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SISOM 2007 and Homagial Session of the Commission of Acoustics, Bucharest 29-31 May

CHARACTERISTIC FREQUENCIES RELATIONSHIPS IN THE RHEODYNAMICS OFSTANDARD LINEAR VISCOELASTIC SOLID

Horia PAVEN

ICECHIM - National Research and Development Institute of Research for Chemistry and Petrochemistry, Bucharest; email:[email protected]

The contribution is dealing with the effect of representation type on the apparent rheodynamic

properties within the realm of standard linear viscoelastic model of solid-like behaviour. Taking

into account the dynamic strain-, and stress-controlled processes, respectively, the characteristicfrequencies relationships corresponding to maximum and/or inflection conditions for welldefined sets of viscoelastic quantities are established in terms of meaningful rheologicalparameters, the consequences being illustrated in case of natural, semi-logarithmic,

logarithmic, and double-logarithmic representations.

Key words: rheodynamics, linear viscoelasticity, representations, characteristic frequencies.

1. INTRODUCTION

The viscoelastic features of operation of structural materials are rather a premium than a weakness if

the know-how and the know-why aspects of rheological behaviour are carefully balanced, the matter being

critical especially in case of polymer materials, where the involved native relaxation/retardation and mixedmechanisms drive in a well defined manner the effective response to different loadings [1]-[3].

Aiming at to point out the influence of the peculiarities of strain- and stress-controlled processes,

respectively on the corresponding response of different viscoelastic-like solids, the flow chart includes the

rheological equations, which provide the background, the expressions of the rheodynamic properties, which

definine the self contained sets of quantities to be considered from the standpoint of characteristic

frequencies, as well as the explicit illustration of using different representations [4]-[7].

2. PHENOMENOLOGICAL PREREQUISITES

The rheological equation in case of standard linear viscoelastic solid is defined as

tt DAADB 101 +=+ (1.1)for a strain()-controlled process [8], [10], [12], and

tt DCCDD 101 +=+ (2.1)

for a stress()-controlled one [9], [11]; , are the natural rheological variables, whereas and

represent the characteristic rheological parameters, and

110 ;, BAA

110 ;, DCC dtdDt /= stands for the first ordertime derivative.

Consequently, in the circumstance of a sinusoidal -controlled process of circular frequency , the

resulting set of rheodynamic quantities comprises the primary ones - the storage modulus, )()1,1(M , loss

modulus, )()1,1(M , absolute modulus, , and loss factor,|)(|*

)1,1( M )()1,1( M - as well as the secondary

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Characteristic frequencies relationships in the rheodynamics of standard linear viscoelastic solid57

ones - the storage compliance, )()1,1( MJ , loss compliance, )()1,1( MJ , and absolute

compliance, | , defined by)(| * )1,1( MJ

)1/()()( 22

1

2

110)1,1( BBAAM ++= , )1/()()(22

1101)1,1( BBAAM +=

2/1221

221

20)1,1(

* )]1/()[()( BAAM ++= , (1.2))/()()( 2110101)1,1( BAABAAM +=

)/()()( 22

1

2

0

2

110)1,1( AABAAJM ++= , )/()()(22

1

2

0101)1,1( AABAAJM +=2/122

1

2

0

22

1

*

)1,1( )]/()1[()( AABJM ++= , (1.3))/()()(2

110101)1,1( BAABAA

MJ+=

where the rheological parameters are given in terms of meaningful quantities

, (1.4.1)00)1,1(lim

0 )( AMM == 11)1,1(lim

/)( BAMM ==

, (1.4.2)000)1,1(lim

0 /1/1)( MAJJM === === MABJJM /1/)( 11)1,1(lim

On the other hand, in the situation of a sinusoidal -controlled process of circular frequency , the

resulting set of rheodynamic quantities contains the primary ones - the storage compliance, )()1,1(J , losscompliance, )()1,1(J , absolute compliance, , and loss factor,|)(|

*

)1,1( J )()1,1(J - as well as the

secondary ones - the storage modulus, )()1,1(JM , loss modulus, )()1,1(JM , and absolute

modulus, , stated as|)(| * )1,1( JM

)1/()()( 22

1

2

110)1,1( DDCCJ ++= , )1/()()(22

1110)1,1( DCDCJ +=2/122

1

22

1

2

0)1,1(* )]1/()[()( DCCJ ++= , (2.2)

