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SISOM 2009 and Session of the Commission of Acoustics, Bucharest 28-29 May

THERMORHEODYNAMICAL CHARACTERIZATIONOF STANDARD LINEAR

VISCOELASTIC SOLID MODEL IN STRAIN-CONTROLLED CONDITIONS.I. DIRECT PRIMARY QUANTITIES

Horia PAVEN*,Sandor POPOVICS**

* National Institute of Research and Development for Chemistry and Petrochemistry - ICECHIM, Bucharest, ROMANIA,e-mail: htopaven@netscape.net

** Drexel University, Philadelphia, 19104-PA, USA

By using an extended model of standard linear viscoelastic solid, the isothermal and isochronal

circumstances are taken into account in case of direct primary rheodynamic quantities including thestorage and the loss modulus in strain- controlled conditions. Correspondingly, the monotonicvariation of the storage modulus as well as the peak-like one of the loss modulus are approached from

the standpoint of maximum and inflection criteria, respectively, the meaning of- and T-characteristicequations being pointed out.

Keywords: thermorheodynamical standard linear viscoelastic model, isothermal/isochronalcircumstances, strain-controlled conditions, direct primary quantities.

1. INTRODUCTION

Extensive theoretical and experimental work has gone into approaching rheological properties of

emergent materials [1] - [6]. Accordingly, the present work is highly motivated by the essential need to pointout the intrinsic peculiarities of frequency, , and temperature, T, dependences of direct primary dynamic

viscoelastic quantities in strain-controlled conditions, including the storage modulus, ),( TM , and the loss

modulus, ),( TM , in the realm of the model of standard linear viscoelastic solid.The general form of the

dynamic rheological equation in case of a sinusoidal strain-controlled excitation, ,~ and resulting sinusoidal

stress, ~ , of same frequency but shifted in time, is expressed as [7]

~)(~)1(hl

MiMi +=+ (0.1)

where represent the low- and high-frequency limit of storage modulus, respectively, the

Arrhenius-like relaxation time being given as

hl MM ,

)/exp()( TTT

= (0.2)

)(

, and stands for a virtual activation temperature defined as the ratio, ( is the

-Arrhenius activation energy, and stands for the universal gas constant).

T RA /

A

KmolJR */314,8=

Typical definitions of extremum(maximum)- and inflection-conditions in case of isothermal

cirumstances, i. e., the frequency dependence at given temperature - );( T , and ofisochronal one, i. e., the

temperature dependence at given frequency - );( T , are used to generate the - and T-characteristic

equations wich provide the values of corresponding independent variables.

2. METHOD AND RESULTS

Storage moduls.

Taking into account the basic definitions (0.1), (0.2), the general expression of the storage modulus is

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Horia PAVEN,Sandor POPOVICS 154

given as

T

T

T

T

hl

e

eMMTM

2

22

222

1

),(

+

+=

(1.0)

Frequency dependence in isothermal circumstances - );( T .

If there is a maximum of the storage modulus, );( TM , at the frequency )};({ TMm , this is the

solution of the first -characteristic equation, i. e.,

]0);([)};({ == TMDsolTMm (1.1.1)

where

);();(

TMTMD (1.1.2)

is the corresponding first order

-derivative, if a (+,-) sequence of corresponding signs exists.The resulting first -characteristic equation is

0= (1.1.3)

i. e., without a 0)};({ > TMm solution.

On the other hand, if there is an inflection of );( TM at the frequency )};({ TMi , it is given as the

positive solution of the second -characteristic equation,

i. e.,

(1.1.4)]0);([)2(

)};({ == TMDsolTMi

where

2

2)2( );();(

TMTMD (1.1.5)

is the corresponding second order-derivative, if there is a (+,-) signs sequence of the second derivative.

The resulting second -characteristic equation is

031

222

=

T

T

e

(1.1.6)

with a 0)};({ > TMi solution

)/()3/3()};({T

T

TMi e

= (1.1.7)

the signs criterium being fulfilled.

Temperature dependence in isochronal circumstances - );( T .

By definition, the maximum of the storage modulus, );(TM , at a temperature ,

represents the positive solution of the first T-characteristic equation, i. e.,

)};({ TMmT

]0);([)};({ == TMDsolT TTMm (1.2.1)

where

T

TMTMD

T

);();(

(1.2.2)

depicts the corresponding first order T-derivative, if a (+,-) signs sequence of this derivative exists.

