THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

64
THE THEORETICAL TREATMENT OF VISCO-ELASTIC PROPERTIES OF SILICONE ELASTOMERS AS A FUNCTION OF TEMPERATURE AND STRAIN RATE by MURTY V. R. N. DINIVAHI, B.Sc, B. Tech. A THESIS IN CHEMICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN CHEMICAL ENGINEERING August, 1988

Transcript of THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

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THE THEORETICAL TREATMENT OF VISCO-ELASTIC PROPERTIES OF SILICONE

ELASTOMERS AS A FUNCTION OF TEMPERATURE AND STRAIN RATE

by

MURTY V. R. N. DINIVAHI, B.Sc, B. Tech.

A THESIS

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CHEMICAL ENGINEERING

August, 1988

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< s

C o/> '^ ACKNOWLEDGMENTS

I express my deep appreciation and gratitude to Dr. Richard Wm. Tock

for his care, concern and invaluable guidance. I thank Dr. Fred Senatore and

Dr. C.V.G. Vallabhan for their interest in this research project and for their

help. I convey my heartfelt warmth to all my well-wishers and friends who

have helped me to complete this project successfully.

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TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

ABSTRACT v

LIST OF TABLES vi

LIST OF FIGURES vii

CHAPTER

L INTRODUCTION 1

n. VISCO-ELASTIC PROPERTIES 3

HI. LITERATURE REVIEW 4

IV. PROBLEM STATEMENT 5

V. THEORETICAL DEVELOPMENT 7

Rubber Elasticity 7

Visco-elasticity-MaxwelTs Model 12

Modified Maxwell's Model 15

VI. BRIEF DESCRIPTION OF APPARATUS 17

VII. EXPERIMENTAL PROCEDURE 20

VIII. RESULTS AND DISCUSSION 22

An Outline 22

Elastic Modulus 27

Cross Link Density .16

iii

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Viscosity 42

Discussion 48

IX. CONCLUSIONS AND RECOMMENDATIONS 50

Conclusions 50

Recommendations 51

BIBLIOGRAPHY 53

GLOSSARY 54

IV

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LIST OF TABLES

1. Factors Which Influence The Mechanical

Properties Of Silicone Sealants 3

2. Description Of Mechanical Properties 3

3. Test Specimen Specifications 19

4. A Summary Of The Experimental Plan 21

5. Elastic Modulus As A Function Of Temperature And Elongation Rate 28

6. Cross-link Density As A Function Of Temperature And Elongation Rate 37

7. Viscosity As A Function Of Temperature And Elongation Rate 43

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A B S T R A C T

Silicone elastomers have been used as s t ructura l sealant components

since the early 1970s. Physical propert ies such as elastic modulus , cross-link

density, viscosity, shear modulus , and the effect of orientation arising from

processing are considered in this study. There are few published mathemat ica l

models which adequately describe elastomer behavior nor is there much in the

published l i terature to verify experimental results. A new model was therefore

developed. Stress-strain da ta were obtained from an Instron machine which

was utilized to conduct experimental studies. It was observed that the elastic

modulus of the two siHcone elastomers used in this study decreased with an

increase in t empera tu re . This interesting phenomenon has not been reported

in the l i terature hi ther to .

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LIST OF F IGURES

1. Random Flight in Three Dimensions 10

2. Maxwell Model(s) 13

3. Sample Configuration of the Test Specimen 18

4. Stress vs. Strain Effect of Elongation Rate on

High and Low Modulus Materials at 24° C 23

5. Stress vs. Strain-Effect of Tempera ture on High and Low Modulus Materials at a

Constant Extension Rate of 1.0 - ^ 24 •mm

6. Stress vs. Strain-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 10 - ^ 25 mm

7. Stress vs. Strain-Effect of Temperature on Low Modulus Materials at a Constant Extension Rate of 10 - ^ 26

mtn

8. Modulus vs. Alpha-Effect of Elongation Rate on High and Low Modulus Materials at 24°C and 90^0 29

9. Modulus vs. Alpha-Effect of Temperature on High and Low Modiilus Materials at a

Constant Extension Rate of 1 -^^^ 30 m t n

10. Modulus vs. Alpha-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 10 ^ ^ 31

11. Modulus vs. Alpha-Effect of Orientation on High Modulus Material at 24°C and a

Constant Extension Rate of 1 ^ ^ '.\2 m t n

Vll

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12. Stress vs. (a —V)-Effect of Elongation Rate on

High and Low Modulus Materials at 24° C 38

13. Stress vs. (a - ^ ) -Ef fec t of Tempera ture on

High and Low Modulus Materials at a

Constant Extension Rate of 1 -—- 39 m t n

14. Stress vs. (a - ^ ) -Ef fec t of Tempera ture on

High and Low Modulus Materials at a

Constant Extension Rate of 10 - ^ 40 m m

15. Relaxation Phenomenon-Effect of Orientation on High and Low Modulus Materials 44

16. Relaxation Phenomenon-Effect of Elongation Rate on High and Low Modulus Materials at 24° C 45

17. Relaxation Phenomenon-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 1 - ^ 46 m t n

18. Relaxation Phenomenon-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 10 -^^^ 47 m t n

VI11

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C H A P T E R I

I N T R O D U C T I O N

Synthetic products have been extensively used for many years in almost

all the walks of modern industrialized society. Elastomeric polymers is one

class of synthetic products which has received extensive use. High modulus

elastomers have been increasingly used as s t ructural sealants since the early

1970s. A s t ructural sealant is actually a connector between gla^s curtain

walls or insulating glass units and the s tructure itself. A structural sealant

is also intended to prevent mass and heat transfer through the joint or seam

region. S tandard definition states that any material that has the necessary

adhesive and cohesive properties to form a seal is to be called a sealant in

terms of the "building construction." Hence, for many years, the primary

concerns with these sealant polymers has been limited to the properties

like adhesive s t rength, ul t imate cohesive strength, etc. Lately, it has been

realized that additional critical factors such as stress-relaxation, strain rate,

and t empera tu re dependence of stress should also be studied in detail.

Low elastic modulus silicone rubbers have atypical high movement capabil­

ity. Therefore, they were considered to be ideal joint connectors. Subsequently,

high modulus silicone rubbers have become popular due to their high working

tempera tures and moisture stability. Structural sealants must also be resistant

to corrosion. S tandard design procedures for construction usually require com­

plete engineering specifications of s tructural sealants. Product specifications

of a manufacturer alone, typically do not meet these requirements. Impor tan t

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visco-elastic properties like stress, elastic modulus, shear modulus, viscosity,

and cross-link density can be a complex function of temperature, and elonga­

tion rate needs to be studied.

Moreover, it is supposed^hat these physical properties will have orienta­

tion effects arising from processing procedures. Samples cut either perpendic­

ular or parallel to the principal processing direction will be taken and studied

with respect to the above mentioned properties.

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CHAPTER n

VISCO-ELASTIC PROPERTIES

Strain rate, absolute temperature (operating temperature), and the type

of cure are among the important parameters (Table 1) which greatly influence

mechanical properties of siHcone rubber sealants. Cure affects cross-link

density. Mechanical properties of practical significance to be considered are

summarized in Table 2.

Table 1

Factors Which Influence The Mechanical Properties Of Silicone Sealants

No. Property Symbol

1. Strain rate ^

2. Absolute Temperature T

3. Cure

Table 2

Description Of Mechanical Properties

Property Symbol Units

Stress <T psi or Pa

Measure of Cross-Link

Density N Ibmol./cu.ft or g/cc

Extension Ratio -f^ = a dimensionless

Elastic Modulus E psi or Pa

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C H A P T E R III

L I T E R A T U R E R E V I E W

Visco-elastic behavior of polymeric materials has been studied by differ­

ent methods . Methods of dynamic mieasurements include the response to sinu­

soidal stress. Static measurements envisage stress relaxation and creep under

constant strain and constant stress.

Visco-elastic behavior of Polyisobutylene under constant rates of elonga­

tion was studied by Thor (1956). His studies were directed toward determining

the visco-elastic properties of rubberlike materials from stress strain curves as

a function of tempera tu re and strain rate. By the generalized Maxwell model

(an infinite number of Maxwell elements connected in parallel) stress strain

curves were obtained from the following equation:

5 '-^"^

where, R is the constant rate of strain, S is the stress, 7 the strain, and M[r)

the relaxation distr ibution function. A/( r ) dlnr is the contribution to the

instantaneous tensile modulus.

Anderson (1985) used a similar equation from Nielson (1962), based on

generalized Maxwell model to obtain the following stress strain relationship:

/

-t-oo r -

r M ( / n r ) ( l - exp( )) dlnr - 0 0

where, a is tensile stress, e is strain, E^o is rubbery flow modulus, A' is strain

rate and r is relaxation t ime. M{ln^) is the increment of distribution of

relaxation timers.

j M[T)r{\-exp{-^))dln(T)

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CHAPTER IV

PROBLEM STATEMENT

Thor (1956) modeled the visco-elastic properties of polyisobutylene at

constant rates of elongation. Experiments were focussed upon the tempera ture

as well as the time dependence of dynamic viscosity. Also, the tempera ture

dependence of stress s train curves was illustrated. An isotherm of 25° C was

the chosen s tandard t empera tu re , in a range of -31 to 85°C. Anderson's studies

describe elastic modulus as a function of elongation rate. Maximum elongation

rate was about 1.7 i n /min .

The present problem deals with modeling based physical properties such

as the modulus of elasticity, cross-link density, shear modulus, and stress-

alpha curves with respect to t empera ture , elongation rate and sample cut-

direction. A test sample can be cut either parallel or perpendicular to the axis

of processing. It was proposed to investigate the dependency of mechanical

properties of a sealant on the sample cut-direction.

A s tandard t empera tu re was chosen as 25° C with a range of 25 to 90° C.

Elongation rate was varied from 0.1 - 10 in /min .

For isothermal conditions, stress-extension ra t io(a) data were reduced and

fitted to a least square third degree polynomial. Plot(s) of stress(cr) vs. a were

made to study and compare the nature of variable interdependence. From th<'

slope of cr vs. a V? ^V(T, a, a, t) was calculated, for a sri of stf-ss - alpha

(lata.

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Equation 18 implies that the stress in the elastomer is the difference

between a recoverable deformation and a retarded, or partially recoverable,

viscous deformation. The latter sometimes being negligibly small. Equation

21 implies that the rate of polymer uncoiling is the difference between straiin

rate and relajcation rate.

Experimental evaluation of equation 17 (the fixed elongation) condition

was achieved by elongating the sample to 150%. Then the sample was allowed

to relax at this strained condition. The stress relajcation data were utilized in

predicting the apparent viscosity of the silicone rubber material. The slope,

( —), of a plot of In (J vs. t was obtained. Then, E obtained at 50% elongation

was used in determining the viscosity of the sample.

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CHAPTER V

THEORETICAL DEVELOPMENT

Rubber Elasticity

By the first law of thermodynamics the energy of a closed system can be

writ ten as

AE = Q-W (1)

where, Q is heat transferred, W is the work done, and AE is the change in

internal energy of a system. Based on the second law of thermodynamics,

d S = ^ (2)

where, dS is the increment of the entropy due to the increment of heat dO.

added reversibly at t empera ture T. K a piece of rubber is stretched by a force

/ through a distance dL then, the differential work done, dW, is given by the

following equation:

dW = (PdV - fdL) (3)

where, PdV is the work done by the rubber in expanding through a volum<'

dV against a pressure P. The product fdL is the work done by the force used

to stretch the rubber distance, dL. Thus, dW is the resultant work. By the

eqns. 1, 2, and 3

dE = TdS - PdV + fdL. ( l )

Assuming that Poisson's ratio is 0.5, i.e., the rubber is incoinpressibl*',

dV - 0. ,v fdL = dF TdS.

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and.

8

CfL T,V OL T,V

By equation 4,

- ( - ) = ( ^ )

Hen ce.

For a system, changes in internal energy are due to the following changes:

(i) Tempera tu re

(ii) Energy stored by bond bending and stretching.