)/()()(2

110110)1,1( DCCCDCJ +=

)/()()( 22

1

2

0

2

110)1,1( CCDCCMJ ++= , )/()()(22

1

2

0110)1,1( CCCDCMJ +=2/122

1

2

0

22

1

*

)1,1( )]/()1[()( CCDMJ ++= , (2.3))/()()(2

110110)1,1( DCCCDC

JM+=

the different rheological parameters being expressed in the form

00)1,1(

lim

0 )( CJJ == , (2.4.1)11)1,1(lim

/)( DCJJ ==

000)1,1(

lim

0 /1/1)( JCMMJ === , (2.4.2) === JCDMMJ /1/)( 11)1,1(lim

3. RELATIONSHIPS AND MODELLING RESULTS

The key relationships providing the necessary conditions of maximum (m) and inflection (i) for distinct casesof independent - dependent variables for different natural, semi-logarithmic, logarithmic and double-

logarithmic representations - result by solving well established equations. As a matter of fact one obtain

- for natural )(, R - representation

maximum: ;0)(=

d

dRinflection: 0

)(2

2

=

d

Rd. (3.1)

- for semi-logarithmic )(,ln R - representation

maximum:

0)(

1

1)(

ln

1)(

ln

)(

ln

)(=====

RR

ddd

dR

d

d

d

dR

d

dR;

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Horia PAVEN 58

inflection: (3.2)

0])()([ln

1)(

ln

)(

ln

)(

ln

)(2

2

=+=

=

=

=

RR

d

dd

Rd

d

d

d

Rd

d

Rd

d

Rd.

- for logarithmic )(ln, R - representationmaximum:

0)(

)()(ln=

=

R

R

d

Rd; (3.3)

inflection: 0)(

)()()()(

)(ln2

2

2

2

=

=

=

R

RRR

d

R

Rd

d

Rd.

- for double-logarithmic )(ln,ln R - representation

maximum:

;0)(

)(

ln

1

ln

)(ln

ln

)(ln

=

=

==

R

R

d

dR

R

d

d

d

Rd

d

Rd

inflection: (3.4)

0)(

)(

)(

)(1

)(

)()(

ln

1)(

ln

)(

ln

ln

)(ln

ln

)(ln

2

2

2

2

=

+

=

+

=

=

=

==

R

R

R

R

R

R

R

RRR

R

R

d

dd

R

Rd

d

R

Rd

d

d

Rdd

d

Rd

The natural characteristic relaxation frequencies corresponding to maximum (m) and/or inflection (i)

conditions are disposed in the well defined sequence (Table 1.1)

}{}{}{}{}{}{MJMJMMMM

imJiJmJiJi

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Table 1.1 Characteristic frequencies of rheodynamic quantitiesforcontrolled processes.

Viscoelastic

Quantity

Frequency Value

[rad/s]M

}{Mi )3/3(

M }{Mm

M }{Mi 3

M}{Mi 1/31)/)(3/3( 20

2

0 + MMMM

M }{ Mm MM /0

M }{ Mi MM /3 0 MJ }{ MJi )/)(3/3( 0 MM

MJ }{ MJm )/( 0 MM

MJ }{ MJi )/(3 0 MM

MJ }{ MJi 1/31)3/3( 220 + MM

MJ }{

MJm MM /0

MJ }{

MJi MM /3 0

Table 1.2. Calculated values of different characteristic frequenciesin case ofcontrolled processes

Viscoelastic

Quantity

Characteristic frequencies

[rad/s]

M1 wi{M1} 0.57735 1.154701 2.886751

wm{M2} 1 2 5M2

wi{M2} 1.732051 3.464102 8.660254

M wi{M} 0.024021 0.048042 0.120105

wm{BM} 0.031623 0.063246 0.158114BMwi{BM} 0.054772 0.109545 0.273861

J1M wi{J1M} 0.000577 0.001155 0.002887

wm{J2M} 0.001 0.002 0.005J2M

wi{J2M} 0.001732 0.003464 0.008661

JM wi{JM} 0.000707 0.001414 0.003536

wm{BJM} 0.031623 0.063246 0.158114BJM

wi{BJM} 0.054772 0.109545 0.273861

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Horia PAVEN 60

Table 2.1 Characteristic frequencies of rheodynamic quantities

forcontrolled processes.

ViscoelasticQuantity

Frequency Value[rad/s]

J }{Ji )3/3(

J }{Jm

J }{Ji 3

J }{Ji 1/31)/)(3/3( 202

0 + JJJJ

BJ }{ Jm JJ /0

BJ }{ Ji JJ /3 0

JM }{ JMi ,)/)(3/3( 0 JJ

JM }{ JMm )/( 0 JJ

JM }{ JMi )/(3 0 JJ

JM }{ JMi 1/31)3/3( 220 + JJ

JMB }{

JMm JJ /0

J

MB }{J

Mi

JJ /30

Table 1.2. Calculated values of different characteristic frequenciesin case ofcontrolled processes.

