NASA Contractor Report 3772

The Nonlinear Viscoelastic Response of Resin Matrix Composite Laminates

C. Hiel, A. H. Cardon, and H. F. Brinson Virginia PoZytechnic Institute and State University B Zucksbu rg, Virginia

Prepared for Ames Research Center under Grant NCC2-7 1

National Aeronautics and Space Administration

Scientific and Technical Information Branch

1984

https://ntrs.nasa.gov/search.jsp?R=19840018682 2020-07-25T19:17:01+00:00Z

ACKNOMLEDGEMENTS

The au tho rs a r e indebted t o the c o n t r i b u t i o n s o f many i n d i -

v i d u a l s who have helped make t h i s work p o s s i b l e . Specia l acknowledge-

ment i s a p p r o p r i a t e f o r t h e f i n a n c i a l support of t h e Na t iona l Aero-

n a u t i c s and Space A d m i n i s t r a t i o n through Grant NASA-NCC 2-71 and t h e

authors a r e deeply g r a t e f u l t o D r . H. G. Nelson o f NASA-Ames f o r t h i s

suppor t and f o r h i s many h e l p f u l d iscuss ions.

F i n a n c i a l suppor t p rov ided by The Free U n i v e r s i t y of Brussels ,

VUB and t h e Be lg ian Na t iona l Science Foundation (F.K.F.O.) i s a l s o

g r e a t l y app rec ia ted and hereby acknowledged.

The ass i s tance o f Andrea B e r t o l o t t i , K ie ra Sword, David F i t z -

g e r a l d and Cather ine Br inson, t oge the r w i t h t h e f a s t ana e x c e l l e n t

t y p i n g of Mrs. Peggy Epper ly, proved t o be i n v a l u a b l e throughout t h i s

work.

iii

TABLE OF CONTENTS

1 . D I S C U S S I O N OF S I G N I F I C A N C E AND SCOPE OF THE UNDERTAKEN STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 1.2 Previous E f f o r t . . . . . . . . . . . . . . . . . . . 1.3 Object ives and O u t l i n e o f t he Present Study . . . . .

2 . INELASTIC TIrlE-DEPEllDENT FIATERIAL B E H A V I O R . . . . . . . . 2.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . 2.2 Representat ions o f C o n s t i t u t i v e Equations i n

. . . . . . . . . . . . . . . . Nonl i near V i scoel a s t i c i t y

3 . SINGLE INTEGRAL REPRESENTATION BASED ON IRREVERSIBLE THERMODYNAMI CS . . . . . . . . . . . . . . . . . . . . . . 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 3.2 . Gibbs Free Energy Formulat ion o f t h e Schapery

Equations . . . . . . . . . . . . . . . . . . . . . 4 . DELAYED FAILURE . . . . . . . . . . . . . . . . . . . . . . .

4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 4.2 Reiner-Weissenberg Formulat ion and I t s Consequences .

5 . R E S I N CHARACTERIZATION: EXPERIMENTAL RESULTS AND DISCUSS I ON . . . . . . . . . . . . . . . . . . . . . . . . 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 5.2 General Resin I n f o r m a t i o n . . . . . . . . . . . . . . 5.3 Experimental Results Obtained on 934 Resin . . . . . .

6 . EXPERIMENTAL RESULTS OBTAINED ON T300/934 Composite . . . . 6.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 6.2 Resul ts and Discussion on [90Ias . . . . . . . . . . . 6 .3 Resul ts and Discuss ion on . . . . . . . . . . .

7 . EXPERIMENTAL RESULTS AND DISCUSSION ON DELAYED FAILURE AND LAMINATE RESPONSE . . . . . . . . . . . . . . . . . . . . . 7.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . 7.2 Delayed F a i l u r e . . . . . . . . . . . . . . . . . . . 7.3 Creep Response on [90/ i45/90]2s Compared w i t h Creep

Response o f [90]8s and [?45]4s a t 320 deg F . . . .

Page

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6

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8

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23

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53 53 65

78

78 81 91

112

112 112

118

i v

Page

7.4 Numerical P r e d i c t i o n of [90/+45/90]2, Laminate Behavior . . . . . . . . . . . . . . . . . . . . . . . 122

8. SUMMARY AND RECOMMENDATIONS FOR FURTHER STUDY . . . . . . . 126

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 130

APPENDIX A . . . , . . . . . . . . . . . . . . . . . . . . 139

APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . 142

APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . 143

V

Chapter 1

DISCUSSION OF SIGNIFICANCE AND SCOPE

OF THE UNDERTAKEN STUDY

1 .1 Introduction

Research and development i n advanced composites techno1 ogy and

design procedures has advanced t o a p o i n t where these materials can

now meet the challenge of highly demanding aerospace, automobile and

other s t ructural a p p l icat ions.

weight savings, component integration and a f ive-fold maintenance re-

duction as compared t o metal design have been demonstrated (Forsch

[ l ] ) and contribute t o the overall picture of an a t t r ac t ive marketable

product.

Performance payoffs i n the form of

However, resin matrix composites have moduli and strength

properties which are viscoelast ic or time dependent.

i s a need fo r methods by w h i c h short term t e s t resu l t s can be used t o

predict long term re su l t s .

As a resu l t there

The current e f fo r t i s par t of a continuing cooperative NASA-Ames - VPI&SU research program directed towards the development o f an

accelerated strength and s t i f fnes s characterization procedure which

generates the design da ta and the confidence leve ls , needed t o insure

s t ructural in tegr i ty f o r any projected l i fe t ime. This process i s

cyeatly compl icated by a number o f environment exposure variables

1

such as temperature, moisture*, UV and space radiat ion, high vacuum

and many others. An exampl’e of a typical thermal prof i le fo r an a i r

carr ier given by Ripley [2] i s given in Fig. 1 . 1 .

h a n d l e such variables by overdesign will penalize the economical

features mentioned above as a knockdown fac tor w i l l seriously l imi t

the full weight reduction potential of advanced composi t e s .

Any attempt t o

1 . 2 Previous Efforts

An accelerated characterization, which i s schematically given in

F i g . 1 . 2 , has been developed and documented by Brinson e t a1 . [4-131.

A f a i r amount of experimental s u p p o r t h a s been accumulated, based on

elements of l inear v i scoe las t ic i ty by Yeow [5 ] .

e la s t i c i ty came into t h i s picture when Gr i f f i th [7] potent ia l ly ident i -

f i e d t h i s behavior as the reason fo r a serious underprediction of

creep rupture data for [90/k60/90]2s laminates.

based on a l i nea r incremental lamination theory developed by Yeow [6].

As a resu l t , a study of nonlinear viscoelast ic characterization pro-

cedures was performed by Gr i f f i th [7] which led towards a valid

graphical Time-Temperature-Stress Superposition (TTSSP) principle for

the generation of modulus master curves.

Nonlinear visco-

The prediction was

Dillard [9] developed a non-linear viscoelast ic incremental

lamination theory and found reasonable correlation between his predic-

t i o n s and experiments. Because of the limited experimental d a t a

* I t has been shown tha t moisture pickup i s generally reversible . When comb1 ned wi t h thermal spikes, however, thi s phenomenon can create i r revers ible darnage as shown by Adamson [31.

2

/- /'

f

0 ro

0 lo

i >

0, 0

i I a v)

3

V

.- L

I

4 I

0 0 O-o-fco P N N

3 o aJn(0JadUal

Q .- a L W - L L 0 0 L - a 0 - C 0 v, L Q, Q 3 m 0

0 L

'c

W - . - Y-

E) a z L W L I- - 0 u Q )r

. -

3

Short Term Lamina ( 0"). ( IO" ,,(90°,

Constant Strain Rate Tests 0, e , T, y ( t , T, M ,... 1

'

lr Short Term Lamina

(IO"),( 90") Creep Compliance Testing

S ( t , 6, T, M, . . . I

TTSSP, Findley, Schapery ,

t

Long Term Lamina Compliance Master

Curve and Analyt ical Consti tu t ive Model

Short Term Lamina ( I 0" 1 , (90" 1

Delayed Failures Testing t , ( d , T, M,...)

Modified Tsai - H i I I -

Zhurkov -Crochet,

v Long Term Lamina

Strength Master Curve and Analyt ical Delayed

Failure Model I I

Incremental Lamination Theory, Finite Element

I

General Laminate Predict ions

Fig. I . 2 Accelerated Characterization Method for Laminated Composite Materials.

4

a v a i l a b l e f rom t h e work o f G r i f f i t h , D i l l a r d used a semiempir ica l non-

l i n e a r v i s c o e l a s t i c model due t o F ind ley t o rep resen t lamina

compl i ances .

1.3 Ob jec t i ves and O u t l i n e of t h e Present Study

The o b j e c t i v e of t he c u r r e n t study was t o c a s t t h e g raph ica l

TTSSP method i n t o an a n a l y t i c a l form.

a n a l y t i c a l framework developed by Schapery i n a s e r i e s o f p u b l i c a t i o n s

c o u l d be adapted f o r t h i s purpose.

I t was b e l i e v e d t h a t t h e

P, l i t e r a t u r e rev iew on n o n l i n e a r

v i s c o e l a s t i c c o n s t i t u t i v e e

thermodynamic bas i s and t h e

s u b j e c t s o f Chapter 3.

An e f f o r t t o r e l a t e v

u a t i o n s i s g i ven i n Chapter 2 , w h i l e t h e

d e r i v a t i o n o f t h e Schapery model a r e t h e

s c o e l a s t i c de fo rma t ion and f a i l u r e on an

energy bas i s i s developed i n Chapter 4.

f e r o f any modulus master curve i n t o a s t r e n g t h master curve as

d e p i c t e d i n F i g . 1.2.

This a l l ows an immediate t r a n s -

Chapter 5 concentrates s o l e l y on 934 neat r e s i n p r o p e r t i e s and

on t h e d i scuss ion o f ob ta ined exper imental r e s u l t s t h e r e t o .

Master curves f o r t ransve rse and shear p r o p e r t i e s of T300/934

as ob ta ined through t h e Schapery ana lys i s a r e discussed i n Chapter 6 .

Resul ts on creep r u p t u r e f o r [90IGs and [6OIas lamina, based on

the energy c r i t e r i o n (Chapter 4 ) , together w i t h a comparison o f an

incrementa l l a m i n a t i o n theo ry p r e d i c t i o n wi th exper imental r e s u l t s on

[ 9 0 / 4 5 / 9 0 ] 2 s a r e d i scussed i n Chapter 7 .

A general d i scuss ion and conclusions a re g i ven i n Chapter 8 .

5

Chapter 2

IflELASTIC TIiI;E-DEPEi.jDE:IT :IATERIAL BEHA’JIGR

2 .1 Introduction

The purpose of t h i s chapter i s t o produce a br ief l i t e r a t u r e

overview o n research i n nonlinear v i scoe la s t i c i ty . The majority of

publications explain how t o deal w i t h Tagnitude nonl inear i t ies , i . e .

they account f o r the s t r e s s degendence observed on creep coRp1iance

versus time as i l l u s t r a t e d i n F i g . 2 . 1 .

A much smaller number of a r t i c l e s a l so account for internode

nonlinearity (nonlinear interaction between d i f fe ren t cocDonents o f the

s t r e s s tensor) a n d a lmst i n v a r i a b l y use a s ingle mechanical i n -

variant, namely the octahedral s t r e s s . I n t h i s resDect i t should be

mentioned t h a t d a t a generated by Cole a n d Pipes [14] shown i n F i g . 2 . 2

indicate the extent o f s t r e s s interaction i n off-axi s unidirectional

boron-epoxy coupons. These experimental r e su l t s show t h a t inel a s t i c

behavior i s not a monotonic function of f i b e r or ien ta t ion . S i m i l a r

behavior has been observed fo r graphite-oolyimide off-axis coupons by

Pindera and Herakovich [15] . They indicate t h a t the inclusion of

residual thermal s t r e s ses and axial s t r e s s interaction a t the micro-

mechanics level are necessary i n order t o predict the correct trends

i n the shear response of o f f -ax is specimens.

An almost never mentioned type of nonlinearity i s interaction

nonlinearity, i . e . nonlinear interaction between events ( i . e . l o a d

appl ication or re1 ease) occurri n q a t d i f fe ren t times.

6

u) u) Q)

v)

L c

Fig

CI .- u) Y Y

u) u)

E c 0

0 Q) z v,

L

Fig.

rlortic I l ineor

0

' 2 ' ' 1

Strain

2.1 Stress- Strain Behaviour o f Elastic and Viscoelastic Materials after two Values of Elapsed Time , t .

7

Shear Strain 2.2 Extent of Stress interaction in Off-axis

Unidirectional Boron- epoxy Coupons - Cole and Pipes [14], 1974.

2 .2 Representations of Constitutive Equations i n Nonlinear

Vi scoel a s t i ci t y

2.2.1 Mu 1 t i p l e Integral Repres enta t i on s

The mu1 t i p1 e integral representation is a very appeal i ng

theoretical concept, since i t i s not limited to a par t icu lar material

o r class of materials. The Volterra-Frechet expansion for a one-

dimensional case i s writ ten a s :

t 6 = 1 D , ( t - T ~ ) ~ c Y ( T ~ ) + i' f D 2 ( t - r l , t - ~ ~ ) d u ( r ~ ) d a ( ~ ~ )

-m -OD -m

+ ... (2 .1 1

The f i r s t integral of Eq. ( 2 . 1 ) describes the l inear viscoelast ic

behavior, defined by the Boltzmann theory and the second and higher

order integrals are representations of both magnitude nonl ineari t y and

interaction nonlinearity, the l a t t e r implying an interact ion e f f ec t

between, e .g . , the s t r e s s increments a t times T~ and T ~ .

If a force i s suddenly applied to a specimen, producing a constant

s t r e s s u , i t follows from 2.1 t h a t

e = D , ( t ) u + D 2 ( t , t ) 0 2 + D 3 ( t , t , t ) u 3 + ... ( 2 .a The functions Dk(t,t,. . . , t ) can t h u s be determined from creep t e s t s .

The number of s t r e s s levels must be equal t o the number of kerne

Gradowczyk [16] showed, however, t ha t the system of 1 i near

t ions w h i c h serves t o determine the unknown functions i s i l l -

conditioned and the procedure of f i n d i n g the kernels D k i s uns

Being unaware o f Gradowczyk's work, t h i s author attempted to use

S Jk.

equa-

able.

the

8

m u l t i p l e i n t e g r a l r e p r e s e n t a t i o n f o r P.V.C. and had t o g i v e up the

at tempts due t o t h e extreme i n s t a b i l i t y o f t h e r e s u l t s ob ta ined [19].

The ex tens ion of t h e Vol ter ra-Frechet expansion t o t h e 3-

dimensional case has been g i ven by Green, R i v l i n and co-workers [17].

Even i f we l i m i t t h e i r expression t o t r i p l e i n t e g r a l s (smal l non-

1 i n e a r i t i e s ) twe lve kernel s s t i 11 must be determined by experiment.

An a1 t e r n a t i ve o f t h e Green-Ri v l i n approach has been formulated

by P i p k i n and Ropers [18] and i s o f t e n r e f e r r e d t o as n o n l i n e a r super-

p o s i t i o n theo ry (NLST) which can be w r i t t e n as

t t t s ( t ) = [ Dl(t - T,a(T));(T)dT +I D 2 ( t - T ~ , C T ( T ~ ) , ~

J - -00 -w

(2 .3 )

I t i s c l e a r t h a t eq. 2.3 i s n o n l i n e a r even on the f i r s t approximat ion.

Thus, t h e a d d i t i v i t y o f incrementa l s t ress e f f e c t s i n t h e Boltzmann

s u p e r p o s i t i o n sense i s preserved.

K inder and S t e r n s t e i n [ Z O ] showed t h a t d i f f e r e n c e i n s t r a i n s pre-

d i c t e d by t h e two t h e o r i e s i s i n t h e t r a n s i e n t p o r t i o n i f the m a t e r i a l

e x h i b i t s f a d i n g memory. Thus,in a s i n g l e creep t e s t , t h e equat ion 2.3

reduces t o

r t e ( t ) = j D,(t - T ,Z (T ) ) ; ( : )~T (2.4)

J -m

which i s t he most general rep resen ta t i on o f n o n l i n e a r creep p rov ided

t h e t r a n s i e n t response from t h e s tep l oad ings has decayed. However,

whether mu1 t i p l e 1 oadi ng and/or unloading programs r e q u i r e a d d i t i o n a l

i n t e g r a l s t o descr ibe t ime i n t e r a c t i o n s o f t h e l o a d i n g h i s t o r y remains

9

an open question.

2 . 2 . 2 Single Integral Representations

i ) Modifications of the Boltzman Theory of Linear Heredity

Two schools of thought have been ident i f ied i n the l i t e r a t u r e .

The f i r s t one s t a t e s t h a t i t i s , i n p r inc ip le , possible t o map a s e t

of nonlinear isochronous s t r e s s - s t r a i n curves in to a l i nea r s e t , ?ro-

v i d e d we nonlinearize the s t r e s s a n d s t r a i n measares as

( 2 . 5 )

Here the equation i s l i nea r b u t only i n the nonlinear s t r e s s and s t r a i n

measures and 2 . This form has been used by Koltunov [ 2 1 ] . A s i m p l i -

f ied form o f E q . 2 . 5 have been proposed by Leadenan [ 2 2 ]

a n d by Rabotnov [23]

( 2 . 7 )

The second school allows f o r a s t r e s s or pressure-deformable time

measure. Indeed, the pressure deDendence o f the mechanical response has

been predicted theoret ical ly on the basis o f the f ree volume approach

by Ferry and Stratton [24] w h o derived the pressure a n a l o g o f the well-

known NLF equation i n the form:

10

where a

r e l a x a t i o n t imes a t pressure po, B i s a cons tan t assumed t o be o f t h e

o r d e r o f u n i t y , fo i s t h e f r a c t i o n a l f r e e volume a t t h e re fe rence

pressure, and K f i s t h e isothermal c o m p r e s s i b i l i t y o f the f r e e volume.

Beuche [25] and O ' R e i l l y [26] proposed t h e equat ion

i s t h e r a t i o of t h e r e l a x a t i o n t imes a t pressure p t o t h e P

where C i s a constant and o t h e r va r iab les a r e as de f i ned above.

Knauss and E m r i [27] p o s t u l a t e a l i n e a r dependence o f f r a c t i o n a l

f r e e volume, on t h e instantaneous values o f temperature, T; water

concen t ra t i on , C; and mechanical d i l a t a t i o n 0 , i . e .

f = f + aAT + y c + 60 (2.10) 0

where a i s t h e c o e f f i c i e n t o f volume expansion, y i s t h e m o i s t u r e co-

e f f i c i e n t o f volume expansion and 6 i s t h e c o e f f i c i e n t o f mechanical

volume expansion. By s u b s t i t u t i o n o f (2.10) i n t h e D o o l i t t l e equat ion

[281

(2.11)

they o b t a i n the f o l l ow ing very general temperature, mo is tu re and pressure

dependent s h i f t - f a c t o r

l o g a = - (2.12)

N e i t h e r equat ion (2.8) , (2 .9) o r (2 .12) has so f a r been s u b j e c t t o

e x t e n s i v e mechanical t e s t i n g on s o l i d polymers.

11

i i ) Single Integral Representations Based on Thermodynamics

In order not t o obscure the main issue of a l i t e r a t u r e review,

we shall not attempt a fu l l treatment of the thermodynamic foundations,

b u t ra ther concentrate on the hierarchy o f di f fe ren t continuum therm-

dynamic theories.

i n a well-documented survey paper by K . Hutter [29].

tha t the main concepts which can be u t i l i zed t o develop const i tut ive

equations fo r i ne l a s t i c materials a r e the rational thermodynamics

approach and the s t a t e var i ab1 e approach (a1 so cal l ed i rreversi b i e

thermodynamics). The l a t t e r w i l l be discussed i n detai l i n Chapter 3.

Coleman and No11 [30] applied functional analysis t o continuum mechanics

i n order t o express current s t r e s s , f ree energy and entropy

as functionals of the themokinematic h is tory . Their analysis

continues t o influence rational thermodynamics research up t o t h i s date .

T h i s hierarchy, shown i n F i g . 2.3, has been reviewed

Figure 2.3 shows

The s t a t e variable approach includes cer ta in internal variables

Constitutive i n order t o represent the internal s t a t e of the material .

equations which describe the evolution of the internal s t a t e are i n -

cluded as part of the theory. Onsager [31] published the concept of

i nternal variables i n thermodynami cs . T h i s formal i sm was used by

Biot [32] i n the derivation of const i tut ive equations o f l inear visco-

e l a s t i c i ty .

viscoelastic materials and his approach has been used by Schapery [34]

and Valanis [35] to develop specif ic theories .

clude viscoplastic materials was made by Perzyna [36].

Ziegler [33] extended t h i s work t o include nonlinear

The extension t o i n -

12

.

