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ON SOME BASIC FREQUENCY EFFECTS WITHIN THE

AB - (2,2) LINEAR VISCOELASTIC SOLID - LIKE BEHAVIOUR.II. STRESS-CONTROLLED RHEODYNAMIC PROCESSES

Horia PAVEN*, Sandor POPOVICS**

* Chemical Research Institute ICECHIM, Bucharest-77208, ROMANIA

** Drexel University, Philadelphia, Pa 19104, USA

Given the intimate peculiarities ofstress-controlledprocess, a well defined hierarchy of different

rheodynamic properties arises in a natural way. The primary quantities, including the storage and

loss compliances, the derived ones, including the dynamic viscoelastic modulus and thecorresponding loss factor, as well as the secondary quantities, including storage, loss and dynamic

viscoelastic moduli, are discussed within Alfrey - Burgers (2,2) - rheological state model. The exact

relationships concern the frequency dependence of rheodynamic quantities of the set of distinct

characteristics, as well as of their first- and second-order frequency derivative. The use of a

powerful symbolic calculus package ensures necessary precision and effectiveness for clarifying the

dependence of characteristic frequencies on rheological parameters as well as the relevant trends for

better control of material multifunctionality.

1. PRELIMINARIES ON STRESS-CONTROLLED RHEODYNAMIC PROCESSES

Stress-controlledrheodynamic processes occur frequently both in characterization and testing as

well as in end-use conditions. In the case of (1,1) linear viscoelastic behaviour, the response, expressed by

means of the components of the complex compliance, including the storage- and loss- compliance, the

dynamic viscoelastic compliance and the corresponding loss factor, evidence significant trends from the

standpoint of frequency dependence. Moreover, the modulus-like quantities calculated on the basis of

inverse relationship of modulus and compliance, outline the significant features, major complications

being expected in case of Alfrey-Burgers (2,2) linear viscoelastic state. The following analysis provides the

exact relationships fully supporting this assertion.

In the case of linear viscoelastic solid-like behaviour, to the (m, n) rheological state corresponds a

well established rheological equation, given in the operator representation as

)()( )()( tmtn DPDQ = (II.0.1)

where

)()2(

210)( ...)(n

tntttn DqDqDqqDQ ++++= (II.0.2.1)

)()2(

210)( ...)(m

tmtttm DpDpDppDP ++++= (II.0.2.2)

depict the linear differential polynomial operators,kkk

t dtdD /)(= represent the k-th order time

derivative, and , are the natural rheological variables, for the sake of simplicity the uni-dimensional

case is taken into account. Consequently, to a sinusoidal stress-controlled input, ~ , corresponds asresponse a steady strain-state, ~ ,

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Horia PAVEN, Sandor POPOVICS 98

)](exp[~)exp(~ 00 ==== titi (II.0.3)

By definition

)(/)()(~/~ )()(*

),( iQiPJ nmnm == (II.0.3.1)

is the complex compliance, and

)()()( ),(),(*

),( nmnmnm JiJJ = (II.0.3.2)

Taking the complex conjugate

)()()( ),(),(*

),( nmnmnm JiJJ += (II.0.3.3)

results for the (2,2) - storage (dynamic elastic) compliance, )(),( nmJ ,

})(Re{)(*

),(),( nmnm JJ = (II.0.4)

whereas for the (2,2) - loss compliance, )(),( nmJ , one obtains

})(Im{)(*

),(),( nmnm JJ = (II.0.5)

As a direct consequence of basic hierarchical reasons, it is natural to consider, in the case ofstress-

controlledstate, )(J and )(J asprimary rheodynamic quantities, and as derivedones - the dynamic

viscoelastic (absolute) compliance, )(J , and the loss factor, )(J , defined as

)](1)[()( ),(),(*

),( nmJnmnm iJJ = (II.0.6)

)](sin)()[cos()( ),(),(),(*

),( nmnmnmnm iJJ = (II.0.7)