Changes in. entropy can be described by statistical thermodynamics .

Changes in entropy in this case are due to the changes in conformation of

the molecular s t ructures . For a stretched string, if all conformations in space

axe equally likely, the stretched condition is the least probable conformation

and henct represents the lowest entropy; while the orientation where the two

ends are nearly together is the most probable conformation and represents the

highest entropy. For an ideal rubber, therefore.

Tha t is, an ideal elastomer responses to external stress only by uncoihng.

The quant i ty (ff") is therefore iiegHgibly small for many cross-linked amor­

phous polymers above T j , the glass-transition tempera ture . Also, polymer

segments can not deform at temperatures below Tg, in the time scale in which

Tg is measured. Hence,

OL T<Tg

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Therefore,

^ = -̂ (ffW- («) By Bol tzmann ' s theory of entropy,

ASn=k]n^ (7) I I I

where, Q denotes the probability of a s ta te . Subscripts 1 and 2 denote the

s ta tes . In the equation

N

the pa ramete r k, is the Boltzmann's constant; and A5i2 is the change in

entropy of the system between the states 1 and 2.

It is impor tan t to s tudy the change in probability of a polymer segment

varying between more probable and less probable states. If a whole piece of

rubber is s tretched, the ends of the segments move to new positions in the

same proport ion as the whole piece. This phenomenon is called the affine

deformation. Due to their bonded s t ructure , chain ends can not assume more

probable positions; consequently the ideal rubber is unable to relax. Once, the

external stresses axe removed chain ends return to their original distr ibution.

For a polymer segment with a fixed end at Q{x,y.,z) and the other end at

r{x,y,z) (refer to Fig. 1),

n{x,y,z) ={^f^exp{-(3'{x' +y' + z')) (8)

where, Q(x,y,z) is the Gaussian probabiHty distribution.

V 2na^

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10

r(x.y .z)

Figure 1

Random Flight in Three Dimensions

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11

and n is the number of bonds each of length a. The maxiimiuni length to which

a polymer can be elongated is limited by the number of bonds. The model

does not impose any limit on the real probability of the chain lengths.

and

— 2 2 TQ = 7ia

ro^ is the most probable value of r^ and TQ is the RMS (root mean square)

value. The most probable value for r^ as well as for x, T/, and z is zero.

Introduce,

x'^ = xa^, y = yocy., and z = za^

As, the initial orientation is random,

2 2 2 ^0

^ 3

By equations 7 and 8,

Hence,

AS= - ^ ( a , 2 + a , 2 + a , 2 ) . (10)

If q is the effective number of chains, then

A 5 = ^ ( a , ^ + a , ^ + a , ^ ) . (11)

But,

OL T,V Lu OCx TV

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12

It is assumed tha t the volume of the rubber remains constant . Consider

the set of equat ions:

1 T ' l i 1 a^aya, = l, ay^ = a,^ = —^ k f = -jr^{a^ ^ ) .

Qlr LI,, OCr '•U " I

We know t h a t .

A, '•u

Combining the above equations and equation 11, a working equation for the

tensile stress as a function of extension is generated.

(T = RTN{a-\). (13) a"

Elquation is valid for an ideal elastomer.

Visco-elasticity-Maxwell 's Model

For an ideal rubber ,

e = (a-l).

For very small deformations, the elastic and viscous na ture of real materials can

be i l lustrated by a combination of Hookean stress-strain and Newtonian viscos­

ity theories. The quali tat ive nature of visco-elastic behavior can be illustrated

by a simple mathemat ica l model called a Maxwell element. Generafization of

the model by the concept of the distr ibution relaxation times makes it adequate

for quant i ta t ive evaluation.

Maxwell 's element (Fig. 2) consists of a viscous parameter and an elastic

parameter . From the view point of mechanics, it is a series arrangement of a

Hookean spring and a Newtonian dashpot .

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Generalized Maxwell Model

^ - ^ Hookean Spnng

I I ^ Newtonian Dashpot

Simple Maxwell Model

13

Figure 2

Maxwell Model(s)

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14

For a Hookean spring

(Ti = Eei.

Dashpot(Newtonian flow is assumed):

d€.2 <T2 = 7)

dt '

and One-Dimensional Stress:

a- = CTi = a-2, Ci 7^ 62

Therefore,

— = — + — (14) dt dt dt

But, dti 1 dai I da de2 (T2 cr

It ~ 'E'dT ~ 'S'dt' ' ~dt ~ '^ ~ T)

Hence, by equation 14,

de \ da (T

~dt " ^ " ^ "̂ ̂ * ^ ^̂

Case(i) - Creep: Strain rate (creep) is given by the equation,

de da

dt ~ dt'

And a = (JQ(constant), at time t = to, hence, "^ = 0

By equation 15,

de (Jo

dt 77

Integrating we obtain,

^(t) ~-MF. ' » ' ). ( l » i )

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15

Case(ii) . - Stress Relaxation: c = CQ, at t ime t = to, and e is independent of

t ime. Hence, ^ = 0. Therefore, by equation 15,

In tegrat ing we obta in ,

da Ea

at 7]

— Et a = aoexp( ). (17)

Case(iii). - Constant Strain Rate:

Hence, by equation 15

E' dt 7]

Upon integration and rearrangement we obtain.

de — = K => € = Kt. dt

A ^da a ^, , , „ -( T ; ) 3 7 + - = ^ - (18)

— Et a(t)=rjK{l-exp{ )). (19)

Modified Maxwell's Model

By case(iii) above, equation 18,

dt 7)

By equation 13,

^ = E(K - '-). (20)

By definition,

da ^ , , . NRT, 1 .. , ^^ , — = E(K ( « - — ) ) • (21) dt 7] a^

da da

dt da

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16

But by equation 13, da 2 — = NRT{\^—). (22) da a^

Also,

d/T da da

dt da dt

Hence, by comparison,

d^ ,^ NRT 1

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CHAPTER VI

BRIEF DESCRIPTION OF APPARATUS

The apparatus consisted of an Instron machine and test specimens. Tensile

tests were performed on the Instron machine. A test sample (Fig. 3) is clamped

to the Instron machine grips. The cross-head speed is set to a preselected speed

for constant-rate elongation tests. Subsequently, stress relaxation data can be

taken for a short interval of time following a fixed elongation. The machine can

also be fitted with a heating chamber to heat the test sample to the desired

temperature inorder that isothermal data can be obtained.

