combining rules for predicting the thermoelastic properties of particulate filled polymers,...

Combining Rules for Predicting the Thermoelastic Properties of Particulate Filled Polymers, Polyblends,

and Foams

:STUART McGEE and R. L. McCULLOUGH

Department of Chemical Engineering and

Center for Composite Materials University of Delaware

Newark, Delaware 1971 1

Various models that have been proposed to predict the properties oi’particulate filled systems are reviewed and compared with experimental data. At filler volume fractions less than -0 .2-4 .3 , these models give essentially equivalent predictions th;tt are within the scatter of experimental measurements. At higher volume fraction of inclusions, significantly different results ,ire obtained from the various models. These predictions ei :her overestimate or underestimate observed properties. Ne w, theoretical combining rules are presented to predict the YOL ng’s modulus, Poisson’s ratio, shear modulus, bulk modulus, and coefficient of thermal expansion in terms of the properties of the matrix and inclusion and the volume fraction concentration of the inclusion. The predictions of these combining rules are in good agreement with experimental data that cover the feasible concentration range of inclusions for a variety of composite materials, ranging from particulate filled thermo$etting resins to thermoplastic foams.

INTRODUCTION e increased interest in the utilization of short-fiber

load-bearing applications has prompt -d renewed interest in constitutive relationships (or “2ombining rules”) which can be used to predict the p-operties of these composite materials in terms of the p-operties and concentration of the components. Constitutive relationships for unidirectional, continuous fiber laminates are reasonably well developed. For thi 5 special material system, the continuity of the fiber assiires that the strain field parallel to the aligned fibers IS essentially uniform. As a consequence of this special condition, the longitudinal properties are given by the simple “rule of mixtures,” while the transverse properties and shear moduli may be adequately estimated by the Halpin- Tsai relationship (1).

The situation for short-fiber and particulate reinforced composites is considerably m Jre complex. The discontinuous nature of the reinforc 2ment introduces significant variations in the stress and strain field so that the simplifying assumption of uniform strain (or stress) is no longer justified. Further difficulties are introduced by partial alignment of the short-fibers during fabrication. Intermediate states of fiber orientation (ranging between random to complei ely aligned) lead

Th and particulate reinforced polymeric materials for

to material anisotropy (i. e., directional dependence) which complicates both the theoretical modeling and experimental characterization of the liehavior of these materials. In addition, many practical short-fiber composite systems incorporate fillers (e.g., calcium carbo- nate particles). Consequently, combining rules are required that are capable of predicting the behavior of multicomponent systems, e.g., fiberifillerlresin.

As a first step toward developing combining rules for multicomponent systems, it is useful to restrict attention to the thermoelastic properties of two-component systems with inclusions of near spherical geometry embedded in a continuous matrix. This focus alleviates some of the theoretical and experimental problems associated with characterizing anisotropic materials. Al- though limited in scope, the resulting models will provide an essential component for the development of combining rules for short-fiber and platelet reinforced composites.

Several models have been proposed to predict the properties of particulate filled systems. These models are reviewed in the following section and compared with experimental data in subsequent sections. It is shown that the various models yield predictions which are essentially equivalent for volume fraction of filler less than -0.2-0.3. At higher volume fraction loadings

POLYMER COMPOSITES, OCTOBER, 1981, Vof. 2, No. 4 149

Stuart McGee and R . L. McCullough

of fillers, the models diverge to either overestim, ‘I t e or underestimate the properties of particulate filled systems.

This review points out the need for relationships capable of predicting thc behavior of multicomponent materials over a wide range of the concentrations of constituents whose individual properties can differ by orders of magnitude. The important concepts which have evolved through the various attempts at predicting the behavior of heterogeneous materials are consoli- dated in the subsequent section through a development of combining rules to predict thermoelastic behavior of materials ranging from particulate filled to foamed polymeric resins.

REVIEW OF MODELS FOR PARTICULATE FILLED SYSTEMS

A variety of approaches has heen proposed to develop relationships to predict the Young‘s modulus, shear modulus, Poisson’s ratio, bulk modulus, and the coefficient of thermal expansion of particulate filled materials in terms of the properties and concentration of the polymeric and filler components. These approaches range from empirical curve fitting techniques to bophis- ticated analytical treatments. This review will focus on the predictive models obtained from the latter approach with the intent of identifying the concentration range for which the model is applicable. Initial attention will be directed at models to predict moduli, since these quantities are required as input for models to predict the coefficient of thermal expansion.

In principle, the effective thermoclastic properties can be obtained by specifying the details of the microgeometry (particle shape, packing geometry, and spacings); the distribution of surface loads; and the connectivity between the particle and matrix phase. The effective bulk response of the material can be determined b y taking appropriate volume averages, thereby relating the volume fraction of the components and their respective properties to the average properties of the composite material. In practice, either simplifying assumptions must be introduced to make a general analysis tractable, or detailed numerical analyses must be performed for special cases. The various theoretical approaches may be distinguished by the nature of the assumptions introduced to obtain tractable solutions. The methods employed may be broadly grouped into the following categories: (i) Mechanics of Materials, (ii) Embedding (or “Self-Consistent Field”), and (iii) Bounding. .411 of these approaches share the following common assumptions:

(i) The phase surfaces are assumed to be in direct contact and bonded (either chemically or physically) so that slip does not occur at a phase interface.

(ii) Attention is directed toward the overall average response of the material to loads (or deformations) rather than localized variations in the material response characteristics.

These assumptions are appropriate for those properties associated with small deformation behavior and hence do not seriously limit application to the predic-

tion of thermoelastic properties. The additional simplifying assumptions which distinguish the various models play a more significant role in establishing the validity of the models.

The mechanics of materials is the simplest and, consequently, the most commonly encountered approach. This approach is based on the assumption that each phase component (or prescribed combination of phase components) is subject to either the same stress or the same strain. These gross simplifications in the internal distributions of stress (or strain) mitigates the influence of the shape, size, and packing of the phase components so that the only descriptors of the phase regions that are retained in this approach are (i) the elastic properties of the components and (ii) the volume fraction of the components.

The classical results of Voigt (2) and Reuss (3) belong to this category. Voigt assumed that each component was subject to the same strain. By analogy to parallel- connected springs, this assumption leads to the following combining rules on the elastic constants:

Voigt (constant strain)

[Cvl = ~,[CInl + Uf[Cfl (1)

where the [Cl’s are the elastic constant arrays for the composite, the matrix, and the filler respectively; vf is the volume fraction of filler and c , is the volume fraction of the matrix with 1 = cf + en,.