2/12

),(

2

),(

*

),(),( )]()([)()( nmnmnmnm JJJJ +== (II.0.8)

)(/)()(tan)( ),(),(),(),( nmnmnmJnmJ JJ == (II.0.9)

Accordingly, the corresponding secondary rheodynamic quantities are defined taking into account

the following definitions for different moduli

)()()( ),(),(*

),( nmJnmJnmJ MiMM += (II.0.10)

)](sin)()[cos()(),(),(),(

*

),( nmJnmJ MMnmJnmJ iMM += (II.0.11)

)(/)()(2

),(),(),( nmnmnmJ JJM = (II.0.12)

)(/)()(2

),(),(),( nmnmnmJ JJM = (II.0.13)

2/12

),(

2

),(

*

),(),( )]()([)()( nmJnmJnmJnmJ MMMM +== (II.0.14)

)()(/)()(tan)( ),(),(),(),(),( nmJnmJnmJMM MMnmJnmJ === (II.0.15)

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II. Stress-controlled rheodynamic processes99

2. ALFREY-BURGERS (AB) LINEAR VISCOELASTIC SOLID-LIKE RHEOLOGICAL MODEL

In order to clarify the peculiarities of different relationships in the case of AB-like rheological

equation

)2(210)2(

210 tttt DpDppDqDqq ++=++ (II.1.1)

corresponding to (2,2) - rheological state, the corresponding characteristic rheological operators are

)2(

210)1( )( ttt DqDqqDQ ++= (II.1.2.1)

)2(

210)1( )( tt DpDppDP ++= (II.1.2.2)

Consequently,

])()(/[])()([

)()()(~/~

2

2

102

2

10

)2,2()2,2(

*

)2,2(

qiqiqpipip

JiJJ

++++=

===(II.1.3)

In virtue of the definition of conjugate of complex compliance

)()()( )2,2()2,2(*

)2,2( JiJJ += (II.1.3.1)

where

})(Re{)(*

)2,2()2,2( JJ = (II.1.4)

})(Im{)(*

)2,2()2,2( JJ = (II.1.5)

Taking into account the above definitions, in case ofprimary quantities result for the storage

(dynamic elastic), )1,1(J , and loss compliance, )1,1(J , respectively

))2(1/(

/))(()(

42

2

2

2

2

1

422

2202110

')2,2(

DDD

DCDCCDCCJ

++

++= (II.1.4.0)

))2(1/(

/))()(()(

42

2

2

2

2

1

3

1221110

"

)2,2(

DDD

DCDCCDCJ

++

+=

(II.1.5.0)

where the characteristic rheological parameres are defined in terms of nominal rheological parameters as

022011022011000 /,/,/,/,/ qqDqqDqpCqpCqpC ===== .

The derived rheodynamic quantities in the case of stress-controlled state, the dynamic viscoelastic

(absolute) compliance, )2,2(J , and the corresponding loss factor, )2,2(J ,

)](1)[()( )2,2()2,2(*

)2,2( MiJJ = (II.1.6)

)](sin)()[cos()( )2,2()2,2()2,2(*

)2,2( iJJ = (II.1.7)

2/12

)2,2(

2

)2,2(

*

)2,2()2,2( )]()([)()( JJJJ +== (II.1.8)

)(/)()(tan)( )2,2()2,2()2,2()2,2( JJJJ == (II.1.9)

result in a natural way the relationships

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Horia PAVEN, Sandor POPOVICS 100

2/142

2

2

2

2

1

2

2

2

20

2

1

2

0)2,2(

)])2(1/(

/))2([()(

DDD

CCCCCJ

++

++=

(II.1.8.0)

))(/(/))()(()(

4

22

2

202110

3

1221110)2,2(

DCDCCDCCDCDCCDCJ

++

+=

(II.1.9.0)