Dow Corning(DC) test samples were utilized to obtain the experimental

data. Test samples were molded from a cross linked silicone polymer. The high

modulus polymer sample was a two component material, whereas low modulus

polymer was a single component material. Samples obtained by parallel as

well as perpendicular to the processing mould direction were considered for

the study. Table 3 summarizes the test sample specifications.

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18

0.5 inches

effective length 3 inches

Figure 3

Sample Configuration of the Test Specimen

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19

Table 3

Test Specimen Specifications

DC983

Silicone Elastomer

DC795

Silicone Eilastomer

Cross-linked Cross-linked

High Modulus Polymer

Two component sealant

No. of samples

Numbered

Thickness

Effective length

Parallel-cut

Perpendicular-cut

10

1-10

11-20

appr. 0.5 in.

3 in.

Nos. 1-5

Nos. 11-15

Nos. 6-10

Nos. 16-20

Low Modulus Polymer

One component sealant

No. of samples

Numbers

Thickness

Effective length

Parallel-cut

Perpendicular-cut -̂

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CHAPTER VII

EXPERIMENTAL PROCEDURE

The cross-head movement is set to a chosen elongation rate , say 0.1

i n /min . For a suitable chart speed the instantaneous force vs time da ta are

recorded as the sample is being elongated. Stress is obtained by dividing the

force with the measured area of initial(original) cross-section of the test sample.

Extension is obta ined by multiplving the elongation rate by the elapsed time.

Subsequently, relaxation da ta axe also recorded. This occurs during a

short interval of t ime, say about 6 to 10 minutes after the test. Stress is

evaluated as described above. Stress - time data axe utilized for determining

the apparent viscosity of the elastomer. The same procedure is followed to

obtain da t a at different strain rates, viz., 1 and 10 in/min.

For tests at other tempera tures , say 34°C, the temperature is maintained

at a constant prescribed value while the data axe recorded. At this tempera ture

both the stress-alpha da ta as well as stress-relaxation data axe obtained for

different strain rates by the procedure as outlined above. Similar prorodures

axe followed to obtain da ta at temperatures , namely 49 and 90° C. Table 4

summarizes the experimental plan.

20

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21

Table 4

. A Summa: ry Of The Experimental Plan

Sample

number

1 - 20

1 - 10, 16 - 20

1 - 20,

6 - 10, 16 - 20

Temperature

deg. C

24

34

49

90

Elongation Rate

in/min.

0.1, 1

0.1

1, 10

10

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C H A P T E R VIII

RESULTS AND DISCUSSION

An OutHne

Visco-elastic propert ies of silicone sealants, such as elastic modulus , cross­

link density, shear modulus , and viscosity as a function of t empera tu re ,

t ime, elongation ra te , and sample cut-direction will be addressed separately.

Theoretical predictions based on the new visco-elastic model are compared

with the exper imental results. Sources of potential error and possible methods

of improvement axe suggested.

Elongation rates chosen for investigation were 0.1, 1 and 10 in /min . All

the tests were carried out for total elongations of up to 50%, At each of these

strain ra tes , tensile tests were conducted on test samples at four different

isothermal t empera tu res , viz., 24, 34, 49 and 90°C. Uniform maintenance of

high t empera tu res was not at all effective with the equipment used. This was

due to the periodic power surge on the heating system which caused the pen

of the recording device to be displaced periodically from its originzJ locus.

Stress-strain (alpha) da ta (Figs. 4 - 7) at different temperatures and

different rates of elongation were used in verifying the validity of the proposed

model. Da t a at different elongation rates can also be useful in predicting the

max imum or u l t imate elongation rate a sample can withstand without having

to yield. Stress vs. (a - ^ ) <iata were used in determining A ,̂ the cross-link

density. Stress relaxation da ta were obtained in short time intervals. Apparent

viscosities were determined utiUzing the relaxation data .

22

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23

" 30

strain

Figure 4

Stress vs. Strain-Effect of Elongation Rate on High and Low Modulus Materials at 24° C

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24

BB

a «4 B

09 ca o

« 4

ca

0.0

! 1

60

^n —

(

40

30

20 i i

1 l o l

0 0. 0

0

— HME. - • HME. - - LME. - - LME.

^^^^ ^*

0

.1 0.2 J 1

Legend

t = 24 deg.C t = 49 deg.C t = 24 deg.C t = 49 deg.C

^^^^^ * « *

^^^ ^

.^

1 1

.1 0.2 s t ra in

0.3 0.4 c 1.5 + 70

!

r6o 1

^ , ^ - ^ ' ^ 5 0

^^, .^-^^^^. ' - ' ' p-0 — " (

^''' -30 1 ^ » - 1

__...--j;_::::i'—-20

Uio

1 n H 0.3 0.4 0

-0 5

Figure 5

Stress vs. Strain-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 1.0 -^^^

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25

a CO o u

0.0 6 0 ! -

50

0 . 1 0.2

^ 40

30;

« 20

0.3 _ j

0.4 0.5

Legend

HME. t = 24 deg.C HME. t = 49 deg.C LME. t = 24 deg.C LME. t = 49 deg.C

0 0.0 0 . 1 0.2 0.3

s t ra in 0.4 0.

Figure 6

Stress vs. Strain-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 10 m t n

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26

0.0 30 (—

25

0 . 1

•S 20 Ok

0) 09

« 10

0.2 I

0.0

Legend

LME. t - 24 deg.C LME. t = 90 deg.C

0.3 0.4 0.5 30

0 . 1 0.2 0.3 s t ra in

0.4 0.5

Figure 7

Stress vs. Strain-Effect of Temperature on Low Modulus Materials at a Constant

Extension Rate of 10 t n • m m

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Elastic Modulus

Elastic modulus for an ideal elastomer at large strain, and at zero strain

can be predicted, respectively by the following equations:

E = RTN, aTid E = 2RTN

Results obtained (Table 5, Figs. 8 - 10) seem to support the developed model

(equat ion 22). Figure 11, i l lustrates interdependence of s t ress /modulus and

elongation (s t ra in) .