This combining rule reduces to the following form for the familiar engineering constants; viz, the bulk modulus, K , and the shear modulus, G:

(2) The corresponding expressions for Young‘s modulus, E , and Poisson’s ratio, v, are somewhat more complicated. These quantities may be determined from the resulting values of K I T and G v from the well known auxiliary relationships for isotropic materials; viz,

P v = 2jmPJn + v f P f (with P = K or G)

E = 9KG/(3K + G)

v = (3K - 2G)/2(3K + G) (3)

(4) Alternately, Reuss assumed that each phase compo-

nent was subject to the same stress. For this situation, an analogy to series-connected springs leads to the following combining rule on the compliance constants:

Reuss (constant stress):

[ S H l = ~n,[Sml + Cf[Sfl (5)

where the [Sl’s are the compliance constant arrays for the composite, the matrix, and the filler, respectively. The compliance constant array and elastic constant array are reciprocal in the matrix sense; viz, [S] [C] = [I], where [1] is the unit matrix.

This combining rule reduces to the following form for the bulk modulus, the shear modulus, and Young’s modulus:

(6)

The estimates of the effective composite moduli determined from the Voigt and Reuss models represent

( ~ / P H ) = (urn/Pnt) + ( v f / P f ) ( P = E , G, K )

150 POLYMER COMPOSITES, OCTOBER, 1981, Vol. 2, No. 4

Combining Rules for Predicting the ‘Thermoelustic Properties of Particulate Filled Polymers, Polyblends, und Foams

extremes in behavior. The Voigt model attributes more significance to the “stiffer” phase; t i e Reuss model emphasizes the compliant phase. Thus if the properties of the components differ by an order cf magnitude, the results obtained from these models ,will differ by an order of magnitude. Several models have been proposed to give results intermediate 1)etween the extremes of the Voigt and Reuss models. In these models certain prescribed combinations of phase components are assumed to be subject to constant stress while other combinations are subject to constant strain. Such spec- ifications introduce additional pararn,.ters which may be adjusted to fit observed data. As pointed out by Hill (4), none of the models based on uniform stress or uniform strain fields can be strictly coi’rect: the interfacial forces are not in equilibrium for constant strain; interfacial discontinuities must exist for constant stress. In consequence, these decorations {I f the primitive Voigt and Reuss models are of little :nerit.

In contrast to the mechanics of mai:erials approach, the embedding (or “self-consistent field”) methods introduce gross simplifications in the microstructure in order to deal in a more realistic manner with the internal distribution of stress and strain. Fnr example, Hill (5) represents the phase geometry of a particulate filled system as a single (typical) spherical papticle embedded in a medium whose properties are taken to be equivalent to the overall average properties of the composite material. An elasticity problem can be formulated for this simple geometry such that a “self-consistent” stress field can be identified and the effectve properties of the medium can be determined.

The difficulties associated with sp’xifying the detailed features of the microstructure as well as the explicit nature of the internal stress-strain field can be avoided by directing attention to the iipper and lower bounds on the response characteristics of heterogene- om materials. The results from thc1:e bounding approaches can serve as practical guide:. to material behavior only if the upper and lower bounds are sufficiently close so as to bracket the true behavior within the experimental error inherent to the experimental characterization of the proper1 ies of the material,

All of these approaches may be reconciled through a general relationship given by Wu and McCullough (6). These workers, following methods pro posed by Hashin and Shtrikman (7) and elaborated by TYalpole (8), employed an improved variational trea:ment to obtain generalized relationships for the effective properties of a wide variety of heterogeneous :naterials. This generalized relationship can serve as a convenient hasis for comparing and unifying the various models.

This treatment leads to constitutive relationships of the form:

where the various terms are 6 x 6 matrix arrays. The components of these arrays for orthotropic and trans- versely isotropic materials are given by Wu and McCul- lough (6). These arrays can be simplified for the special

application to systems which exhibit overall isotropic materials symmetry. In this case, the various 6 x 6 arrays, as represented by the general symbol [A], are of the form:

Ail = A22 = A33

A12 = A21 = Al:j = As, = AZ:{ = AS2

A,, = A 5 z = A66 = &Ai , - A12)

with b = 1/2 for elastic constants or 2 for compliance constants and all other Aij = 0. This reduction emphasizes the fact that only two independent material descriptors (e.g., A, and AI2) are required to specify the response characteristics of an isotropic material.

The term [C*] is the array of predicted elastic constants for the composite material. The term [C,,] represents the elastic constant array for an arbitrary reference material and is the origin of the versatility of the treatment. The term [E, , ] corresponds to a correlation tensor which serves as a measure of two-point correlations. This term is dependent upon the choice of reference material, [C,,], and incorporates information concerning the statistical symmetry. For isotropic materials, the components of [E,,] reduce to the following form

EYi = E& = E::3 = k + (4/3)u

Ey2 = Eii = EY3 = Eii = E& = E i Z = k - (2 /3 )u

EZ = EZ5 = E& = 4u (84

with all other EYi = 0. The terms k and u are related to the bulk modulus, K,,, and shear modulus, G,,, of the reference material through the following relationships.

The term [M,,] is a volume averaged property tensor compensated for correlation effects. This term is related to the volume averaged properties of the a’th component through the relationships:

(9a)

where oa is the volume fraction of the a’th component. The compensated properties for each component are given by

[ m a 1 = ([Ral-l - [ E J - ’ (914 where [R,] is the deviation of the volume averaged elastic constants for the a’th component from the reference material; viz,

(9,)

In the event that the a’th phase is isotropic (e.g., the resin phase or glass particles) the volume averaged elastic constants are given by the isotropic array of elastic constants; viz,

[&I = ([Gal) - [COl

([Gal) = [ G I

POLYMER COMPOSITES, OCTOBER, 1981. Vol. 2, No. 4 151

Stuart McGee und R. L. McCullough

If the a'th phase is anisotropic, appropriate averaging procedures (6) must be used to obtain ([C,]}.

The various elastic constant arrays, [C], are related to the familiar engineering constants (e.g., E , v or K , G) through the compliance array, [S]:

[C] = [S]-l

with

S, , = S22 = SRs = 1 / E

Si2 = S2, = S1:3 = S:3, = S2:j = SR2 = - v / E

s44 = s55 = s66 = 2(sll - SlZ)

= 2(1 + v ) / E = 1/G

with all other S j i = 0. The relationship expressed in E q 7 provides dual

service as (i) bounding relationships or (ii) a model relationship. The choice of these applications depends upon assertions concerning the arbitrary reference material, [C,,].

Bounding In the application as a bounding relationship, the

values assigned to [C,,] are constrained by the following conditions:

Upper bound: [C] - [C,,] 5 [0] (negative definite)

(1Oa)

(lob)

Lower bound: [C] - [C,,] 2 [0] (positive definite)

The simplest choice for [C,,] that assures the conditions ofEq 10a is [C,,] = [m]. Under this selection, E y 7 reduces to the classical Voigt result of E y 1 . Alter- nately, the simplest choice for [C,,] that guarantees the conditions of E q 10h for the lower lmund is [C,,] = [O]. For this choice, E q 7 reduces to the classical Reuss model of E q 5.

These simple results correspond to the bounds determined by Paul (9) and represent the extremes in behavior for the material in the sense that the properties, P*, can never exceed the upper bound (P,,) nor fall below the lower hound (Pf{): P,{ < P* < PI. with P = E , G, K , or v.

