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Unformatted text preview: VT The Adhesion Quality and the Extent of the
Mesophase in Particulates P. S. THEOCARis
Department of Theoretical and Applied Mechanics
The National Technical University of Athvn:
5, Heroes of Palytechnion Avenue.
GR—l57 73 Athens, Greece SUMMARY The threeterm and twoterm unfolding models, which were developed. to describc
mesophase phenomena in ﬁberrcinforced composites, are extended in this paper for
particulate composites. It has been already shown that the thin boundary layer, developed during casting and curing of the matrix around the inclusions in the com
posite, inﬂuences considerably the overall properties of the composite working as a crust between the two main phases of a bimaten'al composite. ' _ However, in a fiberxtomposite the elastic modulus of the composite (E,) consrsts of
three components contributing parallelly in a meridional section, that is the modulus of
the ﬁber, mesophase and matrix, which then contribute additiver to the Efmodulus,
multiplied each one by a weight parameter equal to the particular volume content of
each phase. In a particulate the situation is inversed. Here the compliances of the
typical phases contribute in an equatorial plane superimposrng their contnbutron and
therefore the respective compliances are additive. Based on this fact the threeterm and twoterm unfolding models, introduced for thc
ﬁbercomposites, were established also for the particulates. it was shown that, by measuring the heat capacity jumps of the matrix and the composites, the thickness or
the mesophase layer was experimentally established. The experimental value. of DC
compliancc, combined together with the Dfﬁller compliance and the Dm«matrix corn,
pliance could deﬁne completely the model and gave, furthermore, important indica
tions about the adhesion quality of the phases. by defining the so—called adhesron we} ﬂcient of the composite. INTRODUCTION RIGOROUS MATHEMATICAL REPRESENTATION OF A COMPOSITE SYSTEM
consisting of a polymer matrix, in which particulate ﬁllers have been
dispersed, presents unsurmountable difﬁculties even for the cases where both
constituent phases are considered as elastic. _
Although many geometric, topological, mechanical and physrcal
parameters interfere for the statistic evaluation of the average properties of a
particulate, it has been shown that the representative volume element, con»
sisting of a series of concentric spheres, adequately describes the mechanical 204 Journal of REINFORCED PLASTICS AND COMPOSITES, Vol. 3—July I984 07316844/84/03/0204—28 $04.50/0
@1984 Technomic Publishing Co., Inc. The Adhesion Quality and the Extent of the Mesophase in Particulates 205 and physical behavior of the composite. In the twosphere model introduced
by Hashin [l] for the particulates, a gradation of sizes of spheres is an
ticipated, corresponding to the particular ﬁlling volume content of the com
posite considered. Hashin gave an exoression for the bulk modulus K, of the
composite, considered as the effective homogeneous bulk modulus for the
particulate, in terms of the bulk moduli K, and Km of the filler spheres and the
matrix material enveloping the spheres, as well as of the filler and matrix
volume contents v, and vm (v, + 9', = 1). On the other hand. the homogeneous
shear modulus was determined by bounded values. This, and all other twophase models, assumed ideally smooth and
mathematically described interfaces between phases and perfect adhesion be
tween them, thus accommodating the different mechanical propertics between
phases. Thus, analytic solutions are in general based on variational principles of
mechanics and yield upper and lower bounds of approximations for the effec
tive moduli of the composites. These solutions are valid only for rather low
ﬁller contents, since they ignore, for reasons of efﬁciency, all mechanical in
teractions between neighboring ﬁllers, as well as physicochemical influences
on either phase from the other one. A great number of empirical and semi—
cmpirical expressions for the effective moduli exist, expressing some kind of
law of mixtures. or trying to match theoretical expressions to experimental
data by appropriately deﬁning the existing constants in these expressions. In
all these models perfect adhesion is assumed holding between phases. The most important of these models were described in References [2] and
[3]. They are characterized by the fact that they express the ratio of the
moduli of the composite and the matrix in terms of constants and the filler
volume fraction v,, ignoring completely the inﬂuence of the value of the
elastic modulus E( of the tiller. Only the models presented by Takahashi and coworkers [4] and by
Takayanagi [5] introduced explicitly in the expression of the Ecmodulus the
filler modulus E,. Indeed, in Takayanagi’s models A and B the elements of
the composite, weighed by their respective volume fractions, were arranged in
units in series or in parallel, with one of the units having its elements disposed
in parallel or in series. In this way a combination of units in parallel and in
series was devised, suitable for each case studied. The characteristic of this
model is that the matrix was divided into two parts, the one belonging in the
one unit, and the second contributing together with the inclusion to the other
unit. The relations of these models, however, may be shown [6] to derive from
the exact expressions for the moduli of the components of the composite, by
introducing simpliﬁcations based on the comparison of the values of the
mechanical constants of the constituents (for instance 13{ >> Em > Gm,
vm = v, etc.). Another type of model is referred to multiphase particulates. It assumes
three consecutive phases as concentric spheres and the external one is ex
tended to inﬁnity. The spherical ﬁller is surrounded by a concentric spherical 206 RS. THEOCARIS layer having the properties of the matrix, which in turn is embedded in the in
finite medium with the properties of the composite. The model was intro
duced by Kerner [7]. Van der Peel [8] used a model similar to the Kerner
model, but with different boundary conditions. Finally, Maurer, in his disser
tation, used extenstvely the Van der Poel model and derived interesting results [6]. Although the Kerner and Van der Poel models may be adapted for the
study of the mesophase phenomena developed at the boundary layers be—
tween phases, only Lipatov [9] considered extensively the phenomenon of
creation of the mesophase between main phases in particulates and gave rela
tions interconnecting the heatcapacity jumps at the glass transition tem— peratures of the matrix polymer and the composites with the thickness of this
boundary layer. In all these models the boundaries of the phases were idealized as smooth
surfaces described by exact mathematical expressions. In reality, around an
inclusion a complex state develops, which consists of areas of imperfect
bonding, permanent stresses, due to shrinkage of the polymer phases during
the curing period and the change of the thermal conditions there, high stress»
gradients and stresssingularities, due to the complicated geometry of the in
terfaces, voids, impurities and microcracks, appearing at the vicinity of these
boundaries. Moreover, the interaction of the matrix polymer during its curing period with the surface of the solid inclusion is always a complicated procedure. In deed. the presence of the filler restricts the free segmental and molecular