U

L

c Q)

I u) u

0 C ZI

.- E

0 E 0

rt 0 v) Q)

0 Q) c

I-

.- L

t C

Q) rt w-

z .- D 0 rt

1 3

2.2 .3 Mechanical (Rheological ) Model s

Elast ic behavior i s compared to t h a t of a massless Hookean spring

and the viscous behavior t o t h a t of a Newtonian dashpot. The success-

ful description of the s t r e s s s t r a in behavior depends on the proper

choice and arrangement o f the elements or hypothetical rheological

uni ts . The f inal hardware-system t h a t describes the behavior of a

material system in the l i nea r range can, a t l e a s t i n pr inciple , be

extended into the nonlinear range by the introduction of nonlinear

springs and dashpots. An example i s the nonlinearized generalized

IKelvin element whose mathematical equation f o r creep can be written

as ,

f 2 ( d N 1 e = - U f l ( U ) + - t + f 3 ( u ) 1 E ( 1 - exp ( - t / T i ) ) (2.13) i = l i EO rl

where f i ( u ) i s a nonlinearizing function o f s t r e s s .

t h i s form in his studies on wood. He separated the total creep s t r a in

i n t o instantaneous, delayed-elastic and flow components. The

instantaneous o r glassy response was always independent o f s t r e s s

(thus f l ( u ) = l ) , b u t the delayed e l a s t i c and flow compliances were

independent of s t r e s s only w i t h i n cer ta in l imi t s of s t r e s s , depending

on moisture content and temperature.

Bach [37] used

We will show (Chapter 3 ) t h a t Schapery's thermodynamic model, a t

l eas t under cer ta in circumstances, can be related t o a spring-dashpot-

hardware system.

14

2.2.4 Power Laws

I t can be shown tha t the use o f d i scre te spring-dashpot models

will allow us t o obtain an accurate representation of viscoelast ic

phenomena over a large time scale provided we use a suf f ic ien t number

of elements - typical ly one per decade.

The idea behind the power law representation i s t o s ac r i f i ce

accuracy a t any one time i n order t o obtain a reasonable representation

over the complete time scale between glassy (short term) behavior and

rubbery (1 ong time) behavior.

mation. Landel and Fedors [38] gave a form for spectral repre-

sentation H ( T ) guided by their understanding of polymer mechanics

This i s often cal l ed broad-band approxi -

which may be written a s ,

Clauser aiid Knauss [39] gave exainples of this approximation of H ( T )

which are shown i n F i g . 2 .4 . These resu l t s lead t o the following

broad-band approximation f o r the relaxation modulus:

(2.15)

I

The representation i s seen t o possess four a rb i t ra ry constants,

The exponent n gives the slope including the l i m i t i n g moduli values.

of the relaxation curve ( F i g . 2 . 5 ) t h r o u g h the t rans i t ion region

between glassy and rubbery behavior and -c0, f o r isothermal conditions,

f ixes the charac te r i s t ic re1 axa t ion time.

1 5

2 4 0 0 0

2 2 0 0 0

2 0 0 0 0

I8000

I6000

14000

I 2 0 0 0

IO000

8000

6000

4000

2000

0

I n = , r0= I

0 Numerical Result

Widder - Post ; N = I ---

. . . . . . . . . Widder - P o r t ; N = 2

- 4 - 2 0 2 4 6 8 IO

Log,, T

F i g . 2 . 4 Power Law Approximation for the Relaxation Spectrum H ( t ) - Clauser and Knauss [39], 1969.

h

c Y

W-

E e 0 X 0

Log t

Fig. 2 . 5 Relaxation Modulus E ( t ) vs Log Time

16

Recognizing tha t relaxation and retardation a re basically

reciprocal , one could also postulate a modified power law for the

compliance spectrum, L(r ) as

(2.16)

However, the integration for the creep compliance does not give a

simple form and i t i s more d i r ec t t o assume a reasonable expression as

( 2 . 1 7 )

where n may be the same exponent as i n the relaxation modulus and r;

i s chosen for the best f i t o f the d a t a . A special case resu l t s when

t << T' 0 and C = De - D g ' n

D C ( t ) = D + C 9

A s l igh t ly modified form can be written as

D C ( t ) = D + C* 9

(2.18)

( 2 . 1 9 )

where r; = A U, C* = C / A n y A i s a dimensionless multiplication fac tor ,

and U = 1 i s a time u n i t selected t o accommodate data reduction ( i .e.

1 sec, 1 m i n , 1 h r , e t c . ) . The power law i n this form ( E q . 2.19) i s

often called the "Findley power law" and has been found t o give an

accurate representation of creep behavior o f an extremely broad range

o f amorphous, crystal ine , cross1 i nked glass and carbon-fi ber reinforced

p l a s t i c s and laminates. The equation has been ver i f ied mostly i n

1 7

t e n s i l e t e s t s , b u t i n a few cases i t has s u c c e s s f u l l y descr ibed

behavior i n compression and under combined t e n s i l e and shear s t resses

as w e l l .

Upon conduct ing a screening exper iment on 934 neat r e s i n ( a t room

temperature), we i d e n t i f i e d t h e exponent, n, i n Eq. 2.19 as t h e most

s e n s i t i v e i n d i c a t o r o f exper imenta l e r r o r s . I n a d d i t i o n , n was shown

t o be dependent on t h e d u r a t i o n o f t h e creep t e s t as shown i n F i g . 2.6.

Thus any s h o r t t i m e creep t e s t w i l l r e s u l t i n an underes t imat ion o f n

o r i n o t h e r words a s t a b l e e v a l u a t i o n o f n i s imposs ib le as l ong as t h e

f l o w term (C*tn) i s t o o smal l r e l a t i v e t o t h e i n i t i a l compl iance D o .

The problem i s e a s i l y handled, by per fo rming a smal l a d d i t i o n a l e x p e r i -

ment, which c o n s i s t s o f reco rd ing t h e recovery s t r a i n a f t e r un load ing .

Creep recovery i s mathemat ica l l y g i ven by

= Ae[(1 + A)" - (A)n] €r- (2.20)

t - t l n , A 6 = a-C*t , , and tl i s t h e t ime a t un load ing . t 1

where =

Eq. 2.20 can be ve ry e a s i l y de r i ved us ing t h e power law f o r creep t o -

gether w i t h t h e superpos i t i on p r i n c i p l e .

va lue ob ta ined f rom recovery data i s n e a r l y i n s e n s i t i v e t o t h e d u r a t i o n

o f the recovery exper iment.

F i g . 2 . 7 shows t h a t t h e n

The creep exper iment was repeated on T300/934 carbon epoxy w i t h

t h e load pe rpend icu la r t o t h e f i b e r d i r e c t i o n . The r e s u l t p l o t t e d i n

F ig . 2.6 i s compared w i t h t h e r e s u l t ob ta ined on t h e neat 934 r e s i n

( F i g . 2.8) and shows t h a t t h e composite n-va lue comes w i t h i n 5% of

t h e r e s i n va lue a f t e r 3000 min.

18

c . t c aa c 0

x W

a

0.25

0 .25

0.20

0.1 5

- 0 934 Resin

I O 0 I O 0 0 I0,OOO

Length of Creep Test, Log t (min)

Fig. 2.6 The Variation o f Power Law Exponent with Length of Creep Test.

C - t c aa C 0 a X

W 0.20 o.21 1 0.19 1 I I

I O I O 0 1000

Test Duration,Log t (min)

Fig. 2.7 The variat ion of Power Law Exponent with Length o f Recovery Test .

19

C

t Q) C 0 cr. W X

.25

.23

.2 I

. I 9

.I7

. I 5

D 934 Resin

50 I00 500 1000 10,000

Log t (min)

Fig. 2.8 Comparison of Power L a w Exponent Variation with Length of Creep Test.

An a l te rna t ive power-law type approach was developqd by the

Russian school, mainly under the influence of the ideas published by

Yu. N. Rabotnov [23]. His formulation was based on the viscoelast ic

deformation of polymeric materials which can be described by an

integral equation o f the Bo1 tzmann-Vol t e r r a type

( 2 . 2 1 )

where a ( t ) and e ( t ) a re the stresses and s t ra ins i n a uniaxial s t r e s s

f i e l d , E i s the instantaneous Young's modulus, and K ( t ) i s the t ransient

compliance o r kernel function. Young's modulus can be determined by

the basic quasi s t a t i c measurements, w h i 1 e the kernel must include the

smallest possible number of parameters t h a t need t o be determined

experimentally. To sa t i s fy the l a s t condition, Rabotnov approximated

the s t r a i n , e ( t ) , w i t h a power law and proceeded t o deduce the kernel

function K ( t ) which can be written a s ,

( 2 2 2 )

I n (2 .22) , A ,

gamma function of the argument ( n + l ) ( l + a ) , where 1 + ~1 > 0.

and C( are material parameters, r [ ( n + l ) ( l + a ) ] i s the

The kernel

parameters can be determined graphically (when a f u l l deformation curve

i s obtained) as well as numerically [19].

Functions o f the type 2.22 are c lass i f ied as exponential functions

o f fractional order. They behave as an Abel-type s ingular i ty f o r

instantaneous loading and 1 ike an exponential for the succeeding period

of time. A f a i r l y detailed a lgebra fo r these operators has been

21

developed by Rabotnov and his c o l l a b o r a t o r s .

Equation (2 .21) can be modif ied f o r the c a s e o f creep where u ( t )

i s cons tan t and becomes,

U t c ( t ) = p [l + 1 K(s)ds]

0

For small t , the f i r s t term i n (2 .22 ) dominates , i . e . ,

x t" K(t) = ro'

By s u b s t i t u t i n g (2 .24 ) i n t o ( 2 . 2 3 ) , we o b t a i n :

0 0 x 4 + a 1 e ( t ) = E [ l + r(2+cl)

This again reduces t o the Findley equa t ion (2 .19 )

D ( t ) = Do + C * t n

22

(2 .23)

( 2 . 2 4 )

(2 .25)

and n = 1 + a. I t i s c l e a r aga in t h a t E q . 2 .19 i s

o n l y ab le t o d e s c r i b e a l i m i t e d p o r t i o n o f the f u l l deformation curve.

Chapter 3

SINGLE INTEGRAL REPRESENTATIONS BASED

ON IRREVERSIBLE THERMODYNAMICS

3.1 Introduction

A s ing le integral nonlinear viscoelast ic cons t i tu t ive equation

has been developed by Schapery i n a s e r i e s of publications [34,40-441.

This chapter reviews his thermodynamic development w i t h a special view

t o developing a be t t e r understanding of the nature o f the fou r non-

1 i near iz i ng parameters which appear as we1 1 as thei r physical o r i g i n . The name i r revers i bl e thermodynamics i s we1 1 establ i shed b u t

confusing, since i t suggests geometric i r r e v e r s i b i l i t y . This i s not

generally t rue because thermodynamic i r r e v e r s i b i l i t y r e l a t e s t o entropy

production while a deformation accompanied by a change i n entropy can

be geometrically reversible as f o r example i n rubber e l a s t i c i t y . Thus

a name 1 i ke "process thermodynamics'' or non-equi 1 i bri um thermodynamics

would be m r e related t o the physics of a problem l i k e v iscoe las t ic

deformation.

The consistency of const i tut ive equations w i t h the fundamental

theorems of thermodynamics has been discussed by Truesdell and T o u p i n

[45]. No r e s t r i c t i o n on const i tut ive equations i s given by the f i r s t

and the t h i r d fundamental theorems. The second excludes processes

w i t h negative entropy production and t h u s limits the f i e l d of const i tu-

t i v e equations, b u t not t o the extent desired.

by Biot [46] t h a t the more precise statements about the nature of the

second law, which were made by Onsager [47], are su f f i c i en t t o t r e a t

I t was demonstrated

23

p r o bl erns i n 1 i near v i scoel as t i c i t y .

3.2 Gibbs Free Energy Formulation of the Schapery Equations

This formulation i s par t icular ly useful, since i t allows the

treatment o f temperature and s t r e s s tensor components as independent

variables. The Gibbs f r ee energy g i s defined as

(3.1 1 1 o m m g = u - T s - - Q q

where u i s the specif ic internal energy, T the absolute temperature,

s the specific entropy, P the density, qm ( m = 1 , 2 , . . .,k) a s e t of k

s t a t e variables consisting of observed variables ( s t r a i n s ) , and Q,

( m = 1 , 2 , . . . , k ) i s a s e t of k generalized forces defined by the vir tual

work condition Sw = Qi 6 q i . E q . (3.1) shows tha t f ree energy can be

accumulated by increasing internal energy by decreasing entropy o r

t h r o u g h a potential energy loss of the external loading.

A mathematical expression f o r the strength of the entropy source

as a resul t of " i r r eve r s ib i l i t y" has to be derived f i rs t . We assume

t h a t the c lassical concept of entropy i s extendable into non-

equilibrium s i tua t ions .

called the fundamental equation o f the system

This leads t o the following r e su l t w h i c h i s

SI = S * ( U d i i ) ( i = 1 , 2 , . . . , n ) (3.2)

The index I , ident i f ies the system under study.

q i ( i = k+l ,.. . , n ) represent internal degrees of freedom.* These will

a1 low us t o describe m i cro-structural rearrangements related to

The s t a t e variables

*The number n o f independent thermodynamic variables depends on the nature of the system.

24

straightening and r e l a t ive sTiding of long chain molecules.

variables a re thus eventual ly observable, b u t cer ta inly n o t control - l ab le .

associated w i t h an internal degree of freedom would provide us with a

means t o control t h i s internal coordinate. The entropy-increment

d s I created by a change in du and n variations of dqi i s measured by

the to ta l derivative of the fundamental equation, i . e .

These

For t h i s reason Qi ( i = k+l, ..., n ) = 0 since any force

as du + 1 (3 .3)

where q; means a l l coordinates a re held constant except q i . The incre-

cs t h r o u g h ment du

du

s controlled by the f i r s t law of thermodynam

k

i = l = dh + 1 P-’ Qi dqi (3.4)

T h u s according t o (3.4) any increment in internal energy i s produced

by an infinitesimal amount of heat ( d h ) crossing the boundary and/or

an infinitesimal amount of mechanical work P-’ Q . dqi done on the system. 1

The incremental process i s reversible, when dqi = 0. Then,

du = dh and according to the second law,

dh = TdsI (3.5)

By subst i tut ing (3.5) into (3 .3 ) , we obtain

(3.6) [q 1

qi

I n other words, the temperature measures the sens i t i v i ty of the

internal energy for entropy changes. T h u s (3 .3 ) , written now under

25

the rest r ic t ions o f a reversible process becomes

n Tds I = du + T 1 dqi

i = l aqi u , q ; ( 3 . 7 )

Since the par t ia l derivative in the second term i s an operation on a

s t a t e function s I , i t defines a s t a t e function Qi , R

(3 .8)

Thus

(3.9)

E q . (3 .9) i s a r e su l t known in classical thermodynamics as Gibbs equa-

t ion and describes the associated reversible process, t h a t causes the

same increment of internal energy du as the i r revers ib le process

described by Eq. ( 3 . 4 ) .

By elimination of du between (3 .9 ) and ( 3 . 4 ) , we obtain

(3.10)

w i t h Qi (k+l ,..., n ) = 0.

Thus the entropy of a mass element changes with time for two

reasons.

by this mass element, second because there i s an entropy source due

t o i r revers ible phenomena inside the volume element.

source i s always a non-negative quantity.

F i r s t , because entropy flows into the volume element occupied

This entropy

Since the computational procedure needed for the derivation works

only on isolated systems, we now create such a system by the a r t i f i c e

26

o f adjoining a large heat reservoir I1 t o the physical system I .

Eq. (3.11) s t a t e s tha t the entropy i s an extensive property, i . e . the

entropy of the system i s the sum of the entropies of the subsystems

= d s I + d s I I ds 1+1 I and

(3.11)

(3.12)

Eq. (3.12) s t a t e s t ha t the reservoir plays the role of a reversible

heat source, i . e . heat can be extracted from i t without a change i n i t s

temperature T . A combination of (3.12), (3.11), and (3.10) leads to

n - 1 - - 1 xi 4i ' I+I I P T i= l (3.13)

w i t h X i = Qi - Q i R .

source , s I+I I has a very simple appearance.

each b e i n g a product of a f l u x characterizing an i r revers ib le process

and a quantity, called thermodynamic force.

I t turns out that the strength o f the entropy

I t i s a sum of n terms,

In order t o proceed from here we need t o calculate the ra te of

approach t o equilibrium. This i s only possible by arguments outside

the scope of continuum mechanics.

la ted t o the various i r revers ib le processes t h a t occur i n the system

t h r o u g h a 1 inear phenomenological relation between forces X i and fluxes

4i ca l l ed Onsager I s theorem ,

The entropy source i s exp l i c i t l y re-

(3.14)

This fundamental theorem s t a t e s t h a t when the flow 4 corresponding j y

27

t o the i r r e v e r s i b l e process j , i s i n f l u e n c e d by t h e f o r c e Xi , c o r -

responding t o t h e i r r e v e r s i b l e process i, t h e n f l o w ;li i s then a l s o

i n f l u e n c e d by t h e f o r c e X . through the same i n f l u e n c e c o e f f i c i e n t s ,

bij(Qm,T).

J Eq. (3.14) can be r e w r i t t e n as

I t i s shown i n Appendix B t h a t :

Qi R = P 2%. aq i

(3.15)

i = k+ l , ..., n

The genera l i zed f o r c e QiR assoc ia ted w i t h qi through the Gibbs equa-

t i o n (3.9), i s d e r i v a b l e f rom the Gibbs f r e e ener.gy g (Qm, qr, T )

- - ag - P qm aQm

m = l , Z , ..., k

The problem i s thus reduced t o t h e s o l u t i o n o f two systems o f equat ions,

m = 1,2,...,k (3.16a)

(3.16b) r,s = k+ l ,. . . ,n P ag + brs(QmyT)tS = 0 a q r

A c o n s t i t u t i v e assumption on t h e dependence o f t h e Gibbs f r e e energy

on Q,,,. qr and T has t o be made.

expansion around t h e equi 1 i b r i um s t a t e such t h a t ,

T h i s i s accomplished by a T a y l o r s e r i e s

1 + drs q r 9s (3.17)

where cR, ci and dij may depend on temperature, b u t n o t on qr and 4,.

Note t h a t t h e r e e x i s t s a d i s t i n c t r e l a t i o n between summation index

28

and p h y s i c a l dimension f o r ’ these c o e f f i c i e n t s , i .e.

Coe f f i c i e n t Rep r e sent s Summation Range

Energy

Mechanical s t r a i n m = 1,2 ,...,k

Thermodynamic force r = k+l ,.. . ,n

Mechanical compliance n,m = lY2, . . . ,k

r = k+ l , ..., n; m = l,Z,...,k

Transforms hidden v a r i a b l e s g r i n t o mechanical s t r a i n u n i t s

Thermodynamic modul us r,s = k + l ,. . . ,n

A f u r t h e r approximat ion o f t h e f r e e energy i s found by an expansion

w i t h respec t t o t h e u n c o n t r o l l a b l e v a r i a b l e s qi o n l y , i . e . we can be

f a r away from e q u i l i b r i u m w i t h respect t o Q,.

(3.18)

gRy c; and d i s can be f u n c t i o n s o f t he independent v a r i a b l e s Q,,, and T.

Eq. (3.18) must reduce t o (3.17) under s u f f i c i e n t l y smal l f o rces ,

which leads t o t h e r e l a t i o n s

Q Q (3.19) P ~ R = ‘R + ‘rn Qm + T dm m n

I - (3.20) c y - cy + d,, Qm

(3.21)

The va r ious terms i n equat ion (3.18) have the f o l l o w i n g p h y s i c a l

s i g n i f icance ;

29

0 p g r An energy term t h a t ac t s as a potential for the time inde-

pendent response.

and temperature can be lumped together with cR into a

single nonlinearizing parameter.

The dependence o f cm and dmn on s t r e s s

fined by the energy associated with a simultaneous action

on the internal coordinates q r and the controllable

variables Q, and e . - d' q q This term i s defined by the interaction energy 2 r s r s

associated with the simultaneous action on internal co-

ordinates q r and 9,.

The above items which make u p equation (3 .18) are of par t icu lar i n t e re s t

a s they define the basic interaction mechanisms involved in three d i f -

ferent nonlinearizing functions which appear in Schapery's f inal single

integral equation as developed subsequently.

Substitution of (3.18) into (3.16b) gives

( 3 . 2 2 )

I n order t o reduce (3.22) to a system of n - k d i f fe ren t ia l equations

w i t h c o n s t a n t coeff ic ients , the fur ther assumption i s made t h a t ,

brs(QmYT) = aD(Qm,T)brs (3.23)

This equation implies t h a t in w h i c h brs are now defined as v iscos i t ies .

the f l o w i s Newtonian b u t with s t r e s s and temperature dependent

viscosity.

expl ic i t through a s h i f t function aD(Qm,T).

The implici t s t r e s s and temperature dependence can be made

A l o t of experimental

30

I

evidence e x i s t s f o r t h e Spec ia l case where a l l v i s c o s i t i e s have a

common s h i f t f a c t o r independent o f s t r e s s .

t h e f a m i l i a r aT(T) s h i f t f a c t o r found i n polymer l i t e r a t u r e [28].

I n such cases aD becomes

The f o l l o w i n g form f o r d l s i s guided by (3.23) ,

r ,s = k + l , ..., n (3.24) d Is(QmyT) = aG(QmYT)drs R

where t h e cons tan t symmetric "thermodynamic" modulus , drs R , i s now de-

f i n e d a t t h e re fe rence temperature TR and Q, = 0 as i n d i c a t e d schemat ica l -

ly i n F ig . 3.1 and aG(Qm,T) i s a s h i f t f u n c t i o n as d e f i n e d below.

It fo l l ows from Eq. (3 .13) , (3.14) and (3.23) t h a t aD n o n l i n e a r i z e s

t h e en t ropy p roduc t i on ,

1 'I+II = - PT a D ( Q m ,T) bij {i {j (3.26)

On t h e o t h e r hand aG n o n l i n e a r i z e s t h e t h i r d te rm i n the Gibbs f r e e

energy expansion (3.18) and t h e r e f o r e PQ i s p r o p o r t i o n a l t o t h e above

q u a n t i t i e s as i n d i c a t e d by,

(3 .27)

We cannot make an a - p r i o r i st.atement on t h e dependence o f aG w i t h

s t r e s s , b u t t h e dependence o f dIs(Q,,,) a t a c e r t a i n temperature T can

increase, decrease, o r be independent o f t h e f o r c e Q, ( F i g . 3.2).

A c o n s t i t u t i v e assumption on aG has been developed by Schapery

[48] f o r t h e case where microcrack ing and f l a w growth ( i n t h e opening

mode) a r e t h e dominant s o f t e n i n g (damage) mechanisms.

l / aG, i s an i n c r e a s i n g f u n c t i o n o f t h e q - t h o r d e r Lebesque norm o f

t e n s i l e s t r e s s , $ ' , as s p e c i f i e d by,

The r e c i p r o c a l ,

31

I

R / drs

drs 1 /

T -T,

Fig. 3. I Schematic Definition of “Thermodynamic Modulus”.

I - Qrn

Fig. 3.2 Dependence of Thermodynamic Modulus, d ‘rs, with the Generalized Force ,Qm.

32

, 1 where ,

(3.28)

(3.29)

G(+) i s p o s i t i v e and r e f l e c t s t h e s t a t i s t i c a l d i s t r i b u t i o n o f damage. rQ3

The q u a n t i t y , G(4)d@, can be found from mechanical p r o p e r t y t e s t s J 41 wherein q i s a p o s i t i v e cons tan t .

Extensions of Eq. (3.29) have been made f o r m u l t i a x i a l s t r e s s

c o n d i t i o n s through i n c o r p o r a t i o n o f t h e f i r s t and second i n v a r i a n t s o f

t h e s t r e s s tenso r ( 8 and J2), whi le aging and/or r e h e a l i n g i s i n t r o -

duced through a f u n c t i o n M ( t ) such t h a t ,

1 + I = {I' M" [Ale + A 2 5 Iq d c

0 (3.30)

Here, A1 and A2 a r e constants which s a t i s f y the r e l a t i o n A2 = $ ( 1 - Al) and when they a r e bo th p o s i t i v e t h e r e i s a suppression o f crack

growth by ( e < 0) .

Due t o t h e s i m p l i f i c a t i o n s in t roduced i n (3.23) and (3.24), a

system of n-k d i f f e r e n t i a l equat ions w i t h cons tan t c o e f f i c i e n t s

s o l v a b l e f o r qk+l ,.. . ,qn now remains. These toge the r w i t h t h e k

a l g e b r a i c equat ions ob ta ined by s u b s t i t u t i n g the f r e e energy express ion

(Eq. 3.18) i n t o t h e k equat ions of (3.16a) are,

33

(n-k equa t ions ) (3.31)

( k -equa t ions ) (3.32)

The former equat ions (3.32) can be r e w r i t t e n w i t h t h e reduced

t i m e $ as,

( 3 . 3 3 )

where,

d+ = d t /au and au = aD I aG

The f i n a l system o f equat ions can be e x p l i c i t l y w r i t t e n as,

"'c a',

- - ... 3ci t 1 "it, a'2 a', _ - - -

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

' F t l , k t l ':t1 , k t 2 ' ' ' ,n

d : t 2 , k t l d t t 2 , k t 2 " ' d!+2,n

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

k t l ... 4 . n

0 0 ... 0

0 0 . . . 0

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

0 0 ... 0

-~ ~ ~~ ~

b k + l , k + l b k t l , k + 2 " ' bk+l , n

b k + 2 , k t l b k + 2 , k t 2 " ' b k + 2 ,n

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

bn, k + l bn, k+2 bn .n

q k t l

q k t 2

q n

dqk+2 3%-

'qn dt

4 1

92

qk

- c;t1

- ci*2

- c;

t

(3 .34)

Remark t h a t t h e m a t r i x p a r t i t i o n on t h e upper l e f t hand corner s i d e

has a maximum dimension o f 6 x (n-6) w h i l e t h e two lower p a r t i t i o n s

a r e square mat r ices w i t h dimensions (n-6) x (n-6) where n can be very

l a rge , s ince t h i s i s the number of i n t e r n a l coord ina tes , needed f o r an

adequate d e s c r i p t i o n o f t h e i n e l a s t i c m a t e r i a l phenomena.

I t can be proven tha t a simultaneous diagonalization of d:S and

i s always possible, which means physically t h a t a transformed brs space exists where the n-k internal deformation modes do not i n t e rac t .

T h u s ,

brs = { :r] and drs

and

dqr - c; d r q r + br K - - -

aG The solution of ( 3 . 3 6 ) i s

( 3 . 3 6 )

( 3 . 3 7 )

with T~ = br /d r . Eq. ( 3 . 3 7 ) h a s a fading-memory Kelvin-type kernel and

can t h u s be seen as the evolution equation f o r a Kelvin u n i t associated

w i t h the physical process represented by 9,.

i s t h a t there i s no force Q r associated w i t h the ( n - k ) functions q r

which describe the evolution of t h e internal system. I t appears however

A point o f i n t e re s t here

t h a t these internal mechanisms are driven by forces - c l /aG spiked

i n t o the Kelvin u n i t through a force t ransfer mechanism, which was

mathematically given as a n d includes a thermodynamic force cr due t o

temperature differences ( $ = T - T R ) .

(3 .20)

The t ransfer function ( E q . 3.20) describes how forces Q, associated

w i t h the observed coordinates (ql ,q2,. . . , q k ) d i f fuse down into the

internal material s t ructure where they drive the hidden coordinates

35

This force t ransfer process i s not necessarily l i nea r and introduces

a third nonlinear function Qm t h r o u g h the equation

1 - Q ( Q (3.38) - 'r @ + drm m n

where + i s T - TR and 6, i s the thermal modulus.

Note tha t functions d i s (which show up in the th i rd term of the

f ree energy expansion (3.18)) and d m (which show u p in the second term)

represent two completely d i f fe ren t nonlinear-physical phenomena. This

reasoning leads t o the two d i s t inc t functions aG and Q, respectively. A

Substitution of (3.37) into (3.31) gives

I

q m = - p - + C - ( l agR ac; - e - + / T r ) -- 1 aQm r aQm dr a~

(3.39)

and for time varying forces,

By substi tuting (3.38) in (3 .40) ,

(3.41)

where

( 3.42a )

(3.42b)

( 3 . 4 2 ~ )

(3.42d)

For uniaxial loading where q1 i s the t ens i l e s t r a i n , e , and Q, i s the

t ens i l e s t r e s s , U, E a . (3.41) can be writ ten a s

or

with

- g1 - a9,

(3.44

(3.45b)

( 3 . 4 5 4

(3.45d)

By way o f summary, the nonlinear parameters go, g, , g2 and a.

have the following physical significance.

Nonl i neari zes the el as t i c response and can in pri nci pl e

increase o r decrease with s t r e s s .

as a softening phenomenon while the l a t t e r i s considered

t o be a hardening phenomenon.

go The former i s described

37

0 g, Experimental r e s u l t s i n d i c a t e t h a t g1 i s an i n c r e a s i n g

f u n c t i o n o f s t r e s s which r e q u i r e s Q, t o be a mono ton ica l l y

i n c r e a s i n g f u n c t i o n o f Q1 as seen i n F i g . 3 .3 .

t h a t t he s t resses t h a t a r e t r a n s f e r r e d t o t h e m i c r o s t r u c t u r e

may have a h i g h e r i n t e n s i t y than t h e e x t e r n a l f o r c e . Note

a l s o t h a t when t h e m a t e r i a l i s l i n e a r v i s c o e l a s t i c ,

g1 = 1 and Q1 i s a l i n e a r f u n c t i o n o f Q,.

Th i s means

F ig . 3.3 Dependence o f General i z e d Force Q1 o r Q,

g2 Th is f a c t o r con ta ins two n o n l i n e a r i z i n g f u n c t i o n s , Q, and

aG.

be sp iked i n t o K e l v i n - t y p e m i c r o u n i t s t a k i n g i n t o account

I t can be seen as t h e r a t i o o f t h e f o r c e t h a t c o u l d

t h a t t h e modul i o f t h e sp r ings i n these u n i t s change by a

f a c t o r aG.

so f ten ing (aG < 1 ) . Th is change can be s t i f f e n i n g (aG > 1 ) o r

Represents a s t r e s s and temperature dependent s h i f t

a long t h e t i m e a x i s .

a D . a = - ' aG

Here aD(Qm,T) decreases w i t h

temperature and w i t h increases in stress-induced f ree

volume.

b u t an increase i n s t r e s s would produce more s h i f t .

Also, aG(Qm,T) decreases with temperature

The Kelvin type kernel A D in E q . (3.44) can be approximated by a

power law which was proved t o be valid under res t r ic ted conditions in

Chapter 2 ,

With t h i s approximation Eq. (3 .44) becomes,

For a creep t e s t 0 = u0 H ( t ) , eq. (3.48) reduces t o ,

(3.46)

(3.47)

(3 .48 )

(3.49)

The experimental procedure t h a t should be followed t o obtain the

three l i nea r parameters ( D o , C , n ) and the four nonlinearizing

parameters (go,g, ,g2,a,) has been described in detai l by Lou and

Schapery [49] on the basis of a graphical sh i f t ing procedure based on

creep and creep-recovery data. A computerized procedure tha t avoids

tedious graphical shif t ing was developed d u r i n g the current research

e f f o r t by Bertolot t i e t a1 . [50].

The general equation (3.44) can be simplified t o model specif ic

thermorheological complex materials. For suf f ic ien t ly low s t r e s ses ,

- Q, = Q, = u which means t h a t g1 = 1 and g2 = l / a G . Also, aG and a D

39

a r e on ly functions o f t empera tu re (aD/aG = a T ) .

t o

Eq. (3 .44) now reduces

(3 .50 )

where log + = l o g t - l o g aT. For a = a. H ( t ) t h i s becomes

(3.51)

w i t h DT = E I o0 and D I ( T ) = go(T) D o . T h u s ,

l o g (DT - DI) = l og A D - l o g aG (3 .52 )

Eq. (3.52) shows t h a t mas te r curves can be ob ta ined by r i g i d h o r i z o n t a l

l l og aTI and v e r t i c a l l l og a G / s h i f t s .

Eq. ( 3 .52 ) sugges t t h a t the machinery needed f o r a c c e l e r a t e d

c h a r a c t e r i z a t i o n can on ly be e f f e c t i v e when the v e r t i c a l s h i f t aG(T)

can be separa ted from the hor i zon ta l shif t a T ( T ) .

p o s s i b l e w i t h a d d i t i o n a l t r a n s i e n t t empera tu re t es t s a s sugges ted

from Eq. ( 3 . 4 1 ) . Some a c t u a l d a t a were ob ta ined by Schapery and Mart in

[51] and very r e c e n t l y by Weitsman [52].

form f o r aG(T) which a p p l i e s below and i n the neighborhood o f the

g l a s s t r a n s i t i o n tempera ture Tg was given a s ,

T h i s i s only

In the former , a t h e o r e t i c a l

(3 .53)

where Eq. (3 .53 ) i s based on kinetic

theo ry wherein AH i s the c o n s t a n t a c t i v a t i o n en tha lpy f o r ho le forma-

t i o n . Simple forms f o r aG on the b a s i s of rubber e l a s t i c i t y a r e well

e s t a b l shed a t and above the T g .

= aH/RTR and z = ( T - T R ) / T R .

The resu l t s discussed herein have been applied t o composite

materials by Schapery and his co-workers [49,53].

41

Chapter 4

DELAYED FAILURE

4.1 Introduction

Failure i s most often t reated as a separate issue from the deter-

m i n a t i o n of modulus properties of materials. I n f a c t , most f a i lu re laws

a re derived empirically from observations related t o a catastrophic

event such as yielding or rupture. As a r e su l t , a great deal of

tes t ing and data analysis i s necessary t o es tabl ish a n appropriate

fa i lure law for a material . On the other h a n d , modulus or const i tut ive

laws are derived by more rational means of re la t ing deformations t o the

forces which produce them. For t h i s reason, often much less tes t ing i s

necessary t o define a const i tut ive law for a material especially i f

deformations do not depart from the e l a s t i c o r reversable deformation

range f o r a material .

Failure, however i t i s defined, should be a p a r t o f a complete

consti tutive description of a material . I n other words, the key t o

dealing effect ively w i t h f a i lu re l i e s i n t reat ing i t s behavior as a

termination of a nonl inear viscoelast ic process. Perhaps, for t h i s

reason, a number of investigators have suggested t h a t modulus and

strength laws should be related t o each other f o r viscoelast ic materials

[4,38,54].

The concept of dis tor t ional energy as a measure f o r the c r i t i c a l i t y

of a g i v e n s t a t e of s t r e s s i n an e l a s t i c material (von Mises c r i t e r ion )

cannot be carried over d i rec t ly as a governing cr i te r ion for f a i lu re

i n viscoelastic mater ia ls , mainly because viscoelast ic deformation

42

i n v o l v e s d i s s i p a t i v e mechanisms.

can be w r i t t e n as:

Thus a t any t ime t h e energy balance

T o t a l de format ion = Free ( s to red ) + D iss ipa ted energy energy energy

I n case of a p e r f e c t l y e l a s t i c / v i s c o p l a s t i c m a t e r i a l , i n e l a s t i c

deformat ions occur o n l y a f t e r a l i m i t i n g va lue o f s t r e s s has been

exceeded. Owing t o t h i s assumption, such a y i e l d c r i t e r i o n does n o t

d i f f e r f rom t h e known c r i t e r i a o f the i n v i s c i d p l a s t i c i t y theory .

most app rop r ia te form on t h e l a t t e r i s o f t e n based on t h e energy approach

(Huber-Henkey-von Mises) i n which a c r i t i c a l va lue o f t h e conserved

The

(accumulated) d i s t o r t i o n energy de f ines the t r a n s i t i o n f rom t h e e l a s t i c

t o t h e i n e l a s t i c s t a t e .

I n case o f an e l a s t i c - v i s c o p l a s t i c m a t e r i a l , t h e r h e o l o g i c a l

response occurs f rom t h e beg inn ing o f t h e l o a d i n g process. Under such

c o n d i t i o n s i t seems approp r ia te t o take i n t o account t h e accumulated

energy WE, as w e i l as t h e d i s s i p a t e d power W where bo th q u a n t i t i e s a r e

regarded t o be respons ib le f o r t h e behavior o f t h e v i s c o e l a s t i c m a t e r i a l .

D

The y i e l d c o n d i t i o n i s then g i ven by a f u n c t i o n w i t h two arguments

such t h a t

where t h e s p e c i f i c form i s based on exper imenta l evidence. The s imp les t

p o s s i b l e fo rm i s t h e l i n e a r combinat ion

3

w i t h

43

With CD = 0 we assume t h a t t h e d i s s i p a t e d energy has no e f f e c t on

t h e y i e l d c r i t e r i o n .

p o i n t o f view.

storage c a p a c i t y i s r e s p o n s i b l e f o r t h e t r a n s i t i o n f rom v i s c o e l a s t i c

response t o y i e l d i n d u c t i l e m a t e r i a l s o r f r a c t u r e i n b r i t t l e ones.

They assume t h a t a t h r e s h o l d va lue of t h e d i s t o r t i o n a l f r e e energy,

c a l l e d t h e r e s i l i e n c e o f t h e m a t e r i a l , i s t h e govern ing q u a n t i t y . I t

i s c l e a r t h a t t h e Reiner-Weissenberg c r i t e r i o n a p p l i e d t o a m a t e r i a l

w i t h zero d i s s i p a t i o n ( e l a s t i c m a t e r i a l ) becomes i d e n t i c a l t o t h e von

Mises c r i t e r i o n . When a p p l i e d t o a v i s c o e l a s t i c m a t e r i a l , however, we

have t o keep t r a c k o f t h e f r e e energy which i s h i s t o r y dependent. I f

t h e mechanisms through which t o t a l deformat ion energy i s t ransformed

i n t o d i s s i p a t e d energy a r e so a c t i v a t e d t h a t no f r e e energy can be

accumulated, t h e r e i s p r a c t i c a l l y no l i m i t t o t h e amount o f deformat ion

energy which can be a p p l i e d w i t h o u t i m p a i r i n g t h e s t r e n g t h .

f o r c e s up t o a c e r t a i n magnitude can be a p p l i e d f o r any l e n g t h o f t ime

w i thou t l e a d i n g t o r u p t u r e . I f i t t u r n s o u t t h a t t h e m a t e r i a l cannot

accommodate t h i s energy r e d i s t r i b u t i o n f a s t enough then t h e m a t e r i a l

w i l l s t o r e energy i n i t i a l l y b u t a f t e r a p e r i o d o f t i m e f a i l u r e w i l l

occu r . The f a i l u r e process i s t h e r e f o r e delayed which l i m i t s t h e l i f e

o f a given s t r u c t u r e t o a f i n i t e va lue. The i n s t a n t o f y i e l d i n g i s

t h u s c l e a r l y dependent on t h e f i n a l outcome o f a c o n j e c t i o n between

d e v i a t o r i c f ree energy and d i s s i p a t e d energy.

s t r a i n h i s t o r y on t h e delayed y i e l d i n g phenomenon f o l l o w s from t h i s

model i n a n a t u r a l way. I n t h i s respec t , i t should be noted t h a t bo th

Naghdi and Murch [56] and Crochet [57] have used a m o d i f i e d von Mises

Thi s corresponds t o Rei ne r and Wei ssenberg ' s

Reiner and Weissenberg [55] suggest t h a t t h e energy

That i s ,

Thus t h e e f f e c t o f t h e

44

c r i t e r i a f o r v i s c o e l a s t i c m a t e r i a l s by assuming, ab i n i t i o , t h a t t h e

r a d i u s of t he y i e l d surface Y does depend upon t h e s t r a i n h i s t o r y ,

t h rough a E u c l i d i a n me t r i c , X, i n s t r a i n space. T h e i r f o r m u l a t i o n

can be expressed as,

1 - Y2 = 0 , Y = Y ( x ) (4 .1) si j f = 7 sij

E v E ) ] ' /2 x = [(Eij - f i j ) ( f i j V

- 'ij (4.2)

where x rep resen ts t h e d i s tance between t h e p u r e l y e l a s t i c s t r a i n

response fFj and t h e s t r a i n s t a t e corresponding t o y i e l d eyj as i s

g e o m e t r i c a l l y shown i n F i g . 4.1. The f u n c t i o n a l r e l a t i o n s h i p between

Deformation Tra jectory 7

F ig . 4.1 Geometric i n t e r p r e t a t i o n of t h e E u c l i d i a n m e t r i c x (as de- f i n e d by Crochet) .

Y and x f o r one-dimensional behavior was g i v e n by Crochet [57] t o be,

Y ( t ) = A + Be-Cx (4.3)

where A , B and C a r e m a t e r i a l parameters. The r a d i u s o f t h e y i e l d s u r -

f ace i s thus a con t inuous ly decreasing f u n c t i o n o f the "d i s tance" between

45

E two s ta tes of s t r a i n (ei j and e y j ) .

An ex tens ion o f t h e Naghdi and Murch-Crochet approach t o f i b e r

re in fo rced m a t e r i a l systems i s p o s s i b l e when we can r e l a t e t h e s t r e s s

i n t h e m a t r i x phase tal, t o t h e a p p l i e d e x t e r n a l s t r e s s CUI.

i n t r o d u c t i o n o f s t r e s s c o n c e n t r a t i o n f a c t o r s , which r e l a t e b o t h these

s t r e s s - f i e l d s i s suggested by H i l l [58]

The

and equat ion (4.1) becomes

(4 .4)

(4 .5)

i n which Y i s t h e t e n s i o n y i e l d s t r e s s i n t h e m a t r i x , as d e f i n e d by

eq. (4 .3) . It should be noted, however, t h a t eq. (4 .5) o n l y descr ibes

i n i t i a l y i e l d i n g . The i n c o r p o r a t i o n o f a hardening r u l e , t o g e t h e r w i t h

a f l o w r u l e should be considered i n o r d e r t o model a c c u r a t e l y t h e

phenomena a f t e r i n i t i a l y i e l d i n g up t o f i n a l f r a c t u r e .

