QUANTITATIVE AND QUALITATIVE RHEOLOGICAL EFFECTS IN SOLID BINARY COMPOSITES WITH LINEAR VISCOELASTIC

COMPONENTS

Horia Paven, Chemical Research Institute, Bucharest, Romania Key words: linear viscoelasticity, mixing rules, model composite system, morpho- rheological couplings, quantitative and qualitative effects, rheological operators, rheological parameters, selection rules, solid binary composites. ABSTRACT Aiming at establishing natural, non-conventional relationships for advanced characterization and analysis of deformational properties of composite materials within the framework of viscoelastic behaviour, the intrinsic linear viscoelastic properties of components are considered by using the differential rheological operators technique. Thus, the well known mixing rules for viscoelastic moduli result as direct consequences of composite rheological equations, and furthermore, well defined selection rules are revealed, the potential quantitative and qualitative rheological effects, including the mixing rules for rheological parameters, being evidenced and discussed. INTRODUCTION Despite the impressive effort in establishing lots of mixing rules for different types of composites, it seems some essential tasks remain to be accomplished, Holliday (1), Vinson et al (6), Richardson (8), Hull (12), Janda (15), and Jang (18). In fact, besides the results providing useful relationships for moduli and/or compliances, Ashton et al (2), Crivelli Visconti (4), Jones (5), Tsai et al (11), Hashin et al (13), Gay (16), further relevant information at inner level would be necessary to acquire and process for better fundamental understanding of the behaviour of real composite materials . Moreover, in the case of viscoelastic components distinct features are to be considered, Bland (3), Flugge (7), Bucknall (9), Christensen (10), Tschoegl (14), Nielsen et al (17) and Lakes (19). TRADITIONAL MIXING RULES Given the a/b binary composite system with the components a and b, different mixing rules exppressing the dependence of the elastic modulus of composite on those of components (Ma, Mb), as well as on the volume fraction of the components (va,vb), are obtained as result of dissimilar morphological couplings. As a rule, by using a well known correspondence principle, the relationships established for the linear elastic behaviour are extended in the case of linear viscoelastic behaviour, the complex modulus (M*) being the appropriate quantity for the case of dynamic conditions, Kausch (20), Lipatov (21), Shen et al (22), Schmitt (23), Takayanagi et al (24), Kawai et al (25), Nielsen (26), Popovics (27, 28), Ahmed et al (29), and Paven et al (33). Moreover, for the sake of simplicity we restrict ourselves to the case when the stress, σ, and the strain, ε, can each be represented by a single component.

In the case of the Voigt[V], Π-like, and Reuss[R],Σ-like, basic model couplings, Kausch (20) Lipatov (21), Shen et al (22), Schmitt (23), Ahmed et al (29), and Paven et al (34), the corresponding mixing rules are M*[V]=vaM*a+vbM* [1. 1] and M*[R]=M*aM*b/(vbM*a+vaM*b), [2. 1] respectively.

On the other hand, the Takayanagi-Kawai model coupling is characterized by the mixing rules, Takayanagi et al (24), Kawai et al (25), and Paven et al (35) M*[TK1]=(A[TK1]M*a2+B[TK1]M*aM*b) /(C[TK1]M*a+D[TK1]M*b) [3. 1'] where A[TK1] =1-λ1, B[TK1]=λ1, C[TK1]=1-λ1+vb, D[TK1]=λ1-vb in the case of a Σ-like coupling, whereas for a Π-like coupling one obtain M*[TK2]=(A[TK2]M*a2+B[TK2]M*aM*b) /(D[TK2]M*a+C[TK2]M*b) [3. 1"] where A[TK2]=ϕ2 -vb, B[TK2]=1+vb-ϕ2, D[TK2]=ϕ2, C[TK2]=1-ϕ2 the two model systems being equivalent if λ1=1+vb-ϕ2.

