IM __Jj jVIe Congres International IJU1$ b!iiiU-0 jIu!4

de Rheoicgie - Lion C

4 - 8 Septembre 1972,

Professor of Applied Mechanics and Civil Engineering

Civil Engineering DepartmentCollege of Engineering

University of UtahSalt Lake City, Utah 84112

Reproduced by

NATIONAL TECHNICALiNFORMATION SERVICE

U S Department of CommlfteSpringfield VA 22151

PXVP*4 \-

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-SAIT LAKE CITY, UTAH 84112S. REPORT TITL.E . -

ON THE. Cx.ZNAL "THORY OF STEKLOV AGING HATER-IALS - - .

.1. DESCI4IPT2VE NOTES (2ype of report and Inclfusive detstoj

Scientific InterimS. AUTHORIS) (First name, middle initial, last name)

J EDMUND FITZGERALD

a. =*; _RT OATE 78. TOTAL NO. OF PAGES |.NO, OF REFS

Aug 1972 323. CONTRACT OR GRANT NO. On. ORIGINATOR'S REPORT NUMBER(S)

AFOSR-72-2332 UTEC CE 72-145b . PROJE-CT NO.

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61102F "O - 72-1782C" 681308 _________-72- _178_2

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V3. ABSTRACT

XA broad class of materials possessing both instantaneous ,ionlinqar elasticity anddissipation in addition to fading me.mory with aging effects is 'described. The neasureof the generalized imput function, which is a multiplet in term4 of the deformationgradient, the temperature, the gradient of the temperature, as well as various chemicaaffinities, is given a semi-norm over a Banach Space. The semi-norm includes asummand which is a modification uf the Steklov Average to P " integrable Lebesguefunctions. it is assumed that the generalized response is a nonlinear function ofthe present input and a mate..ial property-history kernel determined by a Steklov-Lebesgue norm. Examples appliea to solid propellant are given.

NOW 1 A Nv 473 UNCLASSIFIEDSecurity Classification

.UNCLASSIFIEDSecu-,ty Classific.tion

14 LINK A LINK B LINKKEY WOROS

ROLL WT ROLE WT ROLE W

3TFKLOV AGING

PROPELIANXT AGING

SOLID PROPELLANT

STEKLOV-ILEBESGUE NORM

VISCOELASTI CITY

II

"ON THE GENERAL THEORY OF STEKLOV-AGING MATERIALS"

J. Edmund Fitzgerald, D.Sc.

Professor of Applied Mechanicsand Civil Engineering

University of UtahSalt Lake City, Utah U.S.A.

SUMMARY

A broad class of materials possessing both instantaneous nonlinearelasticity and dissipation in addition to fading memory with aging effect,is described. The measure of the generalized input function, A, whichis a multi-plet In F, the deformation gradient; 8, the temperature;g = gradO, as well as various chemical affinitios, Ak; is given by asemi-norm over a Banach Space. With the definition of the historyAt - At(s) - (t-s); se[O,w) an4 with the restriction of ,At to pasthistory given by Ar = AI(s) = Ar(t-s)., se(O,oo) the snmi-norm is:

t /P}°)h(°

-P 1=0 P=l r

with lk (s)d }P <-and h(s) (1)

D

and A(i)(O) the present ith time rate of change.

The second summand is a modification of the SteKlov Averaýye[Kantorovlch, 1964) to P - integrable Lebesgue functions.

It is assumed that the generalized response fQ(t) is a nonlinearfuoction(al) of the present input A(t) and a material property-historykernel determined by the Steltlov-Lebesgue norm j i (s)IS such that

S= F[I IAt(s) ISN ]A(t) (2)

Equation (1) shows that as time increases mesD increases andthat the influence of the past history on present response decreases.For a given finite duration input, ite influence decreases both thelonger in the pa6t it occurred and the older the material is. Thislatter effect is termed stecZov-aging. As the age of the materialbecomes very large the past history effects are obliterated and fromEq. (1.)

limllItt(s)ls Il NIA (i)'(o)lI hi(o) (3)

mesD - PtO /

Thus a, StekZov-aing raterial under long term aging approachesthe beh•vior of a nonlinear visoeZta8tic or M&rkoffian type materiaZ.Fk9 very small inputs, A(o) and small time rates of change of inputs,AM')(O), the material, after long term aging, becomes linearlyviscoelastic.

Examples of the theor'y as applied to solid propellants and asand-asphalt concrete are given.

-2-.

1.0 INITRODUCTION

The application 0f the linear atd nr4ninetr theories of viscoelas-

ticity has had wide application in the cheacterization and stress

analysis of solid propellants, highly solids loaded polymers, and

asphait pavement during the past tNo decades.

So long as the characterization ard verification of the selected

constitutive equation was confined to the Lisual loading or straining

conditions such as a constant strain rate, ramp relkxation, or constant

stress creep the comparison of theory with experiment was generally

satisfactory. This result was to be expected since the theory and

experiment are often a curve fitting exercise over a narrow range of

load types.

When viscoelastic theory is applied to the prediction of repeated

loads such as a sawtooth input or multiple ramp input, the predictive

results of the theory are often quite unsatisfactory. The above state-

ment applies to linear viscoelasticity as well as to the several forms

of nonlinear viscoelasticity utilizing the multiple integral approach.

Details of the previous statements are presented in [Farris and

Fitzgerald, 1970), [Fitzgerald and Farrls, 19701, a.nd Chapter XI of

[Fitzgerald and Hufferd, 1971]. A most excellent comparative review

is given in [Stafford, 1969).

Figire I demonstrates the above problem based upon some of Farris'

experiments, using a filled polyurethane propellant.

The use of a repeated sawtooth -train history is typified by the

curves of Figure 2 for a highly solids loaded polybutadiene dcrilo-

nitrile propellant [Bennett, 1971).

-3-

Previous publications by the author and his co-workers employed

the use of homogeneous functions of the strain history made specific

through Lebesgue norms. The Lebesgue norm, lIllLp is essentially the

weighted P-summable integral of the infinitesimal strain history, e

wherein

IkIILp -- S lkea hp(s)ds (4)

The term hp(s) is a positive decreasing funiction of s for fading

memory theories where P is taken as unity. Certain boundedness restric-

tions apply to hp(S).

For P ÷ =, the Chebyshev norm results wherein

IeIL = ess. sup lel (5)

Using a rather simple expression [Farrls, 1970] for stress, T,

versus strain e, wherein

r4t = 3415 .S 12,(t) (6)

the results of Figure 3 were obtained [Farrls, 1970].

It should be noted that the general form of Faq. (6), which is the

L.0 norm divided by the Lp norm to the'nth power (with P > 1), when

multiplied by e1y yields the following results:

-4-

*For a constant strain rate test, ell Rt

I1e•1111 = Rt; i lelliip = Rt - -tV

(p+1)flp= A(P+)n/P Rt'np

fl/p t-n/p

or Tll A(P+I) ellt (7)

-For a step relaxation test e11 = eo

1/pI1lelll = eo0 Iellij p - o t

T11 = A.0 tJnp (8)

The last result predicts an inverse power law for the relaxation

modulus-, A tfn/ so that the value of A and the slope, n/P, can be

determined from a step relaxation test since G(t) relax. = A tn

The constant strain rate test yields the secant modulus

' G(t) secant = (P+I)niP G(t) relax. (9)

from which P+l, hence P and n may be determined.

The predictions of Figure 3 were made using data obtained as described.

A comparison of the above norm expression with linear viscoelasticity for

a polyurethane propellant is given in Figure4EFarris, 1970].

A more general expansion of Farris' earlier exprassions has been

given by [Vakily and Fitzgerald, 1972] wherein the stress function

-5'.

involves a sum of Lebesgue norm ratios and the naturally occurring

inverse, power law 'kernel in- the viscoelastic integral as follows:

Ni N /I ifH I I '1T11t)' ~ Aly T 2) J(t--tY'11'i l l(Cr)dT (10)

i=o J=O -,

The first term of the above expansion, i j = 0 with qo

yields

S[lleiitHl]no l~ )(lT I~t) = Ao I I r~l M It CO nop! 1I

which is the previous Farris expression, Eq. (6).

Including the current value of the strain, jellJ and simplifying

a three term expansion of Eq. (10) results in

It A ,I,]° 11(t) + A2 II1.

(t-T) ýll(T)d.

where we have defi ned IHello = I i.Various other specific forms of the above Lebesgue noris may be

formulated.

Applying Eq. (12) to a series of compression tests on a sand-

asphalt mixture produced the following expression [Vakily and Fitzgerald,

1972] where the constants were evaluated from constant strain rate

"step" relaxatlon tests:

C6I-

-6-

I€

VT- - --

T(t) = 310 4(t) + [i (t-) (,)d (13)

The moduli for a strain of 0.37% and 0.5% is shown in Figs. 5

and 6.

The constant strain rate tests for two different rates are given

• ~in Fig.. ?

The ramp relaxation test results are given in Fig. 8.

Using the expression, Eq..(13), derived from the above data,

predictions shown as open circles and experiments, shown as solid lines,

were conducted for

(1) an interrupted ramo strain input at two different

strain rates, shown in Figs. 9 and 10. A comparison

of the above predictions with those of linear visco-

elasticity is given in Fig. 11

(2) a single constant strain rate "sawtooth" strain,

shown in Fig. 12

(3) a repeated constant strain rate test cycled between

a set maximum strain and zero load, Fig. 13.

Again, it is clear that the sand-asphalt material exhibits a degree

of permanent memory similar to the filled polyurethane propellant and

the PBAMI propellant. Figures 5 through 15 are from (Vakily and Fitz-

gerald, 1972].

An expression for the stress such as

11 T(t) f[U(t), IIU(t)HL (

7-7

-- -_ -, _ L •

produces a stress-strain relation with no relaxation but with the Mullins teffect predominant [Mulllns, 1947].

A mathematical justification for the above norms, and their restric-

tions, is given In [Hufferd and Fitzgerald, 1972] including thermodynamic

implications.

In general, one may express.the Cauchy stress tensor, T, as a

function of the norms, for example

T = frU(t), IIUuIL,. 1 HUIIL (15)

where V is the positive square root of the right Cauchy-Green (finite

strain) tansor. The resultW.,, polynomial function following the well

knrwn Rivlin-Spencer exp,ý.,,os, for initially Isotropic materials will

produce an expressi*1n i to the second power in the U and its norms

with coefficients that are polynomials, or preferably here, rational

fractions in the joint invariants

C2

trU, trU2, - - , trIIUIILp, etc.

Again,.a three-dimensional expression similar to Farris' is derived

for a single term, homogeneous qorm for T, namely

T(t) Lru0 t )(6

Applicatiun of the above tensorial equation to a uniaxial test

produce resilts similar to those previously shown.

-8-

'1

2,.0 AGING EFFECTS AND THE STEKLOV AVERAGE IThe Lebesgue norms previously described are such that they produce

the integral of the strain history on a P - summable basis. The use. of a

non-unit weighting function, hp(s), will provide for fading .-,mory effects.

The Lebesgue norms may have a basis in microscopic theory as presented by

Farris in his doctoral thesis.

Nevertheless, the norms as used herein are siwply continuum postulates

whose use is justified by the results shown here%- and in the various quoted

references.

'With the integrals as used, however, no aging effects are included.

Consider now a class of materials whose response is governed by certain

weighted averages of the past strain history as well as by the present value

of the strain and its several time derivatives.

A norm on such a space may be constructed as follows for P gener-alized

input A

!A ()ls IjN* (o)ij(o) + A (-sjh(s)- r ho (s)dP 1=o P=i [el l -

with h (s)ds/ < and hP(s>0

D

and AMI (o) the present i-th rates of change. (17)

The above is actually a semi-norm unless measU is finit(,

Considering especially the second summand in Eq. (17), we shall call

it. a Stekiov Alverae since it is a generalization of the Steklov-Lebesgue

I average given ir n[Kantorovich and Akilov, 1964).

