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SISOM 2012 and Session of the Commission of Acoustics, Bucharest 30-31 May

BASIC SEQUENCES OF DYNAMIC DISSIPATIVE QUANTITIES INCASE OF LINEAR VISCOELASTIC BEHAVIOR OF POLYMER- LIKE MATERIALS. I.

ISOTHERMAL CIRCUMSTANCES.

Horia PAVEN

National Institute of Research and Development for Chemistry and Petrochemistry - ICECHIM, Bucharest, Splaiul Independentei,

202, ROMANIA; email: htopaven@netscape.net

Given the frequency and temperature dependence of polymer-like materials properties, welldefined quantities including the loss modulus, the corresponding loss factor and the attached loss

compliance,or the loss compliance, the corresponding loss factor and the attached loss modulus, areconsidered in case of frequency dependence at given temperature, both for strain- and stress-

controlled conditions, respectively. Accordingly, the general form of isothermal characterizationcircumstances is presented, the natural restrictions concerning the frequency variations of storagemodulus and storage compliance, respectively, being taken into account and the locations of

characteristic mechanical losses provided.

Keywords: dynamic linear viscoelasticity, dissipative quantities, polymers, isothermal circumstances,characteristic frequencies.

1. INTRODUCTION

As it is well established, the appropriate characterization of dynamic behavior of solid-type

viscoelastic materials can be performed either in strain- or stress-controlled conditions [1 - 3]. If the strain-

controlled conditions are concerned, the primary modulus-like (direct and derivative) quantities, as well as

the attached secondary compliance-like ones result. Moreover, if the stress-controlled are considered, the

primary compliance-like quantities, and the attached secondary compliance-like are obtained [4 - 6].

At present, there are available a lot of theoretical approaches and many empirical or semi-empirical

models useful for describing in an appropriate way the viscoelastic properties. However, if experimental data

are considered, often difficulties may arise, a significant need for the evaluation and appreciation criteria

being a major task. Henceforth, identifying selected relationships intends to provide a deeper meaningful

route towards an error-free approach.

In case of dynamic strain-controlled conditions, in isothermal circumstances, the linear viscoelastic

behavior, is described by means of frequency, , and temperature, T, dependences. On the hand, welldefined properties concern the primary quantities, including

- Storage modulus, ),;( TM

- Loss modulus, ),;('' TM

- Absolute modulus, | );(* TM |,

- Loss factor, );( TM ,

as well as the attached, secondary ones, containing

- Storage compliance, ),;( TJM

- Loss compliance, );( TJM ,

- Absolute compliance, | |,);(* TJM

- Loss factor, );( TJ , which is similar to );( TM ,

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Basic sequences of dynamic dissipative quantities in case of linear viscoelastic behavior. I. Isothermal circumstances.141

where the semi-colon, ;, point out a variable quantity, at left, and a parameter, at right.

On the other hand, in case of stress-controlled conditions, well defined properties arise as the

primary ones, including

- Storage compliance, ),;( TJ

- Loss compliance, ),;( TJ

- Absolute compliance, | |,);(* TJ

- Loss factor, );( TJ ,

as well as the attached, secondary ones, including

- Storage modulus, ),;( TJM

- Loss modulus, );( TJM ,

- Absolute modulus, | |,);(* TJM

- Loss factor, );( TJ , which is identical to );( TM .

2. METHOD AND RESULTS

Taking into account the typical dissipative properties, illustrated either by the set including

);( TM , );( TM , and );( TJM , or the other one, containing );( TJ , );( TJ , );( TMJ , the

corresponding peak-like patterns which are observed in case of dynamic viscoelastic quantities are relevant

from the standpoint of structural parameters.

There is a natural need to clarify the comparative positions of frequencies which correspond to the

maximum peaks, taking into account both the control conditions as well as the circumstances of

characterization.

Strain-controlled frequency-dependent primary-like quantities

On the basis of experimental data obtained in isothermal circumstances, it is remarked that the

storage modulus increases in a monotonical manner as the frequency is raising, i. e.,

0);(

);(

TM

TMD (1)

Given the standard definition of the loss factor as ratio of loss to storage modulus, the corresponding

frequency derivative is

);('

);();();();(

);();(

);(

2TM

TMDTMTMDTM

TMTM

TD M

(2)

The natural target now arising is to establish the variation trend of the loss factor at the frequency

)};('{ TMm , i. e., at that value of frequency for which the loss modulus shows a maximum peak, so that

0|);();({ TMm

TMD

(3)

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Horia PAVEN 142

Consequently, the resulting condition stands

)};({2)};({|);(

);(

);(|);(

TMTMM mmTMD

TM

TMTD

(4)

and finally, given the physical restriction (1), it follows

0|);( )};({ TMM mTD (5)

i. e., it results that the value of frequency at which there is a maximum of the loss modulus, a negative slope of

the loss factor as function of frequency is obtained.

Accordingly, the peak of the loss factor is located at a frequency lower than that corresponding to the

loss modulus peak,

)};({)};({ TMmTm M (6)

Strain-controlled frequency-dependent secondary-like quantities

In order to depict the complete sequence of characteristic frequencies in isothermal circumstances, the

relative locations of attached loss compliance and loss factor peaks are also considered.