mobility of the polymeric matrix, as adsorption interaction between phases occurs. This phenomenon inﬂuences considerably the quality of adhesion
between phases, contributing to the development of a hybrid phase between
main phases, which is called interphase, or better, mesophase. The existence of mesophase was proved experimentally and its extent was
evaluated by the theory developed by Lipatov [9], or it may be calculated by
dynamic measurements of the storage moduli and the loss factors at the
vicinity of Tg’s of the matrix and the composites [10]. Although the extent of mesophase may be determined from ther
modynamic measurements, the influence of this pseudophase to the
mechanical behavior of the composite was not extensively studied. For the ease of ﬁberreinforced and particulate composites the multi
cylinder model was introduced in References [l l] and [12], which studied the
inﬂuence of the mesophase on the properties of the composite. Another
model was afterwards presented, where the variable with polar distance
elastic modulus of the mesophase was expressed as sum of a constant term
and two variable terms, expressed as modiﬁcations of the moduli of the filler
and the matrix by negative power laws [13,14]. The two—term improvement of
the previous model [15], where the variable mesophase modulus was ex
pressed by a negative power term of variation of the E,—modulus and a linear
term for the variation of the Emmodulus, gave better and much more stable
results [16]. The Adhesion Quality and the Extent of the Mnophm in Particulate: 207 All these models were based on an improved law of mixtures, where the
composite modulus was assumed as the sum of the ﬁller, mesophase and
matrix moduli, multiplied, each one of them, by their respective volume frac
tions in the composite. Whereas this law is simple for ﬁber reinforced com
posites, where the components or the moduli are connected in parallel, and
they add themselves, for the case of particulates a new form of improved law of mixtures should be sought, since, in this case, the contributing moduli may
not any more be assumed as acting parallelly. The models of Kerner and Van der Poel, which are valid for the study of
threephase models, may be adapted to derive an improved law of mixtures
for particulates, if one assumes the intermediate phase representing the
matrix in the Kerner model to occupy the place of the mesophase, whereas the
inﬁnite medium possessing the unknown properties of the composite is
restricted to an external phase having the shape of a sphere and the properties
ofthe matrix [16,17]. Some attempts have already been made to derive in a very sittiplified
manner the expression for the elastic modulus Ec of a particulate composite,
by taking into consideration the existence of mesophase. One of these is the
expression given by Lipatov [18], who attributed this simple model to
Sagalaev and SimonovEmeljanov [19]. According to this model, when the composite attains its critical content of
filler, that is the content above which no changes in density of the composite
appear, the matrix phase of the Kerner model becomes a mesophase. In this
case the compliance of the composite is expressed by the sum of compliance;
of the constituent phases, multiplied by their respective volume contents.
Although this model was not tested in the praxis, it seems conceptually in«
compatible with the real behavior of composites, since it is impossible to
assume in the praxis a composite with an overall compliance (Dc) larger than
the compliance attributed to the matrix. Another approximate model, which considers the inﬂuence of mesophase
for the evaluation of the elastic modulus of the composite, was introduced by
Spathis, Sideridis and Theocaris [20], according to which the elastic modulus
of the composite is expressed as the sum of the filler, mesophase and matrix
moduli, multiplied with their respective volume fractions. The term express
ing the contribution of the fillermodulus was further multiplied by a factor
k, taking care of the quality of adhesion. This factor was derived from the ex
perimental determination of the mesophase volume fraction vi and its modulus by measuring the Ecvalues for different volume contents.
Measurements at least at two different filler contents sufﬁce for the evalua tion of the factor k. Further tests with different v,s were used to check the
constancy of the values of k. This modulus gave a lower bound for the com,
posite modulus EC. ln this paper a new model was introduced. relating the composite com
pliance (Dc = l/Ec) to the compliances of the matrix, mesophase and filler.
Then, by measuring the compliance of the composite and the matrix materials
and knowing the compliance of the tiller, the compliance of the mesophase 208 P.S. THEOCARKS can be determined. Introducing the twoterm unfolding model for the com Dliances. as it has been developed for the fiberreinforced composites. the
adhesion coefﬁcient between phases can be determined, by evaluating the mode of variation of the elastic compliance or modulus of the mesophase, to
match the bounds of D’s and E’s of the ﬁller and matrix. The new model gave
reasonable results, as it did the same model for ﬁber composites and thus
characterized the quality of adhesion of the composite. Determination of the Elastic Modulus of the Composite In order to determine the Ecmodulus of the composite, we consider that
the representative volume element consists of three concentric spheres, having
radii r,, ri and rm respectively. The external radius of the RVE was taken such
that this model corresponds to the average properties of the respective par
ticulate. All quantities are normalized to the volume of the outer sphere 4/31rr,’n to be equal to unity.
We assume lurther that all phases are elastic, homogeneous and isotropic, with the exception of the mesophase layer, which has a variable modulus,
changing with the polar distance between the values of Efand E,“ at its bound
aries. However, this layer is also averaged, so that Ei corresponds to its E,“
mean value. Moreover, it was assumed that the fillers are perfect spheres, their distribu
tion is uniform and their intercenter distance large enough to ascertain a
negligible interaction between neighbouring ﬁllers. Finally, we assumed that
the RVE is submitted to a radial pressure of magnitude pm, so that, at the
other boundaries, equal radial pressures of magnitudes pi and pr are exerted, which exhibit the interaction between successive phases. The representative
volume element, as descrlbed above, is presented in Figure 1. Because of the spherical symmetry of the problem we introduce spherical (r,0,¢p)—coordinates with the simpliﬁcation that, from all the components of
the spherical displacement vector, the ur—radial displacements are different than zero, because of the nature of geometry and loading of the problem. (u) (bl (r) ngm 1. Principal sections of the three concentric spheres forming the representative volume
element lRVé‘I of a typical particulate and the forces applied to their boundaries. The Adhesion Qualin and the Extent of the Mesophase in Particulates 209 The solution of the problem may be derived by the stress functions in] f(r) (with j = f,i,m for the filler, mesophase and matrix respectively),
which are expressed by: it, =A,r"+19.r2 (1) with the constants A, and .3, defined by the boundary conditions between
phases, With A, = 0 for ﬁn1teness of stresses at r = 0. The components of displacements are given by: u., = (~A,r*2 + 28,.r)/ZG, (2) with the other components: “or = "av = 0 (3) The component u,, is expressed by: Mr  7, (4) since A J = 0.