A slight variation in the above, which-will be termed the modified S';eklov

Average and which satisfies all the requirements of a norm (or semi-norm) is

given by 1/P

SJAI, p (+kl t(s) hp(s)dj (18)

with k>O

Dropping the fading memory factor hp(s), for simpllcity, results then in

JhAI I-1 t JAI ds] (19)

where we have taken measD = t for the usual rather smooth physical inrputs. A

Looking at the Lebesgue norm ratios of Eq. (10) or Eq. (11) for example

and the definitions Eq.() and Eq. (19) yields the following relatiot between

the Lp norms and the S ne0,Is

-(+kt) (20)I^AI3 [IJ LI IIILpJ

and for Q +,

rhl hl-.I' rill n ° /LSe L/P

-10-

l Llpi IA ,P

It is readily shown that the ratio

T>I VQ>P P>l (22)

since the dominance of LQ over Lp is well known when Q>P [Kantorovich and

Akilov, 1964] and for k>O, the multiplier is also >1

Defining TIL as the stress obtained from Lebesgus norms as in the previous

section 1.0, the stress obtained by substituting modified Steklov norms isTS where

TS = TL (1 + kt)0- (23)

or with Q -

~S L n/P (4T = TL(I + kt) (24)

Consider for example a material governed by a simple permanent memory

norm relation such as Eq. (6)

ST1 1 AL= e.11 1 (t) (25)

For a step relaxation test with strain magnitude co (25) yields

T11L Ac t-n/P(26)

Applying instead the ratio of the modified Steklov norms Sp producess t-n/ ni

TillT1 z A .o (1 + kt*) (27)-11-

For a material which is strained immediately upon being formed, t = t

Otherwise however, t. is the time relative to the beginning of the step strain

whereas t is the actual time from the initidl creation of the material to

the present. The difference in behavlor of the L and S norms is shown

in Fig. 14. A step load-unload input is used with the Ll, 5, and L. = S.o =

e max. curves plotted for an example. After the load occurs, L1 is constant

as is L.. However, the Sl norm reduces with time since it is averaging over

the life of the specimen.

The relation between the L and Sp norms is, from (4) and (19)

IIAIIp = (AIp (1 + kt)flIP (28)P L P

It will also be noticed from Eq. (24).that the exponent of the "aging"

term l+kt* is equal to the positive numerical slope of the inverse power

law relaxation modulus, n/P.

Consider a typical solid propellant with n/P - 0.2 , then

TIl =A C t-0.2 (1 + kt*)0" (29)

If the relaxation modulus increases by 20% over a one-year ambient aging

then k3 x lO 8 lsec

A typical CTPB propellant increases its modulus by 20% in 100 days.

With a slope n/P = 0.2 , hence, the value of k is 15 x 108 sec -1.

It is also to be noted that for large values of kt*, the aging is

described by a straight line on a log-log scale. Further for n/P = 0, no

relaxation and no strain rate effects, the aging is nonexistent.

-12-

Because of the small values of k. the use of the L rather than the S

norms for short time loads or newly formed materials is fully equtivalent.

.urti,,, for short loading times relative to the)"if of the material,

(l+kt*)n/P is essentially a constant and Steklov. aging reduces to the s'-

called homothetic aging process [DeArriaga, 1969J.

For finite strain, the essential results also. hold. Consider an incom-

pressible material subject to a simple step elongatit5nwith a stretch ratio

of X where X is the= stretched length divided by thehitial length. Using

a simple constitutive equation of the form of (1() but with modified Steklov

norms instead of Lebesgue norms as in (21) results in

T(t) A ALX+2I t-n/P (I + kt*)n/P (X- 1) £ (30)

For a material stretched when newly formed ahd then held at the stretch

ratio, X, with t = t

.["X+ "n W ln/T(t) A±.j( 1 + kQn/P (X - 1) (31)

For long times. then as t -o

T(t) ' 1n kn/P (X - 1) (32)

One could obviously extend the expansion to higher order terms in U,

buLý the essential noint to be made is that the material does not fully relax

as t + . For the typical material previously mentioned, if the relaxation

modulus at one second were 1000 psi, the final modulus would reduce to 60 psi.

In addition, thare can be considered an elastic component with no loss of

generality. -13-

3.0 CONCLUSIONS

It has been proposed that the characterization of relaxing, rate sensitive

materials be based upon certain weighted averages of the past history which

are semi-norms called modified Steklov averages.

The rationaleis based upon previously mentioned guidance from molecular

theory but is primarily based upon the postulate that the present response of

a material is governesd by its present deformation gradient and a selected

weighted P-sunmable Steklov average (17).

Since in engineering practice, one seldom knows the detailed past of a

structures thermal and deformation history but usually knows the average and

the maximums, no serious loss of applicability should result.

The justification for the use of the proposed norms is based upon the

several examples given herein where the predictions using norms is shown to

be superior in accuracy to the predictions based upon linear viscoelasticity.