The ex tens ion o f t h e Reiner-Weissenberg approach t o a general

a n i s o t r o p i c m a t e r i a l system can be done q u i t e e a s i l y when we use t h e

general d e f i n i t i o n o f d i s t o r t i o n a l energy, as d e f i n e d by von Mises [59]

- 1 2 - (Sll a; + s c? + . . . + s66 TZ) + S12 axay + . . . "di s t 22 Y

The advantage of t h i s approach i s t h a t t h e onset o f y i e l d i n g o r f a i l u r e

i s c o n t r o l l e d by o n l y one m a t e r i a l cons tan t \ i d i s t y w h i l e the former

46

energy, a s defined by von Mises, has

special case of an isotropic materia

Pagano's comments are unknown a t the

approach (Naghdi e t a1.3 introduces a t l e a s t three parameters.

been argued by Pagano [60], on t h e other hand, t ha t the dis tor t ional

I t has

only physical significance i n the

. The fu l l implications of

present time. A possible a l t e r -

native way t o extend the Reiner-Weissenberg c r i t e r ion , would be t o

evaluate the f ree energy i n the matrix phase.

t h r o u g h H i l l ' s concentration factors .

This could also be done

following i s a brief review of the Reiner-Weissenberg c r i te r ion

ows Brul le r ' s [61-681 extensive investigation along these

The

which fol

1 ines.

4.1.1 Free Energy Accumulation for a Linear Kelvin C h a i n i n Creep

For a s ingle Kelvin unit in se r ies w i t h a spring as shown in

Fig. 4 . 2 ,

F ig . 4 . 2 Linear Kelvin model

47

e = eg + €1

9 D?3 “0

where

e =

el = D1 ao [ l - exp ( - t / T

o r

e ( t ) = [ D + Dl( l - exp 9

(4 .7)

(4 .8)

) l o o

The s t ress , as, i n t h e s p r i n g w i t h compliance D1 i s ,

(4 .9 )

(4 .10)

The s t ress , ad, i n the dashpot i s then,

“d = u 0 exp ( - t /T l ) (4.11)

A phys ica l i n t e r p r e t a t i o n o f f r e e energy would be the energy s t o r e d i n

t h e spr ings D and D1, 9

‘D = a2 3 + It a is d t

S 0

‘springs o

(4.12)

I n t h e same way we can c a l c u l a t e the energy d i s s i p a t i o n i n t h e dashpot,

- 2 Dl - a. 2 [1 - exp ( -2 t / . r l ) ] (4 .13)

The t o t a l energy i n t h e t h r e e parameter system i s g iven by,

48

- - ' t o ta l 'springs + 'dashpot

(4.14)

The ex tens ion of Eq. (4.12), (4.13) and (4.14) t o a n-element K e l v i n

u n i t w i t h a f r e e s p r i n g i s s t ra igh t fo rward .

N Di + -1 2 (1 - exp ( - t / K i ) )

1 =1 's p r i ngs

N Di 2 'dashpots = u o i=l 1 (1 - exp ( -2 t /T i ) )

1 N

i =1 = u2 + 1 Di ( 1 - exp

' t o ta l 0

By t a k i n g t h e l i m i t s f o r

2 lim 'springs = u o t-

2 lim 'dashpots = u o t-

t -f o~ o f these equat ions, we o b t a i n

'D

2 i =1

N Di c 2' i =1

N = G o 2 2 + igl ~ i ]

lim ' t o t a l t-

(4.15)

(4.16)

(4.1 7 )

(4.18)

(4 .19)

(4.20)

Thus h a l f t h e work done by t h e ex te rna l f o rces goes t o inc rease t h e

f r e e energy w h i l e t h e o t h e r h a l f i s d i s s i p a t e d .

4.1.2 Power Law Approximat ions for Free Energy

The f r e e energy express ion i n t h e form o f Eq. (4.15) f o r c e s us t o

i d e n t i f y t h e 2N m a t e r i a l constants Di and T ~ . Th is can be avoided

49

using a power law approximation as follows,

( 4 . 2 1 )

and

( 4 . 2 2 )

This allows us t o obtain the following simple expression for the free

energy

- hi - fai 1 ure D

3 + D ( tn - 2

U

( 2 t ) n )

( 4 . 2 3 )

( 4 . 2 4 )

The nonlinearizing parameters introduced in Eq. ( 4 . 2 4 ) lead t o

J; (4 .25)

The ab i l i t y of Eqs. ( 4 . 1 5 ) and (4.25) t o model experimental d a t a i s

checked in Chapter 7 .

Note on the Z h u r k o v Criterion

The existence of a strength l imit as a real physical characteris-

t i c of a material was seriously doubted by Z h u r k o v [ 69 ] due t o experi-

mental evidence t h a t sol ids may also fracture a t s t resses below the

strength l imi t .

lifetime o f sol ids under a constant t ens i l e s t r e s s . Both t ens i l e

s t r e s s and temperature were widely varied among experiments.

He s e t up systematic and careful measurements o f the

More t h a n

a hundred substances of 'all principal types of sol ids were investigated--

metals, a l loys , glasses, polymers, crystals . I t turned out t h a t the

time t o f racture possessed a surprisingly uniform dependence when

sui tably expressed i n terms of the s t ress (0) and temperature ( T ) .

Analytical treatment of t h i s vast amount o f d a t a led him t o the u n i -

versal formula f o r the temperature and s t r e s s dependence o f l i fe t ime,

r , (except for very small s t r e s s e s ) ,

r = r 0 exp [ ( U o - y a ) / k T ] ( 4 . 2 6 )

where k i s the Boltzmann constant, T~ i s a constant on the order of

the molecular osc i l la t ion period of

each substance regardless o f i t s structure and treatment, and y depends

on the previous treatment of the substance and varies over a wide

range for d i f fe ren t materi a1 s .

Equa t ion ( 4 . 2 6 ) can be formally incorporated i n t o a continuum

sec . , Uo i s a constant for

mechanics analysis i n a modified form t o account for time varying

s t resses a n d / o r temperatures,

= 1 J o exp [ (Uo - y u ( t ) ) / k T ( t ) J ( 4 . 2 7 )

Roylance and Wang [70] used Eq . ( 4 . 2 7 ) i n a f i n i t e element code where

the current value o f the above integral was computed a t each node. The

time and location o f f racture was determined when the integral value

reached unity a t any node.

Williams and his collaborators [71 , 7 2 1 disagree w i t h Eq. ( 4 . 2 6 ) .

They argue t h a t the rupture process i s much more detailed t h a n re-

f lected by t h i s equation. Their experiments on s t r e s s history induced

51

rup tu re l e d them t o t h e conc lus ion t h a t "Zhurkov 's impress ive f i n d i n g s

appear t o be t h e somewhat f o r t u i t o u s r e s u l t o f a p a r t i c u l a r t e s t

met hod. I'

When Eq. (4 .27) i s a p p l i e d t o such h i g h l y heterogeneous m a t e r i a l

systems as composi tes, t h e ques t i on of whether o r n c t f r a c t u r e can be

regarded as a t h e r m a l l y a c t i v a t e d process of damage accumulat ion i s f a r

f rom s e t t l e d . I t i s a l s o obvious t h a t T ~ , Uo and y l o s e t h e i r p h y s i c a l

meaning. Regel e t a l . [73 ] suggest a l i f e o f m i x t u r e t ype p r e d i c t i o n s

f o r these v a r i a b l e s .

52

Chapter 5

RESIN CHARACTERIZATION: EXPERIMENTAL RESULTS

AND DISCUSSION

5.1 I n t r o d u c t i o n

I n o r d e r t o understand exper imenta l l y observed d i s c o l o r a t i o n s o f

t h e F i b e r i t e 934 neat r e s i n due t o p o s t c u r i n g and some anoma l i t i es o f

t h e creep response i n t h e temperature reg ion , 200-250°F (93.3-1 21 "C) ,

i t was decided t o s tudy d e t a i l s o f t h e r e l a t i o n s h i p between r e s i n

chemis t ry , va r ious p o s t c u r i n g procedures, and mechanical p r o p e r t i e s .

General r e s i n in fo rmat ion i s summarized i n 5.2 w h i l e exper iments and

a n a l y s i s r e l a t e d t o t h e mechanical c h a r a c t e r i z a t i o n o f t h e r e s i n a re

r e p o r t e d i n 5.3.

5.2 General Resin I n f o r m a t i on

5,2.1 Resin Chemistry and Cur ing Procedure

Many composi t e s , e s p e c i a l l y carbon-f i b e r based composites, u t i 1 i z e

an epoxy r e s i n m a t r i x .

a r e a v a i l a b l e , o n l y a smal 1 number ( = 10) a r e used f o r composi t e s . epoxy of i n t e r e s t i n t h i s s tudy i s a po l y func t i ona l epoxide based on

t h e t e t r a g l y c i d y l d e r i v a t i v e o f methy lened ian i l i n e (TGDDM), a v a i l a b l e

from Ciba-Geigy under t h e tradename MY 720.

t h e bas i s o f v i r t u a l l y a l l aerospace ma t r i ces f o r carbon f i b r e bo th

here and abroad, i s shown i n F i g . 5 . la . The Ciba-Geigy Eporal hardener

used i n con junc t i on w i t h MY 720 i s 4,4' d iaminodiphenyl su l fane (DDS)

Ciba-Geigy Eporal and i s shown i n Fig. 5 . l b .

From the hundreds of d i f f e r e n t epoxies t h a t

The

Th is epoxide, which i s now

53

/O \ CH, - CH

CH, - \ 0 7

- CH,

N \ / - CH,

CH 2

CH,- /

\ N

CH,-

/O\ CH2

CH -

CH - \

0

a ) Tetraglyci dyl 4 , 4 ' d i ami nodi phenyl methane epoxy TGDDM

H 2 N *i+NH2

0

b ) 4 , 4 ' diarninodiphenyl sulfane (DDS)

F i g . 5.1 Chemical Structure for Epoxy Resins

Because the DDS cure i s slugqish, even a t moderately h i g h tempera-

t u re s , a boron t r i f l uo r ide ca ta lys t i s used ( B F 3 ) i n order t o produce

a more manageable cure cycle.

curing agents often have poor humid i ty res is tance, they are used as a

compromise between mechanical properties a n d manufacturing convenience.

The final epoxy network s t ructure depends n o t o n l y on the chemistry of

the system, b u t also on the i n i t i a l epoxy versus curing agent r a t i o

and cure conditions.

tained three d i f fe ren t panels manufactured with three different c u r i n g

procedures. Cure de ta i l s are given i n Appendix A .

Despite the fac t t h a t these ionic

A t the beginning o f t h i s research program, we ob-

I t i s usually assumed tha t the properties of the cured resin i n

a composite a re the same as those o f the bulk material . However,

McGullough [74], Hancox [75] and Yang, Carlsson and Sternstein [76]

suggest otherwise. The reasons mentioned by these authors a re :

The s t ructure in a small volume of resin may be more ordered t h a n i n the b u l k material , because the exotherm i n the composite w i 11 be limited t o the overall curing temperature by the presence of good conducting f ibers .

The small distance between the f ibers may provide s t e r i c hindrance and inh ib i t epoxy-amine cross-link reactions d u r i n g the l a t t e r stages of cure and l i m i t the overall achievable cross-link density.

Due t o the large amount of surface area of the f ibe r s , adsorption o f epoxy segments a t the f ibe r surface would cause a t i gh te r network.

A s t a t e of negative hydrostatic s t r e s s could be imposed on the epoxy due t o the f ibers , r e s t r i c t ing the volume expan- s ion o f epoxy d u r i n g heating.

No convincing evidence has been produced yet ( t o t h i s author 's know1 edge)

tha t one or a combination of these mechanisms leads t o the resin i n the

composite tha t i s thermorheologi cal ly d i f fe ren t from the neat res in .

As a polymer i s cooled a t a f i n i t e ra te t h r o u g h the glass t rans i -

t i o n into the glassy s t a t e , a sudden increase i n modulus para l le l5 a

rapid decrease i n molecular mobility. The process of molecular re-

arrangements operates on i t s own internal time scale . Equilibrium can

only be achieved i f the cooling rate i s so slow t h a t the molecular re-

arrangements are g iven suf f ic ien t time t o operate. I f the cooling ra te

i s too f a s t , these rearrangements are essent ia l ly quenched i n t o a non-

equilibrium s t a t e , i . e . , a s t a t e characterized by excess entropy,

enthalpy and volume. This gives rise t o a d r i v i n g force towards

55

equilibrium, i . e . , the system densifies t i l l the f ree volume, excess

entropy and excess enthalpy a l l reach zero.

on a scale of days, months o r years and i s known as aging.

This process can operate

Experimental equil i brium has been unexpectedly d i f f i c u l t t o

achieve. Spontaneous changes in the properties of glassy polymers have

been observed long a f t e r v i t r i f i c a t i o n .

Excess entropy cannot be measured d i rec t ly b u t excess enthalpy

The l a t t e r show a i s easi ly measured by calorimetric methods (DSC).

decrease i n enthalpy with time in a s imilar fashion as the time dependent

mechanical properties as reported by Pe t r ie [77,78]. As indicated in

Fig. 5 . 2 , she did not find a one t o one correspondence to specif ic

volume behavior.

I n o u r experiments, we t r i ed t o achieve t h i s equilibrium s t a t e by

a postcuring of 350°F i 10°F (176"C+5"C) for 4 hr ? 15 min. followed

by a slow, controlled cooldown a t a r a t e of 5 O F / h r (2.5"C/hr)(see Fig.

5 . 3 ) .

5.2.2 Thermal Transitions in

Two d i s t i nc t t rans i t ion

the Epoxy System

temperatures a t 140°F (60°C) and 356°F

(180°C) respectively have been ident i f ied by Yeow and Brinson [4] d u r -

ing the i r experiments on T300/934 graphite epoxy.

primary glass-transi t i on temperature , T

secondary t rans i t ion .

continuities in the coeff ic ient of thermal expansion which were measured

using electr ical s t r a i n gages.

The l a t t e r was the

whi 1 e the former was a 9 '

Their measurements were based on detecting d is -

56

L V

lu 0 Q -

- I CT

0 V

- .. I a

0.5 1 1 0.0 I I I I

0 I O 0 200 300

Annealing Time ( h r )

F i g . 5.2 Shear Storage Modulus, G i , Logarithmic Decrement, A , and Enthalpy Relaxation, A H , o f Quenched Atactic PS vs . Annealing Time a t 92°C. Pet r ie [77,78].

G i and A Were Measured a t 0 .6 Hz.

LI c A

V 7 200 0 k 400 r

Y 0 0 W 3001 /T--- Prescribed Cycle

W L 3 c 0

I-

2oo t I Natura l Decay 1 Q, I : Y I

IO0 =I

0 c

I-

T ime ( h r )

F i g . 5 .3 Postcure Cycle Used fo r 934-Resin and f o r T300/934 Composite.

57

frequency

sequent 1 y

a r e i n v a r

method, i

Keenan, S e f e r i s and Q u i l i v a n [79] used o s c i l l a t i n g mechanical

f i e l d s t o c h a r a c t e r i z e t h e dynamic mechanical behavior o f an epoxy

system. T h e i r r e s u l t s a r e shown i n F i g . 5.4 w i t h an w - t r a n s i t i o n

centered around 212°F (100OC) and an a - t r a n s i t i o n a t 464°F (240°C).

However, these d e f i n i t i o n s of T s u f f e r f rom t h e f a c t t h a t t hey a r e

based on dynamic mechanical q u a n t i t i e s which a re q u i t e s e n s i t i v e t o t h e

of exper imentat ion as w e l l as t h e sample hea t ing r a t e . * Con-

9

t h e T values d e f i n e d from dynamic mechanical exper iments 9

n

€ 0

v) \ v) al c )r 0

u

Y

v) 1 1 -0

-

s 0

v) 0

w

.- c

-

I O'O

I o9

a b l y h ighe r than those ob ta ined

e . , d . i l a tomer i c exper iments.

by a more convent ional

I oe -160 -40 80 200 320

Temperature ("C) a) Storage Modulus

F i g . 5.4 Storage Modulus and Loss

I 0'

a w

IO ' - -160 -40 80 200 320

Temperature ("C) b ) Loss Modulus

Modulus a f t e r Keenan e t a1 . [79].

*The conclusions ob ta ined by Keenan e t a l . can be c a r r i e d ove r t o t h e 934 r e s i n , s ince t h i s p a r t i c u l a r member o f t h e TGDDM-DDS epoxy f a m i l y has t h e same bas ic network. one o f degree, n o t i n k i n d .

Any d i f f e r e n c e should be

58

5.2.3 Environmental S t a h l i ' t y

1 ) Thermal S tab i 1 i t y

Different IR a n a l y s i s of an epoxy pos tcured above 350°F (180°C)

r e v e a l s an a d d i t i o n a l broad s o r p t i o n band i n the 5 .5 - 14.3 p r eg ion

acco rd ing t o Morgan [80].

exposure time and i n c r e a s i n g temperature .

5.85 p appears dur ing the i n i t i a l s t ages o f the degrada t ion p r o c e s s ,

w h i c h i n d i c a t e s the formation o f carbonyl groups. Such groups should

result from ox ida t ion o f unreac ted epoxide rings t o form a-hydroxy

The i n t e n s i t y o f th i s s o r p t i o n inc reased with

A s t r o n g IR s o r p t i o n a t

a lo l ehyde and ca rboxy l i c a c i d [81].

Degradation causes a d i s c o l o r a t i o n o f the 934 epoxy a s ind

i n F ig . 5 . 5 and may i n d i c a t e a s i g n i f i c a n t change o f behavior i n

g l a s s y reg ion due t o pos t cu r ing . We observed t h a t hea t ing above

(180°C) f o r 36 hr seemed t o cause embr i t t l ement , i . e . decreased

c a t e d

the

356°F

.ensi 1 e

s t r e n g t h and u l t i m a t e e longa t ion accompanied by a s l i g h t l y inc reased

modulus (F ig . 5 . 6 ) . In a d d i t i o n , aii i i icreased ino is ture abso rp t ion

r a t e ( F i g . 5 .7) was found. This i s most l i k e l y due t o a thermal o x i -

d a t i o n phenomenon.

damping r a t i o 6 f o r the same r e s i n .

F igure 5 . 8 shows the s t o r a g e modulus E ' and the

2 ) Moisture Absorption

One consequence of d i f f u s e d water i s t h a t i t lowers the mechanical

r i g i d i t y o f the resin.

g l a s s t r a n s i t i o n tempera ture by a s much as 212°F (100°C).

of mois ture on the g l a s s t r a n s i t i o n temperature and the r a t e s o f

abso rp t ion and deso rp t ion f o r d i f f e r e n t r e s i n s ( i n c l u d i n g the F ibe r i t e

In extreme cases the effect i s t o lower the

The e f f e c t

59

As delivered

4 hours at 350' F (177OC), slow cooled , 5 O F (2 .5 O C ) per hour.

I5 minutes at approximately 10-15°F ( 5 - 8 " C ) above Tg , then quenched down t o room temperature.

Short overheating a t 500" F (260" C 1, then quenched down to room temperature.

36 hours a t 380°F ( 1 9 3 O C ) , slow cooled , 5 O F - ( 2 . 5 " C ) per hour.

F i g 5.5 Effec t of Cure Cycle on Resin Color

60

A .- cn Y Y

cn cn

3i

I I 2.0 - PO8tCUrOd 380 -

Nonportcurod -

Strain (%)

F i g . 5.6 Ef fect o f Postcur ing on t h e Modulus.

F i g . 5 . 7 E f f e c t o f

0 Postcursd

(36 hr at 38OoF, 193OC)

0 Non - postcured -.- Po s tcured 0.5 -

( 4 hr. at 350°F,

0.3 -

0 2 4 6 8 10

Postcur ing on Mois tu re Absorp t ion Rate o f 934 Resin a t Room Temperature.

61

Temperature, ( deg F )

I O 0 20 0 300 4000 I I I I 1

2 3000 z

2000 I - 3 Q 0 1000 z

0 Postcured 380 F 36 hr.

D Postcured 350 F 4 h r . - 3 200

I 4 100

0 0 0 50 I O 0 I 50 200

Temperature, ( deg C )

Temperature, ( deg F )

IO0 200 300 0. I 5 I 1 I

0 Postcured 380 F 36 hr.

0.10 - A Postcured 350 F 4 hr - M

c 0

l- 0.05 - Y

c. A

0 m

v m

- 0.00 1 I I

0 50 I O 0 I 5 0 200

Temperature, (deg C 1

F i g . 5 . 8 Storage blodulus and Phase Angle (Tangent D e l t a ) as a Func t ion o f TemDerature f o r 934 Resin.

62

934) has been s t u d i e d by De isa i and Whi tes ide [82].

t h e r e l a t i o n s h i p between e q u i l i b r i u m m o i s t u r e c o n c e n t r a t i o n and p r imary

g l ass t r a n s i t i on temperature f o r several epoxy r e s i n s .

shows the e f f e c t o f t he humid i t y i n the t e s t environment on t h e

e q u i l i b r i u m m o i s t u r e con ten t i n t h e same r e s i n s .

t h a t a prolonged postcure p e r i o d could adve rse l y a f f e c t t h e m o i s t u r e

a b s o r p t i o n r a t e s , i n d i c a t i n g a damaged m a t r i x . The specimens were

a l l owed t o p i c k up mo is tu re i n the l a b [75-80°F (24-27°C) and

F i g u r e 5.11 shows t h e i n f l u e n c e o f mois ture on t h e w - t r a n s i t i o n .

i s seen t h a t m o i s t u r e has a s t r o n g e f f e c t e s p e c i a l l y on t h e w i d t h o f t h e

W - t r a n s i t i on.