The components of stresses are given by: =ﬁ+2r1+w 2(1+v
,3 m3” aoj=ow=—A,r3+H’) (1 — 2v.) 3] (5) a,, The boundary conditions yield:
17A! r = r,.' 0,, = 0., = ‘1),
ii) At r = r,: o.. = 0,... =  p. (6)
170A! r = r,,: 0,... = ‘11.. The solution of the system derived from the boundary conditions (6) yields
the values of the constants A,, 3,: AI=(PJ_P1—i)’7"3'i (’73P1_’}1—1P1—1)”_2Vj)
I 2(’f"7‘i) B, = —
2m ~ r; . .i (1 + v.) (7’ Where the index (j + 1) means the next layer of the layer j considered, with
the successron from the ﬁller outwards, so that j = 1 corresponds to the ﬁller, 1:.2 to the mesophase, and j = 3 to the matrix of the particulate, and
n» — r, _ 1 = 0 forj = l, r, = r,,r.r,r, forj = 1,2,3 respectively. 210 P.S. THEOCARIS The equations for the radial displacements are: _ _ P!” _ 2V1), 8
“r! ‘ E, ( )
03”: (PI ‘ Pa) (1 + Vi) (1 " 21h) (1730/ ‘ P9P.)
u . = + r (9)
" 2(r.‘  r;‘)E.r2 E. n‘  I?
__ if: ([7: ‘17...) (1 + V...) (1 ' 2V) (n’pr — lip...) 10
“'“‘ 2(I3.n‘)Ef + E. e—e ’ ( ’
The components of strains in the three layers are given by:
_du,,_i _ _u,,___ﬁ
c"_ dr_G,'£"£W_r G,
du  A 8 ud A, B,
= "=4 V: ,= .=_=_ +i 1
"" dr Ho: + G: ' “" ‘e r 2r’G. G: (1 ’
dun, A," + B... _ _ um _ (A. + Bin
"'4 ' dr ‘ r’G... 0,. ’ ‘W W" r 2140,, 6,. The boundary conditions, implying continuity of displacements at the
boundary between filler and mesophase, yield:
For r = r, 14,, = u,, then: p,(l  2w) _ (I  2vt)_ (P13 p.) + (l + v.»)(pf “pd (12)
E, ‘ E; 1 ~ B 2(1 ~ B)E_. in which the ratio r,:‘/r.3 is replaced by the quantity B where: U!
= 3
B u, + ul. (1 ) Solving Equation (12) with respect to the ratio A,. = p,/p. we have: 3(1 — v,)(u, + u.)E, = 14
2u.E,.(1 — 2v,) + [3u,(l — v.) + u,(1 + v.)]E, ( ) 1,, We apply now the second boundary condition concerning the compatibility
of the radial displacements at the second interface between mesophase and
matrix. In this boundary it is valid that u,, = u". and this yields: = 3(1 — v”) — 2u,,.(1 — 2v”) _ Emu, (1 + vi) 3(1 — v") E, 3(1 — 11..) (15) Air" The Adhesion Quality and the Extent of the Mesaphase in Particulates 21 1 In order to evaluate the elastic modulus Ea of the composite, we deﬁne the
elastic strainenergy balance on the representative volume element. Then we have: 1
V25 Il’gdu‘=I/z§ W,du,+%s W,dul.+%5 Mary"
H c U! U: V». e which yields:
I‘m 171 r! rl.
" r’dr = W,r‘dr + W,r‘dr +
o Kc o r, where Kc is the bulk modulus of the composite, related to the elastic modulus
E: by the well known relationship: r».
er‘dr (16)
r.‘ E K“ = 3(1 — 2m (17) The elastic strain energies of each one of the layers of the particulate may be
deﬁned by the relation: "/1 = "'1‘?! + "9150/ + aw‘w = firm) lntroducing the appropriate values for the components of stresses and strains
for each phase, we may readily derive that: 93%
W/ = TE; ’ (18)
3A2 933k,
W" = r‘G: + 0,? (19)
and
. ,5. B2 m
Wm _ 34 + 9 ,K (20) ' PG... 0}, Introducing now the values for W’s into relation (16) we obtain, after some
algebra, that: 1_u, 1,:1 31;; I—A,.]‘ u; i,._ 1 2
K: ‘ K, (it) + 4BG.u. L + Kiu. .. mm + 3n, _ L ‘ 1 u, _ 2
+ 430,21. [ hm] +K_u... [mm ] (2” 212 PS. THEOCARIS where:
_ a _ I: a = h 2
Aft _ p; ; Ann " p; and a“ ll“ ( and: .
v, + u,
rE/ri, = u, + 1),, ri/rﬁ, = u, and ﬁ/r}= u (23)
1' Equation (21) yields the bulk modulus Kc of the composite in terms of the
bulk and shear moduli of the phases and the stressratios Al, and A“, which, on
the other hand, depend on the elastic moduli and Poisson’s ratios of the
phases. _ Since the mesophase is derived from the matrix material, it is reasonable to accept that:
v. E v" (24) whereas for the Poisson ratio 1/, of the composite we use a modification of the
relationship given by Reference [21] interconnecting the values of Porsson’s
ratios. This relationship is given by: L = a + a + U_~ (25)
1/6 1/, v, v, This equation completes the number of relationships necessary for evaluating
the characteristic properties of the composite. Comparison with Exlstlng Models for Plrticulntes Relation (21), interconnecting the inverses of the bulk and shearmoduli of
the phases and the composite, constitutes an improved law of mixtures for
particulates. It indicates that some relation between compliances, instead of
moduli, should hold for particulates. Indeed, Equation (21) shows that the
composite bulkcompliance (or simply any compliance, since the material is
assumed at a thermal equilibrium, where the reduced elastic relations between
moduli should hold, provided the actual values of Poisson ratios at this
temperature level are known, together with the values of any modulus at the
same temperature or time level) equals the sum of the bulk compliance of the
fiber, the bulk and shear compliances of the mesophase and the bulk and
shear compliances ot the matrix material, all these terms multiplied by con»
venient factors depending on the volume fractions of phases and the ratios of
the elastic moduli of filler and matrix, normalized to the elastic modulus of
the mesophase through the convenient factors 1,, and l,,,.. It is obvious from the structure of relation (21) that this constitutes an im The A dhesion Quality and the Extent of the Mesoplmse in Particulate: 2 l 3 provement of previous assumptions concerning the interrelations between moduli in particulates. indeed, while a simple improved law of mixtures for
fiberreinforced composites exists, which may be expressed as: E‘ = 15,11, + En), + Emu... (26) the same is not valid for the particulates. This is because, if we assume separated the contributions of the moduli in
the RVE of the ﬁber—composite, these component moduli cooperate through
their interfaces, which are assumed parallel to the direction of application of
the external load. this cooperation achieved by the lateral surfaces of
cylinders developing shears between phases. Thus, this model necessitates an
addition of the moduli of the phases multiplied, all of them, by weight
factors which are simply their respective volume contents. In an electrical
analogy scheme then, the weighted moduli constitute capacitances connected
in parallel, which explains the validity of relation (26). In the case of particulates the situation is much different. The component
phases are interconnected through consecutive spherical surfaces of the fiber,
mesophase and matrix. The dominant transverse interconnection through
shear, holding in the fibercomposites, is now insignificant, the adhesion is
now achieved by a continuously varying combination of shear and normal
forces at the interfaces, derived as variable components of the hydrostatic
pressure mainly applied on these surfaces. It is now closer to the real situation
of forcedistribution, to assume that the contributions of phases are done by
superposed layers of the phases, lying the one on the top of the other in the
direction of the application of the external load. in this case the compliances, instead of the moduli, should be added to