It can be readily noted that the modulus at 50% strain approaches less

than (1 /3 ) of its magni tude at 0% strain. Because at 50% strain, a is 1.5 and

hence, - is 0.6667 as opposed to 0, at infinite strain. Therefore, we can observe

tha t modulus at 50% is less than (1/3) of its magnitude at 0% strain.

By equat ion 22, modulus is predicted at 50% elongation. Also, predicted

values of modulus axe plotted against alpha as a function elongation rate,

t empera tu re and sample cut-direction. Experimental values of A'̂ as obtained

from equation 13 were utilized in predicting the value of the modulus. Elastic

modulus at 50% elongation for the high modulus elastomer (Table 5) varied

from about 117 to about 144 psi, with respect to the elongation rate and varied

from about 116 to about 104 psi, with respect to the tempera ture . The elastic

modulus for the low modulus elastomer (Table 5) varied from about 60 to 68

psi (Table 5), with respect to the the elongation rate and varied from about

60 to 30 psi, with respect to the tempera ture . Again these were predicted for

50% elongation.

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28

In the following table, samples numbered 2-10 represented high modulus

elastomer; the rest of them are representative of low modulus elastomer. In

each category, the first set of values represents the parallel-cut elastomer and

the the second one represents perpendicular-cut elastomer.

Table 5

Elastic Mod ulus As A Function Of Temperature And Elongation Rate

Elastic

Sample

No.

2-5

Modulus

Elongation

Rate

in/min.

0.1

at

24

116.11

50% elongation in psi

Temperature

Degrees Centigrade

34 49 90

1.0 117.34 104.24

10.0 143.65 124.0

6-10

11-15

16-20

0.1

1.0

10.0

0.1

1.0

10.0

0.1

1.0

10.0

126.913

60.6

68.39

68.8

67.96

46.79

58.25

36.3'

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29

1.0 350

300 I t

1.1 1.2 1

Legend

1.4 '350

h300 — HME. XHS = 0.1 in/min - • XHS = 1 in/min •- LME. XHS = 10 in/min T — LME. XHS = 1 in/min

h200

Figure 8

Modulus vs. alpha-Effect of Elongation Rate on High and Low Modulus Materials at

24° C and 90° C

Page 38: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

30

1.0 350

1.1 1.2 1.3 1.4

Legend

HME. HME. LME. LME.

t = 24 deg.C t = 49 deg.C t = 24 deg.C t = 49 deg.C

•350

^300

'-250

^200

r l 5 0

-100

•50

Figure 9

Modulus vs. alpha-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 1 tn

m t n

Page 39: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

31

200

r l " 5

alDha

Figure 10

Modulus vs. alpha-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 10 :zrh

Page 40: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

32

1.2 1.3

0.

09

s 3 o 6 u o OB BB O

140 i^

120 i i !

100

• ^ ,

- 1 4 0 I

- 1 2 0

I 4 00

Legend

HM (ParC) E. st'^ess H.y (PerpC) E. stress HM (ParC) E. modulus HM(PepC)E. modulus

Figure 11

Modulus vs. alpha-Effect of Orientation on High Modulus Material at 24° C and a

Constant Extension Rate of 1 TntTT

Page 41: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

33

Effect of temperature. In general, as the temperature increases polymers

soften and tend to flow more freely due to the decrease in viscosity. With

axiditional increases in temperature, polymers tend to attain their melt state.

And, hence their modulus of elasticity undergoes severe reduction as the fluid

state is approached. Results obtained (Figs. 8 - 10) illustrate this physical

property of materials. On the contrary, equation 22 seem to predict that

the modulus of elasticity increases with the increase in temperature; which

is true (Dr. R.W. Tock, TTU, Lubbock, TX) when a sample is heated

from temperatures below 0° C to room temperature. As the viscosity of

silicone rubbers increases with a decrease in temperature, the viso-elastic model

(equation 19), predicts the stress to increase. And, hence the elastic modulus

increases for a fixed strain. Whereas, for a decrease in viscosity with the

increase of t empera tu re , the visco-elastic model requires the stress to decrease.

And, hence the elastic modulus decreases. Results obtained (Table 5) support

the visco-elastic model.

This interesting physical phenomenon can be best understood by con­

sidering the imiqueness of the polymer structure. Celebrated kinetic theory

predicts that as the tempera ture of the system increases, vibrational 'iiergy

at the molecular level increases significantly. Therefore, the functional bonds

resonate between stable and unstable states alternatigly. This s tate is the well

known ' E X C I T E D STATE' for molecules. As the tempera ture is maintained

constant at a part icular t empera tu re , the excited state of the material sus­

tains, thereby releasing the internal stresses. That is, because of \he rxpaiision

(.'^dimensional) stress experienced by a unit volume of the mnt.- ial derreas.-s.

Page 42: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

34

In o therwords , for a given str2dn(elongation) stress decreases (Fig. 7) with in­

crease in t empera tu re . Therefore, to maintain the validity of the definition of

modulus of elasticity, the elastic modulus decreases. By Fig. 7 it can be noted

tha t elastomer tends to be ideal at higher tempera tures due to the decrease of

viscous effects.

Of course, the above described theory, does not hold for those materials

which contract with the increase of tempera ture . But , silicone elastomers

are not known to contract with the increase in tempera ture . Water at 4*^0,

and mercury are the well known fluids which do not expand with increase of

t empera tu re . Therefore, it appears that the established model is quite effective

in predicting the actual elastic modulus observed particularly at the higher

t empera tu re range.