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Thermorheodynamical characterization of standard linear viscoelastic solid model. I. direct primary quantities155

The resulting first T-characteristic equation is

0=TT

e

(1.2.3)

without a positive solution .)};({ TMmT

On the other part, the inflection of );( TM at the temperature is the positive solution of

the second T-characteristic equation, i. e.,

)};({ TMiT

(1.2.4)]0);([)2(

)};({ == TMDsolT TTMi

where

2

2)2( );(

);(T

TMTMDT

(1.2.5)

is the second order T-derivative, if there is a signs inversion of this derivative.

The resulting second T-characteristic equation is

0)()(

222

=+

T

T

eTTTT

(1.2.6)with a presumed positive solution, if the above quoted condition for derivative exists.

Loss modulus.

In virtue of the basic definitions (0.1), (0.2), the loss modulus is given as

T

T

T

T

lh

e

eMMTM

2

2

0

2

0

1

)(),(

+

=

(2.0)

Frequency dependence in isothermal circumstances - );( T .

As is well established, if there is a maximum of the loss modulus, );( TM , at the frequency

)};({ TMm , it is obtained as the positive solution of the first -characteristic equation, i. e.,

]0);([)};({ == TMDsolTMm (2.1.1)

where

);();(

TMTMD (2.1.2)

represents the first order-derivative, if a (+,-) sequence of derivative signs is present.

The resulting first -characteristic equation is

01

222

=

T

T

e

(2.1.3)

with a - positive solution

)/(1)};({T

T

TMme

= (2.1.3a)

obeying to the above quoted signs criterium.

Further, in case of existence of a );( TM inflection at a frequency )};({ TMi , the second

-characteristic equation has a positive solution, i. e.,

(2.1.4)]0);([)2(

)};({ == TMDsolTMi

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Horia PAVEN,Sandor POPOVICS 156

where

2

2)2( );(

);(

TMTMD (2.1.5)

express the corresponding second order T-derivative, if there is a typical (-,+) sequence of derivative signs.

The resulting second -characteristic equation

03

222

=+

T

T

e

(2.1.6)

admits the positive solution

)/(3)};({T

T

TMi e

= (2.1.6a)

and satisfies the referred to signs sequence condition.

Temperature dependence in isochronal circumstances - );( T

If a maximum of the loss modulus, ),;( TM exists at temperature , it results to be the

positive solution of the first T-characteristic equation, i. e.,

)};({ TMmT

]0);([)};({ == TMDsolT TTMm (2.2.1)

where

T

TMTMDT

);();( (2.2.2)

is the first order T-derivative, if there is a (+,-) signs sequence of this derivative.

The resulting first T-characteristic equation is

01

2

22=+

T

T

e

(2.2.3)

with the positive solution

)]/(1ln[)};({

=

TT TMm (2.2.3a)

supposed to fulfill the derivative signs restriction.

Moreover, if there is an inflection of );( TM at temperature , this corresponds to the

positive solution of the second T-charcteristic equation, i. e.,

)};({ TMiT

(2.2.4)]0);([)2(

)};({ == TMDsolT TTMi

where

2

2)2( );;(

);(T

TMTMDT

(2.2.5)

is the second order T-derivative, if there is a derivative signs inversion.

The resulting second T-characteristic equation is

0)2(6)2(

444

222

=++

T

T

T

T

eTTeTTT

(2.2.6)

with expected positive solutions , if the corresponding signs inversion of the second order

T-derivative is fulfilled.

)};({ TMiT

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3. CONCLUSIONS

The use of the proposed frequency and temperature coupled model of standard linear viscoelastic

behaviour point out in a meaningful manner intrinsic features revealed by the direct primary rheodynamic

quantites given in an explicit form, as frequency and temperature effects:

- in case of frequency dependence, in isothermal circumstances, the storage modulus shows amonotonic increase with frequency, as well as an inflection, whereas the loss modulus presents a peak-like

trend, including a maximum and then, at higher frequencies, an inflection, in the sequence

)};({)};({)};({ TMiTMmTMi