derive the compliance of the composite. since now the capacitances of the
respective electrical analogy are connected in series. This argumentation ex
plains the necessity of adding in this case the compliances of phases, instead
of moduli. Figure 2 presents a schematic of the difference of the two cases in a RVE
of fiber (a) and particulate (b) composites. The drawn surfaces between
phases explain the reasoning of the previous ideas. However, in the case of particulates, since the connecting surfaces are not
ﬂat, engendering almost equal distributions of shear forces (except at sin~
guloritics and discontinuities), but spherical, with variable contributions for
the boundary conditions, relation (21) is much more complicated than the
relation (26) holding for fiberreinforced materials. There are several models expressing the E,modulus of the particulate,
which are based on simple relations interconnecting the matrix with the ﬁller
through a mesophase. One of the interesting models is the model introduced
by Sagalaev and SimonovEmiljanov [19], which is expressed by: EfEiEm E6 = mEIEiUM + EmEiUl + (27) a 14 ES. Tuuocmus Flynn 2. Schematic of the difference in the cooperation of the phases in a fing unidirectional
composite (ai and a spherical particulate (bl. This expression may be reduced to:
I), = u,D, + u,D; + u_D (28) which interrelates the extension compliance of the composite (D;) to the com
plianees (D,) of the phases multiplied by their respective volume fractions (11,}
(j = f,i,m). I _ This model is the analogy for particulates of the respective model for the
elastic moduli of fiber reinforced composites, expressed by relation (26), However, as it is already stated here and proved by several authors, this
simple relation is an oversimpliﬁcation of the real situation in particulates Another approximate model was introduced by Spnthis, Sideridis and
Theocaris [20] which constitutes an upperbound for the value of the elastic
modulus of the composite. This model is expressed by: E: = E,u,k + E111; + Emu, (29) where the adhesion factor k between ﬁller and matrix is introduced to take
care of the inﬂuence of the shape (approximately spherical) and number of TheAdhesion Quality and the Extent oft/u. Metsophatm in Particulate: 215 fillers and their interaction, on the value of the average elastic modulus of the
composite. In addition, the existence of the mesophasc layer, contributing to
the quality of adhesion, was put with to take care of the proper inﬂuence of
this pseudophase on the value of [Scmodulus. Finally the study by Mauier is
worthwhile mentioning [6]. This study, based on the Van cler Poel model [8],
has examined the inﬂuence of the existence of a mesophase on the viscoelastic
behavior of a composite containing spherical inclusions. The inﬂuence of
discrete amounts and qualities of mesophases on the overall mechanical prop
erties of the particulate composite was exhaustively studied, concerning the
two principal simple modes of loading, that is hydrostatic pressure and pure
shear, Besides other important contributions, this study has shown the in
fluence of a third phase inbetween the two others, be it a real phase, or
pseudophase developed during polymerization of the composite. It was shown that, for the bulk modulus of a threephase material. the
following relation holds: K _ (KW! + KiUiR + KmUmS) ‘ u, + u,R#+#uM—S— (30) where the factors R and S are functions of the bulk and shear moduli of the
three phases. For the shear modulus of the composite a (10 X 10)matricial expression
for this modulus is anticipated. It can be derived, by comparing relations (29) and (30), that relation (29) is
a simplified expression of relation (30) incorporating into the factor k the in
ﬂuence of the quantities R and S. Then, relation (29) may be considered as a
reasonable upper bound for the K6, or E,»moduli. In completing this discussion about the expression for the moduli in com
posites, it is worth mentioning the KernerKerner model [7], where the
classical Kcrncr model was uscd twice, once for the ﬁllermesophasc material,
and for the second time for the internal heterogeneous material and the
matrix. In this model we have again the expression for the shear modulus G‘
(if the particulate in the form: GU +Guk+Gu1
:_J_I_%1_L___'LL
0‘ (u,+u.k+u,,.l) (3” which is similar in conception with relations (29) and (30). The factors k and l
are complicated expressions of the shear moduli G, and Poisson‘s ratios 11, of the phases. Similar expression is also given by the Kudykina and Pervak
model [22), where the: shear modulus of the composite is given by: _ 0,114+ 0,115+ vat
G“ _ ("If + ms + U...) (32) 216 RS. THEOCARIS All these expressions are more or less similar, displaying the correction fac
tnrs between two of the three phases. Quite different expression for the composite moduli is given by the Takano
and Sakanishi model, which includes the concept of mesophase. The ratio
(PE/Gm of the shear moduli is given by [6]: G, _[r—3/2(u,+u,)r' m]
0., _ [F+(u,+u,)"Q,,,] (32) where the quantities F,l' ' and PWQ, are complicated expressions of the bulk
and shear moduli of the phases. Besides these models incorporating explicitly the existence of a mesophase,
which may be a real one, or may be a pseudophnse developed by the matrix at
its boundary with the ﬁllers, there is a quantity of simple phenomenological
models, based on the interaction of the two main phases. These models have
been exhaustively described in the literature and they )n'eld more or less
reliable results [3]. One important model, where a convenient combination of the two limiting
arrangements, that is in parallel and in series, was introduced to express, in an
effective way, and with a high approximation, the elastic moduli of par
ticulates is the Takayanagi model [5,23]. While the Takayanagi model is designed mainly for two—phase materials, it
could be extended to incorporate the inﬂuence of mesophase [24]. It seems
that a further extension of the concept of using mixedmode connections, that
is in series and in parallel, for the elements contained in the model represent
ing the mechanical properties of the particulate, is Very promising, since it
yields high flexibility in the model to be adapted to the real behaviour of the
substance. However, if one examines relation (21) expressing Kc, he may recognize that the terms in the righthand side of this relation could not be
strictly considered as connected in series, The Unfolding Model for the Mesophase A decisive factor for the physical behaviour of a particulate composite is
the adhesion efﬁciency at the boundaries between phases. In all theoretical
models this adhesion is considered as perfect, assuming that the interfaces en
sure continuity of stresses and displacements between phases, which should
be different because of the proper nature of the constituents of composites.