The fact that these two simple models represent the limits of behavior can be appreciated further by recall- ing that the Reuss model was developed under the assumption of a uniform stress throughout the material. However, if the particles and resin have different moduli, the corresponding strain field cannot be uniform and discontinuities may occur. These discontinuities would be manifest as hypothetical material overlaps (or voids) in the vicinities of the phase liound- aries. In order to eliminate these hypothetical overlaps, additional energy would he required to bend and dis- tort the overlapping regions to suitable positions. Con- sequently, the Reuss model depicts a strain energy lower than the real situation and thereby serves as a lower 1)ound. Similarly, the Voigt model was dc- veloped under the assumption of a uniform strain.

Again because of the variation in the moduli between particles and resin, a constant strain would result in a non-equilibrium stress field which would tend to drive the particles into motion. Consequently, the assertion of constant strain corresponds to an energy state higher than could occur for static equilibrium s o that the Voigt model gives an upper limit to the behavior.

Hashin and Shtrikman (lo), as well as Hill (11) noted that more realistic choices for [C,,], which assure the conditions of E q 10, may be made by assigning to [C,,] the properties of the average (isotropic) particle and isotropic matrix, respectively. These improved choices for [C,,] reflect the fact that the fluctuations in the internal elastic field, [C], can never exceed the values of the most rigid component nor fall below the values of the most compliant component. Consequently, the following improved bounds are obtained:

ECUI = [Cfl + ([Mrl-' + [Erl)-'

[C/I = [ C m I + (LMrn1-l + [Ern])-'

(114

( I lb)

The form of these relationships suggests that the lower bound corresponds to particles embedded in a continuous matrix phase, while the upper bound relationship may be viewed as representing a situation in which the matrix material is embedded in a continuous phase that possesses the properties of the filler. It is reasonable to expect that the behavior of particulate filled polymers will tend toward the lower bound representation.

These relationships, as well as E q s 2 and 6 , can be cast into the compact form of the Halpin-Tsai equation (1); viz,

with

for P = K or G. The Young's modulus, E , and Poisson's ratio, v, can lie computed from the auxiliary relationships given in E q s 3 and 4 . The factor (,, for the Reuss lower bound, the improved lower bound, the improved upper bound, and the Voigt upper bound are sum- marized in Table 1 . Although in a different form, these results correspond to the improved bounds reported by Hill (ll), Hashin and Shtrikman (lo), and Walpole (8). These bounds are the most restrictive that can be made without information (or assertions) concerning the dis- triliutions of size, shape, and packing of the inclusions. While these improved bounds offer a significant im-

Table 1. Values for the 5-Factor

Reuss Lower Upper Voigt Bound Bound Bound Bound

2(1 - 2~,) 2(1 - 2ur)KI

7 - 5v, (7 5vr)Gf

K 0 X

G 0

(1 + .In) (1 + Vi)Km

8 - ~ O V , (8 - 10vf)G,


Combining Rules f o r Predicting the Thermoelastic Properties of Particulate Filled Polymers, Polyblends, and Foams

provement over the primitive Voigt and Reuss bounds, they remain too far apart to brackt t the properties to within the margin of experimental error. This separa- tion in the bounds implies that the properties of filled systems are sensitive to detailed features of the microstructure that are not included in the information used to develop the bounds. Furthix tightening of the bounds would be expected upon the introduction of additional information concerning the microstructure.

Additional information concerni ig the microstructure can be introduced through the use of “n-point” correlation functions. Recently, Nornura (12) has shown that the incorporation of %point correlations (in addition to the 2-point correlations reflected by [E,]) yields some further tightening of the bounds. Unfortunately, these further improvements do not tighten the bounds sufficiently to bracket the behavio- to within experimental error. The incorporation 0’ yet higher order correlations could lead to sufficiently tight bounds. However, the development of the information neces- sary for the determination of such r.-point correlations will require structural data (or structural models) of progressively greater detail. In orller to obtain such information, detailed microstructur 11 characterizations will be required to identify and evaluate the appropriate structural parameters that can 3e used to predict thermomechanical behavior. In efj‘ect, this approach trades direct thermomechanical cl- aracterization of a composite material for more com.?licated structural characterization experiments. Consquently, a theory based on the use of higher order ccrrelation functions will have limited value in practical ;tpplications. These considerations suggest that the search for further improvements in the bounds should be abandoned in favor of a modeling approach that j ields relationships based on limited and readily accessil)le structural data.

Embedding Mode’s In the application ofEq 7 as a model relationship, the

reference elasticity may be viewed ;is a parameter that incorporates the influence of the various features of the microstructure. Indeed, the use of [C,,] in the bounding approach may bc interpreted from this point of view. The assignment to [C,,] the values of [O] and [a] may be viewed as a total lack of information concerning the nature of the composite specimen. Viith no information available, preliminary predictions must be limited to the statement that the properties ;ire bounded from below by [0] and from above by [..I. The use of these preliminary predictions as values for [C,,] give the Voigt and Reuss bounds. Improved bounl3s are obtained by introducing information concerning the properties of the components. With a knowledge of the properties of the components, the preliminary ?rediction can be made that the composite specime:i will be at least bounded from below by the properties of the compliant component and from above by the rigid component. The subsequent use of these preliminary predictions for [C,,] lead to the improved bounds of E q 11.

This interpretation of the role of [C,,] suggests that a further tightening of the brackets on behavior could be

obtained by assigning the value of [C,] and [C,] as preliminary estimates for [C,,] from which tighter upper and lower brackets could be obtained. It should be emphasized that such assignments will not generally satisfy E9 10. Nonetheless, such trials could be proposed as an iterative scheme where the [C,] of the i + 1 iteration is assigned the value of the [C*] computed from the i’th iteration. Continued iterations could force a convergence of the bounds so that the final assignment for [C,,] becomes the unknown effective elastic constant array [C*]. Such an assignment corresponds to the “self-consistent field” model originally proposed by Hershey (13) and Kroner (14). The assertion that [C*] = [C,] leads to the following relationships

+ [E,,l)-l = [OI (13)

where the various elements of [C,,] appear in both [M,,] and [E,,]. This relationship represents a series of coupled equations for the elements of [C,,]. Solution of these relationships gives the elements of [C,,] and thence the elements of [C*]. These solutions for the special case of an isotropic material correspond to the self-consistent field result for spherical inclusions reported by Hill (15) and Budiansky (16). Smith (17) has compared the self-consistent field results for spherical inclusions to experimental values and has shown that the self-consistent field approach can seriously overestimate the effective properties in the practical ranges of volume fraction concentrations.

Self-consistent field models should be restricted to systems in which the phase properties are essentially equivalent. Such a restriction could he satisfied by a one-component “grainey” metal. This restriction is emphasized by Budiansky (16): the treatment of voids, [C,] = [0] and absolutely rigid inclusiods, [C,] = [w] gives absurd results. The self-consistent field model has been criticized by Hashin (18) and Christensen (19). Chris- tensen argues that the self-consistent field approach should not be viewed as a geometric model but rather as an iterative scheme as described in the previous para- graphs. The iterative replacement of [C,] by successive approximations will lead to unique relationships by ar- tificially requiring coalesce of the bounds. There is no a priori reason to believe that the bounds should converge for information limited to the properties and concentration of the components. The self-consistent field scheme remains popular in spite of these deficiencies.