The use of the ' or Lo maximum value norms implies that the material also is

sensitive to the maximum strain value it has ever been subjected to. The

possible extension of the concept to viscoplastic materials is under study.

Farris has shown that for a selected class of solid propellants the

use of the conventional time-temperature superposition integral is valid.

The present author suggests that the tiv. temperature shift integral

t

"f t -T."' (33),=0

for re6•.ed time g with the temperature shift factor aT be used in the modified

Steklov norm for aging effects. It is, of course, equally possible to use an

absolute reaction rate correction for the kt term in (19).

-14-

The use of a rather simple homogeneous fraction involving only the max.

norm and the P-norm of the modified Steklov average (25) produces three

specific results for infinitesmal deformations

.the relaxation modulus is described by an inverse power law, t-n/P

the constant strain r-ate curves are described by a power law whose

exponent is unity plus the relaxation modulus exponent, t 1-n/P

,the relaxation and secant moduli are subject to an age hardenin9

process n/P described by the factor (l+kt)

.for very long term stretching and aging, the material stress response

is nonvanishing as shown in (32).

It has been also shown that for short aging times the form of the

equations reduces to ratios of Lebesgue norms only. For short term loading

relative to the age of the material, a homothetic aging process is produced.

There does not exist a unique inverse for strain as a function of stress,

i.e., a general creep inverse. Indeed, an inverse does not generally exist

unless a nonconstant weighting function, hp(s) is used. Even then, there

results a nonlinear integral equation of the type

ý.'M ofl (t)l hp(S)ds ; O(s)-xAP(s) (34)

with the exponent m a rational fraction whose value will generally be near unity.

Since the present value of stress is governed by certain averages of

the past strain history, uniqueness in the inverse is not to be expected.

It is therefore suggested that a creep law be formulated in the same fashion

as (25), for example

e(t) = f[JITIlJ: , , T(t)] (35)

-15-

made specific as-

e(t) =B£ITs IS. T(t) (36)

which for a constant stress, To$ results in

e(t) = B T0 tm/P (1 + kt)m/P . (37)

The calculation time necessary to use the various norm equations given

herein is much shorter for general inputs than the time needed for visco-

elasticity. This computer time saving results from the fact that the 1.orms

are only a number at the present time whose value changes by the modified

average at each step. If, however, nonconstant weighting functions, hp(s),

are used, no computational advantage results.

-16-

4.0 ACKNOWLEDGEMENTS

The author wishes to acknowledge the past and continuing efforts of

his colleagues R. Farris, W. Hufferd, J. Vakily, and M. Quinlan in pursuing

and contributing to the ideas herein presented.

This work was sponsored in part by the Office of Naval Research under

Contract No. 00014-67-A..0325-0001 and the Air Force Office of Scientific

Research under Contract No. 72-2332.

-17-

REFERENCES

1. Bennett, J.: Privats Cowtunicatir~n, Thickol Chemical Co., 1971.

2. DeArriaoa, F.. o"mothetic Aging Viscr/elastcH Ma.teeials," Int'l. Conf.on Strocture and Mdaterials, Southamptonv 1969,

3. Farris, R. J.: Hwomogeneous Constitutive Equations for Materials witnPermanent Mewry'," UTEC TH 70-083, PROJECT THEMIS Report AFCSR70-1962 TR, University of Utah, June 1970t

4. Farris. -R. J; and Fltzqetrald, J, E.: "Deficiencies of ViscoelasticTheories as Applied to Solid Propellants," Bulletin JANNAF fet-hani-cal Behavior Wrkino Group, 8th eetinog 19-, CPI_ Publication No.

5. Fitzgerald, J. E. and Farris, R. J,: "Characterization and AnalysisMethods for Nonlinear Viscoelastic Materials," PROJECT THEMISReport, UTEC TH 70-204., University of Utah, November 1970.

6. Fitzgerald, J. E. and Hufferd, W. L.,: "Enqlneering Structural Analysisof Solid Propellants," CPIA Publication No. 214, The Johns HopkinsUniversity, July 1971.