F igu re 5.9 shows

F igu re 5.10

F igure 5.10 shows

45% RH].

I t

0 240 I I 0 X 3501 5 al Y

2 20 0 5208 -

0

u) c

.- .- c 120

too F = ‘“1 “-gi

I I 1 I 1 0 0 2 4 6 0 to

Moisture Content, Ol0 ( w t )

F i g . 5.9 Ef fect of Yo is tu re Content on T [82]. 9

63

c4

t

3 Y

c c Q) c t 0 0

I O

9

8

7

6

5

4

3

2

I

0

A 3501.6

NMD2373

0 I O 20 30 40 50 60 70 80 90 100

Relative Humidity ( % I

F i g . 5.10 E f f e c t o f R e l a t i v e Humid i ty on E q u i l i b r i u m Mo is tu re Concen- t r a t i o n i n Epoxy Resin [82].

6

0

0)

c - a C 0 I-

4 L 2

8 6

1

L I. 3 0% i 4l 2

-160 - 40 80 200 320

Temperature (C ) F i g . 5 .11 I n f l u e n c e o f Mo is tu re on t h e o - T r a n s i t i o n [79].

64

The specimens we used f o r creep and creep-recovery l o s t approx i -

mate ly 1.5% o f t h e i r t o t a l weight dur ing pos tcu r ing . For t h i s reason,

t h e f o l l o w i n g precaut ions were taken t o c o n t r o l mo is tu re abso rp t i on :

The specimens were dess ica ted immediate ly a f t e r pos tcu r ing . (The r e l a t i v e humid i t y i n the d e s s i c a t o r was mon i to red and kep t below 20%. )

A l l specimens were t e s t e d w i t h i n a month a f t e r pos tcu r ing when poss ib le .

The specimens t e s t e d a t room temperature were surrounded w i t h dess icant t h a t was kep t i n t h e oven space.

I n conclus ion, o u r own mo is tu re absorp t ion exper iments, t oge the r w i t h

a c a r e f u l s tudy o f t h e l i t e r a t u r e ava i l ab le , l e t us b e l i e v e t h a t t he

precaut ions taken were s u f f i c i e n t t o keep t h e mo is tu re l e v e l w e l l below

.3% WT, a l e v e l t h a t i s be l i eved t o be n e g l i g i b l e .

5.3 Resu l ts and D iscuss ion on 934 Resin

Two 934-res in panels were f a b r i c a t e d by Gra f tech as d e t a i l e d i n

Appendix A.

o f approx imate ly 25" x 20" x .12", wh i le one t h i c k p l a t e o f 10" x 10" x

.39" was made by F i b e r i t e .

These panels , l a b e l e d I and I 1 , were r e c t a n g u l a r p l a t e s

The panels were c u t i n t o 8" x 1 "

r e c t a n g u l a r specimens and l a b e l e d according t o ba tch and p o s i t i o n . I n

o rde r t o have as homogeneous a s e t o f specimens as poss ib le , each was

sub jec ted t o a pos tcu re c y c l e o f 350°F (176°C) f o r 4 h as d iscussed i n

5.1. A screening exper iment d i d n o t revea l s u b s t a n t i a l d i f f e r e n c e s

between t h e creep r a t e s o f specimens c u t f rom d i f f e r e n t panels .

The sequence o f e leva ted temperature creep and creep-recovery

t e s t s , i n d i c a t e d i n F i g . 5.12, was app l ied a t a moderate s t r e s s l e v e l

of J~ = 1600 p s i (11 KPa). The inverse compliance a t t = 1 min, i .e.

65

0 0 m t

LL m Q V

v

0 2 - 10 hr I hr + + . . . . . . . . . . . . -

- Time

F i g . 5.12 Creep ( 1 h r ) and Creep-Recovery (10 h r ) Test Sequence a t D i f f e r e n t Temperatures.

C J ~ / ~ (1 min) i s p l o t t e d i n F i g . 5.13.

t e s t s a r e p l o t t e d on t h e same graph.*

Moduli ob ta ined from I n s t r o n

The creep and creep-recovery curves ob ta ined d u r i n g t h e t e s t

sequence given i n F i g . 5.12 a r e ve ry a c c u r a t e l y model led by the power

1 aw equation,

D(T) = Do(T) + D1 tn

and t h e r e s u l t i s shown i n F i g . 5.14.

(2.19)

*The I n s t r o n t e s t r e s u l t s were ob ta ined us ing a 2 - i n extensometer, whereas the creep and creep-recovery r e s u l t s were ob ta ined us ing M i cro-Measurements WK-06-125AD-350 s t r a i n gages.

66

Temperature ( deg C 50 I O 0 I50

I I I

0 Creep Test I / S ( 1 m i n ) A lnstron Test . 0 5 in /min

0 , - e

A " %- A

5 0 Q W - fn 3 3 0 0

-

I

0 200

Temperature ( d e g F 1 300

F i g . 5.13 Neat Resin Modulus Versus Temperature.

The temperature dependence o f Do(T) , ( F i g . 5.15), i n d i c a t e s a

thermorheologically-complex m a t e r i a l .

s imple power law [Eq. (5 .21) ] breaks down above 300°F (149°C). The

temperature dependence o f D1 (T) versus temperature i s shown i n F ig .

5.16. The i r r e g u l a r i t y i n t h i s curve i d e n t i f i e s t h e o - t r a n s i t i o n . The

t r a n s i t i o n i s centered around 200°F (93.3"C), which i s i n agreement

w i t h t h e r e s u l t s obta ined by Keenan e t a l . [79], i . e . 212°F (lOO°C), on

t h i s t y p e o f epoxy as w e l l as those r e p o r t e d by G r i f f i t h [7 ] f o r t h e

o - t r a n s i t i o n o f t he T300/934 composi t e system, 220°F (104°C).

va lue ob ta ined by Yeow [ S I , 140°F (60"C), c o u l d be s imply due t o t h e

combined e f f e c t s o f mo is tu re absorpt ion and u n c e r t a i n t i e s i n t h e i den -

t i f i c a t i o n o f a small d i s c o n t i n u i t y i n t h e thermal expansion versus

F igu re 5.15 a l s o shows t h a t t h e

The

67

3.5 /

3.0 ’ 2.5 ’ 2.0 ’ 1.5 ’ I .O

12

7 5

I/ I

0.0 I .O 2.0 3.0

Log Time ( m i n )

F i g . 5.14 Temperature Dependence o f Compliance.

Tempera ture ( d e g c )

50 I O 0 I50 2.0

I .5 I I I I O 0 200 300

Tempera tu re ( d e g F )

29

25

22

c

(0

I

0 X

0 Q Y \

- Y c

- Y - h

I- Y

no

Fig. 5.15 Temperature Dependence o f Initial Compliance Do(T).

68

Temperature (deg c

15

10

5 I

50 I O 0 I50 I I 1

200 300 I O 0

Temperature ( d e g F 1

F i g . 5.16 Temperature Dependence of D, ( T ) .

temperature curve.

The master curve, o b t a i n e d by horizontal and vertical shif t ing

of the data in F i g . 5.14 i s shown i n F i g . 5.17 while the amount of

ver t ical versus horizontal s h i f t i s shown i n F i g . 5.18. This l a t t e r

f igure can be approximated by two s t r a igh t l ines w i t h slopes a and 3

respectively.

change from one activation mechanism t o another.

the horizontal s h i f t i n the glassy region can be represented by an

Arhenius type of temperature dependence

The discontinuity i n the slope i s an indication of the

I t i s well known t h a t

where A F i s the activation energy (per mole), R i s the universal gas

constant, and T R i s an a rb i t ra ry reference temperature. Figure 5.19

69

A

CD I

0 3.0 -

2.5 -

2.0 -

c 4 0.4 7 I 0 -

X I =

t - 1 0 . 2

1.0 I I I 1 I I I J - I .o 0.0 I .o 2.0 3.0 4.0 5.0 6.0

Log T i m e ( r n i n )

F i g . 5 .17 Room Temperature Master Curve f o r 934 Resin.

Q) V c 0 - .- 1.0

o w

._ n

E g

t L Q) >

I I I

I I

/ I- / I

b I- / -

c

(0 I 0 - X Y

0.5 - 0 & Y \ c Y

0 I 2 3 4 5

Hor izonta l S h i f t ( l og a T 1

Fig . 5.18 Verti cal (Downward) Compl iance Shi f t Versus Horizontal Sh i f t .

70

0

I- O

CT 0

L - I 0 0 0 LL

- - c

c Y- .- r v,

-2

3 4 5 6 7 8 9

1 / T X

Fig. 5.19 Horizontal Shi f t Versus the Reci procal of Temperature.

reveals a discontinuity in the log aT versus T-l curve. This again i s

a n indication of a changing mechanism. The act ivat ion energy below

220°C was found t o be 4.76 kcal/gm-mole while above 220°F (104°C) a

value of 23.50 kcal/gm-mole was found.

Ins t ron s t r e s s - s t r a i n curves (Fig. 5.20) revealed a bri t t l e-

duc t i le t rans i t ion in the temperature range 200-300°F* (93449°C).

All curves have an i n i t i a l l y l inear s t r e s s - s t r a in behavior. The

values of s t r e s s and s t r a in a t t h e "linear-nonlinear t rans i t ion point"

are plotted versus temperature in Fig. 5 .21 . Based on these r e su l t s ,

a t e s t program was developed.

specimen was forced through a series of creep and creep recovery t e s t s

A t a given temperature, the same

*Probably a f t e r the o-transit ion has decayed.

71

h

.- cn n Y

cn v)

?! t tn

F i g

I0885

7257

3 0 2 8

I I I 1 1

0 I 2 3 4 5

Strain o/o

5.20 S t r e s s - S t r a i n Curves f o r ' 9 3 4 Resin Postcured a t 350 (177 C )

2

o\"

E l 5

c .-

0

I I I

0 Stress Level

A Strain Level

0 I O 0 200 3 0 0

Temperature ( d e g F 1

50

c cu E E \

25 Y

cn cn ?! 4-

tn

0

75

50 = Q r

?!

Y

cn cn

25 fj

0

F

c .- cn Y Y

cn cn ?! t tn

Fig. 5.21 Values of S t r e s s ( 0 ) and S t r a i n (A) a t the Linear - Nonlinear T r a n s i t i o n P o i n t .

72

as i n d i c a t e d i n F i g . 5.22.

temperature.

w i t h a t l e a s t e i g h t subsequent creep and creep-recovery t e s t s a t each

temperature, up t i l l 320°F (160°C) d i d n o t show evidence o f n o n l i n e a r

v i s c o e l a s t i c behavior . As an example, F i g . 5.23 shows t h a t t h e

compliance versus t ime e v o l u t i o n a t t h e h ighes t temperature, 320°F

(160°C), i s s t r e s s independent.

l ower temperature t e s t s .

A f resh specimen was used f o r each new

Test r e s u l t s obta ined a t f i v e d i f f e r e n t temperatures,

S i m i 1 a r r e s u l t s were ob ta ined f o r t h e

When we proceeded through the load ing-un load i ng cyc les ( F i g . 5.22)

we observed an accumulat ion o f nonrecoverable s t r a i n ( F i g . 5 .24 ) . The

importance o f t h i s permanent s t r a i n accumulat ion process w i t h respec t

t o t h e v i s c o e l a s t i c de format ion process can be judged when we r e p l o t

cn cn

c m

0 0

IO hr- hr

Time

F i g . 5.22 Test Sequence o f Creep and Creep Recovery Temperature and D i f f e r e n t S t ress Leve ls .

- Tests a t a S i n g l e

73

Q

0 V € 1

1629 psi

w I .35 - X

.30 - Y

0 Q i s \

.25 - Y i 0 3956 psi

1.5 1 - I .oo 0.00 I .oo 2.00 3.00

Log Time ( m i n ) F ig . 5.23 Comparison o f Resin Compliance f o r Low (1.63 k s i (11.2 MPa))

and H igh (3.95 k s i (27.3 MPa)).

(MPa)

0 I O 20 30 40 200

I50

I O 0

50

O Z 0 1 2 3 4 5

Stress ( K s i 1

F ig . 5.24 Permanent S t r a i n Accumulation Versus S t ress ( f o r 934 Resin a t 246°F (119°C)).

74

Fig . 5.24 w i t h a vert7cal scale R , being defined as:

t o t a l permanent se t accumulated after k creep x 100% and creep-recovery cycl es

the time dependent deformation a t s t r e s s level uk

=

T h i s f igure (5.25) shows t h a t the permanent s e t accumulation tends t o

30% o f the t ransient deformation a t the highest s t r e s s levels . The

nonlinearizing parameters g and a, became s t r e s s independent when we 2 subtracted the accumulated s t ra in (Figs. 5.26 and 5.27). This leads

t o the conclusion t h a t the matr ix material behaves 1 inear viscoelastic

a t lower s t r e s s levels and l inear viscoelast ic-plast ic a.t higher s t r e s s

levels .

I O 20 30 40 c.

o\" r I I 1 1

I Y

a

v)

c Q) c 0

c

E L

2 0) > 0

.- c - Q) a

30

20

10

1

I I I I I I 2 3 4 5 6 I 2 3 4 5 6

Stress ( K s i )

Fig . 5.25 Relative Importance o f Permanent Deformation Compared t o the Viscoelastic Deformation fo r Different Stress Levels.

75

0 20 30 40

1.0 ' 0 O O v

tu a,

L

c a

Stress ( Ksi 1

Fig. 5.26 (a) Nonlinear Parameter g for 934 Resin with Subtraction o f Accumul ated Permangnt Strain.

N 0

1.0 L

c a a E - 0, 0.5 0 a

a 5

0 0 I 2 3 4

Stress ( K s i 1

Fig. 5.26 (b) Nonlinear Parameter g for 934 Resin Without Subtrac- tion of Accumulated Pgrmanent Strain.

76

1.0

0.5

0

b 0

I I 1

0 0 " " - 'oe

- -

I I I I I

F ig . 5.27 ( a ) S h i f t Fac to r f o r 934 Resin w i t h S u b t r a c t i o n o f Accumulated Permanent S t r a i n .

b 0

'c t

( MPa 1 0 IO 20 30 40

I I 1

1 .O

0.5 -

0. I I I I I

0 I 2 3 4 5

Stress ( K s i )

Fig. 5 .27 ( b ) S h i f t Fac to r f o r 934 Resin Without S u b t r a c t i o n of Accumulated Permanent S t r a i n .

77

Chapter 6

EXPERIMENTAL RESULTS AND DISCUSSION F U R T300/934 COMPOSITE

6.1 I n t r o d u c t i o n

Only creep measurements were made i n our p rev ious s tud ies [4-131.

The Schapery non l i nea r v i s c o e l a s t i c method descr ibed e a r l i e r cou ld n o t

be used as i t requ i res bo th creep and creep recovery data. The present

i n v e s t i g a t i o n was undertaken t o o b t a i n t h e proper i n f o r m a t i o n on

T300/934 laminates such t h a t t h e Schapery a n a l y s i s cou ld be developed.

We a l s o wished t o compare t h e r e s u l t s o f o u r measurements w i t h new

batches o f T300/934 f rom a new source, F i b e r i t e (see Appendix A f o r

c u r i n g deta i 1 s ) , and e a r l i e r batches suppl i e d by Lockheed-Sunnyvale.

A f u r t h e r o b j e c t i v e was t o compare t h e n o n l i n e a r v i s c o e l a s t i c p roper -

t i e s o f the T300/934 composite w i t h those o f t h e 934 r e s i n g i ven i n

Chapter 5.

The p lane s t r e s s c o n s t i t u t i v e p r o p e r t i e s o f a l i n e a r o r t h o t r o p i c

ma te r i a1 i s comple te ly c h a r a c t e r i zed by f o u r independent p r o p e r t i e s . These may be t h e compliances, Dll, D12 = D21, D22 and D 6 6 .

two, Dll and D12, can be determined f rom a t e n s i l e t e s t o f a u n i -

d i r e c t i o n a l laminate t e n s i l e coupon w i t h t h e f i b e r s i n t h e l o a d d i r e c -

t i o n (see F i g . 6 .1 ) . D Z 2 and D66 can be determined from t e n s i l e t e s t s

o f a u n i d i r e c t i o n a l laminate w i t h t h e f i b e r s t ransverse t o t h e l o a d

a x i s and the f i b e r s a t 10" t o t h e l o a d a x i s r e s p e c t i v e l y . Performing

creep t e s t s t o express these parameters as a f u n c t i o n o f t ime permi ts

t h e c a l cul a t i on o f o r t h o t r o p i c 1 i near v i scoel a s t i c response f o r an

a r b i t r a r y d i r e c t i on o f t h e f i bers .

The f i r s t

78

Q 2

F i g . 6 .1 Off-Axis Test Geometry and Internal Stress State .

Normally, because the graphite f ibers of a GI/E laminate are very

s t i f f and nearly time independent, creep t e s t s t o determine time

dependent responses a re on ly performed fo r the 90" and 10" cases

respecti ve 1 y . I f an orthotropic material i s nonlinear, D2* i s not only a func-

t ion of a2 b u t may be a function of al and T , ~ as well [ ls] .

r i s e t o an interaction among s t r e s s components not present fo r a

This gives

l inear ly viscoelast ic orthotropic material.

t h i s interact ion e f fec t , various methods have been employed usually

based on the introduction of a flow rule. Such an approach i s not

I n order t o account f o r

always straightforward due to the possibi l i ty of various s t r e s s i n t e r -

actions a t the microlevel. A simple method t o account fo r interact ion

79

e f fec t s i s t o replace the one-dimensional s t r e s s by the octahedral

shear s t ress . This stems from the in tu i t i ve hypothesis t ha t creep

caused by the shearing 05 the polymer molecules past one another.

A somewhat more i n t r i c a t e approach was proposed for f ibe r re

forced materials by Lou and Schapery [49] and was i n t u r n used by

D i 11 ard [9]. Their procedure u t i 1 i red the octahedral shear s t r e s s

i n the mat r ix phase as a nonlinearizing parameter. The relationsh

i s

n -

on1 y

P

between the octahedral s t r e s s i s the matrix ( T~~~ ] and the plane s t r e s s

components ( u ~ , u ~ , T ~ ~ ) shown i n F i g . 6 .1 was given through a micro-

mechanics model. The compliance model proposed by Dil lard [9] was

given as ,

sion

I

€1

€2

Iy1 i

Dl 1

D l 2 0

0 D l 2 D22 (t,T;ctj 0

0 D66 It "!ct] An a l te rna te

of Eq. (3.18)

or s , r e s componen

approach would be the use of the f ree energy expan-

khich could accommodate higher orders of the s t r a in

s or t h e i r invariants. Hahn and Tsai [83] used a

fourth order s t r a in expansion of the complementary energy density t o

handle shear nonl inear i t ies . I n other words, temperature was excluded

from the Gibbs f ree energy o f the system. Me feel t h i s approach should

be extended and explored fur ther .

Nonlinear viscoelastic e f f ec t was not observed i n the transverse

During the f inal cornp la t ion of direction for our T300/934 material .

the results i t was realized t h a t t h i s resu l t should be reexamined,

since there i s some indication tha t the nonlinear mechanism could

80

opera te on such a l o n g t ime scaTe t h a t i t becomes imposs

d i s c r i m i n a t e between m a t e r i a l s response and exper imental

b a s i s o f s h o r t t ime acce le ra ted c h a r a c t e r i z a t i o n r e s u l t s

6.2 Resul ts and Discussion on [90]8s

b l e t o

e r r o r on t h e

The lam ina te used i n t h i s i n v e s t i g a t i o n c o n s i s t e d o f s i x t e e n u n i -

d i r e c t i o n a l p l i e s , w i t h about s i x t y pe rcen t volume o f T300 g r a p h i t e

f i b e r s i n a m a t r i x o f F i b e r i t e - 9 3 4 epoxy r e s i n .

T e n s i l e specimens were c u t from a u n i d i r e c t i o n a l p l a t e by means

o f a diamond c u t t i n g wheel, postcured a t 350°F (177°C) f o r 4 hours and

dess i ca ted as discussed e a r l i e r .

creep and creep-recovery t e s t s were performed i n t h e same manner as

those f o r t h e r e s i n shown i n Fig. 5.12. The s t r a i n response was

A s e r i e s o f e l e v a t e d temperature

t h e e f f e c t s o f bend-

e c c e n t r i c i t y ( e ) o f

on as shown i n F i g .

which i s 6% o f t h e

a i l in Appendix C . )

measured us ing back t o back s t r a i n g a u g e s t o avo id

i n g . P a r e n t h e t i c a l l y , i t i s e a s i l y shown t h a t an

1% o f t h e depth ( d ) o f t h e rec tangu la r cross sec t

6.2 causes a bending s t r e s s a t t h e gauge l o c a t i o n

average s t r e s s u ~ . (Th is i s discussed i n more de

F i g . 6.2 Poss ib le Bending o f T e n s i l e Specimen

81

The e c c e n t r i c i t y i n creep t e n s i o n t e s t s has been es t ima ted by

Isaksson [84] t o be o f t h e o r d e r o f 5% o f t h e depth ( d ) causing a 30%

d e v i a t i o n o f t h e average s t r e s s a2 .