Further progress in developing theoretical relationships to predict the properties of particulate filled materials can be achieved b y the introduction of specific geometrical models to take into account the influence of the microstructure. The composite sphere model proposed by Hashin (20) provides a rigorous basis for this approach.

The basic element of the composite sphere model is a spherical inclusion of radius a embedded in a sphere of the matrix material of radius h. The ratio df a3 to h3 is taken to be independent of the absolute size of the element and equal to the volume fraction of particles. Hashin (20) further proposed that a particulate filled

POLYMER COMPOSITES, OCTOBER, 1981, V d . 2 , No. 4 153

Stuurt McGee and R. L. McCullough

system could be modeled as a collection of such elements with a particular gradient in the size of the spheres so that the collection of spherical elements could be space filling. Consequently, this model presumes a broad distribution of particle size with a range extending to infinitesimal particles.

Bounds on the effective moduli for this model struc- ture were obtained by variational techniques. The upper and lower bounds on the bulk modulus, K , coin- cide and are given by the relations obtained from the improved lower bound; viz, E q Llb. Since the improved lower bound relationship was developed without recourse to specific geometrical models, this corre- spondence suggests that the bulk modulus is insensitive to the shape of the inclusion.

Unfortunately, the bounds on the shear modulus (and hence on the Young’s modulus and Poisson’s ratio) remain distinct. The form obtained for the upper and lower bounds on the shear modulus is somewhat involved with the lower bound of the composite sphere model falling below the improved lower bound of E q I lb .

The three-phase (or “doubly” embedded) model introduced by Kerner (21) and van der Pol (22) provides the basis for an alternate version of the self-consistent field treatment. In this model, the phase geometry is represented by a single (typical) spherical inclusion of radius a embedded in a sphere of the matrix material with a radius b. The ratio a3/ h3 is taken to be equal to the volume fraction of filler. The composite sphere is in turn considered to be embedded in a medium that is assigned the unknown effective properties of the filled material. As with the “singly” embedded model, an elasticity problem can be formulated for this simplified geometry such that a self-consistent stress field can be identified and the effective properties of the medium determined.

The results reported by Kerner (21) for this model correspond to the relationships given for the improved lower bounds, E q 11 b. As a consequence of this corre- spondence, it could be argued that the specific geometrical features of the three phase model are unimportant since the lower bound relationships were developed without recourse to assumptions concerning a specific phase geometry. Unfortunately, the treatments by Kerner (21) and van der Pol (22) are in error. Smith (23) and Christensen and Lo (24) have pointed out the specific error van der Pol made in his work. Christensen (19) comments that the explicit error in the Kerner work cannot “be pinpointed because of the brevity of the derivation.”

The correct treatment of the three phase model, as reported by Smith (23) and Christensen and 1x1 (241, results in involved expressions that yield values for the shear modulus which are slightly higher than those obtained from the lower bound, E 4 Llb. These analyses show that the assumption of a specific geometrical model enters into the prediction of the properties and add support to Hashin’s (18) argument that structural information, in addition to the volume fraction of

spherical inclusions, is required for the prediction of properties.

Finally, mention should be made of the “statistical” models reported in the Russian literature as sum- marized by Sendeckyj (25). Sendeckyj shows that the reported polynomial expressions can be reduced to more compact forms. At volume fractions of particles greater than -0.4, the results from the statistical models are essentially equivalent to the singly embedded models of Hill (15) and Budiansky (16); at volume fractions below -0.25, these models give results which fall below the lower limits established hy the Reuss lower bound and in some cases negative values for the moduli.

The dependence of the Young’s modulus, E , on the volume fraction of particles as computed ffom the various models is shown in Fig. 1 . These numerical results were obtained for a particulate filled system with properties typical of glass spheres in a thermosetting resin; viz, E f = 10.5 X 106psi, v f = 0.25; E , = 0.ij x 106psi, v,, = 0.35.

In Fig. 2 , the predictions from selected models are compared with the data reported by Ishai and Cohen (26) for a natural silica filler (Ef = 10.6 x lo6 psi, vf = 0.25) with a narrow distribution of particle size (mean diameter 0.4 mm) embedded in an epoxy resin (E,n = 0.3 x lo6, u,, = 0.4). This comparison shows that the singly embedded (self-consistent field) models of Hill (15) and Budiansky (16) tend to seriously overestimate the Young‘s modulus at volume fractibns of filler greater than -0.3. Reference to the ordering shown in F i g . 1, shows that the improved upper bounds of E 4 l l a and the primitive Voigt bound of E q 1 give progressively larger overestimates of the Young‘s modulus.

o l I 0 00 0.20 0.40 0.60 0.80 1.00

VOLUME FRACTION FILLER

Fig. 1. Compurison of the predictions f o r Young’s modulus, E , us u function of volume fruction filler, c,, f rom theoreticul models: (V) Voigt upper bound, E q 1; (ub) improced upper bound, E q l l a ; ( H B ) singly embedded model, Hill (15) und Budiansky (16); (CS ,,) composite spheres model upper bound, Hushin (20); (SCL) three-phase model of Smith (23) and Christ- ensen und Lo (24); (Ib) improz;ed lower bound, E9 I lb; (CSJ composite spheres niodel lower bound, Hushin (20); ( R ) Reuss lower bound, E q 6. The numericul results ure based on the following propertie.s f o r thefiller und mutrix respecticely: E l = 10.5 x IO‘psi, ul = 0.25; E,,, = 0.5 > 10fipsi , v,,, = 0.35.


Combining Rules for Predicting the Thermoelastic Properties of Particulate Filled Polymers, Polyblends, and Foams

0 0 0.00 0 10 0 20 0 30 0.41) 0.50 0.60


Fig. 2. Comparison of the data of lshai and (:ohen (26) with the value of Young's modulus, E , predicted by .:elected theoretical models: (HB) singly embedded model, Hill I 15) and Budiansky (16); [CS,,) composite spheres model upper bound, Hashin (20); (16) improoed lower bound, Eq 11 b. The pr,?dictions are based on the properties reported by Ishai and C o h w (26) f o r a natural silicafiller in un epoxy resin: E , = 10.6 x 10' psi, v, = 0.25; Em = 0.3 x loti psi, vn, = 0.4.

Further reference to Fig. 1 shows -hat the doubly- embedded model of Smith (23) and Christensen and Lo (24), the improved lower bound results, Eq lib, the lower bound of the composite sphere model (20), and the primitive Reuss lower bound, progressively underestimate the Young's modulus for vol ime fractions of filler in excess of -0.4. As illustrated . n Fig. 1 , all the models tend to coalesce to within experimental scatter for volume fractions of filler less than -0.3.

