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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-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 extended model of linear viscoelastic solid can provide in a meaningful way, in case of

stress-controlled conditions, the frequency dependence, in isothermal circumstances as well as thetemperature dependence, in isochronal circumstances, for the derivate primary rheodynamicquantities, includig the absolute compliance and the loss factor. The quantitative trends, pointed onthe basis of maximum or/and inflection criteria, are presented from the standpoint of resulting

characteristic frequencies and temperatures.

Keywords: thermorheodynamical standard linear viscoelastic model, isothermal/isochronal

circumstances, stress-controlled conditions, derivate primary quantities.

1. INTRODUCTION

As a direct consequence of typical relaxation/retardation mechanisms acting in case of viscoelastic

materials, the use of dynamic mechanical thermal analysis appear to be one of the most useful ways, able toreveal the full set of viscoelastic quantities [1]-[5]. Accordingly, in order to complete the spectrum of

intrinsic features of frequency, , or temperature, T, dependences of primary dynamic viscoelastic properties

in strain-controlled conditions, besides the presented primary quantities, the derivate ones, including the

absolute compliace, |),(*| TJ , and the loss factor, ),( TJ , are considered in the coupled frequency

and temperature realm of standard linear viscoelastic solid [6].

Aiming at to generate the - and T-characteristic equations providing the characteristic values of

corresponding independent T, variables, the basic definitions of extremum(maximum)- and inflection-

conditions in 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, too.

In the considered case of standard linear viscoelastic solid, if there is a sinusoidal stress-controlled

condition, ,~ the resulting sinusoidal strain, ~ , is of same frequency, but shifted in time, obeying to thetypical rheological model equation [6]

~)(~)1(

hl JiJi +=+ (0.1)

As a first step, it is supposed to be coupled with the Arrhenius definition of retardation time [7]

)/exp()( TTT = (0.2)

where represent the low- and high-frequency limits of storage compliance, respectively,

while

hl JJ ,

( ) , and depicts a virtual activation temperature defined as the ratio, ( express

the well-known Arrhenius activation energy, and

T RA / A

KmolJR */314.8= is the universal gas constant).

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Thermorheodynamical characterization of standard linear viscoelastic solid model. II. Derivate primary quantities123

2. METHOD AND RESULTS

2. 1 . Absolute compliance

In virtue of basic relations (0.1), (0.2), the general expression of the absolute compliance is

2/1

222

22222

)

1

(|),(*|

T

T

T

T

hl

e

eJJTJ

+

+=

(3.0)

Frequency dependence in isothermal circumstances - );( T

By definition, if the absolute compliance, |);(*| TJ , presents a maximum at the frequency

|});(*{| TJm , this frequency is the solution of the first -characteristic equation, i. e.,

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

TJDsolTJm (3.1.1)where

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

TJTJD (3.1.2)

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

The resulting first -characteristic equation is

0= (3.1.3)

and this means it is without a positive solution |});(*{| TJm .

In the case of an inflection of the absolute compliance, |);(*| TJ , at the frequency|});(*{| TJi

, it

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

(3.1.4)]0|);(*|[)2(

|});(*{| == TJDsolTJi

where

2

2)2( |);(*|

|);(*|

TJTJD (3.1.5)

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

subsists.

The resulting second -characteristic equation is

032

4244

22222

=++

T

T

hT

T

ll eJeJJ

(3.1.6)

with a positive solution

)/(1/31)/)(3/3(22

|});(*{|T

T

lhhlTJi eJJJJ

+= (3.1.6a)

satisfying the above quoted (-, +) sequence of signs criterium.

