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SISOM 2010 and Session of the Commission of Acoustics, Bucharest 27-28 May

THERMORHEODYNAMICAL CHARACTERIZATION

OF STANDARD LINEAR VISCOELASTIC SOLID MODEL IN STRESS-CONTROLLEDCONDITIONS 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

The extended model of the standard linear viscoelastic solid is considered in case of stress-controlled

conditions, the direct primary rheodynamic quantities, including the storage and loss compliances,being taken into account. Accordingly, the trends of these thermodynamic quantities are evidenced

from the standpoint of maximum or/and inflection criteria for the frequency dependence in isothermalcircumstances, as well as in the case of temperature dependence, if the isochronal circumstances are inview.

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

1. INTRODUCTION

The present work is motivated by the need to point out the intrinsic peculiarities of frequency, , and

temperature, T, dependences of direct primary dynamic viscoelastic quantities in stress-controlled conditions,including the storage compliance, ),( TJ , and the loss compliance, ),( TJ , in the realm of the standard

linear viscoelastic solid model [1] -[5].

Accordingly, the general form of the dynamic rheological equation in case of a sinusoidal

stress-controlled excitation, ~ , and a resulting sinusoidal strain, ,~ of same angular frequency, but shifted in

time, is expressed as [6]

~)(~)1(hl JiJi +=+ (0.1)

where are the low- and high-frequency limit of storage compliances, respectively, the Arrhenius-like

retardation time being given as [7]

hl JJ ,

)/exp()( TTT

= (0.2)

while )(

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

( represents the well known-Arrhenius activation energy, and

T RA /

A KmolJR */314.8= is the universal gas

constant).

In order to carry out a self-contained extended approach, typical definitions of extremum (maximum)-

and inflection-conditions in the case of isothermal cirumstances, i. e., the frequency dependence at given

temperature - );( T , and ofisochronal ones,

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

equations which provide in a natural way the values of corresponding independentvariables.

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2. METHOD AND RESULTS

2. 1Storage moduls

By using the basic definitions (0.1), (0.2), the general expression of the storage compliance results as

T

T

TT

hl

e

eJJTJ

2

22

222

1

),(

+

+=

(1.0)

Frequency dependence in isothermal circumstances - );( T

If the the storage compliance, );( TJ , shows a maximum at the frequency )};({ TJm , this is a

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

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

where

);();(

TJTJD (1.1.2)

indicates the corresponding first order-derivative, if there is a well defined (+,-) sequence of the signs of

corresponding derivative.

The resulting first -characteristic equation is

0= (1.1.3)

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

On the other part, if there is an inflection of the storage compliance, );( TJ , at the frequency

)};({ TJi

, it is in fact the positive solution of the second

-characteristic equation, i. e.,(1.1.4)]0);([

)2(

)};({ == TJDsolTJi

where

2

2)2( );(

);(

TJTJD (1.1.5)

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

The resulting second -characteristic equation is

031

222

=+

T

T

e

(1.1.6)

with a 0)};({ > TJi solution

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

T

TJi e

= (1.1.7)

the (-, +) signs criterium being fulfilled.

Temperature dependence in isochronal circumstances - );( T

By definition, if there is a maximum of the storage compliance, );( TJ , at a temperature ,

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

)};({ TJmT

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

TJDsolT TTJm (1.2.1)where

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Thermorheodynamical characterization of standard linear viscoelastic solid model. I. Direct primary quantities119

T

TJTJDT

);();( (1.2.2)

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

The resulting first T-characteristic equation is

0=TT

e

(1.2.3)

without a positive solution .)};({ TJmT

On the other hand, the inflection(s) of the storage compliance, );(TJ , at the temperature ,

is(are) the positive solution(s) of the second T-characteristic equation, i.e.,

)};({ TJiT

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

)};({ == TJDsolT TTJi

where

2

2)2( );(

);(T

TJTJDT

(1.2.5)

is the second order T-derivative, if there is(are) (+,-) inversions of derivative signs.

The resulting second T-characteristic equation is

0)()(

222

=++

T

T

eTTTT

(1.2.6)

with positive solutions, if the above quoted condition for the second derivative exists.

2. 2Loss compliance

Given the basic relationships (0.1), (0.2), the loss compliance is expressed as

T

T

TT

hl

e

eJJTJ

2

221

)(),(

+

=

(2.0)

Frequency dependence in isothermal circumstances - );( T

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

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

]0);([)};({

== TJDsolTJm

(2.1.1)

where

);();(

TJTJD (2.1.2)

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

Thus, the resulting first -characteristic equation is

01

222

=

T

T

e

(2.1.3)

with a positive solution

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)/(1)};({T

T

TJm e

= (2.1.3a)

In the case of the existence mpliance,

which obeys to the above quoted (+, -) order of signs criterium.

);(TJof an inflection of the loss co , at a defined frequency,

;({Ji )}T , the second

(2.1.4)

where

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

]0);([)2(

)};({ == TJDsolTi J

2

2)2( );(

);(

TJTJD (2.1.5)

ere is a typical (-,+) sequence of second derivative

signs.

uation

express the corresponding second order T-derivative, if th

The resulting second -characteristic eq

032

22=+

T

T

e

(2.1.6)

has a

positive solution

)/(3)};({T

T

TJi e

= (2.1.6a)

and satisfies the referred (-, +) signs sequence condition.

Temperature dependence in isochronal circumstances - ( );T

),;(TJIf a maximum of the loss compliance, exists at temperature it resu

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

)};( TJmT { , lts to be the

positive ({ ]0);([)}; == TJDsolT TJ Tm (2.2.1)

where

TTJDT

TJ );( );(

(2.2.2)

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

tion isThe resulting first T-characteristic equa

01

222

=+

T

T

e

(2.2.3)

ive solutionwith the posit

)]/(1ln[

)};({

=

TJm

supposed to fulfill a frequency restriction, too.

T

(2.2.3a)

M (s) of storage compliance,

T

oreover, if there is(are) inflection );(TJ , at tempe

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

where

rature )};({ TJiT ,

it(they) corre

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

)};({ == TJDsolT TTi

2);(

T

TJDT

2)2( );;( TJ

(2.2.5)

is the second order T-derivative, if there is(are) inversion(s) of derivative signs.

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The second T-characteristic equation, with expected positive solution , is)};({ TJiT

2( + TT

0)2(6)

444

222

=+

T

T

T

T

eTTeT

(2.2.6)

if appropriate signs inversions take place.

r inflectio arise,

quoted frequencies being in the seq

3. CONCLUSIONS

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

behaviour points 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 the case of frequency dependence, in isothermal circumstances, the storage compliance shows a

monotonic decrease with frequency, with a typical inflection, while the loss compliance presents a peak-like

form, with a maximum at intermediate frequencies and then, at higher values, a clea n the

uence

)};({)};({)};({ TJiTJmTJi