The Nielsen[N] model coupling is defined by the mixing rule, Nielsen (26) M*[N]=(A[N]M*a2+B[N]M*b2+D[N]M*aM*b)/ /(C[N]M*a+E[N]M*b) [4. 1] where A[N]=v"av'b, B[N]=v'av"b, C[N]=v'av"a+v'bv"b+v'a+v'b, D[N]=v'b, E[N]=v'a

The composite averages model generate, Popovics (27, 28), in the case of [A] model coupling, the mixing rule M*[A]=(Avavb(M*a2+M*b2)+(1-2Avavb)M*aM*b)/ /(vbM*a+vaM*b) [5. 1] while for the [H] model coupling the resul is M*[H]=(vbM*aM*b

2+vaM*a2Mb*)/

/(Hvavb(M*a

2+M*b2)+(1-2Hvavb)M*aM*b)) [6. 1]

The different relationships [1. 1]-[6. 1] thus obtained correspond, in fact, to well defined model

composite systems, the choose of the "better" approximation depending on the realistic evaluation of the morphological features of composite. Unfortunately, the common use of mixing rules disregard the significant facts involving the intrinsic peculiarities of rheological properties of components upon potential quantitative and qualitative rheological effects. RHEOLOGICAL EQUATIONS In order to reveal the "hidden" information regarding the basic manner in which the rheological behaviour of composite depends on that of linear viscoelastic components, the use of rheological operators technique is considered. Accordingly, the P, Q differential rheological operators are used, Bland (3), Flugge (7), and Tschoegl (14). In the case of the a/b binary composite system the Pa, Qa and Pb, Qb rheological operators, respectively, will be involved. Taking into account the distinct morpho-rheological couplings of rheological states of components in the composite system, the rheological state of composite can be deduced in a straight way, the rheological

operators for well defined model composite systems resulting in terms of those of component ones and the corresponding volume fractions .

The basic morpho-rheological couplings - Voigt[V] and Reuss[R] - are characterized by the rheological operators, Paven et al (30, 34), P[V]=PaPb, Q[V]=vaPbQa+vbPaQb [1. 2] and P[R]=vaPaQb+vbPbQa, Q[R]=QaQb [2. 2] respectively . On the other hand, in the case of the Takayanagi-Kawai model coupling the corresponding rheological operators are, Paven et al (30, 35), P[TK]=C[TK]Pa2Qb+D[TK]PaPbQa, Q[TK]=A[TK]PbQa2+B[TK]PaQaQb, [3. 2] whereas for the Nielsen[N] model coupling result the rheological operators, Paven et al (32, 38), P[N]=D[N]Pa2PbQb+E[N]PaPb2Qa, Q[N]=A[N]Pb2Qa2+B[N]Pa2Qb2+C[N]PaPbQaQb, [4. 2] where A[N]=va"vb', B[N]=va'vb", C[N]=va'vb"+va"vb'+va'+vb', D[N]=vb', E[N]=va'

In the case of the composite averages model, Paven et al (31, 39), for the [A] coupling, the

rheological operators are P[A]=vaPa2PbQb+vbPaPb2Qa, Q[A]=Avavb(Pa2Qb2+Pb2Qa2)+(1-2Avavb)PaPbQaQb [5. 2] while in the case of the [H] coupling the rheological operators are given as P[H]=Hvavb(Pa2Qb2+Pb2Qa2)+(1-2Hvavb)PaPbQaQb, Q[H]=vaPaQaQb

2+vbPbQa2Qb [6. 2] The different form [1. 2]-[6. 2] of the dependence of composite rheological operators on those

of components for distinct morphologies represents the natural ground of characteristic effects arising in the case of composite systems with linear viscoelastic components, both for the rules enabling the selection of the allowed rheological state of the composite, as well as the for the well defined sets of mixing rules for the rheological parameters. SELECTION RULES The composite rheological equations enable us to reveal the corresponding morpho-rheological selection rules, Paven (36). Thus, taking into account the order of rheological states of a and b components as being (ma|na) and (mb|nb), respectively, the order of composite rheological state is given by well defined relationships.

