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Fiber distribution effects in linear viscoelastic behavior of polymer matrix composites: Computation modelling and experimental comparison
Mechanics Restad thmmmicdtiona. W. 2.5.No. 1,pi. 105-115.1998 copyrightQ1998ElswiersdavrLtd RintedintheUSk Anlights-cd 0093~6413/98 $19.00+ .OO
PII SOO!I3-6413(98)00013-5
FIBER DISTRIBUTION EFFECTS IN LINEAR VISCOELASTIC BEHAVIOR OF POLYMER
MATRIX
EXPERIMENTAL
COMPOSITES
: COMPUTATION
MODELLING
AND
COMPARISON
A. Agbossou & M. Lagache Laboratoim Mat&iaux Composites (LaMaCo), ESIGEC - Universitd de Savoie, 73376 Le Bourget du Lac - Chambdry, France. (Received 19 May 1997;acceptedforprint
IS September 1997)
Viilastic properties of composites with polymeric constituents am strongly temperature dependent. This behavior manifests itself in different ways : creep under constant load, stress relaxation under constant deformation, damping of dynamic response, etc. Recently it has been pointed out that the tune-dependent properties may be used to explain and chamcte& the interphase in polymer composite. The efficient design and utilisation of such composites in structurally related applications requites a good understanding of viscoelasticity. Property analysis of viscoelastic composites is closely mlated to the analysis of elastic composites via Lee-Man&l’s or Biot’s correspondence principle. Somewhat arbitrarily, all available predictions of mechanical properties am classiii into thme large classes : micm mechanical, analogical and statistical. The micro mechanical models have as an objective the establishment of a functional dependence between the random, heterogeneous microstructure and the macro-response. The statistical methods examine in their turn the validity of certain assumptions introduced into the mictomechanical models to enhance their tractability. The analogical methods provide simple and practice models. Relevant hterature which refers to viscoelastic behavior uses the micromechanical models. Among these models, several propose a particular space distribution of components. The two mictomechanial methods recommended for modelling composite elastic behavior are the periodic homogenization model and the general self-consistent model which produce satisfactory results for high and low reinforcement contents respectively. In actual composites them is local variation of fiber density, which could gives rise to a considerable discrepancy between theoretical and experimental results. Recently, Saffre et al. [ 1] have proposed using De Larrard and Le Roy’s [2] geometric model to along with Herve and Zaoui’s [3] n+l layered models in order to better define the morphological characteristics of a twophase composite. Although the proposed model improves the abiity to predict the mechanical characteristics of the high particulate composites with different distributions of particle sizes, the approximation still requires improvement, especially for fiber composites. The aim of the present work is to experimentally point out the effects of fiber distribution in the transversal viscoelastic behavior of two-phase composites and to propose a model based on
105
106
A. AGBOSSOU and M. LAGACHE
combination of periodical homogenization model (PHM) and the general self-consistent approach @SCM) to account for these effects. In section 1 we present the experimental results. Section 2 is devoted to present the PHM and GSCM micromechanical models and to propose a model which combines the two to take into account the effects of fiber distribution. Finally the paper concludes by applying this model to the specific composites studied, and by extending it to other fiber composites.
1 - Materials
In order to demonstrate the change in viscoelasctic behavior of a composite due to fiber distribution, two composites samples of viscoelastic epoxy reinforced by elastic continuous glass fibers were provided by Vetrotex International. These samples were made by filament winding, and were composed of approximately 50 % in volume fraction of (DY 026) matrix and 50 8 of E glass fiber (Et = 73 GPa and Vt = 0.22). ‘Ihe first group of samples (Sl) had a homogeneous fiber distribution (FIG. l-a). The second group (S2), which also contained the same fiber volume fraction, presented heterogeneous fiber distribution (FIG. l-b).
m
o”& 0 o”oooo
0~0000000 000 o(yoooo 0000 0 000
~~&~“~-f~o
000000
00
Composite (S I)
(a)
Composite (S2) lb)
FIG. 1 Fiber distribution in the analysed composite ; (a) hwpneous fiber distribution (b) heterogeneous fiber distribution. This
distribution, obtained by changing the winding velocity during manufacturing, was approximately 64 % at the top and 31 % at the bottom of the samples. All samples were cured at 100° C for 1 hr followed by post curing at 160’ C for 5 hr. After curing, the composite were cooled at room temperature in the oven. The winding samples were furally cut perpendicular to the fiber direction. The fiber’ volume fraction was determined by burning the composite samples at 600’ C for 2 hr. 2 - Dynamic mechanical spectrometxy Dynamic mechanical spectrometry was done on a Viscoanalyser (Metravib Company, France). This set-up provides the real (E’) and imaginary (E”) parts of the dynamic stress modulus and the internal friction tan $I( = E/E’) as a function of the temperature or as a function of the frequency under isothermal conditions. