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Revista de Chimie, 2005: 56(2), 173-176.

BASIC PHENOMENOLOGICAL MODELLING OF RHEOLOGICAL BEHAVIOUR

OF TERNARY PHASE IN PHASE POLYMER COMPOSITE SYSTEMS.

II. UNIFORM STRESS APPROACH

Horia PAVEN, Chemical Research Institute-ICECHIM, Bucuresti, Splaiul Independentei, nr. 202

([email protected])

Aiming at to reveal the intrinsic peculiarities of phenomenological principles of modelling

ternary phase in phase composite systems with linear viscoelastic polymeric components,

the second basic case of uniform stress approximation is considered. In the proposed

approach the rheological behaviour laws-selection rules-mixing rules hierarchical path is

followed, the Reuss-like morpho-rheological interactions underlying the form of the

dependence of composite rheological variables on those of component ones. The resulting

selection rules and the corresponding quantitative and qualitative effects, as well as the

primary mixing rules for rheological parameters, are pointed out and illustrated for

distinct cases of mono- and bi-relaxant (retardant) rheological behaviour cases.

Keywords: Rheological phenomenology, linear viscoelastic behaviour, ternary phase in

phase composite systems, polymer components, uniform stress modelling.

Extensive industrial demands on new materials such as high polymers and polymeric based

composite systems are to be emphasized, with particular attention to sensitive questions about the

modelling, numerical analysis and characterization of the effective response of materials /1-7/.

Given the intricate spectrum of morpho-rheological interactions in polymeric composite

systems, i.e., the morphologically controlled dependence of composite rheological variables on those

of components, the natural need to model and analyze the intrinsic manner in which the mixing rules

for the rheological quantities in the framework of linear viscoelastic behaviour is becoming growing

imperative from both academic and engineering technics standpoint /8-10/.

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Accordingly, in addition to the earlier presented case of Voigt-like, uniform deformation

approach, that of Reuss-like, uniform stress one, is considered /11/.

Basic [3R] inter-relationships

Let a / b / c the ternary phase in phase composite system with a, b and c linear

viscoelastic components in the {a}, {b} and {c} rheological states, respectively, given by the

corresponding rheological equations

aaaa QP = (2.1)

bbbb QP =

(2.2)

cccc QP = (2.3)

where the component rheological operators are defined as

(2.1.1)aa

m

amaaaa DpDpDppP ,2

,2,1,0 ...++++=

(2.1.2)aa

n

anaaaa DqDqDqqQ ,2

,2,1,0 ...++++=

(2.2.1)bb

m

bmbbbb DpDpDppP ,2

,2,1,0 ...++++=

(2.2.2)bb

n

bnbbbb DqDqDqqQ ,2

,2,1,0 ...++++=

(2.3.1)cc

mcmcccc DpDpDppP ,

2,2,1,0 ...++++=

(2.3.2)cc

n

cncccc DqDqDqqQ ,2

,2,1,0 ...++++=

and ccbbaa ,;,;, represent the component natural rheological variables - the stress and

strain, respectively, and Dk =dk/dtk is the k-th order time derivative.

In the case of Reuss-like, uniform stress, composite model, the [3R] basic morpho-rheological interaction, i.e., the [3R] morphologically controlled interaction of component

rheological states, can be given symbolically in the form

{a} +{b} +{c} {a / b / c} ]3[ R [3R] (2.4.1.1)

(ma, na) +(mb, nb) +(mc, nc) (m]3[ R

[3R], n[3R]) (2.4.1.2)

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As a direct consequence of the relationships between the composite and component

characteristic rheological variables

cbaR === ]3[ (2.4.2.1)

ccbbaaR vvv ++= ]3[ (2.4.2.2)

where ]3[]3[ , RR are the [3R] model composite rheological variables, and depict the

component volume fractions,

cba vvv ,,

)1( =++ cba vvv , the corresponding [3R] rheological equation,

which expressthe [3R] model composite behaviour law, is

]3[]3[ RR QP = (2.5)

the composite rheological operators being given in terms of those of components as

bacccabbcbaaR QQPvQQPvQQPvP ++=]3[ (2.6.1)

(2.6.2)cbaR QQQQ =]3[

Accordingly, the [3R] selection rules are given in the form

=Ord.{Q]3[ Rm [3R]} =max ( + + ,am bn cn

+ + , (2.7.1)bm an cn

+ + )cm an bn

=Ord.{P]3[ Rn [3R]} = + + (2.7.2)an bn cnand the morpho-rheological interaction parameters are presented in table 2.1

-------------------------------------------------------------------------

Table 2.1

[3V] morpho-rheological interaction parameters

--------------------------------------------------------------------------

Furthermore, the corresponding[3R] specific mixing rules for the viscoelastic moduli are,

in the case of complex moduli,

(2.8)**** ]3[ ////1 ccbbaaR MvMvMvM ++=

whereas for storage- and loss- modulus yield

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22

*2

*2

*]3[ /}///{ XMMvMMvMMvM cccbbbaaaR ++= (2.8.1.1)

22*2*2*]3[ /}///{ XMMvMMvMMvM cccbbbaaaR ++= (2.8.1.2)

and

2/122*2*2*

22*2*2*

}]///[

]///[{

cccbbbaaa

cccbbbaaa

MMvMMvMMv

MMvMMvMMvX

+++

+++=

(2.8.2)

[3R] morpho-rheological effects

Aiming at to illustrate the different quantitative and qualitative forms of [3R] morpho-

rheological effects, one consider for components three simple rheological models, e.g.,- Hooke model:

{1}: (m =0, n =0), rpP ,0}1{ = , rqQ ,0}1{ = ; r =1, 2, 3;

- Kelvin-Voigt model:

{2}: (m =0, n =1), rpP ,0}2{ = , DqqQ rr ,1,0}2{ += ; r =1, 2, 3;

- Poynting-Thomson-Zener model:

{3}: (m =1, n =1), DppP rr ,1,0}3{ += , DqqQ rr ,1,0}3{ += ; r =1, 2, 3;

the results being presented in table 2.2.

---------------------------------------------------------------

Table 2.2

[3V] morpho-rheological effects

--------------------------------------------------------------

It is useful to point out that given the typical symmetrical form of composite rheological

operators from the standpoint of component permutation, a well defined invariance property arise.Moreover, significant consequences are observed from the point of view of morpho-rheological

effects, as follows.

Firstly, it appears that there are well defined cases in which the composites have the same

rheological state, i.e., they are of the same rheological potential:

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- {1} rheological state, for the {1}/{1}/{1} composite;

- {2} forbidden rheological state ;

- {3}rheological state, for {1}/{1}/{2}, {1}/{1}/{3} , {1}/{2}/{1}, {1}/{3}/{1}, {2}/{1}/{1}

and {3}/{1}/{1} composite;

- {4} forbidden rheological state;

- {5}, for {1}/{2}/{2}, {1}/{2}/{3}, {1}/{3}/{2},{1}/{3}/{3}, {2}/{1}/{2}, {2}/{1}/{3},

{2}/{2}/{1}, {2}/{3}/{1}, {3}/{1}/{2}, {3}/{1}/{3}, {3}/{2}/{1} and {3}/{3}/{1} composite;

- {6}, for {2}/{2}/{2};

- {7} rheological state, for {2}/{2}/{3}, {2}/{3}/{2}, {2}/{3}/{3}, {3}/{2}/{2},

{3}/{2}/{3}, {3}/{3}/{2} and {3}/{3}/{3} composite systems.

Secondly, it is possible that the rheological order of the composite rheological equation to

be equal to that of the highest component one, typical quantitative morpho-rheological effects

resulting as follows:

- {1} rheological state, for {1}/{1}/{1};

- {3}, for {1}/{1}/{3}, {1}/{3}/{1} and {3}/{1}/{1}model composites.

Finally, well definedqualitativemorpho-rheological effects arise, when the rheological

order of composite is higher than those of all components:

- {3} rheological state, for {1}/{1}/{2}, {1}/{2}/{1} and {2}/{1}/{1};

- {5}, for {1}/{2}/{2}, {1}/{2}/{3}, {1}/{3}/{2}, {1}/{3}/{3}, {2}/{1}/{2}, {2}/{1}/{3},

{2}/{2}/{1}, {2}/{3}/{1}, {3}/{1}/{2}, {3}/{1}/{3}, {3}/{2}/{1} and {3}/{3}/{1}composite

systems;

- {6}, for {2}/{2}/{2};

- {7} rheological state, for {2}/{2}/{3}; {2}/{3}/{2}, {2}/{3}/{3}, {3}/{2}/{2}, {3}/{2}/{3},

{3}/{3}/{2} and {3}/{3}/{3} composites.

[3R] primary mixing rules

Aiming at to exemplify the way in which the composite rheological parameters are

obtained, as a direct consequence of behaviour law, in terms of the component rheological

parameters and composition, two specific situations are taken into account for the components:

- the (1, 1) - rheological model, i.e. the mono-relaxant (retardant) , and

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- the (2, 2) - rheological model of bi-relaxant (retardant) behaviour.

In the first case, of (1, 1) components, the corresponding rheological operators are

,DppP aaa ,1,0 += DqqQ aaa ,1,0 += (2.9.1)

,DppP bbb ,1,0 += DqqQ bbb ,1,0 += (2.9.2)

,DppP ccc ,1,0 += DqqQ ccc ,1,0 += (2.9.3)

and the composite rheological operators are given by

(2.10.1)332

210]3[ DpDpDppP R +++=

(2.10.2)332

210]3[ DqDqDqqQ R +++=

where the resulting rheological parameters are

bacccabbcbaaR qqpvqqpvqqpvpp ,0,0,0,0,0,0,0,0,0]3[,00 ++= (2.10.1.1)

])([

])([

])([

,0,0,1,1,0,1,0,0

,0,0,1,1,0,1,0,0

,0,0,1,1,0,1,0,0]3[,11

bacabbacc

cabaccabb

cbabccbaaR

qqpqqqqpv

qqpqqqqpv

qqpqqqqpvpp

+++

++++

+++=

(2.10.1.2)

)]([)]([

)]([

,1,0,1,0,1,1,1,0

,1,0,1,0,1,1,1,0

,1,0,1,0,1,1,1,0]3[,22

abbacbacc

accabcabb

bccbacbaaR

qqqqpqqpvqqqqpqqpv

qqqqpqqpvpp

+++

++++

+++=

(2.10.1.3)


and

(2.10.2.1)cbaR qqqqq ,0,0,0]3[,00 =

acbbccbaR qqqqqqqqqq ,1,0,0,1,0,1,0,0]3[,11 )( ++= (2.10.2.2)

abccbcbaR qqqqqqqqqq ,1,1,0,1,0,1,1,0]3[,22 )( ++= (2.10.2.3)

(2.10.2.4)cbaR qqqqq ,1,1,1]3[,33 =

In the second case, of (2, 2) components, the rheological operators are defined

as

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, (2.11.1)2,2,1,0 DpDppP aaaa ++=2

,2,1,0 DqDqqQ aaaa ++=

, (2.11.2)2,2,1,0 DpDppP bbbb ++=2

,2,1,0 DqDqqQ bbbb ++=

, (2.11.3)2,2,1,0 DpDppP cccc ++=2

,2,1,0 DqDqqQ cccc ++=

the resulting composite rheological operators are

(2.12.1)665

52

43

32

210]3[ DpDpDpDpDpDppP R ++++++=

(2.12.2)665

52

43

32

210]3[ DqDqDqDqDqDqqQ R ++++++=

where


])([

])([

])([

,0,0,1,1,0,1,0,

,0,0,1,1,0,1,0,0

,0,0,1,1,0,1,0,0]3[,11

bacabbacoc

cabaccabb

cbabccbaaR

qqpqqqqpv

qqpqqqqpv

qqpqqqqpvpp

+++

++++

+++=

(2.12.1.2)

])(

)([

])(

)([

])(

)([

,0,0,2,1,0,1,0,1

,2,0,1,1,2,0,0

,0,0,2,1,0,1,0,1

,2,0,1,1,2,0,0

,0,0,2,1,0,1,0,1

,2,0,1,1,2,0,0]3[,22

bacabbac

abbabacc

cabaccab

accacabb

cbabccba

bccbcbaaR

qqpqqqqp

qqqqqqpv

qqpqqqqp

qqqqqqpv

qqpqqqqp

qqqqqqpvpp

+++

++++

++++

++++

++++

+++=

(2.12.1.3)

)](

)()([

)](

)(

)([

)](

)(

)([

,1,0,1,0,2

,2,0,1,1,2,0,1

,1,2,2,1,0

,1,0,1,0,2

,2,0,1,1,2,0,1

,1,2,2,1,0

,1,0,1,0,2

,2,0,1,1,2,0,1

,1,2,2,1,0]3.[33

abbac

abbabac

babacc

accab

accacab

cacabb

bccba

bccbcba

cbcbaaR

qqqqp

qqqqqqpqqqqpv

qqqqp

qqqqqqp

qqqqpv

qqqqp

qqqqqqp

qqqqpvpp

++

++++

+++

+++

++++

+++

+++

++++

++=

(2.12.1.4)

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)](

)([

)](

)([

)](

)([

,2,0,1,1,2,0,2

,2,1,2,1,1,2,2,0

,2,0,1,1,2,0,2

,1,2,2,1,1,2,2,0

,2,0,1,1,2,0,2

,1,2,2,1,1,2,2,0]3[,44

abbabac

abbacbacc

accacab

cacabcabb

bccbcba

cbcbacbaaR

qqqqqqp

qqqqpqqpv

qqqqqqp

qqqqpqqpv

qqqqqqp

qqqqpqqpvpp

+++

++++

++++

++++

++++

+++=

(2.12.1.5)

)]([

)]([

)]([

,2,1,2,1,2,2,2,1

,2,1,2,1,2,2,2,1

,2,1,2,1,2,2,2,1]3[,55

abbacbacc

accabcabb

bccbacbaaR

qqqqpqqpv

qqqqpqqpv

qqqqpqqpvpp

+++

++++

+++=

(2.12.1.6)


and

(2.12.2.1)cbaR qqqqq ,0,0,0]3[,00 =

cabbacbaR qqqqqqqqqq ,0,1,0,1,0,1,0,0]3[,11 )( ++= (2.12.2.2)

cabbaba

cabbacbaR

qqqqqqq

qqqqqqqqqq

,0,2,0,1,1,2,0

,1,1,0,1,0,2,0,0]3[,22

)(

)(

+++

+++=(2.12.2.3)

cabba

cabbaba

cabbaR

qqqqq

qqqqqqq

qqqqqqq

,0,2,1,2,1

,1,2,0,1,1,2,0

,2,1,0,1,0]3[,33

)(

)(

)(

++

++++

++=

(2.12.2.4)

cbacabba

cabbabaR

qqqqqqqq

qqqqqqqqq

,0,2,2,1,2,1,2,1

,2,2,0,1,1,2,0]3[,44

)(

)(

+++

+++=(2.12.2.5)

baccabbaR qqqqqqqqqq ,2,2,1,2,2,1,2,1]3[,55 )( ++= (2.12.2.6)

(2.12.2.7)cbaR qqqqq ,2,2,2]3[,66 =

are the composite rheological parameters in terms of component ones.

It is useful to draw attention on the fact that, as a consequence of inner logic of [3R]

morpho-rheological interactions, the dependences of the composite rheological operators versus

those of components point out, in the case of composite Q operators, relationships including only

similar operators of components, whereas the composite P operators are expressed as a mixed

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dependence of both P and Q operators of components. Naturally, as can be seen, the rules

are similar for the rheological parameters.

Conclusions

The uniform stress approach for ternary phase in phase composie systems with linear

viscoelastic components including polymeric materials, accomplished within the realm of of

rheological behaviour laws - selection rules - mixing rules hierarchical procedure, affords relevant

facts on the evaluation of rheological response.

In the case of Reuss-like morpho-rheological interactions, the expressions providing the

composite rheological operators, appearing in the behaviour laws, versus those of components can

be derived in a meaningful way.

The selection rules and the mixing rules resulting as direct consequences are pointed out

and illustrated for components defined by well defined established rheological models, in the case

of mono- and bi-relaxan (retardant) rheological states.

The comparison of effects arising in Voigt - uniform deformation, and Reuss - uniform

stress approach, respectively, shows significant dissimilarities outcoming in virtue of intrinsic

features of morpho-rheological interactions.

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References

1. JONES, R. M., Mechanics of Composite Materials, Wiley, New York, 1975.

2. CRIVELLI-VISCONTI, I., Materiali Compositi. Tecnologie et Progettazione,Tamburini, Milano, 1975.3. COOPER, S. L., ESTES, G. M., (eds.), Multiphase Polymers, Am. Chem. Soc.,

Washington, 1979.4. TSAI, S. W., HAHN, H. T., Introduction to Composite Materials, Technomic, Westport,

1980.5. BERTHELOT, J. -M., Composite Materials. Mechanical Behavior and Structure

Analysis, Springer, New York, 1999.6. NAKATANI, A. I., HJELM, R. P., GERSPACHER, M., KRISHNAMOORTI, R., Filled

and Nanocomposite Polymer Materials, Mater. Res. Soc., Boston, 2001.7. SCHAPERY, R. A., J. Compos. Mater., 1, 1967, p. 228.

8. CHRISTENSEN, R. M., J . Mech., Phys. Solids, 38, nr. 3, 1990, p. 379.9. BRINSON, L. C., KNAUSS, W. G., J. Mech. Phys. Solids, 39, nr. 7, 1991, p. 859.10. HASHIN, Z. V., J. Mech. Phys. Solids,40, nr. 4, 1992, p. 767.11. PAVEN, H., Rev. Chim., 55, nr.6, 2004, p. 444.

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Table 2.1

[3 R] morpho-rheological interaction parameters

[3 R] ma

na

mb

nb

mc

nc

m1 1 0 0 1 0 1

m2 0 1 1 0 0 1

m3 0 1 0 1 1 0

n1 0 1 0 1 0 1

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Table 2.2

[3R] morpho-rheological interactions

Crt. a/b/c Components

No. Composite a b c [3 R]

m n m n m n m1 m2 m3 n1 m n

1 {1}/ {1}/ {1} 0 0 0 0 0 0 0 0 0 0 0 0

2 {1}/ {1}/ {2} 0 0 0 0 0 1 1 1 0 1 1 1

3 {1}/ {1}/ {3} 0 0 0 0 1 1 1 1 1 1 1 1

4 {1}/ {2}/ {1} 0 0 0 1 0 0 1 0 1 1 1 1

5 {1}/ {2}/ {2} 0 0 0 1 0 1 2 1 1 2 2 26 {1}/ {2}/ {3} 0 0 0 1 1 1 2 1 2 2 2 2

7 {1}/ {3}/ {1} 0 0 1 1 0 0 1 1 1 1 1 1

8 {1}/ {3}/ {2} 0 0 1 1 0 1 2 2 1 2 2 2

9 {1}/ {3}/ {3} 0 0 1 1 1 1 2 2 2 2 2 2

10 {2}/ {1}/ {1} 0 1 0 0 0 0 0 1 1 1 1 1

11 {2}/ {1}/ {2} 0 1 0 0 0 1 1 2 1 2 2 2

12 {2}/ {1}/ {3} 0 1 0 0 1 1 1 2 2 2 2 2

13 {2}/ {2}/ {1} 0 1 0 1 0 0 1 1 2 2 2 2

14 {2}/ {2}/ {2} 0 1 0 1 0 1 2 2 2 3 2 3

15 {2}/ {2}/ {3} 0 1 0 1 1 1 2 2 3 3 3 3

16 {2}/ {3}/ {1} 0 1 1 1 0 0 1 2 2 2 2 2

17 {2}/ {3}/ {2} 0 1 1 1 0 1 2 3 2 3 3 3

18 {2}/ {3}/ {3} 0 1 1 1 1 1 2 3 3 3 3 3

19 {3}/ {1}/ {1} 1 1 0 0 0 0 1 1 1 1 1 1

20 {3}/ {1}/ {2} 1 1 0 0 0 1 2 2 1 2 2 2

21 {3}/ {1}/ {3} 1 1 0 0 1 1 2 2 2 2 2 2

22 {3}/ {2}/ {1} 1 1 0 1 0 0 2 1 2 2 2 2

23 {3}/ {2}/ {2} 1 1 0 1 0 1 3 2 2 3 3 3

24 {3}/ {2}/ {3} 1 1 0 1 1 1 3 2 3 3 3 3

25 {3}/ {3}/ {1} 1 1 1 1 0 0 2 2 2 2 2 2

26 {3}/ {3}/ {2} 1 1 1 1 0 1 3 3 2 3 3 3

27 {3}/ {3}/ {3} 1 1 1 1 1 1 3 3 3 3 3 3

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Table 2.3

[3V] and [3R] morho-rheological effects

Crt.

No. Components

a / b / c

Composite

a b c [3 V] [3 R]

1 {1} {1} {1} {1} {1}

2 {1} {1} {2} {2} {3}

3 {1} {1} {3} {3} {3}

4 {1} {2} {1} {2} {3}

5 {1} {2} {2} {2} {4}

6 {1} {2} {3} {4} {5}

7 {1} {3} {1} {3} {3}

8 {1} {3} {2} {4} {5}

9 {1} {3} {3} {5} {5}

10 {2} {1} {1} {2} {2}

11 {2} {1} {2} {2} {4}

12 {2} {1} {3} {4} {4}

13 {2} {2} {1} {2} {4}

14 {2} {2} {2} {2} {4}

15 {2} {2} {3} {4} {6}

16 {2} {3} {1} {4} {4}

17 {2} {3} {2} {4} {6}

18 {2} {3} {3} {6} {6}

19 {3} {1} {1} {3} {3}

20 {3} {1} {2} {4} {5}

21 {3} {1} {3} {5} {5}

22 {3} {2} {1} {4} {5}

23 {3} {2} {2} {4} {6}

24 {3} {2} {3} {6} {7}

25 {3} {3} {1} {5} {5}

26 {3} {3} {2} {6} {7}

27 {3} {3} {3} {7} {7}

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