Data points in Fig. 2 which fall below the limits established by the improved lower bound can be rationalized in terms of fabrication pi.ocedures (e.g., void formation) that may degrade thcb properties. In contrast, the fact that a significant number of data points fall above the limits of the upper bound of the composite sphere model points to the iinporta ice of the assertion of a wide distribution of particle size inherent to this model. Evidently, the composite, sphere model should be restricted to systems known to possess a wide range of particle size.

Comparisons to the data reported by !imith (17), Ken- yon and Duffey (27), and Richard (28) f >r glass spheres in thermosetting resins yield similar trends. These comparisons show the deficiencies of the existing theoretical models at high volume fractions of particles.

Semi-Empirical Models The complexities and inadequacies of the theoretical

models has prompted recourse to semi-empirical relationships. An evaluation of the host of semi-empirical relationships which rely upon the deter nination of adjustable parameters by curve fitting tecmhniques is be- yond the intent of this work. Rather, attention w i l l be directed to those forms for which the parameters can be fixed by other considerations so that the resulting relationships can serve in a predictive capa2ity.

The Halpin-Tsai form, Ey 12 , has eme .ged as a popular semi-empirical model. In this applic,ition of E q 12,

POLYMER COMPOSITES, OCTOBER, 1981, Vol. 2 , No. 4

the symbol P denotes an appropriate property (P = K , G, as well as E ) ; r;, is the volume fraction of the rigid component (i.e., Pf > Pm); and 4 can be treated as an adjustable parameter. The central feature of the Halpin-Tsai relationship is the recognition that the properties of a composite material must lie between the extremes of the primitive Voigt upper bound and the Reuss lower bound. As pointed out in Table 1 , the parameter [ serves to adjust the effective properties between the limits of the Reuss (4 = 0) and Voigt (5 = w) bounds.

In the preliminary application to continuous fiber composites (I), values of [ were obtained by fitting the results of detailed numerical analyses for rectangular fibers (width a and thickness b) packed in a rectangular array. These comparisons suggested that the [-factor associated with the transverse Young's modulus and shear modulus were dependent upon the cross- sectional shape of the fiber while the 4-factor for the longitudinal properties could be assigned the value tp,, = 00. Specifically, these relationships were of the form tET = 2(a/ b) and tc = (a/ b)-. For circular fibers a = b so that tET = 2, and tG = 1. This simple geometrical interpretation of the [-factor has prompted the use of the Halpin-Tsai Equation for aligned short-fiber systems (29) with eb;T = 2(1/d) (where 1 is the length of the fiber and d the diameter), and Ec = 1. The extension (29) to particulate filled systems (with l = d) corresponds to tfi; = 2 and [(; = 1. As shown in Fig. 3 , the Halpin-Tsai relationship (with eF; = 2) tends to underestimate the Young's modulus for volume fractions of filler in excess of -0.4.

In an analogy to the Mooney equation for the viscosity of suspensions, Lewis and Nielson (30) proposed a modification of the Kerner equation to account for the

001 I 0 00 010 0 20 0 30 040 0 50 06

VOLUME FRACTION FILLER ,

Fig . 3. Compurison of the dutu of lshai and Cohen (26) with the oalue of Young's niodulus, E , predicted by selected models: (HB) singly embedded model, Hill (15) and Budiunsky (16); (HT,,,) the Halpin-Tsai Equation, E q 12 with 5 = 2, with the Lewis-Nielsen modification; (LN) the Lewis-Nielsen Equation, E q 14; (S), the S-Combining Rule, Eq 21; (HT) the Halpin-Tsai Equation, Eq 12 with € = 2; (SCL) the three-phase model of Smith(23) and Christensen and Lo (24); (lb) the iniprotied lower bound, E q l l b . The predictions are based on the properties reported by lshai und Cohen (26) f o r (I naturul silicafiller in un epoxy resin: E , = 10.6 x IO'psi, v,, = 0.25; E,,, = 0.3 X 10'psi; v,,, = 0.4.

155

Stuart McGee and R . L. McCullough

limits imposed by the maximum packing that can be achieved for uniform spherical particles. Since the Kerner equation corresponds to the improved lower bounds, the Lewis-Nielson relationship can be expressed as a modification of Ey 12:

with the factor tp, given in Table 1 under the improved lower bound column. The term 6 is introduced to com- pensate for the fact that close packed spheres cannot exceed a certain maximum packing fraction, ufm’”. For example, hexagonal and face-centered close packing are both limited to a maximum packing fraction of 0.74; body-centered packing is limited to 0.60; while simple cubic packing is limited at vfmar = 0.52. In a real particulate system, it may be assumed that these various packing forms occur at random and with equal probability so that the maximum volume fraction for such “random” close packing is given by the average of these four forms; viz, (ofmas) = 213.

To specify 6, Lewis and Nielson required: (I) i;uf = 0 at Gf = 0, (ii) d(Gt~~)ldzj~ = I at uf = 0, and (iii) Guf = 1 at of = ufmus. They proposed the following form (among others) that would satisfy these conditions:

(1 - ufm“”) I ( u y ) z 1.1 1;= 1 +

As illustrated in Fig. 3 , the Lewis-Nielsen relationship (with fips = 213) correlates with the data over the range of available data points. However, the sensitivity of this relationship in the vicinity of of = 0.5 to small changes in the concentration of filler is disturbing. For example, this relationship predicts that a 10 percent increase in the filler (to of = 0.55) could provide a 50 percent increase in the Young’s modulus. This sensitivity appears to be an artifact due to overcompensation for the influence of packing geometry.

IMPROVED COMBINING RULES FOR MULTI-PHASE (ISOTROPIC) MATERIALS

The purpose of this section is to consolidate the concepts reviewed in the previous section and obtain improved relationships to predict thermomechanical behavior. Considerations will be directed to the class of composite materials comprised of inclusions (of near spherical geometry) embedded in a continuous matrix phase. This class of materials may be further distinguished as: (i) those systems with the inclusion more rigid than the matrix and (ii) those systems with the inclusions more compliant than the matrix. The first category corresponds, for example, to particulate filled polymers with Pr > P,,,; the second category corresponds to rubber-toughened polymers and foams in which the properties of the inclusion are less than those of the matrix, P i <

The “S-Combining Rule” for Rigid Particles in a Compliant Matrix

A classic approach to developing constitutive relationships is based on series expansions in concentrations (31). The application of such asymptotic relationships is usually restricted to the extremes of concentration (e.g., uf - 0). However, as noted by Christensen (19), the range of applicability can be extended by ajudicious choice for the basis of the expansion. It was shown in the previous section that the properties of a continuous matrix filled with rigid particles tend to fall closer to the values predicted by the Reuss model, Ey 5, than to the Voigt model. In the Reuss model, the effective compliance is linear in volume fraction. This observation suggests that a series expansion in terms of volume fractions that is based on the compliance constants would require the retention of fewer terms than an expansion based on t h e elastic constants. Con- sequently, the compliance array serves as a convenient basis for developing an improved “S-Combining Rule” for particulate filled systems.