7. Hufferd, W. L. and Flt;gerald, J. E.: "A Thermodynamic Theory of Materialswith Permanent Memory," Proc. 1971 Int'l. Conf. on Mechanical Behaviorof Materials,_ Vol. HI_,T Rp--p. •.-kO ~c.F(r-T- c-• pn 92•-

8. Kantorovicho L. V. and Akiloy, G. P..: "Fun.tional Analysis in NormedSpaces," Pergamon Press, New York, 1964 (translated from the 1959Russian original).

9. Mullins, L. J.: "Effect of Stretching on the Properties of Rubber,"J. Rubber Res., Vol. 16, pp. 275-289, 194;'.

10. Stafford, R. 0.: "On Mathematical Forms for the Material Functions inNonlinear Viscoelasticity," J, Mech. Pbhs. Solidst Vol. 17, pp. 339-358,1969,

11. Vakili, J. and Fitzgerald, J. E.: "Nonlinear Characterization of SandAsphalt Concrete by Means of Permanent Memory Norms," UTEC CE 72-079,Univ. Utah, 1972 (submitted to SESA for publication).

-18-

Linear VlMcoelastic Predicted

.

/

• // /

4J4 /

90

,10

• * Time

e co

Strain, e

Figure 1 - STRESS FOR AN INTERRUPTED CONSTANTSTRAIN RATE TEST

/6

'I

.... .. ..... . .. .... .

- -- -~ ~R-

r

Cycle

-asWý TimFigue 2- UNAXIL COSTAT RAE CCLIN OFPBA

PRPLLN

c)I

fi

'--ý

'

__W_.

~ 1*~' ~ _ _'mill

1-4

t)

4JJ4-* I 0

I w2w

40 (t0 wo~

04J

U-

4.30

0) *

UY)

co K-Is Im--4si

zW-••" • ,,' , °'"> • Y•- -- -~ •- •. . -"-• •° - " 7 - -. "- *•% i

L i ._ _ _ _ _ _ _ _ __._

16 0 _ -- ' . . ,• -

{: " ; I :.037 rainexperimental /i/.

120 experimental constant strain rate /---------- predicted if linea.r

/

// /-/ I /

/ I /- A/ V

/I/

•*,.1 u• 40 / / /"

• Hi e11(t) .

I--~

05 time, t,minutes0 '.t2 -- ,4 . . . .h.08 .)o

Strain, e11,

Figure 4 - STRESS OUTPUT FOR INTERRUPTED CONSTANT STRAIN RATE TEST

- - -~ C.,. -- . ,,. . -f.-,

LU

CDI-

M-

I-

L/J

CL LO)

LA.

(19d)09/(1 4)I0tsminsoI)

I.I00

9 ~LI

L I- -L -

dis ,jlvm'0'

O0 ,,,

• ' -..-

()I--

0 I 0 is

0- o: 0

0w000

CIO.

_ , i

0

a (csm c;Ia '$S~uz

C 0C

4- w

Iii i

010

(1d (4S SS)1

LUJ

0 0

L- 00 0 -L

00

M 06

U.U

LU.

co

0 00

(Id)(4S'S~W I

fld

* w

"ILL

0 0 i ORC!

N Id (4S'A31

-- .l- - - 0

-------------------

IOD ci~:20 OR

4- CL

ci V

MII1"0

' 0

U0 0 0 0

(lad) (4)S 'SS38.WS

f -

0 0 C%4

CCA,

O~ccmcoo I-

p II I II I,, Li

0 i0

4O-c0

4

o0

ow

8) z( jfi0 = 0

0 L00

0 C)0

(n -

Ito-

0 it) 0 0t

-t (Isd) (4)S ISS381S

"-_,' _ _ _ __ _ _ _

0 Calculated Points

30 - Experimental Curves

25

-** 20

cliLw 15---" UT,

TIME, t (minutes)

SI0

5

00 0.002 0.004 0.006 0.008 0010

STRAIN, e81(t)

Figure 12 - COMPARISON OF CALCULATED AND OBSERVED STRESS-STRAIN

BEHAVIOR OF SAND-ASPHALT

30

CC)

0))0L

tDV

a-J

- U)co

b-4

CLi

CD )

r-4

0V w

0 0m4-4

E 00In C..)LI.

0 0

9 tO 0

(led) (4)S 'SS38.IS

-tt

.- 0040

Tim

z -!

ho /

F e - E N/ S -

Figure 14 - TIME VS. NORMS

I-!