Temperature dependence in isochronal circumstances - );( T

The maximum of the absolute compliance, |);(*| TJ , at the temperature means that

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

|});(*{| TJmT

]0|);(*|[|});(*{| == TJDsolT TTJm (3.2.1)

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Horia PAVEN,Sandor POPOVICS 124

where

T

TJTJDT

|);(*||);(*| (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 .|});(*{| TJmT

Moreover, the existence of an inflection of |);(*| TJ , at the temperature , means it is

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

|});(*{| TJiT

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

|});(*{| == TJDsolT TTJi

where

2

2)2( |);(*|

|);(*|T

TJTJDT

(3.2.5)

is the corresponding second order T-derivative, if the (+, -) signs inversion is present.

Consequently, the resulting second T-characteristic equation is

0)(2

])2()2([)(2

4244

222222

=+

++++

T

T

h

T

T

lhl

eJTT

eJTTJTTJTT

(3.2.6)

with a solution, which obeys to the above quoted criterium of signs inversion .0|});(*{| >TJiT

2. 2Loss factor

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

T

T

hl

T

T

hlJ

eJJ

eJJT

222

)(),(

+

=

(4.0)

Frequency dependence in isothermal circumstances - );( T

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

)};({ Tm J , the value of this frequency is the positive solution of the first

-characteristic equation, i. e.,

]0);([)};({ == TDsol JTm J (4.1.1)

where

);();(

TTD JJ (4.1.2)

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

Accordingly, the first -characteristic equation is

0

222

=+

T

T

hl eJJ

(4.1.3)

resulting a positive solution

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Thermorheodynamical characterization of standard linear viscoelastic solid model. II. Derivate primary quantities125

)/()};({T

T

h

lTm e

J

JJ

= (4.1.3a)

which respects the (+,-) sequence of derivative signs restriction.

Furthermore, if an inflexion of );( TJ exists at the frequency )};({ TiJ

, it represents the

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

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

)};({ == TDsol JTi J

where

2

2)2( );(

);(

TTD JJ (4.1.5)

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

The resulting second -characteristic equation is

032

22=+

T

T

hl eJJ

(4.1.6)

with the positive solution

)/(3)};({T

T

h

lTi e

J

JJ

= (4.1.6a)

satisfying the (-,+) signs sequence criterium.

Temperature dependence in isochronal circumstances - );( T

The existene of a peak-value of the loss factor, );( TJ , at the temperature , indicates the

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

)};({ Tm JT

]0);([)};({ == TDsolT JTTm J (4.2.1)

where

T

TTD JJT

);();(

(4.2.2)

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

The corresponding first T-characteristic equation is

0

222

=

T

T

hl eTJJ

(4.2.3)

with the necessary positive solution

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

=

hlTm

JJ

TT

J(4.2.3a)

satisfying the above sign inversion requirement is respected.

Finally, the existence of a inflection of );( TM

, at the temperature , denotes the

existence of a positive solution

)};({ Ti JT

]0);([)2(

)};({ == TDsolT JTTi J (4.2.4)

where

2

2)2( );(

);(T

TTD JJT

(4.2.5)

is the corresponding second order T-derivative, if the criterium of signs inversion arise.

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The resulting second T-characteristic equations is

0)2(6)2(

4244

2222

=++

T

T

hT

T

hll eJTTeTJJJTT

(4.2.6)

with two positive solutions satisfying the above quoted criterium of derivative signs.

3. CONCLUSIONS

The extended standard linear viscoelastic solid model affords in a well manner different typical

features of derivate primary rheodynamic quantities, both in terms of frequency and temperature effects:

- in the case of frequency dependence, in isothermal circumstances, the absolute compliance is a

well definite cumulate contribution of a monotonic decrease of the storage compliance with frequency, and

that of a peak-like one of the loss compliance; the overall result for the absolute compliance provides, on the

hand, higher values than for storage compliance, and on the other hand, a monotonic variation, while, for the

loss factor, which reflects rather an viscoelasticity index than solely a measure of a dissipative mechanisms,

the new peak-like effect being identified by a typical sequence of characteristic frequencies as

)};({)};({)};({)};({|});(*{|};({ TiTmTJiTJmTJiTJi JJ