I t was further demonstrated that the improved bounds ofEy 11 provide better estimates of the limiting behavior than do the primitive Reuss and Voigt bounds. As an additional point, it was noted that the difference between the improved upper and lower bounds serves as a measure of the influence of the microgeometry (coupled with the properties of the components). In view of these notions, it is reasonable to propose that the effective properties may be modeled as a series expansion in terms of the difference between the upper and lower bounds. For convenience, only the linear term will be retained: viz,

s* = s, - $ ( S , - S,) + . . . (15)

where S is any one of the diagonal elements of the compliance array (S = S,ii, j = 1, 2 , . . . 6). These elements may be obtained from the relationships given inEy 1 1 through matrix inversion: [S ,] = [C,]-’and [S,] = [CJ’. The proportionality factor, $, is taken to be a function of the volume fraction of filler, uf. Since the behavior of the actual material must fall between the extremes of the improved upper and lower bounds, the term $(of) must satisfy the condition 0 5 $(uf) 5 1. As illustrated in Figs. 1 and 2, as approaches 0, the actual properties tend toward the improved lower bound. Alternately, as qapproaches 1, so that the filler assumes the hypothetical role of the continuous phase, the behavior should tend to the upper bound. This condition can lie assured by requiring that +(O) = 0 and $(I) = 1. Since the compliance is used as a h i s , it is reasonable to assume that +(cf) can be adequately approximated by a truncated series expansion in of; viz,

,

-

$(Of) = ff + puf + yv; + . . . (16)

with

$(O) = ff = 0

and

$(l) = a! + p + y = 1

156 POLYMER COMPOSITES, OCTOBER, 1981, Vol. 2 , No. 4

Combining Rules for Predicting the l‘hermoelastic Properties of Particulate Filled Polymers, Polyblends, and Foams

Under this approximation for @(vf), EG 15 takes on the form of an “S-Mixing Rule”;

s* = oms/ + UfS, + yUfvm(S/ - S,) (17) In order for Su 5 S* I SI for all cor.centrations, the parameter y must fall within the 1imi:s -1 < y < 1.

The surviving parameter, 7 , may tle fixed by con- sidering the limiting aspects of phase continuity. As previously noted, the maximum volume fraction that can be achieved for equivalent close packed spheres corresponds to hexagonal (or face centered) close packing with oy” = 0.74. Any value for i:f > 0.74 would correspond to a continuous filler p ‘lase containing polymeric inclusions. Alternately, for a hypothetical system of equivalent spherical particl YS of the matrix material embedded in a continuous “f Iler” phase, the maximum packing fraction would correspond to vzar = 0.74. Any volume fraction of matrix i i excess of 0.74 would correspond to a system in which the polymeric phase can be treated unequivocally as the continuous phase. In terms of the notion of “cor tiguity,” as described by Hashin (20), the situation for v, > v ~ ~ ” corresponds to ‘‘zero” inclusion contiguity in the sense that the average contact area of neighboiing inclusions is zero; the situation for vf > v y ” corresponds to “zero” matrix contiguity. Intermediate states of contiguity are obtained between these limits of volume fraction; viz, 1 - v y < Vf < u p a x .

These considerations imply that at some critical volume fraction, &, a transition in phase contiguity must occur so that in the neighborhood of zf = 4*., the behavior tends toward the limits of thc upper bound, while in the vicinity of v , = &, the beliavior will tend toward the limits of the lower bound. It is reasonable to require that for volume fractions of matrix material v, = 4c (or of = 1 - +,.) that S* is within E of S /. Similarly, for v f = &, S* will be within an equivalent range of S,; viz,

(18) s* = S,(Vf = 1 - &) - E

s* = S,(Vf = +*.) + E

These conditions lead to the following expression for the parameter, y :

If the properties of the components are reasonably close, the improved upper and lower hounds tend to coalesce so that the correction term of .Fq 15 becomes superfluous. Clearly, the major contributions from the correction terms will occur for the case in which the properties of the components differ by at least an order of magnitude; viz, Pf >> P,. For this important case, E q 19 simplifies to

24c - 1 4f. Y =

If the critical volume fraction, +*., i:i taken as the maximum packing fraction for “random ’ close packing ($J~ = 2/3), the “S-Combining Rule” is jpecified as

1 2 s* = v,s/ + UfS, + - V,U,(S/ - S,,) (21)

The results ofEq 17 can be alternately arranged into a form similar to the Lewis-Nielson modification of the Halpin-Tsai relationship:

and

(0 5 of 5 1) (2%)

with the quantities tpu and f p I given in Table 1 . Again, for the case of interest with P f >> P , and vftpu >> 1, the correction factor $ reduces to

$ r= 1 + (Vrn/+c)(vj+c + v m ( 1 - 4c)) (of > 0)

For c$c = 213, the correction term becomes

as compared to the Lewis-Nielsen correction of

6 = 1 + (3/4)0f

Although the two correction factors give essentially equivalent values for the compensated volume fractions (Cur or $of) for 0 < of 5 %, the Lewis Nielsen factor causes iTcf to increase more rapidly than $ofin the region vf > ?h. This behavior is a consequence of the rapid adjustment ofuto satisfy the requirement that Gof = 1 at vf = I#+.. Thus requirement was imposed to normalize the volume fraction of filler with respect to the maximum packing fraction. It is not evident that such a simple scaling reflects the role of the microstructure; indeed, it appears to overemphasize the contribution of the filler at higher volume fractions.

The predictions of the “S-Combining Rule,” E q 21, are compared to experimental data for Young’s modulus in Figs. 3-6. The results from the Hill-Budiansky model , t h e Halpin-Tsai equat ion, t h e Smith- Christensen-Lo model, and the improved lower bounds are included for reference. With the exception of the Halpin-Tsai equation, values for the Young‘s modulus were generated from the predicted values of K* and G* through the auxilliary relationship of Eq 3 ; viz, E* = 9K*G*/3K* + G*. The Halpin-Tsai value for E* was computed fromEq 12 with P = E and tE = 2 (29).

Also exhibited in Fig. 3 are the results from the Lewis-Nielsen equat ion and t h e Lewis-Nielsen modification applied to the Halpin-Tsai equation (32), e.g., E q 14 with P = E and tE = 2. As would be expected from the preceding discussion, the Lewis-Nielsen equation and the “S-Combining Rule” are comparable for vf I ?h and diverge for of > %. It is clear fromFig. 3 , that a Lewis-Nielsen modification of the Halpin-Tsai relation leads to a significant overestimate of the Young’s modulus for v , > -0.4.

POLYMER COMPOSITES, OCTOBER, 1981, Vol. 2 . No. 4 157

Stuurt McGee und R. L. McCullough

/ 15, I

0 0 000 010 0 20 0.30 040 0 50


Fig. 4 . Comparison of the data of Smith (1 7 ) with the value of Young’s modulus, E, predicted by selected models: (HB) singly embedded model, H i l l (15 ) and Budiansky (16); (S) the S-Combining Rule, E q 21; (HT) the Halpin-Tsai Equation, Eq 12 with 6 = 2; (SCL) the three-phase model of Smith (23) and Christensen and Lo (24); (lb) the improved lower bounds, E q 1 1 b . The predictions are based on the properties reported by Smith (1 7) f o r glass spheres in an epoxy resin: E , = 1 1 x 1 O‘psi, v, = 0.23; E m = 0.389 x lo‘psi, v,,, = 0.39.