However, such conditions are hardly fulﬁlled in reality, leading to imperfect
bonding between phases and variable adhesion between them. The introduc—
tion of the mesophase layer has as function to bring together, in a smooth
way, the differences on both sides of interfaces. The model for the representative volume element of a particulate consists
of a unit of three concentric spheres with respective radii r,, r, and r,,,, whose
volume contents are expressed by: The Adhesion Quality and the Extent of (he Mesaphase in Particulates 217 _ 51‘ _ 5:0“, _ [ti
u,— I: ,u.— n," andum— ,7" Assuming the appropriate boundary conditions between the internal sphere
and any number of layers surrounding it, in the RVE of the composite, which
assure continuity of radial stresses and displacements, we may establish, by
an energy balance between phases, a relationship, interconnecting the com
pliances, or the moduli between phases and composite. Such a relationship is
given by Equation (21). In this relation the quantity E, corresponds to the
average value of the modulus of the mesophase and, in the following, it will
be denoted as E‘.'. However, this effective or average value of the mesophase modulus, necessary for introducing the contribution of the mesophase to the
value of the modulus of the composite, does not really exist, except in one very thin spherical layer. In order to approach closer to reality we assume that the mesophase layer
consists of a material having progressively variable mechanical properties, in
order to match the respective properties of the two main phases surrounding
the mesophase. We can then define a variable elasticmodulus for the
mesophase, which, for reasons of symmetry, depends only on the polar
distance from the fibermesophase boundary. In other words, we assume that the mesophase layer consists of a series of elementary layers, whose constant
mechanical properties differ to each other by a quantity (small enough), defined by the law of variation of E,(r) . In this way the elastic modulus of the
mesophase is unfolding between two limits, from the E, to [Em—modulus.
The unfolding model for the mesophase may be expressed in two forms:
In the ﬁrst form, the E,(r)modulus is expressed by the sum of three terms,
i.e.: i) a constant one and equal to the Emmodulus. ii) a variable one, depending on the modulus of the filler (E,), which should
be added to the ﬁrst one, and iii) a third variableone, which should depend on the Enrmodulus, and which
should be antagonistic to the second term. The two variable terms should yield very abrupt variations in the
E.(r)modu1us, since the generally large differences between the moduli of the
fillers and the matrix must be accommodated in very thin layers for the
mesophases. The appropriate functions for such steep variations are power
functions of r with large exponents. Then, the E,(r)vmodulus may be ex pressed by:
E,(r) = E," + E, ( ‘ —E,,, 1 (33) Expression (33) may be interpreted by the fact that the variable
E,(r).modulus, which connects two phases with highly different mechanical 218 P.S. THEOCARIS properties and elastic moduli, must interconnect and span these differences.
Indeed, for hardcore composites it is valid that EI >> Em, whereas for
rubbery ﬁllers we have the opposite relation E, >> E,. However. it never
happens that E, = 5,. Relation (33) has been already established for expressing the mesophase
variable modulus for ﬁberreinforced composites with excellent results [14 to
17]. In relation (33) the second righthand term expresses the contribution of
the Ermodulus to the variation of EMUmodulus, whereas the third right
hand term deﬁnes the counterbalancing contribution of the Em—modulus, to
correct the contribution of the E,modulus, and insert the inﬂuence of the
matrix to the outer layers of the mesophaseannulus. From the compatibility conditions for the moduli at the boundaries be
tween ﬁller, mesophase and matrix it may be readily derived that for r = r,
relation (33) yields E,(r) = E, and therefore satisﬁes automatically the bound
ary condition at the ﬁllermesophase interface. On the other hand, at the other boundary between mesophase and matrix,
that is for r = n, we obtain E.(r) = E... if the following relation holds [16]: logE,/E,,, A = (m — "1) = logr/r (34) The constant A = (n,  m), which depends on the ratio of the moduli of
the two phases and the ratio of the radii of the mesophase and the ﬁller,
characterizes the adhesion quality of the composite, because for large values
of A the extent of the mesophase layer is reduced and the adhesion is more ef
fective. The constant A is called the adhesion parameter. Higher values ofﬁiindicatg ibegeLa’dhgsign‘for a particular composite,
because’irnilysmalldifferencesinthe valuescfrt'anﬂ r',". For’haidcore
materials, where E, >> 5,, the radius r, must be always larger than r,, and
E, >> E," , therefore the logarithm of the ratio r,/r, is a positive number and
since log(E,/E,..) > 0 the values of A are always positive. This means for a
hardcore composite it is always valid that n, > 7);. On the contrary, for
rubbercore composites log(E,/E,.) is now negative and therefore it is valid
that Y], < m. A simple and more stable version of the previous model for the variation of
the E.(r)modulus is given by the twoterm form of the previously described
model. Since the third term of the righthand side of Equation (33) takes care of
the inﬂuence of the matrix modulus E... to the variation of E, (r)modulus and
since always for strongcore composites this contributiOn is secondary, rela
tion (33) may be somehow relaxed, by assuming that this third term varies
linearly with the radius r along the interphase. This means, in other words,
that the exponent n; may be assumed equal to unity without loosing gene
erality, and imposing to the ﬁrst righthand side term to take care of the
totality of the change of slope of the E,(r)modulus. The Adhesion Quality and the Extent of the Mesaphase in Particulates 219 Then, relation (33) may be written by: ’1 2" _ fl 1" (V _ 7'1) Ex”)  E! (r) +{E— El<rl) }(ri _ r!) (35)
Now relation (35) contains only two terms in its righthand side, since the
contribution of the Enconstant term in the Equation (33) is now incor—
porated into the second linear righthand side term of Equation (35). It is easy to show that the boundary conditions for this equation are
automatically satisﬁed. Indeed, for r = r, we have the second righthand term
of Equation (35) equal to zero and the ﬁrst term equal to the E,modulus, as
it should be. Moreover, for r = r,, Equation (35) yields automatically E,(r) =
E, and this satisﬁes the exterior boundary condition. Relation (35) has the advantage to contain only one unknown exponent
and therefore simplifies considerably the evaluation of the unknown quan
tities in the deﬁnition of the variable E,(r)modulus. Moreover, it was established during the numerical evaluation of the
unknown quantities in relations (33) and (35) that the deﬁnition of the two ex—
ponents m and m in relation (3 3) is rather unstable, depending fraily on small
variations of the value of the E,—modulus. 0n the contrary, the single
unknOWn Znexponent, deﬁning relation (35), yields rather stable and reliable
results. In the twoterm unfolding model the 2nexponent is the characteristic