Viscoelastic

Quantity

Characteristic frequencies

[rad/s]

J1 wi{J1} 0.57735 1.154701 2.886751

J2 wm{J2} 1 2 5

wi{J2} 1.732051 3.464102 8.660255

J wi{J} 0.707107 1.4142 3.536BJ wm{BJ} 31.62277 63.246 158.114

wi{BJ} 54.77226 109.545 273.861

M1J wi{M1J} 577.35 1154.701 2886.751

M2J wm{M2J} 1000 2000 5000

wi{M2J} 1732.051 3464.102 8660.254

MJ wi{MJ} 24.02118 48.0423 120.105

BMJ wm{BMJ} 31.62277 63.246 158.114

wi{BMJ} 54.77226 109.545 273.861

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Figure 1. -controlled rheodynamic quantities vs. frequency in different representations.

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Horia PAVEN 62

Figure 2. -controlled rheodynamic quantities vs. frequency in different representations

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4. CONCLUSIONS

The self contained evaluation of viscoelastic behaviour needs, even in the framework of linear

approximation, the consideration of well established full sets of modulus- as well as of compliance-like

quantities.

The characteristic frequencies are obtained in terms of rheological parameters and point out meaningfulinformation on the intrinsic inter-relations underlying the effective response of the material.

It is necessary to distinguish between the true effects and possible artefacts arising due to different

representations.

REFERENCES

1. SPERLING, L. H.,Introduction to Physical Polymer Science, Wiley, New York, 2006.

2. PAVEN, H.,Efecte reodinamice in compozite polimerice binare - I. Cuplajul morforeologic "Pi", Revista de Chimie, 49(9), 878-883, 199 1998.

3. PAVEN, H.,Efecte reodinamice in compozite polimerice binare - II. Cuplajul morforeologic "Sigma", Revista de Chimie,49(12),910-915, 1998.

4. PAVEN, H., POPOVICS, S., Efecte reodinamice fundamentale in sisteme compozite binare cu componenti cu comportarevascoelastica liniara - I. Modulul de inmagazinare, SISOM, Bucharest, 173-184, 1999.

5. PAVEN, H., POPOVICS, S., Efecte reodinamice fundamentale in sisteme compozite binare cu componenti cu comportare

vascoelastica liniara - II. Modulul de pierderi, SISOM, 185-196, 1999.6. PAVEN, H., Efecte reologice cantitative si calitative in cazul compozitelor solide binare cu componenti cu comportare

vascoelastica liniara, Proc. IXth European Conference on Composite Materials "Composites-from Fundamentals toExploitation", IOM, London, X01-X10 , London, 2000.

7. PAVEN, H., Restrictii intrinseci asupra regulilor de selectie reologice pentru compozite binare cu componenti cu comportare

vascoelastica liniara, Proc. XIIIth International Congress on Rheology, Vol. 4, 18-20, BSR, Cambridge, 2000.8. PAVEN, H., POPOVICS, S., Interdependente reodinamice fundamentale in cadrul comportarii vascoelastice liniare standard a

solidului PTZ.- I. Procese cu deformare controlata , SISOM, 277-286, 2002.9. PAVEN, H., POPOVICS, S., Interdependente reodinamice fundamentale in cadrul comportarii vascoelastice liniare standard a

solidului PTZ.- II. Procese cu tensionare controlata, SISOM, 287-296, 2002.10. PAVEN, H., Relatii exacte in analiza fenomenologica a proceselor reodinamice de relaxare/retardare in vascoelasticitatea

liniara a sistemelor polimerice. I.1 Marimi caracteristice primare in cazul proceselor reodinamice de deformare controlata,Materiale Plastice, 40(4), 171-176, 2003.

11. PAVEN, H., Relatii exacte in analiza fenomenologica a proceselor reodinamice de relaxare/retardare in vascoelasticitatea

liniara a sistemelor polimerice. II.1 Marimi caracteristice primare in cazul proceselor reodinamice de tensionare controlata,Materiale Plastice, 41(2), 56-61, 2004.

12. PAVEN, H., Relatii exacte in analiza fenomenologica a proceselor reodinamice de relaxare/retardare in vascoelasticitatea

liniara a sistemelor polimerice. I.1 Marimi caracteristice secundare in cazul proceselor reodinamice de deformare

controlata, Materiale Plastice, 43(4), 345-351, 2006.

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