The change o f t he gauge f a c t o r w i t h temperature was neg lec ted

because i t was t y p i c a l o n l y 1.5% f o r a temperature change from room

temperature up t o 320°F (177°C).

Another exper imental n e c e s s i t y was t o avo id heat leakage through

t h e l ower oven l o a d t r a i n entrance as i l l u s t r a t e d i n F ig . 6 .3 . E f f o r t s

t o i n s u l a t e t h i s ent rance i n t h e same way as t h e top l o a d t r a i n

ent rance caused i n t e r f e r e n c e w i t h t h e recovery data. The r o d t h a t was

used t o connect and d isconnect t he specimen w i t h the lower and movable

end o f t h e l o a d t r a i n needed t o be complete ly f r e e . A s a t i s f a c t o r y

s o l u t i o n f o r t h i s problem was t o p lace a heavy mass o f metal a t t he

l ower entrance as shown i n F i g . 6.3b. T h i s "heat s i n k " conta ined an

F i g . 6.3 Loading and Oven I n s u l a t i o n D e t a i l s

82

opening t r o u g h which t h e cqnnect ing r o d c o u l d move f r e e l y , b u t a t t h e

same t i m e a l lowed a uni form oven temperature t o be mainta ined.

The t ransve rse compliance as a f u n c t i o n o f t ime f o r d i f f e r e n t

temperatures i n t h e range 200-320°F (93-160°C) i s shown i n F i g . 6.4.

An expanded v e r t i c a l sca le was used as compared w i t h t h e s i m i l a r r e s i n -

compliance p l o t o f F ig . 5.3, because the cont inuous i n t e r r u p t i o n s of

t h e m a t r i x by t ime independent f i l a m e n t s p r o h i b i t e d a l a r g e t ransve rse

creep s t r a i n b u i l d u p .

A d d i t i o n a l t e s t s a t d i f f e r e n t s t r e s s l e v e l s were performed f o r t h e

t ransve rse compliance as a f u n c t i o n o f temperature as shown i n F ig . 6.4.

However, these a r e n o t shown as the compliance appeared t o be inde-

pendent o f s t r e s s l e v e l , i . e . , was l i n e a r , f o r t h e t ime sca le o f our

t e s t s as w i l l be discussed subsequently.

n

I - .- a i.0 r Y

(u tu n Q) u C 0.9 I

0.8

0.7

0.6

r

I., 0.0 I .o 2 .o 3.0

Log Time ( rn in )

F ig . 6.4 Transverse o r 022 Compliance o f T300/934 as a Func t i on o f Time and Temperature

83

The inve rse compliance a t t z 1 min i s p l o t t e d versus temperature

S i m i l a r r e s u l t s ob ta ined by Yeow [5] and G r i f f i t h [7 ] a r e i n F ig . 6.5.

p l o t t e d on t h e same graph, t o g e t h e r w i t h a micromechanics p r e d i c t i o n ,

based on the p r e v i o u s l y measured r e s i n modulus as a f u n c t i o n o f

temperature.

Temperature ( d e g C

I I

IO

9 -

8 -

- n

cn 1-

- = Y

cn 3 3 D

- 2 Q, cn Q, > cn c

L

z I-

-

50 I O 0 I50 I I I I 1

I .5

Micromechanics Analysis

4- Halpin - Tsa i ( 5 4 )

E xper I men t

0 Current

a Grif f i th (7)

0 Yeow (6)

0 i 1.0 I I 1 1

I O 0 200 300

Temperature ( d e g F

T300/934 1 /D22 (1 min) Versus Temperature

The micromechanics equa t ion used t o produce F i g . 6.5 was t h e

Hal pin-Tsai equa t ion [54]

where Ef/Em - 1 ' = Ef/Em + 5

i n which 5 i s a measure o f re in fo rcemen t and depends on t h e f i b e r

geometry, pack ing geometry and 1 oading condi t i o n s as d i scussed i n [541.

84

I

Equation 6 .2 i s valid only fo r an e las t ic material b u t can be extended

t o the viscoelast ic problem us ing the so-called "quasi-elastic"

approach.

are found , an approximate viscoelastic solution can be obtained by re-

placing e l a s t i c constants by time dependent modul i .

t h a t f o r a creep time of several hours, this quasi-elast ic approach

deviates 1 i t t l e from the more theoretically justif ied Laplace transform

inversion approach.

E ( t ) = l / D ( t ) , which, according t o Aklonis and Tobolsky [86], does not

introduce an e r ror larger t h a n o u r experimental uncertainty i n deter-

mi n i ng the compl i ance.

I t means tha t i f the solutions t o the e l a s t i c f i e l d equations

Sims [8S] showed

I n F i g . 6 .5 i t was a lso t a c i t l y assumed tha t

As may be observed, good agreement was o b t a i n e d between the Halpin-

Tsai micromechanics equation and current and previous experimental work.

Such i s the case even though the model i s based on isotropic properties

on both f ibers and matrix even though the graphite f ibers are known

to be very anisotropic. This confirms a statement by Whitney [87]

which indicates t h a t a straightforward subst i tut ion of appropriate

isotropic constants appears t o be adequate t o account fo r transversely

isotropic f ibe r properties.

Micromechanics solutions based upon an exact or approximate solu-

t ion to the f i e l d equations have been developed, however, which include

complete anisotropy of the f ibe r s .

bound method of Bulavs e t a l . [go] and Hashin [91]. Experimental

studies have been conducted by Ishikawa e t a l . [88] and Thompson [89]

which show the Thornel 300 h i g h strength f ibers t o be greatly aniso-

t rop ic . These resul t s are shown i n Table 6 .1 .

These include the upper and lower

85

Table 6.1

Ish ikawa e t a l . [85] Thompson [89] (T-300A) (T-300, WYP 1 5 )

32.5 (MSI) 1

E22 2.93 (MSI)

.2

.42

w1 2

'23

6.82 (MSI) G1 2

A comparison o f Halp in-Tsai equat ions (6 .2 ) , Hash in 's upper and

lower bound method w i t h a n i s o t r o p i c p r o p e r t i e s , Hashin's equat ions w i t h

equ iva len t l o n g i t u d i n a l (Efl) and t ransverse (Ef t ) f i b e r p r o p e r t i e s ,

Bu lav 'S

F i g . 6.6.

f i rmed .

Square a r r a y method and ou r c u r r e n t exper iments a re g i ven i n

As may be observed, N h i t n e y ' s conc lus ion i s once aga in con-

Hashin 's upper and lower bound r e s u l t s a re shown i n F i g . 6.7 as

a f u n c t i o n o f t h e r a t i o o f l o n g i t u d i n a l t o t ransverse f i b e r modu l i .

Equat ion 6.1 was modi f i e d t o i n c l ude t h e m i cromechani cs approach

Using the q u a s i - e l a s t i c approach mentioned pre- g i ven by Eq. 6.2.

v i o u s l y p r e d i c t i o n s o f t h e t ime dependent t ransverse modulus were made

and these a re g i ven toge the r w i t h exper imenta l r e s u l t s i n F i g . 6 .8.

Good agreement was ob ta ined which would i n d i c a t e a s imple t ime dependent

micromechanics ana lys i s t o be a reasonable approach t o represent o u r

da ta over t h e t ime range invo lved .

86

.- s n n

c .- E c

Y

ru ru a \ c

Y

cn 3 3 = -

s a cn a > cn c

L

+

2.0

I .5

n

cn .- 5 Y

cn

I .o

I I

P

‘P + Hashin (91) ( upper q’ and lower bound 1 4- Bulavs et a 1 . ( 9 0 )

( Square array 1 -a- Hashin ( 9 1 )

(E , , = E,, = 2 . 9 3 Ms

+ Halpin-Tsai ( 5 4 ) I I Current ExDeriment

15

IO

5

a u)

a > cn c

L

c I-

F i g . 6 . 7 E v o l u t i o n t h e Rat io

0.0 0.5 I .o 1.5

Fiber Volume ( V , )

I .7

I .6

I . 5

F i g . 6 .6 Modulus P r e d i c t i o n Based on D i f f e r e n t Micromechanical Models

I

87

I I

Upper Bound

Lower Bound

11.5

11.0

10.5

0 a E

0 X

h

lo

- Y

0.0 0.5 I .o

E f l ’€ft

o f Upper and Lower Hashin Bound as a Func t ion o f o f Long i tud ina l and Transverse Modul i

+ 1.6 - - Experimental 1 / D,, ( t 1

0 Micromechanics ( Halpin -Tsai eqn )

11.0 1 % n

In 0

X - Y

0 - 10.5 A

1.5 . 1 I I I 1 1 0.0 10.0 20.0 30.0 4 0.0 50.0 60.0

F ig . 6.8 Comparison Between Exper imental ( - ) and Micromechanics ( A ) P r e d i c t i o n o f E22 Versus Time (T300/934

An at tempt was made t o model t h e creep response w i t h a Schapery

t ype power law equa t ion

where a l l q u a n t i t i e s a r e as d e f i n e d p r e v i o u s l y i n Equat ion 3.49.

upper index T i n d i c a t e s t h a t t h e n o n l i n e a r i z i n g f u n c t i o n s a r e evaluated

i n t h e t ransverse f i b e r d i r e c t i o n .

The

A s e r i e s o f 40 (one hour) creep and ( t e n hour) creep recovery t e s t s

a t e i g h t d i f f e r e n t s t r e s s l e v e l s and f i v e d i f f e r e n t temperatures, us ing

a f r e s h specimen a t each new temperature, d i d n o t show any dependence of

t h e t ransverse compliance on the s t r e s s l e v e l . I n o t h e r words,

which means t h a t t h e t ransve rse response was l i n e a r w i t h i n our t ime

frame and w i t h i n exper imental e r r o r .

88

The stress-strain diagram (Fig. 6 . 9 ) ob ta ined from an I n s t r o n

t es t a t 100°F (37°C) a l s o shows l i t t l e d i s s i p a t i o n and a f r a c t u r e s t r a i n

of about

- lu E E Z \

Y

v) v)

c cn

F i g . 6 . 9

5000 m i c r o s t r a i n . \

h .- v) Y Y

X Rupture

Strain (%)

S t r e s s - S t r a i n Curve (Sol id Line) f o r [go]& T300/934 Laminate a t Room Temperature (0.05 in /min) Compared with Linear Response (Dashed L i n e ) .

Equation (6 .3 ) reduces t o

D ( t ) = Do + D1 tn ( 6 . 4 )

F igure 6 .10 shows the tempera ture dependence o f the f i r s t term ( D o ) i n

Eq. ( 6 . 4 ) . I t i s c l e a r t h a t the dependence i s a lmost p e r f e c t l y l i n e a r

i n the tempera ture i n t e r v a l which we s t u d i e d . T h e tempera ture dependence

o f the c o e f f i c i e n t D1 has a pronounced n o n l i n e a r v a r i a t i o n a s can be

seen i n Fig. 6.11.

the nea t r e s i n ( F i g . 5 . 2 ) i s a l s o c l e a r l y v i s i b l e i n Fig. 6.11.

the min imum i n the curve i s now approximately 20°F ( = 10°C) h ighe r .

Notice t h a t t h e w-mechanism w h i c h was i d e n t i f i e d i n

However

89

0.74

h - I

u) 0.72

Y

n

I- - Oo 0.70

Temperature ( d e g C

I O 0 I50

0.105 lo 0 X

0 Q z \

- Y

h

c Y

0. IO0

0.68 2 00 300

Temperature ( d e g F )

Fig. 6.10 Ins t an taneous Transverse Compl i ance a s a Function o f Tempera t u re

0.04

n

I

v)

- .-

0.03 Y - I-

0

Y - 0.02

Temperature (deq C I O 0 I50 I I

I 0.12

6

n

5 Q, I

0 - X Y

4 n

0 Q Y \ v 3 Y

2

200 300

Temperature (de9 F )

Fig. 6.11 D1(T) a s a Funct ion o f Temperature

90

Such differences have been reported f o r the primary t r ans i t i on by a

number of invest igators [ 7 6 , 7 9 , 9 2 ] , b u t the cause i s unclear.

The long time creep da ta plotted i n F i g . 6 .12 reveals t h a t ou r

experimental time scale for short time creep ( 1 hour) was probably too

shor t t o be able t o discriminate between s t r e s s dependence o f the

compliance curve and small experimental i r r e g u l a r i t i e s such as mis-

alignment i n the load t r a i n , f luctuation i n temperature, e t c .

other words, the shorter the t e s t period, the more precise the t e s t

conditions need be i n order t o allow s t r e s s dependent discrimination of

the compliance curve. Thus our e a r l i e r conclusion abou t the l inear

v iscoe las t ic or s t r e s s independence of our transverse compliance data

f o r T300/934 needs t o be reexamined over a longer time sca le .

Or, in

Howeve!-

- 0 X -

I I I I - 0.2 A Current Research (5.18 K S I )

-. c

I- I I 1 I : 0.0 1

4 0 -1.0 0 0 I .o 2.0 3.0

Log Time ( m i n )

F i g . 6 . 1 2 Comparison of Transverse Compliance Versus Time for Moderate ( 2 . 7 6 Ksi) 1 9 Mpa) and High (5.18 Ksi = 35.7 Mpa) Stresses

91

f o r a f i rs t approximation, l i n e a r would be a reasonable assumption.

6.3 Results and Discussion on [10]8s

6.3.1 Experimental Background

An e f f i c i e n t intralaminar ( in plane) shear characterization tech-

nique known as the 10-deg off-axis t e s t has been proposed and analyzed

by C . C . Chamis and J . H . S inc la i r [93].

major contribution t o f rac ture comes from intralaminar shear, i .e. the

configuration ac t iva tes a long continuous shear path, along w h i c h load

diffusion i n t o the reinforcing f ibers occurs. One d i f f i c u l t y w i t h the

off-axis t e s t i s t h a t shear coupling effects a r e a function of the load

t o fiber a n g l e o r the aspect r a t i o .

coupling analysis was used t o generate F i g . 6.13.

gest t h a t f o r an aspect r a t i o or off-axis angle of l o " , the shear

modulus, G,2)approx. 9 i s approximately 10% higher than the true shear

modulus G,*.

temperature p l o t of Fig. 6.14 w h i c h indicates a measured value some-

what above .8 MSI a t room temperature, while a widely p u b l i s h e d value

i s c loser t o .7 MSI.

Their analysis shows t h a t the

The Pagano-Halpin [94] shear

Their analysis sug-

This is confirmed by the inverse shear compliance ver:us

Two back-to-back 45 deg rose t te s t r a i n gages [WK-06-062R-350]

were used t o eliminate out of plane bending from test r e su l t s . The

orientation of the gages w i t h respect t o the f iber ax i s was checked

under the microscope. Mi sal ignments were taken in to account during

subsequent data reduction. Figure 6.15 shows the or ientat ion of the

s t r a i n gage rose t te a s well a s the manner i n w h i c h aluminum tabs were

adhesively bonded t o each specimen.

92

I .o

u) 2 1.0

n

r

Y

.- E c

Y

rD CD 0

c

u) 3 3 U

, 0.5

- r" L

g 0.0

0.5 10 20 30 40 50 60 70 80 90

i i I +

9 - 5

- 0 Experiment

1 I I 0

O f f - A x i s Angle

Fig. 6.13 Ratio o f Actual t o Apparent Shear Modulus i n O f f - A x i s Test.

A

0 a c3 Y

Fig . 6.14 Shear blodulus a s a Function o f Temperature.

93

0.50" - 12.7 m a

t

- 9.0" - 6.0' - 228.6 am 162.4 am

L I

0.075" - t 1.9 an

' C

Grippinq reqian }

*

-

/

- \

Uniaxial Gage

F i g . 6.15 Specimen, Strain Gage and Tab Configuration.

Each specimen was postcured with the same temperature cycle as

were the transverse and neat resin coupons.

The shear compliance D s 6 ( t ) for a 10" specimen loaded i n tension

as illustrated schematically in Fig. 6.1 i s given by the following

expression

w.h-(-.598 sx ( t ) + 1.879 E 4 , = ( t ) - 1.282 8 ( t ) ) D66(t) = ,171 ax

where e , ( t ) , ~ ~ ~ ( t ) and s ( t ) are the strains measured along the

longitudinal , 45" and transverse directions of the specimen , respectively.

The angle between the measured strain directions and the fiber-direction

of the material are 10" , 45" and 100" respectively.

Y

W and h represent

the w i d t h and thickness of the tens i le coupon.

along the x and 45" direction were posi t ive, while the s t ra in measured

along the y-direction was negative.

i f the rectangular rose t te i s mounted w i t h the 90" and 45" arms d i f -

fe ren t than shown i n Fig. 6.15, different signs will be obtained and

the preceding equation for D66 must be a l tered appropriately.

compliance 066 i s t h u s eas i ly obtained t h r o u g h simultaneous measurement

o f e x ( t ) , ~ ~ ~ ( t ) and E ( t ) b u t caution should be taken w i t h respect t o

the gage or ientat ion.

The s t r a ins measured

I t should be noted, of course, t ha t

The shear

Y

6.3.2 Experimental Results

The shear compliance versus log t curves obtained a t f o u r d i f -

ferent temperatures a re shown i n Figs. 6.16a, b , c and d .

indicate the behavior t o be dependent upon s t r e s s .

nonlinear v i scoe las t ic i ty , as discussed e a r l i e r i n Chapter 3 , t h i s means

tha t a s e t o f molecular processes which a re sensi t ive t o s t r e s s must

a lso contribute to the observed deformation i n addition to those nor-

mally associated w i t h l inear viscoelast ic i ty .

They c lear ly

I n the context of

A def in i t e accumul ation o f nonrecoverable permanent s t r a i n was

observed d u r i n g the creep and creep-recovery test sequences.

o b t a i n e d a t 246°F (119°C) i s shown as an example i n F i g . 6 .17 .

molecular s t ructure moves o u t o f the positions which i t occupied on

release o f the s t r e s s b u t does not reach i t s former spat ia l arrangement

under the action of the restoring forces. One poss ib i l i ty i s t ha t t h i s

former spatial arrangement i s unattainable due t o a "p las t ic i ty" type

s t ructural change i n the matrix phase. Thus, for our material i t seems

The amount

Thus the

95

- I .-

2.5 1 I I I I 1 u) I 0 X

Q

- 2.0 '

(D (D

5160 psi 0 3557 psi

A 4758 psi 0 2756 psi A 4358 psi 0 1955 psi + 3950 psi 0 I154 psi

- A

3.0 'f 0 X - Y I c

Y

I 1 1 I I a -1.0 0.0 I .o 2.0 3.0 4.0 c cn

Log Time ( m i n )

Fig. 6.16a Shear Compliance Versus Time a t 246°F (119°C).

0 5110 ps i 0 4317 psi 0 3523 psi 4 3.0

2.0

Log Time ( m i n )

A

(D I 0

X

0 c1 * \

- Y c

c U t 1 1.0

0.5 -I .o 0.0 I .o 2.0

Fig. 6.16b Shear Compliance Versus Time a t 300°F (149" C ) .

96

I I I I 2.5

1 I 1

- + 5210 psi

.- 0 4401 psi tn 3.0 - n 0 3592 psi

0 3187 psi - 0 0 2378 psi

0 1569 psi x 2.5 - 0

I

cg l

- u) u)

al -

-

2.0

I .5

0 3521 psi

0 3125 psi 0 2331 psi 0 1538 psi I

4.0

3.0

2.0

I I I I I I - I .o 0.0 I .o 2.0 3.0 4.0

Log Time (min )

F i g . 6 . 1 6 ~ Shear Compliance a t 320°F (16OOC).

-1.0 0.0 1.0 2 .o 3.0 4.0

Log Time ( m i n )

A

(D I

0 X - Y A

0

z \

n

c

Y

n

(D 1 0 - X Y

n

0

Y \

a

c

Y

Fig . 6.16d Shear Compliance Versus Time a t 335OF (168°C).

97

- c .- \ c .- =t Y

a w

400

300

200

I O 0

0

6.0 12.0 18.0 24.0 30.0 36.0

0 14.91 sinh ( .0007637 T

6.0 12.0 18.0 24.0 30.0 36.0 I I I I I I

P

0 14.91 sinh ( .0007637 T

- 0.0 I .o 2.0 3.0 4.0 5.0 6.0

7,Z ( k s i 1

Fig. 6.17 Permanent Strain Accumulation Dur ing 10 Deg. Off Axis Test a t 246°F (119°C).

not possible fo r nonl i near v i scoel a s t i c i ty t o occur without i nvol v i ng

other i r revers ible deformation mechanisms. This conclusion disagrees

w i t h resul ts obtained by Ferry [28] on PMMA. Shear creep deformation

u p t o 5% were fu l ly recoverable although the l i nea r viscoelast ic l imi t

was below 1%.