Finally, the secondary rheodynamic quantities, in this case the related moduli are defined as follows

)()()(/1)( )2,2()2,2(*

)2,2(

*

)2,2( JJJ MiMJM +== (II.1.10)

)(/)()( 2 )2,2()2,2()2,2( JJMJ = (II.1.11)

)(/)()( 2 )2,2()2,2()2,2( JJMJ = (II.1.12)

2/12

)2,2(

2

)2,2(

*

)2,2()2,2( )]()([)()( MMMM JJ +== (II.1.13)

)()(/)()(tan)( )2,2()2,2()2,2()2,2()2,2( JJJMM MMJJ === (II.1.14)

and, consequently, the final expressions for these secondary rheodynamic quantities are

))2(/(

/))(()(

42

2

2

20

2

1

2

0

4

22

2

202110

'

)2,2(

CCCCC

DCDCCDCCMJ

++

++=(II.1.11.0)

))2(/(

/))()(()(

42

2

2

20

2

1

2

0

3

1221110

"

)2,2(

CCCCC

DCDCCDCMJ

++

+= (II.1.12.0)

2/142

2

2

20

2

1

2

0

42

2

2

2

2

1)2,2(

)])2(/(

/))2(1[()(

CCCCC

DDDMJ

++

++=(II.1.13.0)

3. BASIC (2,2) - FREQUENCY EFFECTS

The exact relationships obtained for different rheodynamic quantities as well as for their first- and

second-order frequency derivative , RD and RD)2(

, respectively, are summarized as follows.

II.A. Primary (2,2) - rheodynamic quantities

II.A.P.1. The storage (dynamic elastic) compliance is defined as

)1/()()(4'

05

2'

04

4'

03

2'

02

'

01

'

)2,2( JJJJJJ ++++= (II.1.4.0)where

22

'052

21

'0422

'0320211

'020

'01 ,2,,, DJDDJDCJDCCDCJCJ =====

and the corresponding first- and second-order frequency derivative are24'

05

2'

04

4'

13

2'

12

'

11

'

)2,2( )1/()()( JJJJJJD ++++= (II.1.4.1)

34'

05

2'

04

8'

25

6'

24

4'

23

2'

22

'

21

'

)2,2(

)2()1/()()( JJJJJJJJD ++++++=

(II.1.4.2)

II.A.P.2. The loss compliance is given as

)1/()()(4'

05

2'

04

2"

02

"

01

"

)2,2( JJJJJ +++= (II.1.5.1)where

1221"

02110

"

01 , DCDCJCDCJ ==

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II. Stress-controlled rheodynamic processes101

and the corresponding first- and second-order frequency derivative are

24'

05

2'

04

6"

14

4"

13

2"

12

"

11

"

)2,2( )1/()()( JJJJJJJD +++++= (II.1.5.1)

34'

05

2'

04

8"

25

6"

24

4"

23

2"

22

"

21

"

)2,2(

)2(

)1/(

/)()(

JJ

JJJJJJD

++

++++=

(II.1.5.2)

II.B. Derived (2,2) - rheodynamic quantities

II.B.D.1. The dynamic viscoelastic (absolute) compliance is defined as2/14'

05

2'

04

4

03

2

0201)2,2( )]1/()[()( JJJJJJ ++++= (II.1.8.0)where

2

20320

2

102

2

001 ,2, CJCCCJCJ ===

the corresponding first- and second-order frequency derivative being given as

])1()/[(

/)()(

2/34'05

2'04

2/1403

20201

4

13

2

1211)2,2(

JJJJJ

JJJJD

++++

++=(II.1.8.1)

])1()/[(

/)()(

2/54'

05

2'

04

2/34

03

2

0201

12

27

10

26

8

25

6

24

4

23

2

2221)2,2(

)2(

JJJJJ

JJJJJJJJD

++++

++++++=

(II.1.8.2)

II.B.D.2. The loss factor is defined as

)/()()(4'