/ /

0 0 0 0 0 0 I0 0 20 0 30 040 0.50


Fig . 5 . Comparison of the data of Kenyon and Duffey (27) with the value of Young’s modulus, E , predicted b y selected m~idels: (HB) singly embedded model, Hill (15) and Budiunsky (16); ( S ) the S-Combining Rule, E q 21; (HT) the Halpin-Tsai Equation, E q 12 with 5 =2; (SCL) the three-phase model of Smith(2S)and Christensen and Lo (24); (lb) the impror;ed lower bounds, E q l l b . The predictions are based on the properties reported b y Kenyon and Duffey (27) f o r glass spheres in un epoxy resin: E , = 10.2 X IO‘psi, v, = 0.25; E,,t = 0.48 x IO’psi, Y,,, = 0.35.

With the exception ofFig. 6, the “S-Combining Rule” is in good agreement with the available data. The comparisons shown in Fig . 6 are based on the properties of the pure matrix are reported by Richard (28); viz, En, = 0.245 x lo6 psi (with a range of 0.129 to 0.262 x lo6 psi) and v, = 0.444 (with a range of 0.422 to 0.459). Extrapo- lation of the particulate data in Fig. 6 to vf = 0 gives an intercept at E , = 0.3 x lo6 psi. The predictions of the “S-Combining Rule” based on this value for E m (and v, = 0.4) are compared with Richard’s data in Fig. 7. The improved agreement afforded by this adjustment in the matrix properties supports the view that the in s i tu properties of the resin phase may differ from the values measured for the pure resin (32). Similarly, the values

0 0 0.00 0 10 0 20 0 30 0 4 0 0 50


Fig. 6 . Compurison of the data of Richard (28) with the ljalue of Young’s modulus, E , predicted by selected models: (HB) the singly embedded model, Hill (15) and Budiansky (16); ( S ) the S-Combining Rule, E q 21; (lb) the improced lower bound, E q l l b . The predictions are based on the properties reported by Richard (28) f o r glass spheres in a polyester resin; E , = 10.2 x lo‘psi, v, = 0.21; E,, , = 0.245 X IO‘psi, v,,, = 0.44.

0 0 0.00 0 10 0.20 0.30 0.40 0.50


Fig. 7. Compurison of the data of Richard (28) with the Galue i~f

Young’s modulus, E , predicted by selected models based on udjusted mcitrix properties: (HB) the singly embedded model, Hill (15) and Budiansky (16); (S) the S-Combining Rule, E q 21; ( lb) the improved lower bounds, E9 l l b . The predictions ore based on the properties reported b y Richard (28) for glass spheres with udjusted propertiesforthe matrix; E , = 10.2 x lo8 psi, v, = 0.21; E,,, = 0.3 x IO‘psi, v,,, = 0.4.

predicted by the “S-Combining Rule” for Poisson’s ratio, v*, are consistent with the data range reported by Smith (17) for glass spheres in an epoxy matrix as well as the data range reported by Richard (28) for glass spheres in a polyester matrix.

Rosen and Hashin (33), Schapery (34), and Levin (35) have shown for the special case of two-component composites that the bounds on t h e thermal stress coefficient, 1 = [C*b, converge. As a consequence, the thermal expansion coefficient, a , can be uniquely specified in terms of the effective elastic constants of the composite. Thus for near spherical inclusions,

158 POLYMER COMPOSITES, OCTOBER, 1981, Vol. 2 , No. 4

Combining Rules f o r Predicting tha: Thermoelustic Properties

and

where ai is the appropriate coefficienj of thermal expansion, KJ is the appropriate bulk modulus of the components, and K* is the effective bull: modulus of the composite material.

The application of the “S-Combining Rule” for the prediction of K* (and thence a*) is ccimpared with the experimental data reported by Sato, e t al. (36) inFig. 8. Again, good agreement is obtained 3etween the predicted and observed values.

The “C-Combining Rule” for Compliant Inclusions in a Rigid Matrix

Important examples of materials in which the continuous phase is more rigid than the ir. cluded phase are provided by certain polyblends and 3olymeric foams. The behavior of these materials should tend toward the limits of the upper bound since uppt:r bound models mimic the situation of compliant inclusions in a rigid continuum. Under the circumstances, series expansions based on the elastic constants, C , s h d d require the retention of fewer terms than an expansion based on the compliance constants. Thus for polybl ends and foams, the elastic constant array serves as an iippropriate basis for an improved “C-Combining Rule.”

The expansion analogous to E q 15 is given by: c* = C , - n(c, - C,) + . . . (24)

where C is anyone of the diagonal elements of the elastic constant array.

Following the procedures used to obtain the “S- Combining Rule” leads to the following relationships:

P* = v,,,P, + viPl + hpviv,(P, - P 1 ) (25a) or equivalently

P* = ~ , ( l + A,u~) P , + ~ i ( l - Apvm) PI

VOLUME FRACTION RESIN

Fig. 8 . Compurison of the ciutu of Suto, et u1. (36) with the oulue of the coefficient of thermul expunsion, a, lireclicted by the S-Combining rule, Ey 21 i n conjunction wi thEq 23. The predictions ure bused on the properties reported by S ato, et ul. (36) for “C”-glass spheres i n un epoxy resin: E , = 10.5 :: IO‘psi, u, = 0.3, a, =8.5 x 10~-fi,ii,iilnim”C;E,,, =0.5 X 1Ofipsi, u,, =0.3, a,,, = 5 6 x 1 0-6 mnilmm%.

of Particulate Filled Polymers, Polyblends, und Foums

where P = K or G and vi(= 1 - v,) is the volume fraction of the compliant inclusions.

Phase contiguity arguments yield the following specification for the parameter A,,

As before, the quantity 4,. is the critical volume fraction; e.g., 4,. = % for “random”c1ose packing. The quantities qpj are given by the following relationships

(26a) 3KJ = 1 + v, ”IKJ = 3K, + 4G,

where vj is the Poisson’s ratio of the matrix phase (j = m) or the inclusion (j = i).

For the present treatment, it is useful to arrange the upper and lower bound results ofEq 11 in an alternate form.

(27a)

q ( P , - Pi) P , = P m { l - [

with P = K or G. In the event the properties of the matrix and inclusion

are on the same order of magnitude (e.g., certain polyblends), P , = P , , so that the “C-Combining Rule” reduces to the simple rule of mixtures. For the more interesting situation in which P J P , < 1, the “C- Combining Rule” takes on the form:

P* = &P,(1 - ~ ) , @ p , ) + P,(l - t;,@p,) (28)

with the correction factors, @, given by

P P ? a-. =-

and

= V p J / ( ’ - VPJ)

where againj = rn denotes properties P for the matrix and j = i denotes properties for the inclusion.