Effect of strain ra te . It is observed that the elastic modulus increased

with the elongation rate. Results (Fig. 8) axe consistent with theoretical

predictions and the developed model. Stress experienced by the sample is

directly proport ional to the speed of the cross-head (elongation rate) . Also, for

a fixed strain (final elongation), stress experienced by the sample between the

end states is relatively higher at higher elongation rates. For a fixed strain, the

increase (or expansion) in volume or viceversa is the same at any elongation

rate. But, as a material experiences a higher force per unit area at higher

elongation rate , then this induces a higher elastic modulus. .Mtcrnatively.

at low elongation rates Newtonian relaxation is greater than the Hookean

elongation, which results in a less Hookean stress as compared to liigher strain

rates.

Page 43: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

35

As the Hookean effect is predominant in silicone elastomers, less elastic

modulus is displayed at lower elongation rates.

Effect of sample cut-direction. At an elongation rate of 1 in/min. and a

temperature of 24** C perpendicularly-cut elastomer has an elastic modulus

of 126.913 psi whereas a parallel-cut elastomer has an elastic modulus of

116.11 psi. This difference being less than 10% was not considered significant.

Differences of this magnitude could be due to experimental error, to the

non-uniformity in obtaining the sample or they could be due to the lack of

perfectness in the numerical methods . Therefore, it was concluded that the

sample cut-direction (Fig. 11) does not have a significant effect on elastic

modulus of silicone sealants.

Effect of alpha. Elastic modulus displayed mixed trends with respect to

elongation ratio (alpha which is a function of time). Initially, elastic modulus

decreases with the increase of elongation. As the sample at tains the maximum

preset s train, the modulus seems to have increased. As the elongation rate

is constant , at the final elongation condition, the cross-head experiences a

seemingly sudden restricting force. Thus, the sample experiences a reverse

expansion or compression. Consequently, the resultant force on the sample is

more than the expected value, which results in the observed increase of the

elastic modulus .

These results conform to the expectations that arise by observing the

stress strain (alpha) curves (Figs. 4 - 7). Slopes at smaller (initial) elongations

and at 50% elongation seem to be higher when compared to the relatively flat

middle portif)n of the curve. Ar(ordino;ly, modulus decreases from a hi^h initial

Page 44: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

36

value ( m a x i m u m in almost all the samples) and then increases. In some cases,

at 50% elongation, the sample nearly regained its initial value.

Cross-Link Density

Cross-Hnks affect bo th the physical as well as the chemical properties of

organic compounds . Physical propert ies such as melting point, boiling point,

viscosity, density, and elastic moduli , etc. , are greatly influenced by the number

and the position of cross-links. The greater the cross-link density the larger

Young's modulus (elastic modulus) is expected to be. Some of the impor tant

factors tha t influence cross link density are type of cure, water immersion,

shelf life of the sample, and environmental aging-temperature , UV radiation,

oxygen, ozone, humidity, etc.

Cross-hnk density of the high modulus elastomer (Table 6) was found to

lie in the range of 14.3 x 10"^ - 17.1 x 10~^ lb mol/cu.ft , and that of low

modulus elastomer (Table 6) was in the range of 6.7 x 10 ~^ - 8.0 x 10"^ lb

mol/cu.f t . Slopes of a vs. {a - -^) curves (Figs. 12 - 14) was utiHzed in

evaluating the cross-link density, A .̂

At 50% elongation, cross link density of the high modulus elastomer varies

from about 16.7 x 10~^ to about 17.1 x 10"^ lb mol/cu.ft , with respect to

elongation rate and varied from about 16.7 x 10"^ to about 14.6 x 10~^ lb.

mol/cu.ft with respect to tempera ture .

At 50% elongation cross link density of the low modulus elastomer varied

from about 7.6 x 10"^ to about 8.0 x 10"^ lb mol/cu.ft , with respect to

elongation ra te .

Page 45: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

37

In the following table, samples numbered 2-10 represent high modulus

elastomer; the rest of them represent low modulus elastomer. In each category

first set represents the parallel-cut elastomer and the second one represents the

perpendicular-cut elastomer.

Table 6

Cross-hnk Density As A Function Of Temperature And Elongation Rate

Cross-hnk Density at 50%) elongation, multiply by 10"^ lb mol/cu.ft

Sample Elongation Temperature

No. Rate Degrees Centigrade

in/min. 24 34 49 90

2-5

6-10

11-15

16-20

0.1

1.0

10.0

0.1

1.0

10.0

0.1

1.0

10.0

0.1

1.0

10.0

16.7

18.2

17.1

8.63

7.96

7.02

7.6

7.99

14.3

14.6

.03

6.62

Page 46: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

38

60

50

^ 40 o.

0.1

OB 9 «

30

20

0.2 0.3

Legend

HME. XHS = 0.1 in/min - ! • • XHS = 10 in/min - - - LME. XHS = 1 in/min -^- LME. XHS = 10 in/min

0.2 0.3 0.4 (alpha - 1/alpha sq.)

Figure 12

Stress vs. (a — ^)-Effect of Elongation Rate on High and Low Modulus Materials at 24° C

Page 47: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

39

«B

a

oa sa 0)

u

0.2 0.3 (alpha - 1/alpha sq.)

Figure 13

Stress vs. (a - ^ ) -Ef fec t of Temperature on

High and Low Modulus Materials at a

Constant Extension Rate of 1 i n rntn

Page 48: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

40

0.0 60 i—

50

£ 40

0.1 0.2 0.3 I

0.4 I

Legend

KME. t = 24 deg.C --K- HME. t = 49 deg.C -*- LME. t = 24 deg.C — - LME. t = 49 deg.C

0.5 —^60

• -̂̂ •^•+

> 5 0

0.0 0 . 1 0.2 0.3 (alpna - l /alpna sq.)

Figure 14

Stress vs. (a V)-Effect of Temperature on

High and Low Modulus Materials at a

Constant Extension Rate of 10 -^-

Page 49: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

41

Cross-Hnk density varied from about 6.63 x 10 ~^ to about 7.6 x 10 ~^ lb

mol/cu.f t . , with respect to t empera tu re . The effect of t empera ture seems to

be more significant than that of elongation rate; which was expected for the

ranges covered.

For both high and low modulus elastomers variation in cross Unk density

with t empera tu re is about a max imum of ± 1 3 % . Considering the accuracy of

strain measurements and t empera tu re control (experimental errors), processing

of sample material and possible errors associated with numerical methods, the

variation was not considered significant.

