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

THERMORHEODYNAMICAL CHARACTERIZATION

OF STANDARD LINEAR VISCOELASTIC SOLID MODEL

IN STRAIN-CONTROLLED CONDITIONS.II. DERIVATE 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 use of the frequency and temperature extended model of linear viscoelastic behaviour to

describe both the frequency dependence in isothermal circumstance as well as the temperaturedependence in isochronal one, is proposed for derivate primary rheodynamic quantities represented

by the absolute modulus and the loss factor for strain-controlled conditions. Accordingly, the

monotonic trends of the absolute modulus as well as of the typical loss factor peak are evaluated byusing the resulting - and T-caracteristic equations.

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

1. INTRODUCTION

The large range of variation of end-use conditions of viscoelastic-like model materials needs deeper

know-why understanding of control mechanisms providing meaningful scientific and practical-technical

data [1]-[11]. Correspondingly, to complete the spectrum of intrinsic peculiarities

of frequency, , or temperature, T, dependence of primary dynamic viscoelastic quantities in strain-

controlled conditions, the derivates ones, including the absolute modulus, |);(*| TM , and the loss factor,);( TM , in the coupled frequency and temperature realm of standard linear viscoelastic solid, are

considered [12].

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

cirumstances, i. e., the frequency dependence at given temperature - );( T , and of isochronal ones, i. e., the

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

equations providing the values of corresponding independent T, variables.

In the situation of a sinusoidal strain-controlled condition, ,~ and resulting sinusoidal stress, ~ , of

same frequency, but shifted in time, the typical rheological equation

~)(~)1(

hlMiMi +=+ (0.1)

is coupled with the Arrhenius definition of relaxation time

)/exp( TT = (0.2)

where represent the low- and high-frequency limit of storage modulus, respectively,hl MM , )(

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

activation energy, while

T RA / A

KmolJR */314,8= express the universal gas constant).

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Horia PAVEN,Sandor POPOVICS 160

where

T

TMTMDT

|);(*||);(*| (3.2.2)

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

The resulting first T-characteristic equation is

0=TT

e

(3.2.3)

without a positive solution .|});(*{| TMmT

The existence of an inflection of |);(*| TM at the temperature , indicates this is given

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

|});(*{| TMiT

(3.2.4)]0|);(*|[)2(

|});(*{| == TMDsolT TTMi

where

2

2)2( |);(*|

|);(*|T

TMTMDT

(3.2.5)

is the corresponding second order T-derivative, if the signs inversion is present.The resulting second T-characteristic equation is

0)(2

])2()2[()(2

4244

222222

=

+++

T

T

h

T

T

lhl

eMTT

eMTTMTTMTT

(3.2.6)

with presumed solution, obeying to the signs inversion criterium.0|});(*{| >TMiT

Loss factor

In virtue of the basic definitions (0.1), (0.2), the general expression of the loss factor is defined as

T

T

hl

T

T

lh

M

eMM

eMMT

222

)(),(

+

=

(4.0)

Frequency dependence in isothermal circumstances - );( T .

If the loss factor, );( TM , is characterized by the presence of a maximum at the frequency

)};({ Tm M , the value of this frequency is given as the positive solution of the first -characteristic equation,

i. e.,

]0);([)};({ == TDsol MTm M (4.1.1)

where

);();(

TTD MM (4.1.2)

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

The first -characteristic equation is

0

222

=+

T

T

hlM (4.1.3)eM

providing a positive solution

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Thermorheodynamical characterization of standard linear viscoelastic solid model. II. derivate primary quantities161

)/()};({T

T

h

lTm e

M

MM

= (4.1.3a)

satisfying the (+,-) sequence of signs restriction.

Further, if an inflexion of );( TM

exists at the frequency )};({ TiM

, it represents the positive

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

(4.1.4)]0);([)2(

)};({ == TDsol MTi M

where

2

2)2( );(

);(

TTD M

M (4.1.5)

stands for the corresponding second order-derivative, if the (-,+) sequence of the derivative signs subsists.

The resulting second -characteristic equation is

03

222

=+

T

T

hleMM

(4.1.6)

with the positive solution

)/(3)};({T

T

h

lTi

eM

MM

= (4.1.6a)

satisfying the (-,+) signs sequence.

Temperature dependence in isochronal circumstances - );( T .

The arise of a peak-value of the loss modulus, );( TM

, at the temperature indicates the

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

)};({ Tm MT

]0);([)};({ == TDsolT MTTm M (4.2.1)

where

T

TTD MMT

);();(

(4.2.2)

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

The resulting first T-characteristic equation is

0

222

=

T

T

hl eTMM

(4.2.3)

with the positive solution

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

=

hlTm

MM

TT

M(4.2.3a)

satisfying the signs sequence requirement.

At the end, the presence of a inflection of );( TM at the temperature , needs the

existence of a positive solution

)};({ Ti MT

(4.2.4)

]0);([)2(

)};({ == TDsolT MTTi M

where

2

2)2( );(

);(T

TTD M

MT

(4.2.5)

is the corresponding second order T-derivative, if there is an signs inversion of the corresponding derivative .

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The resulting second T-characteristic equations is

0)2(6)2(

4244

2222

=++

T

T

hT

T

hll eMTTeTMMMTT

(4.2.6)

with a presumed positive solution satisfying the above quoted criterium of derivative signs sequence.

3. CONCLUSIONS

The standard linear viscoelastic solid model provides in a well perceptive way typical peculiarities of

derivate primary rheodynamic quantities expressed in terms of both frequency and temperature influences:

- in case of frequency dependence, in isothermal circumstances, the absolute modulus represents a

well definite cumulate contribution of a monotonic increase of the storage modulus with frequency, and that

of a peak-like one of the loss modulus, the overall result for the absolute modulus being, on the hand, higher

values, and on the other hand, monotonic variation, such that for the corresponding characteristic

frequencies

)};({|});(*{| TMiTMi <

while, for the loss factor, reflecting rather an viscoelasticity index than exclusively a measure of a dissipativemechanism, a new peak-like effect identified by the ensuing frequencies

)};({)};({)};({)};({ TMiTMmTiTm MM