These correction factors may be reasonably approximated as

The associated values for the Young‘s modulus and

POLYMER COMPOSITES, OCTOBER, 1981. Vol. 2 , No. 4 159

Stuart McGee and R. L. McCullough

Poisson’s ratio may be computed from auxiliary Eqs 3 and 4 .

For systems, such as foams, for which P , >> Pi, the Young‘s modulus may be approximated by the relationship

with (30)

E* r VgEm(l - vi@Ern)

The largest value that QEnI can assume is -0.5 (at #+ = %, Y, = Yz) so that for ui < %, the correction contributes no more than 25 percent. Accordingly, the Young’s modulus of a foam can be estimated by the simple relationship:

E*IE, = & (31)

Progelhof and Throne (37) have shown that the Young‘s modulus for a variety of thermoplastic foams can be correlated by a density squared relationship; viz, E*IE, = (d*Id,)*, where d,, is the density of the thermoplastic matrix, and d* is the density of the foam. The average density is given by d* = vmdnr + vidi where di is the density of the inclusion. For foams, d i << d , so that d*ld, is simply the volume fraction of matrix, urn. Hence E q 31 corresponds to an established empirical correlation for thermoplastic foams over a density range corresponding to -0.1 5 u,,, 5 1. In addition, E q 31 gives excellent agreement with the data reported by Ishai and Cohen (26) for porous epoxy specimens which cover the range from 5 to 70 percent (by volume) void content.

Combining Rules for Polyblends The term “polyblend” is used to denote incompatible

combinations of materials in which the identity of each component is retained on a sub-macroscopic scale. Rubber toughened thermosetting resins are an example of such polyhlends.

I t was pointed out in the previous section that if the properties of the a and /3 components of a polyblend are essentially equivalent (Pa P,J, then the Combining Rules reducc to the simple “rule of mixtures”; viz,

P* = c,P, + uBPp (with P = K or G) (32)

In the event that the Poisson’s ratios of the components are equal under the criteria that (v, - vo)’ = 0 , E y 32 , can he applied for P = E , the Young’s modulus.

Any further refinement of Ey 32 requires the identification of the apparent continuous phase through qualitative techniques such as microscopy. If the phase LY is continuousand P, < Po, the “S” combining rule, E y 21 can be used to refine the predictions of Ey 32. Alternately, if phase a is continuous and P, > Po, the “C” combining rule can be used as a refinement of Ey 32. Upon identification of the appropriate combining rule for the prediction of the effective bulk modulus, K * , E g 23 can be used to predict the coefficient of thermal expansion.

SUMMARY AND CONCLUSIONS The comparison given in the previous section shows

that the “S” and “C” combining rules provide predictions in agreement with experimental data over a wide range of concentrations and for a variety of materials; viz, particulate filled thermosetting resins and thermoplastic foams, These combining rules are developed by making use of the symmetry of the improved bounding expressions, Eq 11, with respect to the interchange of the role of the components as the continuous phase: the lower bound expressions, E q l l b , describe the limiting behavior when the compliant component is the continuous phase, the upper bound E q l l a , describe the limiting behavior when the compliant component is the continuous phase.

The notion of a critical packing concentration, &., is used to designate transitions in phase continuity (or “contiguity”). When the volume fraction of either component is greater than this critical packing concentration, it is possible to unambiguously identify this phase as the continuous phase. Under these conditions of “zero” contiguity for the second component, the material system tends toward the behavior described by the bounding expressions in which the first (major) component is assigned the role of the continuum. The behavior of the material at intermediate concentrations is assumed to follow a smooth transition between the limiting behavior when either phase is identified as the continuum.

In actual materials, the polymeric component is re- garded as the continuous phase. For particulate filled systems in which the polymer component is more compliant than the particulate inclusion (Pf > P,J, the material tends toward the lower bound behavior emphasized by the “S-Combining Rule,” E q 21. Conversely, for foams in which the continuous polymer phase is more rigid than the gas inclusions (P , > P,), the material tends toward the upper bound behavior reflected by the “C-Comhning Rule,” E q 25.

An assignment of c$(. -- 2/3 (corresponding to “random” close packing) appears to be an appropriate choice for the critical packing fraction. Since the contiguity of the particulate inclusions is “zero” for uf < 1 - +,., this choice suggests that the polymeric component can be assigned i~nequivocally the role of the continuous phase for uf < %. It is interesting to note that all models developed under the primary assumption that the compliant phase is the continuous phase give essentially equivalent predictions for volume fractions of filler in the range 0 5 cf 5 %. This observation suggests that in this concentration range, the specific shape assigned to the particle has little influence on the behavior of the material. Indeed, the “S” (and “C”) combining rules utilize bounding results which avoid the specification of an explicit shape for the inclusion. The major fcature of the geometry of the inclusion is that the inclusion be near spherical so that the local material symmetry can be treated as essentially isotropic. Such conditions would be satisfied by irregularly shaped inclusions Lvhose length, width, and thickness are essentially equal. However, inclusions for which one dimension

1 60 POLYMER CoMPosim, OCTOBER, 1981, voi. 2, N ~ . 4

Combining Rules for Predicting the l‘hermoelastic Properties of Particulate Filled Polymers, Polyblends, and Foums

differs significantly from the others (e.g;. , short-fibers or platelets) would induce local anisotropy and thereby violate the important assumption of isc tropy. Although the relationships reported in this work are not applicable to such systems, the underlying approach can be extended to treat short-fiber and platelet reinforced materials.

As illustrated inFigs. 2 and3, most oi‘the models will predict adequately the behavior for Farticulate filled systems in the concentration range, 0 5 vf 5 %;however only the “S-Combining Rule”, E9 21, and the Lewis- Nielsen equation, E q 14, are in reasorable agreement with the data at higher volume fraction concentrations of filler. A clear distinction between the milder dependence on filler concentration predicted by the S-Mixing rule and the substantial dependence predicted by the Lewis-Nielsen equation could be made with data for particulate filled systems at high voliime fraction of fillers (of = 0.5 - 0.6) for which the properties of the components differ by a wide margin (e.g., diamonds particles in an elastomeric matrix).

In conclusion, it should be pointed o i t that the comparison of experimental results with several of the theoretical models can provide usefu 1 insights. For example, data points which fall below i he limits of the improved lower bounds, Ey I l b [orEq 2 with 6 = ( p , ] ,

would indicate (i) a broad distribution of particle size for which the composites sphere model rnight be applicable, and/or (ii) processing induced degradations in the properties such as void formation or incomplete cure of the resin. Data points that fall below the limits of the primitive Reuss lower bound, E q 6, :.re clear indica- tions of process induced degradations in the properties.

ACKNOWLEDGMENT’ This work was supported by a grant f -om the Rogers

Corporation, Rogers, Connecticut.

1.

2.

3. 4. 5 .

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combining rules for predicting the thermoelastic properties of particulate filled polymers,...

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