parameter defining the quality of adhesion and therefore this exponent may
be called the adhesion coefficient. This quantity depends solely on the ratios
of the mainphase moduli (E,/E,,.), as well as on the ratio of the radii of the
ﬁller and the mesophase. The Thickness of the Mesophase It has been observed that, for the same volume fraction 11, of the filler, an
increase of T, indicates an increase of the total surface of the ﬁller [18]. This
is because an increase in T, may be interpreted as a further formation of
molecular bonds and grafting between secondary chains of molecules of the
matrix and the solid surface of inclusions, thus restricting signiﬁcantly the
mobility of neighbour chains. This increase leads to a change of the overall
viscoelastic behaviour of the composite, by increasing the volume fraction of
the strong phase of inclusions. A considerable amount of experimental work indicates an increase of T, in
composites with an increase of the ﬁller content [18]. The degree, however, of
this variation and the character of its change may differ from composrte to
composite and, also, for the same composite, is depending on the method
used for its measurement [25,26]. Moreover, in many cases, a shift of T, to lower values of temperature has
been detected, but in these cases the quality of adhesion between phases may
be the main reason for the reversing of this attitude [18,27]. If calorimetric 220 RS. THEOCARIS measurements are executed in the neighbourhood of the glass transition zone,
it is easy to show that lumps of energies appear in this neighbourhood. These
jumps are very sensitive to the amount of ﬁller added to the matrix polymer, and they were used for the evaluation of the boundary layers developed
around fillers, The experimental data Show that the magnitude of the heat capacity (or
similarly of the specific heat), under adiabatic conditions, decreases regularly
with the increase of filler content This phenomenon was explained by the
fact that the macromolecules appertaining to the mesophase layers are totally
or partly excluded to participate in the cooperative process taking place in the
glasstransition zone, due to their interactions with the surfaces of the solid
inclusions. It was shown [28,29] that, aﬂhe‘ fuller7vplume’fraption“ismincreasedtthe proportiongt macromgggggspartigpaghg‘n’mis boundary layers with
reagentmobmmathenymjﬂiﬁ} Bmwules
participating in the T,process is reduced. This is eguivalepttgamlatlye in
crease :9f 11,. ‘ "
W patov [18] has indicated that the following relation holds between a
weight constant A, deﬁning the mesophase volume—fraction 11,, and the jumps
of the heat capacity AC], of the ﬁlledcomposite and AC}, of the unfilled
polymer for particulate composites: Act i=1—Aq (36) where A is a real constant, which multiplies the filler volume fraction u,, in
order to take into consideration the contribution of the mesophase volume
fraction u, to the mechanical behaviour of the composite. In order to define the volume—fraction u, of the mesophase for the par
ticular composite studied, which was an iron—epoxy particulate, a series of
dilatometric measurements were executed in a differential scanning
calorimeter (DSC) over a range of temperatures including the glass transition
of a pure epoxy polymer used as matrix, and a series of samples of com
posites containing different amounts of iron—particles of three different
diameters d, = 150,300 and 400m” varying between u, = 5 percent to
u. = 25 percent. The graphs of all the data, which were similar to the heatcapacity graphs
for ironepoxy particulates, shown in Reference [27], presented shapes, which
were qualitatively similar to one another. They consisted of two linearly in—
creasing regions separated by the glass transition zone. While the glassy linear
regions presented a positive and signiﬁcant slope, the rubbery linear regions
were, all of them, almost horizontal. The AC,‘s were calculated by ignoring the smooth protrusions B 'BC ap
pearing in the CP = f(T) curves, as indicated in Figure 3 where the variation
of the speciﬁc heat, and its jumps at the region of glasstransition
temperatures Tgc, versus temperature was plotted for an ironepoxy par Cp Ital, gr‘, "[4] —> 50 75 100 125 150
T (“C l —> Figum 3. The variation of heat capacity, C,, 0/ immepaxy particulates plotted against tempera
fare, [or Iaur different lid/er volume Iraot‘ime, e,, and for a particle diamator d, = 0.40 x 103m, ticulate with d, = 400nm and for various u,’s. Then, we measured the distance
of the intersection A of the tangents AA ’ of the glassy curve and AB of the
transition curve from the horizontal tangent CC' of the rubbery part of the
curve AC, = f(T). The values of chs for the different composites are in
dicated, among others, in Table l. in order. now. to deﬁne the radius r. of the spherical layer corresponding to
the mesophase, we express it as r. = (r, + Ar) and we use the respective rela
tion, given by Lipatov [18] for particulates, which is given by: (r,+Ar)3 _1_ Au,
7; lUl (37)
which yields: hﬁ‘ﬁ‘T—VF—T) (38)
The real constant B depends only on the ﬁllervolume fraction and the coef f i
cient A, and it is critical for evaluating the exponents n, and m. Introducing
the values of A from Table l we can define the values of Ar for the various
fillervolume contents u, for the ironepoxy particulates. Then, it is easy to
evaluate the volume fraction u. for the mesophase layer. Figure 4 presents the variation of the differences, Ari, of the radii of the
mesophases and inclusions, (Ar, = (r. — r,)), versus the fillervolume content
u, for the three different types of iron—epoxy particulates with diameters of
ﬁllers d, = 150,300 and 400m, as they have been derived from relation (37), Figure 5 presents the variation of the heatcapacity jumps, AC1, at the
respective glasstransition temperatures of the particulates, versus the ﬁller 15 Zn
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’5 = § 3§§§§ §8§§§ @Niﬁi unvn—e ﬂ oo—Nm oo—Nm DDFNlﬂ Figure 5. The venetian of the speciﬁc heat jumps at glasstransition temperature of ironepoxy
Particulate composites, versus the filler volume content u,. The values for the factor A, the  .n o m 8 a m o m a g: m c, m a a mesophaseluil, and the matrix (u..)volume fractions versus 1),, as derived from the values of the
" " " respective AC,’s are also plotted. 222 223 214 P.S. Tneocmus volume content u,, for the three different diameters of the ﬁllers
(d, = 150,300 and 400ym). In the same figure the variation of the coefficient A and the volume fractions for the mesophase and the matrix, versus u, were
plotted, as they have been derived from relation (38). It is apparent from
these graphs, that the niesuphasevolume content 11, for the three different
diameters of inclusions were varied only insigniﬁcantly and, therefore, they
may be assumed as independent of the diameters of the fillers. It may be concluded from Figure 5 that the variation of the heat—capacity
jump curves are smoothly decreasing curves, as the fillervolume content is
increased. This behaviour is logical, since addition of iron particles makes the
composite more rigid [18]. It is also interesting noting that, for various
diameters of the fillers, the ACT, = f(u,)curves differ only slightly, with the respective curves for different fillerdiameters. mutually intertwining. This