In this case, the storage compliance decreases if the frequency increases, i. e.,

0);(

);(

TJ

TJD MM (7)

The typical definition of the loss factor given as the ratio of the loss compliance to storage compliance,

results in

);(

);();();();(

);(

);(

);(

2TJ

TJDTJTJDTJ

TJ

TJ

TD

M

MMMM

M

M

JM

(8)

In order to establish the sequence of the attached loss factor and the loss compliance peaks positions,

the variation trend of the suitable loss factor at the frequency )};({ TJm M , where the loss compliance

presents a maximum,

0|);()};({ TMJm

TJDM

(9)

is given as

)};({2)};({|);(

);(

);(|);(

TJM

M

MTJJ MmMmM

TJDTJ

TJTD

(10)

The use of physical restriction (6) leads to the relation

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Basic sequences of dynamic dissipative quantities in case of linear viscoelastic behavior. I. Isothermal circumstances.143

0|);( )};({ TJJ MmM TD (11)

which means that at the value of frequency where the loss compliance has a maximum, i.e., the resulting

frequency derivative vanishes, the attached loss factor shows a positive slope, and consequently the

corresponding typical maximum peak is located at higher frequency compared with that of loss compliance,

)};({)};({ TmTJmMJM

(12)

As a direct consequence of (6) and (12), the complete sequence of corresponding characteristic

frequencies in strain-controlled conditions and isothermal circumstances is given as

)};({)};({)};({ TMmTmTJm JM (13)

Stress-controlled frequency-dependent primary-like quantities

The basic considered restriction concerning the storage compliance is

0);('

);('

TJ

TJD (14)

The typical definition of the loss factor given as the ratio of the loss compliance to storage compliance,

results in

);('

);();();();('

);(

);(

);(

2TJ

TJDTJTJDTJ

TJ

TJ

TD J

(15)

In order to identify the sequence of loss factor and loss compliance peaks positions, the variation trend

of the loss factor at the frequency )};(''{ TJm , where the loss compliance presents a maximum for

0|);()};({ TJm

TJD

(16)

is given as

)};({2)};({|);(

);('

);(|);( TJTJJ mm TJDTJ

TJTD

(17)

The use of physical restriction (14) leads to the relation

0|);( )};({ TJJ mTD (18)

for the value of frequency where the loss compliance has a maximum, i.e., the resulting frequency derivative

vanishes, the loss factor presenting a positive slope, the corresponding loss factor maximum peak being

located at a higher frequency compared with that of loss compliance,

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Horia PAVEN 144

)};({)};({ TmTJm J (19)

Stress-controlled temperature dependent secondary quantities

The experimental data obtained in isothermal circumstances reveals that the storage modulusincreases in a monotonic way with the frequency, i. e.,

0);(

);(

TM

TMD JJ . (20)

Given the well known definition of the loss factor as ratio of loss to storage modulus, the

corresponding frequency derivative results in

);('

);();();();(

);(

);(

);(

2TM

TMDTMTMDTM

TM

TM

TD

J

JJJJ

J

J

M

(21)

In order to find the variation trend of the loss factor at the frequency )};('{ TMm J , i. e., at that

frequency for which the loss modulus shows a maximum peak, so that

0|);();({ TJMm

TMD J (22)

Consequently,

)};({2)};({|);(

);(

);(|);( TMJ

J

JTMM JmJmJ

TMDTM

TMTD

(23)

and given the physical restriction (20), it follows

0|);( )};({ TMM JmJ TD (24)

i. e., a negative slope of the loss factor is obtained for the frequency where a maximum peak of the loss

modulus arises.

Accordingly, the peak of the loss factor is located at a frequency lower than that corresponding to theloss modulus peak,

)};({)};({ TMmTm JJM (25)

and taking into account the relations (19) and (25), the complete sequence of corresponding characteristic

frequencies in stress-controlled conditions and isothermal circumstances is given as

)};({)};({)};({ TMmTmTJm J (26)

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Basic sequences of dynamic dissipative quantities in case of linear viscoelastic behavior. I. Isothermal circumstances.145

3. CONCLUSIONS

Given the features of frequency dependence of storage modulus, and the standard definition of loss

factor, considered in isothermal circumstances, the complete sequence of characteristic frequencies

corresponding to maximum peaks of lossesquantities is pointed out.

The frequency at which the loss factor peak appeared, is lower than that obtained in case of the lossmodulus one; however, it is higher than frequency at which is located the peak of loss compliance.

The different dynamic linear viscoelastic quantities in isothermal circumstances

are suitable for identifying meaningful criteria to be proposed for applications regarding the damping of

mechanical vibration.

The approach is a model-free one.

REFERENCES

1. BRINSON, H. F., BRINSON, L. C., Polymer Engineering Science and Viscoelasticity. An Introduction, Springer, New York,2008.

2. van KREVELEN, D. M., te NIJEHUIS, K., Properties of Polymers, Elsevier, Amsterdam, 2009.3. LAKES, R., Viscoelastic Materials, Cambridge University Press, Cambridge, 2009.

4. PAVEN, H., Dual Thermorheodynamical Approaches of DMTA Data in the Framework of Standard Viscoelastic Behavior, IstCentral and Eastern European Conference on Thermal Analysis and Calorimetry, Craiova, 7 - 10, Sept., 2011.

5. PAVEN, H., VULUGA, Z., Characteristic Frequencies and Temperatures Underlying 2D - 3D DMTA Data in Case of Strain-

and Stress-Controlled Conditions, NANOTOUGH, PC - NMP - 21346, Copenhagen, 21 - 24, Sept., 2011.6. PAVEN, H., VULUGA, Z., NICOLAE, C. A., IORGA, M., GABOR, R.,Localizations of Dissipative Energetical Effects in

Polymer Maerials.I. Characteristic Frequencies in Isothermal Circumstances, PRIOCHEM- ICECHIM, 27 - 28 Oct., 2011.