The basic Voigt[V], pi-like direct coupling is characterized by the selection rules, Paven (40), m[V]=ma+mb, n[V]=max.(ma+nb, mb+na) [1. 3] while for the basic Reuss[R], sigma-like direct coupling, results, Paven (41)

m[R]=max.(ma+nb, mb+na), n[R]=na+nb [2. 3] On the other hand, in the case of Takayanagi-Kawai coupling the selection rules are, Paven

(36) m[TK]=max.(2ma+nb, ma+mb+na), n[TK]=max.(mb+2na, ma+na+nb) [3. 3] whereas for the Nielsen[N] model coupling one obtain, Paven et al (38), m[N]=max.(2ma+mb+nb, ma+2mb+na), n[N]=max.(2ma+2nb, 2mb+2na, ma+mb+na+nb) [4. 3]

In the case of the composite averages model, the resulting selection rules are given as, Paven et al (39), m[A]=max.(2ma+mb+nb, ma+2mb+na), n[A]=max.(2ma+2nb, 2mb+2na, ma+mb+na+nb) [5. 3] for the [A] -like coupling, while for the [H]-like one the mixing rules are m[H]=max.(2ma+2nb, 2mb+2na, ma+mb+na+nb), n[H]=max.(ma+na+2nb, mb+2na+nb) [6. 3]

It is noteworthy to point out the significant dissimilarities of the morpho-rheological selection rules, [1. 3] - [6. 3], both from the standpoint of given components, and different morphologies, as well as for a given morphology, but different rheological states of components. Indeed, in principle, one can obtain for the composite a rheological equation of the same order as of the one or another of components, or of both components, but also of higher order than of both components. Consequently,one must be aware of the extent in which the modification of morphology and/or rheological states induce quantitative and even qualitative rheological effects, Paven et al (42, 43). QUANTITATIVE AND QUALITATIVE RHEOLOGICAL EFFECTS In order to illustrate the manner in which the morphology of the composite and the peculiarities of rheological properties of components act on the composite properties, the case of standard linear viscoelastic components is considered. Accordingly, if the P, Q rheological operators of a and b components are given as Pa=p0a+p1aD, Qa=q0a+q1aD; Pb=p0b+p1bD, Qb=q0b+q1bD; respectively, where p0, p1; q0, q1 stand for the rheological parameters of componens (evidenced by the subscripts a and b), and D is the time derivative operator, the resulting rheological equation for different model composite systems can be obtained by using the suitable relationships.

In the case of the basic couplings, [1. 2] - [2. 2], for the Voigt[V] coupling one obtain the composite rheological equation p0[V]σ+p1[V]Dσ+p2[V]D2σ=q0[V]ε+q1[V]Dε+q2[V]D2ε, [1. 4] where p0[V]=p0ap0b, p1[V]=p0ap1b+p0bp1a, p2[V]=p1ap1b, [1. 4'] q0[V]=vap0bq0a+vbp0aq0b, q1[V]=va(p0bq1a+p1bq0a)++vb(p0aq1b+p1aq0b), q2[V]=vap1bq1a+vbp1aq1b [1. 4"] whereas for the Reuss[R] coupling, results the rheological equation p0[R]σ+p1[R]Dσ+p2[R]D2σ=q0[R]ε+q1[R]Dε+q2[R]D2ε, [2. 4] where p0[R]=vap0aq0b+vbp0bq0a, p1[R]=va(p0aq1b+p1aq0b)+vb(p0bq1a+p1bq0a), p2[R]=vap1aq1b+vbp1bq1a; [2. 4'] q0[R]=q0aq0b, q1[R]=qoaq1b+q0bq1a, q2[R]=q1aq1b; [2. 4"]

On the other hand, in the case of the Takayanagi-Kawai coupling, [3.2], the rheological equation is

p0[TK]σ+p1[TK]Dσ+p2[TK]D2σ+p3[TK]D3σ= =q0[TK]ε+q1[TK]Dε+q2[TK]D2ε+q3[TK]D3ε, [3. 4] where p0[TK]=C[TK]p0ap0bq0a+D[TK]p0a2q0b, p1[TK]=C[TK](p0ap0bq1a+(p0ap1b+p0bp1a)q0a+ +D[TK](p0a2q1b+2p0ap1aq0b), p2[TK]=C[TK]((p0ap1b+p0bp1a)q1a+p1ap1bq0a)+ +D[TK](2p0aq1b+p1aq0b)p1a, p3[TK]=C[TK]p1ap1bq1a+D[TK]p1a2q1b; [3. 4'] q0[TK]=A[TK]p0bq0a2+B[TK]p0aq0aq0b, q1[TK]=A[TK](2p0bq1a+p1bq0a)q0a+B[TK](p0a(q0aq1b+ +q0bq1a)+p1aq0aq0b), q2[TK]=A[TK](p0bq1a+2p1bq0a)q1a+B[TK](p0aq1aq1b+ +p1a(q0aq1b+q0bq1a)), q3[TK]=A[TK]p1bq1a2+B[TK]p1aq1aq1b; [3. 4"]

In the case of the Nielsen[N] model coupling the rheological equation is given as, [4. 2], p0[N]σ+p1[N]Dσ+p2[N]D2σ+p3[N]D3σ+p4[N]D4σ= =q0[N]ε+q1[N]Dε+q2[N]D2ε+q3[N]D3ε+q4[N]D4ε, [4. 4] where p0[N]=D[N]p0ap0b2q0a+E[N]p0a2p0bq0b, p1[N]=D[N](2p0ap0bp1bq0a+(p0aq1a+p1aq0a)p0b2)+ +E[N](2p0ap0bp1aq0b+p0a2(p0bq1b+p1bq0b)), p2[N]=D[N](2(p0aq1a+p1aq0a)p0bp1b+p0ap1b2q0a+p0b2p1aq1a)+ +E[N](2p0ap1a(p0bq1b+p1bq0b)+p0a2p1bq1b+p0bp1a2q0b), p3[N]=D[N]((p0aq1a+p1aq0a)p1b2+2p0bp1ap1bq1a))+ +E[N](2p0ap1ap1bq1b+(p0bq1b+p1bq0b)p1a2), p4[N]=D[N]p1ap1b2q1a+E[N]p1a2p1bq1b; [4. 4'] q0[N]=A[N]p0b2q0a2+B[N]p0a2q0b2+C[N]p0ap0bq0aq0b,

q1[N]=2A[N]p0bq0a(p0bq1a+p1bq0a)+2B[N]p0aq0b(p0aq1b+p1aq0b)+ +C[N]((p0ap0b(q0aq1b+q0bq1a)+(p0ap1b+p0bp1a)q0aq0b), q2[N]=A[N]((p0bq1a+p1bq0a)2+2p0bp1bq0aq1a)+ +B[N]((p0aq1b+p1aq0b)2+2p0ap1aq0bq1b))+ +C[N](p0ap0bq1aq1b+(p0ap1b+p0bp1a)(q0aq1b+q0bq1a)+ +p1ap1bq0aq0b), q3[N]=2A[N](p0bq1a+p1bq0a)p1bq1a+2B[N](p0aq1b+p1aq0b)p1aq1b)+ +C[N]((p0ap1b+p0bp1a)q1aq1b+p1ap1b(q0aq1b+q0bq1a)), q4[N]=A[N]p1b2q1a2+B[N]p1a2q1b2+C[N]p1ap1bq1aq1b; [4. 4"]

The composite averages model generates in the case of [A] -like coupling,[5. 2], the rheological equation p0[A]σ+p1[A]Dσ+p2[A]D2σ+p3[A]D3σ+p4[A]D4σ= =q0[A]ε+q1[A]Dε+q2[A]D2ε+q3[A]D3ε+q4[A]D4ε, [5. 4] where p0[A]=vap0a2p0bq0b+vbp0ap0b2q0a, p1[A]=va((p0a2(p0bq1b+p1bq0b)+2p0ap0bp1aq0b)+ +vb((p0aq1a+p1aq0a)p0b2+2p0ap0bp1bq0a)), p2[A]=va(p0a2p1bq1b+2p0ap1a(p0bq1b+p1bq0b)+p0bp1a2q0b)+ +vb(p0ap1b2q0a+2(p0aq1a+p1aq0a)p0bp1b+p0b2p1aq1a, p3[A]=va(2p0ap1ap1bq1b+(p0bq1b+p1bq0b)p1a2)+ +vb((p0aq1a+p1aq0a)p1b2+2p0bp1ap1bq1a), p4[A[=vap1a2p1bq1b+vbp1ap1b2q1a; [5. 4'] q0[A]=Avavb(p0a2q0b2+p0b2q0a2)+(1-2Avavb)p0ap0bq0aq0b, q1[A]=2Avavb(p0aq0b(p0aq1b+p1aq0b)+p0bq0a(p0bq1a+p1bq0a)+ +(1-2Avavb)(p0ap0b(q0aq1b+q0bq1a)+(p0ap1b+p0bp1a)q0aq0b), q2[A]=Avavb(((p0aq1b+p1aq0b)2+2p0ap1aq0bq1b)+ +((p0bq1a+p1bq0a)2+2p0bp1bq0aq1a))+ +(1-2Avavb)(p0ap0bq1aq1b+ +(p0ap1b+p0bp1a)(q0aq1b+q0bq1a)+p1ap1bq0aq0b),

q3[A]=2Avavb(p1aq1b(p1aq0b+p0aq1b)+(p0bq1a+p1bq0a)p1bq1a)+ +(1-2Avavb)((p0ap1b+p0bp1a)q1aq1b+p1ap1b(q0aq1b+q0bq1a), q4[A]=Avavb(p1a2q1b2+p1b2q1a2)+(1-2Avavb)p1ap1bq1aq1b; [5. 4"] while for the [H]-like coupling, [6. 2], the resulting rheological equation is p0[H] σ+p1[H]Dσ+p2[H]D2σ+p3[H]D3σ+p4[H]D4σ= =q0[H]ε+q1[H]Dε+q2[H]D2ε+q3[H]D3ε+q4[H]D4ε, [6. 4] where p0[H]=Hvavb(p0a2q0b2+p0b2q0a2)+(1-2Hvavb)p0ap0bq0aq0b, p1[H]=2Hvavb((p0aq1b+p1aq0b)p0aq0b+p0bq0a(p0bq1a+p1bq0a)+ +(1-2Hvavb)(p0ap0b(q0aq1b+q0bq1a)+(p0ap1b+p0bp1a)q0aq0b), p2[H]=Hvavb(((p0aq1b+p1aq0b)2+2p0ap1aq0bq1b)+ +(p0bq1a+p1bq0a)2+2p0bp1bq0aq1a))+ +(1-2Hvavb)(p0ap0bq1aq1b+(p0ap1b+p0bp1a)(q0aq1b+q0bq1a)+ +p1ap1bq0aq0b)), p3[H]=2Hvavb((p0aq1b+p1aq0b)p1aq1b+(p0bq1a+p1bq0a)p1bq1a+ +(1-2Hvavb)((p0ap1b+p0bp1a)q1aq1b+p1ap1b(q0aq1b+q0bq1a)), p4[H]=Hvavb(p1a2q1b2+p1b2q1a2)+(1-2Hvavb)p1ap1bq1aq1b; [6. 4'] q0[H]=vap0aq0aq0b2+vbp0bq0a2q0b, q1[H]=va(2p0aq0aq0bq1b+(p0aq1a+p1aq0a)q0b2)+ +vb(2p0bq0aq0bq1a+(p0bq1b+p1bq0b)q0a2), q2[H]=va(p0aq0aq1b2+2(p0aq1a+p1aq0a)q0bq1b+p1aq0b2q1a)+ +vb(p0bq0bq1a2+2(p0bq1b+p1bq0b)q0aq1a+p1bq0a2q1b), q3[H]=va((p0aq1a+p1aq0a)q1b2+2p1aq0bq1aq1b)+ +vb((p0bq1b+p1bq0b)q1a2+2p1bq0aq1aq1b), q4[H]=vap1aq1aq1b2+vbp1bq1a2q1b; [6. 4"] Taking for the two components a and b of a/b binary composite system well-estalished model rheological states including the elastic Hooke, (0|0), as well as the viscoelastic Kelvin-Voigt, (0|1),and Zener, (1|1) ones, the corresponding rheological states for diferrent well-defined morphologies - [V], [R], [TK], [N], [A] and [H] can be established, the results being presented in Table 1.