fact indicates that the size of diameter of the filler plays only a secondary role on the inﬂuence of the heatcapacity jumps, which are primarily inﬂuenced
by the ﬁllervolume content. A similar behaviour is expected for the variation
of the coefﬁcient A, as indeed it is indicated in Figure S. It is worth indicating that the values for u,'s fitted excellently a third degree
curve expressed by: u, = Cu,J (39) where the constant C was evaluated experimentally to be equal to C = 3.5. The values of ms, together with the respective values for EC ’5, were intro
duced in the respective models and gave excellent coincidence with their ex
perimental values. based on the cubic relation between it.» and u,. The values
of the characteristic quantities for the threeterm unfolding mode], as derived
by this procedure, were included in Table i. In order now to evaluate the one exponent n. of the model. we make
recourse to the law of mixtures given for the spherical particulates by relation (2]). Evaluation of Adhesion Coefﬁcient A for the ThreeTerm Unfolding Model As soon as the Ar’s were determined and the values of as are found, the
values of the adhesion parameter A may he readily deﬁned, by using relation (34). The values of A's for the different fillervolume contents are given in Table l for the ironepoxy particulates with different amounts of ﬁllers, up to
25 percent [27]. In order to deﬁne the m and nIexponents it is necessary to dispose a sec
ond equation, besides relation (38), for the evaluation of nradius and rela
tion (34) for the deﬁnition of the difference (qrIn). For this purpose we use
relation (33), in which the value for E}, as it has been derived from relation
(21) was used. Indeed, in relation (21) every other quantity is either known, or
measured, as it is the bulk modulus K. of the composite and its Poisson's
ratio v. from very meticulous experiments. However, the values for the t The Adhesion Quality and the Extent of the Mesophase in Particulates 225 Poisson’s ratios of the various composites may be found by using relation
(25) and applying the values of v’s for the known phases and composites. It has been derived that the values of Poisson's ratios for the mesophase v. are
close to v... and they may be found when the Poisson ratio for the composite vc
is evaluated. For known values of E: derived from Equation (21) the variation of the
mesophase modulus along its thickness may be deﬁned by using relation (33)
for the threetenn undolding model. It is valid that: « —ir'E +E '4 "—E 1“ r‘dr (40)
[Vi—r: rl m l I. m r integrating relation (40) from r = r, to r = r. we ﬁnd: . .— jE’i _(ﬂ.+_L an” ~ 1) Ely[BIL]: ll.'l >11: _
E,u.—(2n_3){l 2n_3)3 }+ 4 {1+3 +3 }
_ Ezv/ {1+Bl/J + 3—2/3 _,3B»I} (4]) Relation (41) uses the value of E7, derived from Equation (2), and it con
stitutes one equation interrelating the two unknown exponents n, and n,.
Then, relations (34) and (41) form a system of two equations and two
unknowns, which can be solved and yields the values of the exponents n, and
q. and their difference A, which expruses the quality of adhesion and it was
already called the adhesion parameter. The values of the exponents n, and n2,
as well as of their difference are given in Table l and plotted in Figure 6 for
the iron—epoxy particulate composites. Figure 7 presents the variation of the terms E,(r,/r)"I and E,(r,./r)"2 in the
mesophase layer for a 25 percent ironepoxy particulate, as they have been
derived from Equation (40). It is worthwhile indicating the smooth transition
of the E.—modulus to the Enmodulus at the region r E r.. Similar behaviours
present all other compositions. It is interesting plotting the variation of the E‘_(r)morlnlus, versus polar
distance around a typical filler. Figure 8 presents this transition of the moduli from the ﬁbers to the matrices, exemplifying the important role played by the
mesophasc layer to the overall mechanical behaviour of the composite. Figure 9 presents the variation of the various moduli of the composite and
its constituents for various volume fractions of the series of iron—epoxy par—
ticulates. It is of interest to point out the small variation of the average value
of the variable E,(r)—modulus of the mesophase, which reﬂects the uniformity
of the adhesion quality of these series of composites, which is also indicated
by the almost linear variation of the longitudinal composite modulus, versus
the filler—volume content. 226 PS. THEOCARIS E. (r) =EmoE, (r, Ir)“‘ E,,.lr,lrl"i
uf=015
115% 111:12 E;(r .10" NM" —> 2.0 Arxlo‘m —’ Figure 6. The mode of evolution a! the variable terms l:',(/,/rl"1 and Emlrﬂrl'”, contributing to
the deﬁnition of the mesophese modulus, versus the polar distance r from the ﬁller boundary It” a
25% iron~epoxy particulate composite, Evaluation of the Adhesion Coefﬁcient 2:1 for the TwoTerm
Unfolding Model For the case of the twoterm unfolding model we have to replace the in
tegral in the righthand side of Equation (40) by the integral corresponding to
the twoterm model derived from relation (35). In this case we have: i r. i in L In r”
Ems r; SUEC) +{E, E,(n) }ri_r, r‘dr (42) In this relation the only unknown is the exponent 2n, since the value for E: is
derived from relation (21). Integrating relation (42) we find: EN! = _ anls— ]_ % (EﬂI _ Esznn)‘:1 + Bi/a + Bzn _ 33]
(43) Relation (43) yields the value of 2n, which is also tabulated in Table l for all
ironepoxy particulates studied. Figure 6 contains also the plot of the adhe The A dhesion Quality and the Extent of the Mesaphase in Particulate: 227 2000 0 0.05 0,10 0.15 0.20 025
u) l°/o) —> Noun 1. The van'nhbn a! 2;» adhesion pmmm A for the thmnrm unfolding model, and the
adhesion coefﬁcient 2r, for the twomm modal. versus the We! volume fraction v,. sion coefﬁcient 2n, versus ﬁller volume fraction for all ironepoxy par
ticulates studied. It is worth mentioning here that the three 2r] = f(u,) curves for the three
different diameters of the ﬁllers are almost coincident. Furthermore, there is an equivalence between the mode of variation of the two adhesion coefﬁ
cients A and 21, corresponding to the two versions of the unfolding model. Figure 10 presents the variation of the interphase moduli E,(r) for the
various inclusion—volume fractions, versus the extent of the interphase Ar,
normalized to the highest inclusionvolume fraction of 25 percent. This was
done in order to show the similarity of variation of the Ermodulus for the
various values of u, for this series of composites, possessing the same adhe
sion properties between them. These normalized patterns are equivalent with
those presented in Figure 9 for the three—term unfolding model, since the dif— ferences between corresponding values of the two versions of the model are
insignificant. CONCLUSIONS All theoretical models, describing the physical and mechanical properties
of composites, consider the surfaces of the inclusions as perfect mathematical
surfaces. In this way the transition of the mechanical properties from the one
phase to the other is done by jumps in the characteristic properties of either L0 [1,104 Nn‘l —>
m
c N
o D DDS 010 015 0,20 025
u, (°/ol —>
Figure 8. The variation of the moduli of the particulate composite Er, versus the fillervolume fraction, u,, and the mode of variation of the average mesophose modulus, E, as derived from
the models.