An increase of t empera tu re to sufficient levels can affect the internal

s t ruc ture greatly. Even at higher elongation rates, at high temperatures cross­

links merely slip and slide along till they give away. Polymers are capable of

retaining a high percentages of their original s t ructure if the strain is within the

yield point , even at high elongation rates. On the contrary, the tempera ture

effect tends to permanency. Removal of heat source will restore the sample

to its original condition but to a much lesser degree. The decrease in elastic

modulus can be compared to that in cross-hnk density, with the increase of

t empera tu re . Thus the results conform to theoretical expectat ions.

Differences between theoretical and experimental moduh can be explained

in terms of retentivity of cross-hnks. Low modulus elastomers can not re­

tain the cross-link s t ructure as much as the high modulus elastomers can

for a given elongation. Hence, the difference between the experimental and

predicted values was expected to be greater for the low modulus elastomers.

Page 50: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

42

Due to their proport ional dependency cross-hnk density and modulus of

elasticity exhibit similar t rends .

Viscosity

Results given in Table 7 show that the apparent viscosity of test samples

increased as the elongation rate increased. The results shown in Figs. 15 - 18

confirm tha t silicone elastomers conform to the established material property,

as the t empera tu re increases viscosity decreases.

Viscosity of high modulus parallel-cut elastomer varied from about 6.9425

X 10*̂ to about 0.2834 x lO'̂ psi min with respect to elongation rate and varied

from 6.9425 x 10^ to about 0.2555 x 10^ psi min with respect to temperature ,

viscosity of high modulus material with a perpendicular-cut varied from 0.897

X 10*̂ to 0.20886 X 10^ psi min with respect to elongation rate and varied from

0.20886 X 10^ to about 0.055623 x 10^ psi min with respect to tempera ture .

Viscosity of the low modulus perpendicular-cut elastomer varied from

about 0.8556 x 10^ to about 0.124275 x 10^ psi min with respect to elongation

rate and varied from about 0.124275 x 10^ to about 0.085157 x lO^with respect

to t empera tu re . Viscosity of the parallel-cut elastomer varied from about

0.7473 X 10^ to about 0.057944 x 10^ psi min with respect to elongation rate

and varied from about 0.7473 x 10^ to about 0.6449 x 10^ psi min with respect

to t empera tu re .

As the rate of elongation is increased, the velocity gradient - ^ between

different layers of elastomers decreases. V being, the velocity of a freely

flowing (top) layer with respect to a fixed (assumed to be along central axis

•^TH

Page 51: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

43

of sample) layer. In the following table, samples 2-10 numbered represent the

high modulus elastomer; the rest of them represent the low modulus elastomer.

In each category, the first set represents the parallel cut samples and the second

one represents the perpendicular-cut samples.

Table 7

Viscosity As A Funct ion Of Temperature And Elongation Rate

. . <-> r

Viscosity

Sample

No.

2-5

6-10

11-15

16-20

at 50% elongation

Elongation

Rate

in/min.

0.1

1.0

10.0

0.1

1.0

10.0

0.1

1.0

10.0

0.1

1.0

10.0

24

694.25

81.782

28.34

89.77

20.886

74.73

5.7944

85.56

75.11

12.4275

in psi-minA

Temperature

Degrees

34

90.805

Centigrade

49 90

78.80

25.55

5.5623

64.49

K.5157

-ran

Page 52: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

44

0 1 2 3 4 5 6 7 8 9 iOO

n n o u n

00 o

10 H

•80

•70

-60

'[-50 I

-40

-30

1—I—r

•-1-20

iio 0 1 2 3 4 5 6 7 8 9

time in min.

Legend

- ^ HM(ParC). XHS = 0.1. t = 34 -- HM (PerpC). XHS = 0.1. t = 34 --- LM(ParC). XHS = 1 . t = 24 — LM (PerpC). XHS = 1 . t = 24

Figure 15

Relaxation Phenomenon-Effect of Orientation on High and Low Modulus Materials

Page 53: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

45

100 h 4-60

CQ o u

^^ 03

eto o

rr:

k

10 ^

%

1 3 4 t ime in min.

-50

'r-40

30

r20

•^10

Legend

HN'iE. XHS = 0.1 m/min HME. X-S = 10 m/min .

- - - LME. XHS = 0.1 in /mm LME. XHS = 10 in/nin.

Figure 16

Relaxation Phenomenon-Effect of Elongation Rate on High and Low Modulus Materials at 24° C

Page 54: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

46

5 e

CQ

P.

1 CQ CQ 4) U CQ

' ^ O

r 40

^30

-20

Hy:E. _ - . HME. t . . . LME. t ,—LME. t

Legend t = 24 deg.C

= 49 deg .C = 24 deg.C = 49 deg.C

2 3 4 time in mm.

Figure IT Fffect of Temperature on

Relaxation ^^^;r:TZfL Materials at a

Page 55: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

47

0 . 0 100 f-

Cfl

CQ OQ O u

u _ __

0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 —̂ ' \ 1 1 U90

•80

70

1-60 - - - J

J L-OS ! -Mrz." • o

10 h -0 .0

•50

T 0.2 0.4 0.6 0.8

time in min.

-40

^^'-'-"--7-30

^20

1 r l O 1.0 1.2

Legen6.

HME. t = 24 deg.C --• HME. t = 90 deg.C --- LME. t = 24 deg.C —-- LME. t = 90 deg.C

Figure 18

Relaxation Phenomenon-Effect of Temperature on High and Low Modulus Materials at a

Constant Extension Rate of 10 t n m t n

Page 56: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

48

Hence,

a

dx

increases for a given stress. Therefore, at higher elongation rates, the ideal

Therefore, at higher elongation rates , the ideal elastomer effects predominate

over the Newtonian viscous effects.

Discussion

The developed model seems to reasonably describe classical theoretical

predictions within experimental errors. .The predicted elastic modulus was

observed to be high at very low elongations. Also, for each of the modulus

vs. a lpha curves the first few readings of the predicted values axid those close

to a = 1.5 differ significantly from the experimental values. But, both the

predicted and experimental values are high initiaUy.