A number of investigators mechanically condition specimen prior

to creep tes t ing.

cer ta in amount of deformation takes place which does not recover upon

removal o f t h e s t r e s s .

a t the same or smaller load there i s no such nonrecoverable creep. The

s t ress -s t ra in relationship thus becomes repeatable. The material i s

therefore considered “mechanically conditioned. ‘I

The reason i s tha t d u r i n g the f i r s t creep t e s t a

In subsequent creep t e s t s on the same specimen

Leaderman [22] showed

98

t ha t the permanent s e t i n f ibers disappears when the f ibe r was steamed

under no load and then dried. Thus no real s t ructural change o r

p l a s t i c i t y was introduced by the conditioning process.

T h i s author f ea r s , however, t h a t a s ignif icant volume of the

matrix material might be yielded by the very inhomogeneous s t r e s s f i e l d s

caused by the geometrical nature of the composite material . Evidence

of such a matrix yielding phenomenon i n shear deformation, based on a

f i n i t e element analysis , was published by Foye [95].

i n F i g . 6.18a give some indication of the progress of the p las t ic zone

boundaries due to shear loading i n repeating elements of a g r a p h i t e -

epoxy composite w i t h 50% f i b e r by volume. Yielding i n i t i a t e s a t the

interface i n an area enclosed by p las t ic zone boundary number 1 . The

p l a s t i c zone boundaries numbered 2 through 8 correspond t o increasing

postyielding load levels .

i n transverse deformation, yielding f i rs t occurs a t a p o i n t i n the

matrix midway between two adjacent f ibers .

yielding of the composite during the transverse deformation mode occurs

a t s l i gh t ly lower average normal s t ress levels t h a n the bulk matrix.

I n the shear deformation mode the composites yielded a t much lower

average shear s t r e s s levels than the b u l k matrix.

His resu l t s shown

Figure 6.18b, on the other hand, shows tha t

Foye also showed tha t i n i t i a l

In conclusion, preconditioning was n o t made p a r t o f the tes t ing

program since i t was feared tha t the s t ress -s t ra in relationship would

become repeatable a t the expense of changing the matrix phase to an

en t i r e ly new material i n the sense of a p l a s t i c i ty type shakedown model.

99

Matr ix

Fiber

Fig. 6.18a Inhomogeneous Stress Distribution Due t o Shear Deformation Due t o Shear Deformation After Foye [95].

Mat r i x

Fiber

F i g . 6.18b Inhomogeneous Stress Di s t r i bution Due t o Transverse Deforma- t ion After Foye [95].

The creep response in the shear deformation mode was modeled with

the power law

8 00

The upper index (S) indicates t h a t the nonlinearizing functions a re

evaluated i n the shear mode. Plots o f the nonlinearizing functions

9, (’) 9 91 9 92 (’), and a T obtained a t 246°F as a function of s t r e s s are

shown in Figs. 6.19a t o 6.19d.

temperatures. I t can a l so be seen t h a t g i S ) and ghs) vary i n such a way

w i t h s t r e s s leve l , tha t t he i r product i s nearly u n i t y and equation (6 .5)

S imi la r curves are obtained a t higher

for creep response becomes approximately

I

6.3.3 Time-Stress Superposition Principle (TSSP) I

I A very useful viscoelast ic concept i s t h a t time and s t ress are 1

equivalent for describing viscoelast ic behavior [7]. This i s due t o the

fac t tha t an increased t ens i l e s t r e s s s h i f t s the creep spectra towards

shorter times as shown i n Fig. 6.20a. Thus, as shown i n Fig. 6.20b,

creep observed for short times a t a given s t r e s s T~ i s identical with

creep observed for longer times a t a lower s t r e s s TO provided tha t these

d a t a are shif ted on a logarithmic time axis . Similarly, portions of the

P creep curve for T = r can be observed for s t resses T = T~ , T ~ , . . . , as 0

indicated by the curved segments shown i n Fig. 6.20b. These curved

segments can then be shif ted along the log time axis t o construct a t composite curve o r the sigmoidal master curve applicable fo r the given

I s t r e s s level T ~ , extending over many decades of time. I

Sheor Stress ( M P O )

0 10 20 30

1

0 I 2 3 4 5

Shear Stress ( K s i 1

F i g . 6.1 9a Nonl i n e a r Parameter go ( T ) .

Shear Stress (MPo)

0 I O 20 30

0.5 41

0 I 2 3 4 5

Shear Stress ( K s i )

F ig . 6.19b Nonlinear Parameter g, ( T ) .

102

Shear Stress ( M P a )

0 IO 20 30

0 I 2 3 4 5

Shear Stress ( K s i )

Fig. 6 . 1 9 ~ Nonlinear Parameter g2(T).

I .o

a * 0.5

Shear Stress (MPa)

0 IO 20 30

1 1 I I I I I 0 I 2 3 4 5

Shear Stress ( K s i 1

Fig . 6.19d Nonlinear Parameter a (T). T

103

Log Time

Fig. 6.20a In f luence of S t r e s s on Reta rda t ion Spectrum.

Log Time

Fig . 6.20b Dependence o f Compliance D on log t and S t r e s s T .

104

The loga r i thm of the sh i f t f a c t o r aT f o r a p a r t i c u l a r curved

segment i s the hor i zon ta l shif t necessary t o a l low i t t o j o i n smoothly

i n t o the mas te r curve.

be d e s c r i b e d by

Each of the curved segments i n F ig . 6.20b can

(6 .7 )

T h i s same equat ion has a limited c a p a b i l i t y t o represent a master

cu rve us ing the form,

The reason f o r this l i m i t a t i o n i s due t o the approximate n a t u r e o f the

power law used and was d i scussed e a r l i e r i n Chapter 2 .

The t ime-s t r e s s - supe rpos i t i o n principle (TSSP) described above

was used t o conve r t the d a t a o f Figs . 6 .16a, b , c and d t o mas te r

cu rves f o r each tempera ture and are given i n F igs . 6 .21a , b , c and d .

Also, equa t ion (6 .8 ) was used t o a n a l y t i c a l l y de te rmine each mas te r

cu rve and the r e s u l t i n g s h i f t parameters .

mas te r curves produced g r a p h i c a l l y wh i l e the s o l i d l i ne represents the

a n a l y t i c a l l y produced master curve.

found by the two methods a r e shown i n F igs . 6 .22a , b and c . A three-

dimensional ve r s ion of these results i s shown i n F ig . 6 .23 . As may be

observed , exce l 1 ent agreement was ob ta ined between g raph ica l l y and

a n a l y t i c a l l y produced mas ter curves .

The symbols r e p r e s e n t the

Comparison o f the s h i f t f a c t o r s

105

v) p 2.5

(D I

0 - X

D Q)

c

co 2'o co

u .o 1.5

2 0 V

0 1.0 L

F ig . 6.21a s66 Master Curve f o r T300/934 a t 246°F (119°C).

I I I I

5160 psi A 4758 psi

0 4358 psi - 3958 psi

0 3557 psi 0 2756 psi 0 1955 psi 0 1155 ps i

- 0.3

- - 0.2

I I I I

- I .- I I I I I I I

CL

(D I

2.5 -

- 0 2.0 - X

(0 A 5904 psi 0 51 IO p s i - 0 4317 ps i Q)

0

E 1.0 - 0 3523 p s i

0 31 26 ps i ; 0 2333 p s i

1.5 -

.- - -

0.5 - L 0 al c 0.0 - 1 I I I I I I d

Log t (min)

0.3

0.2

0.1

n

(0 I 0 X

0 a x \

- Y CI

c

Y

n

(0 I 0 X - Y

LI

c

Y

Fig . 6 . 2 1 b s66 Master Curve f o r T300/934 a t 300°F (149°C).

106

.- rn Q

2.0

10 I 0 X

Q

- CD co

I I I 0 3521 psi 0 3125 psi

- 0 2331 psi - 0.3

L 0 Q) c cn

1.5 i o.2 . .-

-I .o 0.0 I .o 2.0 3.0 4 .0

Log Time ( m i n )

n

10 I

0 X - v A

0 Q Y \ c Y

Fig . 6 . 2 1 ~ s66 Master C u r v e f o r T300/934 a t 320°F (160°C).

- I

fn -- n - 10

I 0 X

3 -

al 0 c 0 2 Q E 0

.- -

u

0 1569 psi

0 2378 psi - 0 3187 psi

0 4401 psi + 5210 psi

4 0.5 - (0

10.4 h X Y n

0 0.3 ,"

\ - Y I 0.2

L 0 I Q) -1.0 0.0 1.0 2.0 3.0 4.0 c cn

Log t ( m i n )

Fig. 6.21d S66 Master Curve for T300/934 a t 335°F (168°C).

107

c .- cn Y U

cn cn Q) L c cn 0 Q) r cn

L

5.0 -

4.0 -

3.0 -

2.0 -

- 30

- 20

- 10 0 C o m p u t e d

G r a p h i c a l S h i f t

0.0 1.0 2.0

A

0 a I Y

cn cn

3j L 0 Q) E cn

S h i f t Factor

Fig. 6.22a Cornpari son of Graphical and Ana ly t i ca l Obtained Horizontal S h i f t Fac to r .10 deg - off - a x i s for 246°F (119°C).

n

cn Y

.-

Y

tn cn

t cn

0 a L: m

L

6.0 -

5.0 -

4.0 -

3.0 -

2.0 -

1.0 1 0 C o m p u t e d

G r a p h i c a l S h i f t

S h i f t

4 0

30

20

IO

c

0 a I

0.0 1.0 2.0 3.0

Shi f t Factor

F i g . 6.22b Comparison o f Graphical and Ana ly t i ca l Obtained Horizontal S h i f t Fac to r .10 deg - o f f - a x i s fo r 300°F (149°C).

108

A .- u? Y Y

v) u? E! t v)

0 Q, c v)

L.

4.0 -

3.0 -

2.0 - 0 Computed .

Shift

Shift I .o W Graphical

30

20

n 0 a z

L

Y

u) cn

t v)

0 Q) c v)

L

0.0 1.0 2.0 3.0

Shift Factor

Fig . 6 . 2 2 ~ Comparison o f Graphica l and A n a l y t i c a l Obtained H o r i z o n t a l S h i f t Fac to r . 10 deg - o f f - a x i s f o r 335OF (168°C).

Shear Stress ( Ksi )

F ig . 6.23 Graphica l and Computed S h i f t Func t ion Sur face f o r T300/934.

109

Often, a ver t ical s h i f t as well as a horizontal s h i f t i s neces-

sary t o produce a smooth master curve.

individual compliance curves may be necessary. The type o f sh i f t ing

needed depends on the behavior of the l imiting compliances DU and DR

as a function of s t r e s s as shown schematically i n F ig . 6.24.

vertical sh i f t of 0: - DuTO produces the dotted durve Di(t) which can

be superimposed on D ' O ( t ) .

I n f a c t , even a rotat ion of

A

A rotation would be necessary when the l imiting compliances a re

affected d i f fe ren t ly by the s t r e s s in tens i ty , i . e . , when

D R T - DuT +

DRTo - D U T O (6 .9 )

r. r 0 5

S t r e s s Log Time ( m i n )

Fig. 6.24 I1 1 ustration of the Stress-Shifting-Procedure (Based on McCrum e t a l . [96] ) .

I n our case, a small ver t ical s h i f t was found necessary a t 246°F. Th

shif t however was nonexistent a t higher temperatures as shown i n F igs

S

110

6.25a, b and c , where the vertical sh i f t i s plotted versus the hori-

zontal s h i f t . I t was t h o u g h t i n i t i a l l y tha t the t rue ver t ical s h i f t

( D u T - D u T o ) , g iven i n F i g . 6 .24,

however leads t o an overprediction due t o the f a c t tha t the sigmoidal

compliance curves change location w i t h increasing s t r e s s . The impli-

cation i s tha t even when the limiting compliance values are completely

insensi t ive t o s t r e s s (which turns out t o be the case i n this inves t i -

gation) we s t i l l would record a go value d i f fe ren t from unity since the

retardation spectrum involves rapid creep on the time scale of micro-

seconds. In conclusion, very accurate short time limiting compliance

data i s needed t o obtain a proper estimate of the t rue vertical sh i f t .

As a f inal conclusion, a long time control t e s t remains the only

was equivalent t o (go - l)Do. T h i s

c r i t e r ion for checking a parti cul a r accelerated t e s t method.

long term creep t e s t s a r e currently being carried o u t and will d i c t a t e

the adaptations necessary t o achieve a proper mathematical modeling fo r

the prediction o f long term properties from short term t e s t r e su l t s .

These f ina l modeling equations should i n some way r e f l ec t the mechanism

of i ne l a s t i c phenomena operating a t the microstructural level .

These

111

n

I - .- f 0.5 Y

c * .- 0.0 v)

0 0

- .- c L

>" 0.0 -1.0 -2.0 -3.0

Horizontal Shi f t

Fig. 6.25a Cross-plot of Hor izonta l and Vertical S h i f t a t 246 Deg F (119°C)

n - I .-

0.5 I I I I .07 f A

(0 I 0 -

.- c 0.0 -1.0 -2.0 -3.0 L

8 Horizonta I Shift

- Y

Fig , 6.256 Cross-p lo t o f Horizonta l and V e r t i c a l S h i f t a t 300 Deg F (149°C)

c - I

- 0 0 .- c 0.0 -1.0 -2.0 -3.0 L

8 Horizontal Shift

Fig . 6 . 2 5 ~ Cross -p lo t o f Hor izonta l and Ver t i ca l S h i f t a t 320 Deg F (160°C)

112

Chapter 7

EXPERIMENTAL RESULTS AND DISCUSSION O N DELAYED FAILURE

AND LAMINATE RESPONSE

7.1 Introduction

The usual concept of e l a s t i c deviator ic s t r a i n energy i n which

the Tsai-Hill anisotropic f a i l u r e c r i te r ion is based cannot be carr ied

over d i r ec t ly t o v iscoe las t ic materials without modifications t o

account for diss ipat ion mechanisms. However, a be t t e r method t o i n -

clude d iss ipa t ion as well as the en t i re his tory of deformation i s the

concept of f r ee energy which can be used t o def ine a " c r i t i c a l " stress

s t a t e a t the point of f a i l u r e . A delayed failure model based upon

these concepts was developed i n Chapter 4.

The delayed f a i l u r e model requires information on both creep

and on creep rupture.

and [60]8s or ientat ions of T300/934 was avai lable from the work of

Gr i f f i t h [7].

[60]8s creep response t o compliment the [60]8s rupture data can be

inferred from the [90Igs and

i t was decided t o obtain the needed [60]8s creep response from new

specimen because inferred response would introduce a hypothetical

octahedral s t r e s s interact ion factor , TOCt.

An ex is t ing creep rupture data base fo r [90]BS

[90]8s creep data i s given in Chapter 6 . The needed

data given i n Chapter 6. However,

m

7.2 Delayed Fai lure

7.2.1 Analysis on Results

The compliance versus time curve a t 320°F (160°C) and 1580 psi

(10.88 MPa) ( F i g . 6.8) was f i t t e d w i t h a six-element Kelvin model,

113

using a collocation scheme proposed by Schapery [97].

compliance D ( t ) can be expressed as :

The resul t ing

on the power law parameters ( D o , D 1 , n ) which made i t possible t o

evaluate Eq. (4 .17) as a function of time.

def in i te advantage in t h a t i t avoids a collocation scheme.

T h i s procedure has a

~ A comparison of the experimental creep-rupture resu l t s and the

D ( t ) = .71 + .041 [l - exp ( - t / .Ol ) ] t

+ .041 [l - exp ( - t / l ) ] + .060 [l - exp ( - t / lO)]

+ ,049 [l - exp (-t /100)] + .164 [l - exp ( - t / l O O O ) ]

+ .158 [l - exp (-t/lOOOO)] x psi-’ (7.1 1

Based on t h i s ident i f ica t ion scheme, we can use Eqs. (4.9 - 4.11)

t o calculate the total creep energy, stored f ree energy and dissipated

energy as a function of time as shown i n F i g . 7 . 1 . A gradual increase

of the total dissipation w i t h time originates from the energy con-

sumption in the subsequent viscous elements.

be subtracted from the to ta l creep energy, which i s in t h i s case equal

This dissipation has to

to the potential energy loss of the external loading, i n order t o ob-

t a in the to ta l stored f r ee energy.

An identical evolution of the f r ee energy was also obtained based

f ree energy based prediction shows tha t very good agreement was ob-

tained as shown in Fig. 7.2.

constant f ree energy l i ne .

from creep compliance data a t the expense of only one additional

parameter ( w ) .

to ta l creep-energy with creep to rupture data , gave very poor agreement.

T h e fu l l l i n e (+) represents a given

Thus a creep rupture r e su l t i s obtained

Earl ier attempts, in which we t r i ed t o correlate the

114

cv * * n v) u)

2 t u)

U Q)

Q Q 0

\

.- - Y

)r

Q) c W

Y

F ig . 7.1

n .- y" Y

v) u)

L ti Q) tn L Q) > u) c z I-

Fig . 7.2

I 4 Tota l Creep Energy I 0 Tota l Stored Free Energy

Dissipated Energy

0.0 1 1 I I I .o 2 .o 3 .O 4.0

Log Time (min)

T300/934: E v o l u t i o n of Tota l Creep Energy, To ta l Stored Energy and D iss ipa ted Energy as a Funct ion o f Time f o r [90]8s a t 320 deg F (160" C ) .

5 .O

0 Experiment

F r e e Energy Based Prediction

30

20

IO

rr 0 a 5 Y

0 I I I 0.0 0.0 I .o 2 .o 3 .o 4.0-

Log Time (min)

T300/934: P r e d i c t i o n w i t h Experimental Resu l ts Obtained on a t 320 deg F.

Comparison of Free Energy Based Creep t o Rupture

115

7 . 2 . 2 Analysis of [60IBs Results

Long time creep and creep recovery data were obtained on a 60-deg

off axis specimen.

ing were identical w i t h those already discussed i n Chapters 5 and 6 .

The de ta i l s on postcuring, s t r a i n gaging and t e s t -

A plot o f the compliance change w i t h log t a t a s t r e s s level of

72% of ultimate i s shown i n F i g . 7.3.

[7] on the same orientation b u t a t 55% of ultimate i s a lso shown i n

t h i s figure.

s t r e s s intensi ty . I n addition, however, a ra ther large ver t ical s h i f t

would be needed t o superimpose both curves.

matrix p l a s t i c i ty , since a ra ther large permanent deformation remained

a f t e r creep recovery ( F i g . 7 . 4 ) .

such resul ts which a re reproduced i n F i g . 7.5. The nonrecoverable

creep deformation i s seen t o behave a s an increasing-decreasing func-

t i o n w i t h f i b e r angle. These bell-shaped curves a re parameterized by

a number tha t indicates the s t r e s s intensi ty as a f ract ion of ultimate.

Obviously a large nonrecovered s t r a in was observed which varies very

nonlinearly w i t h s t r e s s intensi ty .

A r e su l t obtained by Griffith

The curvature of these plots i s c lear ly affected by the

T h i s could be due to

Marti rosyan [98] gave evidence of

The creep response curve a t 72% of ultimate was again modeled

w i t h a six element Kelvin model, w i t h an additional f ree dashpot i n

s e r i e s , as shown i n F i g . 7.6.

The free energy a t rupture i s seen to be constant f o r delayed

fa i lures i n excess o f 30 m i n . as shown i n F i g . 7.7. Exactly the same

f ree energy evolution w i t h time was found again through the power law

approximation.

a f t e r subtraction of the f ree dashpot contribution.

The power law parameters were evaluated, however,

116

n

I - 2.0 .- z 2

E

Y

CD 1.5

a, 0

0 c 1.0 .- -

0.5 0 Q a, 2 0 0.0

0 Current Research 4665 psi

0 Griff i th ( 7 ) 3593 psi ( 32.27 MPa )

(24.75 MPa)

0.0 I .o 2.0 3.0 4.0

Log Time (min)

n

(0

0.2 I 0 X - Y n

0 a 0. I \ c Y

Fig. 7.3 Long Time Creep Compliance o f [60]8s T300/934 at 320°F (160°C).

Y

L

0

Creep Time X I O 3 ( m i d

0.0 L I I 1 0.0 5 .O 10.0

Fig. 7.4 Typical Creep Curve for [60]8s T300/934 at 320°F (160°C).

117

I i I 0.8

F i g . 7 .5

> 0 0 W L c 0 z

Fiber Anq le

Nonrecoverable Creep of G l a s f i b e r Reinforced P l a s t i c w i t h Respect t o F i b e r Angle and Percent o f U l t i m a t e Stress.

Fig.. 7.6 K e l v i n Element Used t o Model Creep Response f o r [60]8s a t 320°F (160°C).

118

20

10

0 0

I I

F i g . 7 .7 Free Energy a t Rupture f o r [60]8s T300/934 a t 320°F (160°C).

A reasonable f r ee energy based creep-rupture prediction was

obtained, as seen in Fig. 7.8. Final conclusions however cannot be

made y e t , due t o the limited number o f experimental data points which

a re currently avai 1 able.

7 . 3 Creep Response on [90/+45/90]2,, Compared with Creep response o f

[go]& and [t45]4< a t 320°F (160°C)

I t can be expected, t ha t a [90/+45/90]2s laminate configuration

will operate a t a lower creep ra te t h a n a [f45]4s, since the l a t t e r

contains only two f i b e r orientations which allows scissoring action

as may be visualized in Fig. 7.9. The former contains three f ibe r

or ientat ions and t h u s a f i b re t russ network i s formed as shown in

Fig. 7 .9 which prevents scissoring.