03

2'

02

'

01

2"

02

"

01)2,2( JJJJJJ +++= (II.1.9.0)

and the corresponding first- and second-order frequency derivative are

24'

03

2'

02

'

01

6

14

4

13

2

1211)2,2(

)/(

/)()(

JJJ

D JJJJJ

++

+++=(II.1.9.1)

34'

03

2'

02

'

01

8

25

6

24

4

23

2

2221)2,2(

)2(

)/(

/)()(

JJJ

D JJJJJJ

++

++++=

(II.1.9.2)

II.C. Secondary (2,2) - rheodynamic quantities

II.C.S.1. The storage (dynamic elastic) modulus is given as

)/()()( 4032

0201

4'

03

2'

02

'

01

'

)2,2( JJJJJJMJ ++++= (II.1.10.0)

the corresponding first- and second-order frequency derivative being

24

03

2

0201

4'

13

2'

12

'

11

'

)2,2(

)(

/)()(

JJJ

MMMMD JJJJ

++

++=(II.1.11.1)

34

03

2

0201

6'

25

6'

24

4'

23

2'

22

'

21

'

)2,2(

)2(

)/(

/)()(

JJJ

MMMMMMD JJJJJJ

++

++++=

(II.1.11.2)

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Horia PAVEN, Sandor POPOVICS 102

II.C.S.2. The loss modulus is given as

)/()()(4

03

2

0201

2"

02

"

01

"

)2,2( JJJJJMJ +++= (II.1.12.0)

while the corresponding first- and second-order frequency derivative are

24

03

2

0201

6"

14

4"

13

2"

12

"

11

"

)2,2( )/()()( JJJMMMMMD JJJJJ +++++= (II.1.12.1)

34

03

2

0201

8"

25

6"

24

4"

23

2"

22

"

21

"

)2,2(

)2()/()()( JJJMMMMMMD JJJJJJ ++++++=

(II.1.12.2)

II.C.S.3. The dynamic viscoelastic (absolute) modulus is given as

2/142

2

2

20

2

1

2

0

42

2

2

2

2

1)2,2(

)])2(/(

/))2(1[()(

CCCCC

DDDMJ

++

++=(II.1.13.0)

whereas the corresponding first- and second-order frequency derivative are

])()1/[(

/)()(

2/34

03

2

0201

2/14'

05

2'

04

4

13

2

1211)2,2(

JJJJJ

MMMMD JJJJ

++++

++=(II.1.13.1)

])()1/[(

/)()(

2/54

03

2

0201

2/34'

05

2'

04

12

27

10

26

8

25

6

24

4

23

2

2221)2,2(

)2(

JJJJJ

MMMMMMMMD JJJJJJJJ

++++

++++++=

(II.1.13.2)

4. CONCLUSIONS

The consideration of frequency effects needs to establish exact relationships for the evaluation of

both common and distinct features of strain- and stress-controlleddynamic processes, respectively, for

different rheodynamic quantities, including theprimary, derivedand secondary ones, and outlines the basic

and practical significance of using in the analysis of the concept ofcharacteristic frequencies.

ACKNOWLEDGEMENTS

This research suite was supported in part by the Grant no.6158-2000 from ANSTI/MEC and by

Contract CERES no. 144-2001 from MEC, respectively.

REFERENCES

1. SIMS G., Supporting Composites Standardization, Reinforced Plastics, 46(12), 48-50 (2002).

2. ISO 6721, Plastics. Determination of Dynamic Mechanical Properties (2001).

3. ASTM E756, Standard Test Method for Measuring Vibration Damping Properties of Materials (1998).

4. PAVEN H., POPOVICS S., On some Basic Rheodynamic Relationships within the PTZ-Standard Linear Viscoelastic Solid

Behaviour, Proc. Ann. Symp, Inst Solid Mech., Romanian Academy, 277-286, 287-296(2002).