It is observed that the elastomer exhibits a behavior best described by

the 'B ingham' plastic model. During the initial period, the elastomer seems

to require a threshold stress to set the molecules in motion. This threshold

stress gives rise to a high initial modulus. Differences between the theoretical

predictions and the observed experimental values and some possible errors

could be due to the following reasons.

(i) Even from the very beginning, the elastomer was elongated at a

constant ra te . As per the Bingham model, the sample should not start

elongating until a certain stress level is reached. This molecular phenomenon is

taken into consideration by the theoretical model. Experimentally the sample

s tar ts moving along with the cross-head.

Page 57: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

49

(ii) At the maximum extension, that is when a = 1.5, the cross-head

actually is not in motion. The theoretical model can not take this fact into

account.

(iii) Data were obtained for different samples of slightly different dimen­

sions. Then the data are reduced using a cubic least-square polynomial. Wher­

ever density of the data is the greatest the accuracy should be better. In other

words, the fitted curve does not pass through all the points.

(iv) Experimental errors include: (a) nonuniform temperature control, (b)

non-homogeneity of sample preparation, (c) repeated experiments with the

same sample at different experimental conditions, and (e) approximate strain

measurement. Larger errors would arise from inconsistencies in parts (a) and

(b).

Page 58: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

C H A P T E R DC

CONCLUSIONS AND R E C O M M E N D A T I O N S

Conclusions

Visco-elastic propert ies of silicone elastomers have been successfully mod­

eled. These propert ies can be useful in the prediction of the properties of

silicone sealants, used extensively as a s tructural component. The rigor of

numerical methods determines the accuracy of the fitted data, and hence ac­

curacy of the modeling. Lack of published da ta for verification is a handicap in

determining the accuracy of the results. Yet, results were found to be consis­

tent on a quali tat ive scale. Conclusions based on short-durat ion observations

are,

(i) Elastic modulus can be well described as a function of elongation rate,

extension or t ime and tempera ture by the developed model.

(ii) Cross-link density can be estimated as a function of elongation, rate,

elongation or t ime. Tempera ture dependency is also successfully monitored.

Cross-link density is fairly constant for a sample. It is established that

cross-hnk density does not vary significantly either with tempera ture or with

elongation ra te . Difference between the predicted and experimental values of

high modulus elastomers is less due to high cross link density.

(iv) Results of viscosity measurements are in excellent agreement with the

theoret ical predictions.

(v) Sample cut-direction does not affect the visco-elastic properties of

these silicone elastomers significantly.

50

Page 59: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

51

Recommendat ions

I think tha t this project gives a preHminary insight regarding the impor­

tan t role of visco-elastic propert ies of s t ructural siHcone elastomers. It would

be advantageous to extend the study of s t ructural sihcone sealants in a broad

sense. I think tha t the following points deserve consideration:

(i) By changing the na ture of cure, visco-elastic properties can be varied

to decide an op t imum curing procedure. Curing procedures are classified

according to mechanism tha t takes place. Some of the conmiercially important

curing mechanisms which deserve consideration are: Free-radical cross-linking,

cross-linking of Hnear or Hghtly branched polymer chains with reactive end

groups, and cross-linking of polymer types having different functional groups

which react with one another at a controllable rate (also, known as adjustable

cure).

(ii) I think that the constant stress condition would make the most ideal

research topic. Because, from the practical s tandpoint neither the rate of

elongation nor the max imum strain of sealant can be possibly monitored. 1

think tha t high velocity wind gusts can exert a constant stress over a period of

t ime on s t ructura l components . It would be useful to model the visco-elastic

propert ies of sealants under such conditions. Rigorous practical experience

suggests tha t stress exerted by high velocity winds fluctuates between certain

ex t remum. For example, according to typical weather reports , if wind speed

is about 20 mph, then the actual wind speed would be 20 ± 5 mph.

Page 60: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

52

Hence, the effect of such varying stress provides necessary practical

information. One can vary loading rate on an Instron machine to achieve

this variable stress condition.

m^^^mmmmmew « r \

Page 61: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

BIBLIOGRAPHY

Anderson, J.B., "A Method For Finding Engineering Properties Of Sealants," Journal Of Engineering Mechanics, ASCE, Vol. I l l , No. 7, July, 1985, pp. 882-892.

Instron Corporation, "Instron Universal Testing Instrument Model 1122 Op­erating Instructions," Instron Corp., Canton, Mass., 1975.

Robert S.M., "Measurement Of Dynamic Properties Of Rubber," Industrial And Engineering Chemistry. Vol. 44, No. 4, April 1952, pp. 696 -704.

Rodriguez, F., Principles Of Polvmer Systems. 2nd Ed., McGraw-Hill Book Company, New York, NY, 1982.

Thor L.S., "Viscoelastic Behaviour Of Polyisobutylene under Constant Rates Of Elongation," Journal Of Polvmer Science. Vol. XX, 1956, pp. 889-100.

V.\

Page 62: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

GLOSSARY

Table nos. 1 and 2 provide a complete description of the symbols used in

the text. Objective of providing this glossary is to explain some of the terms

used in explaining results and discusssion by way of figures.

HME

LME

XHS

m •mtn

ParC

PerpC

High Modulus Elastomer

Low Modulus Elastomer

Cross Head Speed

ambient temperature (plots only)

in: inches, min: minutes

viscocity

strain

Parallel Cut

Perpendicular Cut

54

Page 63: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

With Reverence To My Beloved Mother

Page 64: THE THEORETICAL TREATMENT OF VISCO-ELASTIC …

PERMISSION TO COPY

In presenting this thesis in partial fulfillment of the

requirements for a master's degree at Texas Tech University, I agree

that the Library and my major department shall make it freely avail­

able for research purposes. Permission to copy this thesis for

scholarly purposes may be granted by the Director of the Library or

my major professor. It is understood that any copying or publication

of this thesis for financial gain shall not be allowed without my

further written permission and that any user may be liable for copy­

right infringement.

Disagree (Permission not granted) Agree (Permission granted)

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y

:"r L f 7 >

Date Date