It is interesting to point out, first, the effects arising, for given pairs of components, as result of morphological changes, [1. 4] - [6. 4]. Thus, in the case of Hooke/Hooke composite, when both components are elastic, the resulting morphological effects are quantitative, the order of

composite rheological state being identical with those of components, and consequently the composite behaves in an elastic manner too. However, even for a "slight" modification of the rheological state of a component, as in the case of Hooke/Kelvin-Voigt or Kelvin-Voigt/Hooke composite, the morphological variations induce either quantitative, [V], or qualitative effects of increasing complexity depending of the morphological features. Moreover, in the case of Hooke/Zener and Zener/Hooke composite systems, the morphological effects are well different versus the above quoted ones . Furthermore, for the Kelvin-Voigt/Zener, Zener/Kelvin-Voigt and Zener/Zener composite systems, only qualitative effects are present, and they are stronger in the last case.

Second, the analysis of the order of rheological states of composite systems, for a given morphology, with respect different component rheological states, evidences the "composite-component" rheological effects, pointing out a "spectrum" of allowed rheological states.

Third, the data summarized in Table 1 enable us to identify the conditions (morphology and components) for which a given rheological state of composite emerges.

Fourth, the dissimilar character of morphological couplings is well illustrated by the unlikeness of non-symmetric [TK], and the other, symmetric ones. Indeed, the rheological states of a/b and b/a composite systems are not identical in the former case, but "invariant' with respect the component permutation in the latter one. CONCLUSIONS The common practice to consider solely the conventional relationships between composite and component moduli fails to provide us with a self-contained background that would tell us how and why the morpho-rheological effects influence not only quantitatively but also qualitatively upon the potential behaviour of composite materials.

The use of rheological operators technique appears to be a powerful manner for obtaining significant complementary information concerning the overall response of composite materials.

It is expected that the identification of rheological selection rules and of the set of mixing rules for the rheological parameters, as well as further refinements of the presented considerations will provide new opportunities for relevant developments in the field. REFERENCES 1. Holliday L (ed), 'Composite Materials', Elsevier (Amsterdam), 1966 2. Ashton J E, Halpin J C, Petit P H,' Primer on Composite Materials: Analysis’, Technomic (Westport), 1969 3. Bland R D, 'The Theory of Linear Viscoelasticity', Academic Press (Oxford), 1971 4. Crivelli-Visconti I, 'Materiali Compositi. Tecnologie e Progettazione', Tamburini (Milano), 1975 5. Jones R M, 'Mechanics of Composite Mayerials', Wiley (New York), 1975 6. Vinson T K, Chou T W, 'Composite Materials and their Use in Structures', Appl Sci Publ (London), 1975 7. Flugge W, 'Viscoelasticity', Springer (Berlin), 1976 8. Richardson M E W (ed), 'Polymer Engineering Composites', Appl Sci Publ (London), 1977 9. Bucknall C B, 'Toughened Plastics', Appl Sci Publ (London), 1977 10. Christensen R M, 'Mechanics of Composite Materials', Wiley (New York) 1979 11. Tsai S W, Hahn H T, 'Introduction to Composite Materials', Technomic (Westport), 1980 12. Hull D, 'An Introduction to Composite Materials', Cambridge Univ Press (Cambridge), 1981 13. Hashin Z, Herakovich (ed), 'Mechanics of Composite Materials', Pergamon Press (New York) 1983

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