119

8

n

7 .- ln Y Y

ln fn 6 Q, L c m

5

4

I I I

0 Experiment

- Free Energy Based Prediction

0 0

Fig. 7.9 Schematic of [ i45] and [90/ Truss Network i n L a t t e r .

120

50 ; a H

L 5

Y

v)

40

30

IO I O 0 1000

Rupture Time (m in )

Fig. 7.8 Free Energy a t Rupture P r e d i c t i o n s f o r [60]8s T300/934 a t 320°F (160°C).

Fig. 7.9 Schematic of [ i45] and [90/ Truss Network i n L a t t e r .

120

'+45/90] Laminates wi th F ibe r

Figure 7.10 compares the creep responses of three laminates. I t

is qua l i t a t ive ly c l ea r t h a t the [k45]4s f iber or ientat ion shows the

most t r ans i en t creep.

power law coef f ic ien ts , D1, which are a function o f stress, as given

This can also be quantified when we compare the

i n F i g s . 7.11. A comparison of the [ 9 0 / ~ 4 5 / 9 0 ] ~ ~ laminate r e su l t s

w i t h those f o r the [90Igs and [k45]4s laminates reveals t h a t the creep

r a t e f o r the former i s an order of magnitude less than t h a t fo r the

other or ientat ions.

35% as compared t o the [ t45]4s orientation.

The i n i t i a l compliance however decreased only by

n

I - .- r" Y

Q) 0 E 0 .- - e 0 0 a Q)

0 2

F i g . 7.10

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

I

(9018,

ea----- ( 90/+45/90)2s

0 .o I .o 2.0

Log Time ( min 1

0.10

n

(D I

0 X - Y h

0 3.05 2

\ c

Y

3.00

Creep Compliance for Three T300/934 Laminates.

121

h 0.06 - I

- - - - - - -

.-

I .2

I .o

0.8

f 0.04 U -

0.02

- 0.06 - I .- ~ 5 0.04 U -

0.02

- .008 - I .- v) z Y - ,006 n

Stress (MPa) IO 20 30 40 I I I I

2 4 6

Stress ( K s i 1 Stress (MPa)

IO 20 30 40 I I I I

Nonrecoverable Strain in Load Direct ion Nonrecovrrable Strain

0- Tranrverc to the Lood Direction

I O

5

IO

5

2 4 6

Stress ( K s i 1 Stress (MPa)

50 IO0

I I I I .4

t

+

0 500 -

X i"' U

h

\ c I O 0 Y I 3 O 0

I I I 1 0.6 .004 I 5 I O 15

Stress ( K s i

Fig . 7.11 Comparison o f the Power Law C o e f f i c i e n t D Obtained Three Different Laminates a t 320°F (160°C 3

122

In conclusion, adding 90's t o a + 45 orientation increases the

modulus by only 35% b u t decreases creep sens i t i v i ty by an order of

magnitude,

Prior t o tes t ing the laminate, we also p u t a s t r a in gage

This information allows us t o transverse t o the loading direction.

o b t a i n the shear compliance [99] which i s given by

I t was observed, however, t ha t e , ( t ) and e ( t) had approximately the

same magnitude.

consistencies in the creep recovery data.

Y This led t o untolerable large errors and even i n -

7 .4 Numerical Prediction of [90/+45/90],_ Laminate Behavior

The incremental nonlinear viscoelastic lamination theory-program,

which was written and documented by Dillard [ 7 ] d u r i n g an e a r l i e r

phase of our research program, was used t o generate the creep response

of a [90/+45/9012, 1 aminate.

A comparison of the numerical prediction w i t h the experimental

resu l t a t 70% of ultimate a t 320°F (160°C) i s shown, on an expanded

sca le , i n F i g . 7.12. Reasonable agreement between predicted and

measured compliance was obtained.

even more seriously underpredicted, indicating a basic deficiency of

the numerical approach which l i e s in i t s i nab i l i t y t o r e f l ec t the

microstructural e f fec ts of the laminate. This microstructural behavior

was deduced using edge replicas [loo], as shown i n F i g . 7.13.

Creep rupture-time, however, was

An

123

n 0 . 4 5 I - .-

- -b Experiment, 20,144 psi ( 1 3 8 . 8 M P a ) Experimental, 14,500 psi (99.9 M P a )

I 4- Predict ion, 14,500 psi ( 9 9 . 9 M P a ) 4

I I I I I

0.30 4 i

0 . 0 6 5

0 . 0 6 0 - 0

(0 I

0 . 0 5 5 Y

A

0 n 0 . 0 5 0 \ c Y

0 . 0 4 5

0 200 400 600 800 1000 1200

Fig. 7.12 Creep Compliance o f a [90/~45/90]~~ T300/934 Laminate at 320°F (160°C).

I z II Edge replica a f te r 7 I23 minutes a t 1 4 , 5 0 0 Psi a t 20,144 P s i

Edge repl ica a f te r 15,840 minutes

F i g . 7.13 Edge Rep1 i cas o f [90/k45/90]2s Lami nate.

124

attempt t o model the stress transfer mechanism involved by means of

a shear-lag model [ l o l l , which was simple enough fo r subsequent

implementation i n the laminate analysis program, turned o u t t o be un-

successful.

Comparison of the creep response a t 97% of ultimate, also as

shown i n Fig. 7 .12, with the response a t 70% o f ultimate indicates a

large vertical shif t . The edge replica, obtained af ter unloading ,

however, revealed a very different crack spacing between results a t

the two stress levels.

lamina i s considerably less t h a n i n the previous case and some

cracking in the 45-deg. oriented lamina became visible a t the higher

load level.

The crack spacing in the 90-deg. oriented

Figure 7.14 shows the irregularities i n the strain reading a t

the beginning of the creep rupture experiment a t 97% of ultimate.

This indicates a sequence of events a t the microstructural level,

which could be deterministic i n the selection process of the stress

transfer mechanism.

More understanding i n this area should be generated, including

the answer on the question of whether or no t crack healing sets i n

under certain unloading o r strain-recovery conditions.

125

n

c .- 0,

e c v)

0 .- E Y

c 0

0

0 a2 c3

.- c

E L

rc

8000

6000

4000

200c

0 I 2 3 4

Creep Time (mim)

F ig . 7.14 Creep Curve a t 97% o f Ultimate.

126

Chapter 8

SUMMARY AND RECOMMENDATIONS FOR FURTHER STUDY

Different const i tut ive formulations able t o model material non-

l i n e a r i t i e s i n v iscoelast ic systems have been discussed. Nonlinear

v i scoe las t ic i ty models are often quite complicated, creating great

experimental d i f f i cu l t i e s for the determination of material parameters.

I t was concluded t h a t the approach advocated by Schapery i s suf f ic ien t ly

general t o account f o r stress-activated nonlinear behavior i n polymers

and composites.

phenomena were eas i ly modeled:

Particularly the following experimentally observed

An i n i t i a l e l a s t i c creep-recovery which i s greater than the

i n i t i a l e l a s t i c creep response

A recovery behavior which i s overpredicted by 1 inear visco-

e l a s t i c considerations ( i . e . the actual recovery curve i n the

nonlinear range is f l a t t e r t h a n i n the l i nea r range)

Creep and creep-recovery experiments which were carried o u t on

the neat 934-resin did n o t reveal any s t r e s s dependent change i n

materials response other than p las t ic i ty . The analysis of time-

dependent response, which i s exclusively connected w i t h specif ics of

the molecular s t ructure revealed the existence of a secondary t rans i -

tion-temperature located between 200 and 250°F (93-121°C).

Essentially the same ser ies of experiments were carried o u t on

the T300/934 graphi te-epoxy system.

was found t o be very s t r e s s insensit ive.

on the other hand, turned o u t to be a sensi t ive function of s t r e s s .

T h e transverse compl i ance D Z 2 ( t )

T h e shear compliance D66(t) ,

127

The role of yielding and subyielding o f the matrix phase, due t o d i f -

ferences i n loading path (transverse loading vs. shear loading) was

discussed. Two uncoupled equations were f i t t o the data D Z 2 ( t ) and

D 6 6 ( t ) respectively. The ca l ibra t ion of the material parameters

(Do,C,n,go,gl,g2,a ) was done by a computer routine. I t i s well known

however tha t mathematical models f i t data well when used on the s t r e s s

paths which were applied f o r parameter cal ibrat ion purposes. In order

t o account f o r s t r e s s interaction along general loading paths, the

principle of octahedral stress, was borrowed from nonlinear approaches

i n isotropic viscoelast ic i ty .

needs t o be careful ly checked.

found t o be .23 and independent o f s t r e s s and temperature.

secondary t rans i t ion , which was ident i f ied i n the res in , was a l so

identified i n the composite, b u t a t a s l i gh t ly higher temperature.

The va l id i ty of this approach however

The power law creep exponent ( n ) , was

T h e

Short time data could be shif ted graphically into a unique "master

curve." The horizontal s h i f t was found to be i n very good agreement

w i t h the analytical sh i f t (log a T ) .

and became even nonexistent a t higher temperatures,

vertical s h i f t i n g i s re la ted t o the behavior of the l imiting compliances.

An interesting p o i n t i n this context i s t h a t Urzhumtsev [lo21 presents

evidence t h a t the time-stress-superposition pr inciple , when used to

sh i f t short term compl iance curves obtained on porous polyurethane i s

i n excellent agreement w i t h a long time control experiment whereas the

time-temperature superposition leads t o very serious overpredi c t i ons

of long time compliance data.

The ver t ical s h i f t was very small,

I t was shown tha t

128

I t should be noted tha t no mechanical conditioning was applied,

since conditioning should be used t o i so l a t e delayed e l a s t i c response,

which i s something tha t has never been proven f o r fiber reinforced

composites ( a t l e a s t not t o t h i s author's knowledge).

Creep compliance data were merged into a f ree energy-based

f a i l u r e c r i t e r ion and reasonable creep-rupture predictions were ob-

tained.

be an order of magnitude l e s s than the creep response of [90]8s and

[245]4s , while the modulus increased only by a fac tor of 1.35.

reasonable creep compliance v e r s a time prediction was obtained by means

The creep response of a [90/?45/9012, laminate was found to

A

of a nonlinear viscoelast ic laminate analysis program I t i s f e l t ,

however, t ha t a damage model t h a t operates on the m i n mechanics leve

of the problem would give a be t te r agreement than the cumulative

damage model which was actual ly used [9].

direction were published recently by Nuismer e t a l . [103]. They

developed a r e a l i s t i c and simple scheme tha t accounts f o r influences

Promising ideas i n t h i s

on the const i tut ive behavior of given matrix dominated laminas through

the lamina-laminate interaction a f te r damage i n i t i a t i o n . Another, more

fundamental , approach tha t describes the influence of damage through a

continuum vector f i e l d has been pursued, w i t h i n a continuum thermo-

dynamics framework by Talreja [104].

Based on this current research e f f o r t , the following recommenda-

t ions fo r fur ther study seem appropriate:

The f e a s i b i l i t y of mechanical conditioning, i n case of composites, should be thoroughly checked.

129

Test programs f o r boron-epoxy by Cole and Pipes [14] and fo r graphite polymide by Pindera and Herakovich [15] produced evi- dence tha t the octahedral shear s t r e s s i s not a v a l i d representation of s t r e s s mode interact ion. Herakovich showed t h a t , i n order t o get access t o the actual s t r e s s interact ion, 3 off-axis t e s t s were needed. An ident i f icat ion of the actual representation fo r s t r e s s mode interaction i n case of graphite-epoxy T300/934 would a t l ea s t give indications on i t s re la t ive importance w i t h respect t o nonl ineari t i e s i n transverse deformation and shear.

Pindera and

T h e f r ee energy based f a i lu re c r i t e r ion together w i t h the Tsai-Hill c r i t e r ion , could be developed to predict delayed off-axis f a i l u r e , based on transverse and shear compliances only .

The necessity of t ransient temperature t e s t s , i n order t o i so l a t e the exact ver t ical sh i f t , and t h u s obtain maximum predictive power, should be explored.

Reversibil i ty is always dependent on the choice of a well- defined eas i ly reproducible reference s t a t e . I n composites, however, i n i t i a l s t resses a r i s e as a r e su l t of physio- chemical processes, techno1 ogical operations and mechani cal and thermal incompatibility. T h e natural s t a t e , i . e . a body f ree of s t resses and s t r a i n s , i s nonexistent. The importance of an "eigenstress'l-field on the viscoelast ic behavior should be checked.

130

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~

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i

7. Gr i f f i th , W . I . "The Accelerated C h a r a c t e r i z a t i o n o f V i s c o e l a s t i c Composite M a t e r i a l s " , Ph.D. D i s s e r t a t i o n , VPI & SU, Blacks- burg, VA, 1980; a l s o VPI Report VPI-E-80-15, w i t h D. H. Morris and H. F. Brinson, 1980.

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9. D i l l a r d , D. A. "Creep and Creep Rupture o f Laminated Graphi te / Epoxy Composites", P h . D . D i s s e r t a t i o n , VPI & SU, Blacksburg, V A , 1981; a l s o VPI Report VPI-E-81-3, w i t h D. H. Morris and H. F . Brinson, 1981.

10. D i l l a r d , D. A . , Morr i s , D. H. , and Brinson, H . F . " P r e d i c t i n g V i s c o e l a s t i c Response and Delayed Fai 1 ures i n General Laminated Composites", ASTM S i x t h Conference on Composite M a t e r i a l s : Tes t ing and Design, Phoenix, A Z , May 1981.

11. D i l l a r d , D. A . , D . H . Morr i s , and H. F. Brinson. "Creep Rupture o f General Lami n a t e s o f Graphi te/Epoxy Composites" , Proceedings o f 1981 Spring Meeting , SESA, Dearborn, June 1981 .

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12. D i l l a r d , D . A . , and H. F . Brinson. "Nonlinear Creep Compliance o f General Laminated Graphi te/Epoxy Composites" , Proceedings SESA Spring Meeting, Honolulu, HI, May 1982.

13 . Brinson, H . F . "Experimental Mechanics Applied t o the Accelera ted C h a r a c t e r i z a t i o n o f Polymer Based Composite M a t e r i a l s " , i n New Physical Trends i n Experimental Mechanics, ( J . T. Pinde ra , e d . ) , CISM Courses and Lectures, No. 264, 1981.

14 . Cole, B. and Pipes, R . "F i lamentary Composite Laminates Sub- j e c t e d t o Biaxia l S t r e s s Fie1 d s " , AFFDL-TR-73-115, 1974.

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139

APPENDIX A

1 .

2 .

3.

4 .

*5.

*6.

*7.

*8.

*9.

Cure Procedure f o r T300/934 Composite Panels

Vacuum bag - insert lay-up i n t o au toc lave a t room t empera tu re .

Apply f u l l vacuum. ( A min imum o f 25 inches o f Hg.) Hold for 30 minutes a t room tempera ture .

Maintain f u l l vacuum throughout en t i re c y c l e .

Raise tempera ture t o 250 F (+5 . - 10 F) a t 2 ? 5 F per minute.

Hold a t 250 F (+5. - 10 F) f o r 15 ? 5 minutes, apply 100 psi (+5. - 0 p s i ) .

Hold a t 250 F (+5. - 1 0 F) and 100 psi (+5. - 0 psi) f o r 45 k 5 minutes.

Raise tempera ture t o 350 F ( + l o . - 0 F) a t 2 ? 5 F per minute.

Hold a t 350 F ( + l o . - 0 F ) f o r two hours 15 minutes.

Cool under pressure and vacuum t o below 175 F.

*Pressure was 85 psi.

140

~-

Curing Procedure fo r Resin Panel #1

1 . Resin heated u n t i l f l u i d and poured i n t o heated mold.

2 .

3.

4.

5 .

6.

7.

8.

9.

Mold vacuum bagged and i n s e r t e d i n t o au toc lave .

F u l l vacuum a p p l i e d and held 30 m i n .

Raise tempera ture t o 250 F (+5. -10 F) a t 2 ? 5 F per minute.

Hold a t 250 F (+5 -10 F) for 15 t 5 minutes. Apply 100 psi (+5 - 0)

Hold a t 250 F (+5 -10 F) and 100 psi (+5 - 0 psi) for 45 ? 5 m i n .

Raise tempera tuer t o 350 F ( + l o -0 F) a t 2 ? 5 F per minute.

Hold a t 350 F ( + l o -0 F) for t&o hours ? 15 minutes.

Cool under pressure (85 ps i ) and vacuum t o below 175 F .

Fab r i ca t ion Procedure f o r Resin Panel #2

1 . Mold hea ted t o 200 F.

2 . Resin hea ted t o 160 F and degassed i n bel 1 j a r f o r 5 minutes.

3. Degassed resin poured i n t o one corner o f t i l t e d and heated mold. Air gun used t o keep mold warm.

4 . Resin then cured i n oven w i t h same schedule as prepreg t a p e , i . e . steps 4-9 above.

141

The Curing Procedure f o r Resin Panel #3

934 Resin Cure Procedure - S o l u t i o n Form

1 . Break o f f chunks o f s o l i d ( c o l d ) r e s i n .

2 . Place i n beaker o f ace tone o r methyl e thyl keytone.

3 . Slowly warm and s t i r s o l u t i o n up t o 80°F (26°C) and u n t i l a l l resin i s d i s so lved t o t a f f y cons i s t ency .

4. Pour i n t o t e f l o n coa ted bread pan and p l a c e i n vacuum oven. Heat t o 150°F (66°C) under fu l l vacuum. Observe u n t i l bubbles cease (about 2 hour s ) .

5. Remove from vacuum oven and p u t i n 212°F (100OC) oven ove r n i g h t ( roughly 3:OO p.m. - 7 : O O a .m. ) .

6 .

7 .

Remove from oven i n morning.

P u t i n oven a t 325°F - 350°F (163°C - 177°C) f o r two ( 2 ) hours .

8. Turn o f f oven and a l low t o cool g r a d u a l l y .

142

APPENDIX B

w i t h

and

a G a G + - dQ,

a G aT aq, aQ, dG = - d T + -

(B.4) and (B.5) -+

a G - a F aT a T - - -

a G aQm

- - - P - -

143

APPENDIX C

INFLUENCE OF EXCENTRIC LOAD APPLICATION [ 1051

Suppose t h a t t h e l o a d F i s a p p l i e d w i t h a smal l e x c e n t r i c i t y e.

Th i s causes n o t o n l y t h e homogeneous s t r e s s s t a t e which we in tended

t o produce, b u t a l s o t h e bending moment.F-e as i s shown i n F ig . C . l .

F F @ = f i + p F i g . C.1 Superpos i t ion o f Normal S t ress and Bending Due t o Excen t r i c

Load A p p l i c a t i o n

The s t ress , a t t h e gage l o c a t i o n uGL can then be w r i t t e n as,

w i t h uU = - , the uni form s t r e s s , which we in tended t o apply . w t

144

I

19. Security Classif. (of this report)

UNCLASSIFIED UNCLASSIFIED 20. Security Classif. (of this page) 21. NO. of Pages

1 5 2

2. Government Accession No. 3. Recipient's Catalog NO. I. Report No.

NASA CR-3772 1. Title and Subtitle 5. Report Date

J u l y 1984 6. Performing Organizationcode The Nonlinear Viscoelastic Response Of Resin Matrix

Compos i te Laminates - 8. Performing Organization Report NO.

VPI-E-83-6 10. Work Unit No. T-4241 11, Contract or Grant No.

7 . Author(s)

C . Hiel, A.H. Cardon, a n d H.F . Brinson

9. Performiriy Organization Name and Address

Department of Engineering Science and Mechanics

Blacksburg, VA 24061 Virginia Polytechnic Ins t i t u t e and State University

2. Contractor Report

NCC2-71 13. Type of Report and Period Covered -

Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D . C . 20546 505-33-21 5. Supplementary Notes

14. Sponsoring Agency Code

P o i n t of Contact: Howard G . Nelson, MS: 230-4, Ames Research Center, Moffett Field, CA 94035 415-965-6137 o r FTS-448-6137

22. Price*

A0 S

6. Abstract

A review of possible treatments of nonlinear viscoelastic behavior of materials is presented. i s discussed i n detai l and i s used as the means to in te rpre t the nonlinear viscoelastic response of a graphite epoxy laminate, T300/934.

A thermodynamic based approach, developed by Schapery

Test data t o verify the analysis for Fiberite 934 neat resin as well as transverse and shear properties of the unidirectional T300/934 composited are presented.

Long time creep charac te r i s t ics as a function o f s t r e s s level and temperature a re generated. graphical and the current analytical approaches a re shown.

Favorable comparisons between the t r ad i t i ona l ,

A free energy based rupture c r i te r ion i s proposed a s a way t o estimate the l i f e that remains i n a s t ructure a t any time.

18. Distribution Statement

Time-temperature-stress superposition, Unclassified - unlimited Creep, Relaxation, Accelerated tes t ing , Life prediction. I----- STAR Category - 24

K e y Words (Suggested by Author(s))

*For sale by the National Technical Information Service, Springfield, Virginia 221 61 MASA-Langley , 1984