l i ‘1 2L0 180 T A 11,:150 um
+ d.=300 um
as u u,:noo um
f 120 l
'9
.1
L:—
60
050,05
U,:0‘lO' . 0.3 0a 05
10 L0 60 3.0 100
2 a 5.6 6 I. 11.2 1L0
Ar x 105 m ——>
ﬁgure 9. The variation of the mesophase moduli, versus polar distance r for different filler
volume contents u, for the throeterm unfolding model. 228 The Adhesion Quality and the Extent of the Mesophase in Particulates 229 2L0 l , z
o nV :rm
A u,=o,2o
15° + u,=015
«it u,=010
A u,=0.05 120 E,x10‘9 "m" —> 60 0
0 10 20 10 L1) 50 Arxloim ——> Figure 10. The variation of the mesophase moduli, normalized to the mesophase thickness Al for the 25 percent filler~volume content composite, versus polar distance I for the twoterm unfo/ding
model phase. This fact introduces high shear straining at the boundaries, which is an
unrealistic fact. in order to alleviate this singular and unrealistic situation, two versions of
models were presented, in which a third pseudophase was considered, as
developed along a thin boundary layer between phases during the polymeriza
tion of the matrix, and whose properties depend on the individual properties
of the phases and the quality of adhesion between them. The two versions of this unfolding model consider that the mesophase
layer, lying between the main phases, possesses varying physical and
mechanical properties, assuring a smooth transition from the properties of
the fillch to the properties of the matrix. Then, it is achieved in a very short
distance, corresponding to the thickness of this boundary layer, the pro~
grcssive change of its mechanical properties from the filler to the matrix. By using Lipatov’s theorv. interrelating the abrupt iumps in the speciﬁc
heat of composites at their respective glass transition temperatures with the
values of the extents of these boundary layers, the thickness of the mcsophase
was accurately calculated Assuming in the one version of the model a three—term representation of
the unfolding value for the elastic modulus of the mesophasc. where each
term is expressed as a negative power function of the polar distance from the
inclusion, the variation of the elastic modulus of the mesophasc was ac
curately determined. The second version of the unfolding model uses, instead of three, two
terms, one of which is a negative power function ofthe polar distance, acting
upon the inclusion modulus, and the other~one, acting upon the matrix
modulus, is expressed as a linear function of r. 230 P.S. THEOCAius Both versions of the unfolding model gave excellent results for the evalua
tion of the variation of modulus of mesophase, with the second one (the two
term model) yielding always more stable results derived in an easy way. The difference in the exponents n, and n1 of the two terms in the threeterm
model (the third term was a constant), and the exponent Zn in the single
powerterm or the twoterm model, gave a measure of the quality 01’ adheslon
between phases and they were called the adhesion parameters or coefﬁcients. Higher values of these coefﬁcients characterized better adhesion, whereas,
as these values were reducing, the adhesion became less and less effective. Both types of the unfolding model describe satisfactorily the state of transi
tion of the mechanical and physical properties in the particulate composite
from its fillers to the matrix. with the twoterm version yielding more stable
and therefore more reliable results than the threeterm model. REFERENCES I. Hashin, Z., Intern. I. 50!. and Slrucl, 6, p. 539 and p. 797 (1970).
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(1982). 4. Takahashi, K.,1keda, M., Hunkwe, K., Tlnalrn. K., and Sakai, T., J. Pol. Sci., 16, p. 415
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625 (1978). 12. ‘I'heocan's, P.S., and Papanicolaou, 6., Fib. Sci. Tech, 12, p. 421 (1979). 13. Papanicolaou, 0.. Theocaris, P.S., and Spathis, (1.. Call. Polymer Sell, 258, p. 1231 (1930).
14. Theocaris, P.S., Proc. Nat. Acad, Athens 591, p. 327 (1984). 15. Theocaris, P.S., Coll. and Polym. Sci, 260, (1984). 16. Theocaris, P.S., “New Developments in the Characterization of Polymers in the Solid
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Verlag (1984). 17. Theocaris, P.S., Proc. intern. Cont. lnlerface—lnterphase in Composite Materials, J.
Liegeois and S. Okuda Editors, SPE publication. p. TlTZB (1983). 18. Lipatov, Yu, "Advances in Polymer Science," 22, p. 31 (1977).
19. Sagnlnev GA, and SimonovEmeljanov. 1., Plastinassy N2, p. 48 (1973). 20. Spathis, G., Sideridis, E., and Theocaris, P.S., Intern. J. Adhesion and Adhesives, l, p. 195
(1931). 21. Jones, R., Mechanics of Composite Materials, McCrraw»Hill Publ., New York (1975).
22. Kudykinn. T.A., and Pervak, 1.G., UkraiIery Flzicherky Zhurnal, 20, p. 1664 (1975).
23. Takayanagi, M., Uemura, S., and Minami, 8., Jill Polymer Sclenne, Part C, No. 5, p. 113. PP‘HPMV' The Adhesion Quality and the Extent of (he Mesophase in Particulates 231 7A.
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27. as.
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This note was uploaded on 07/22/2011 for the course EMA 6166 taught by Professor Staff during the Fall '08 term at University of Florida.
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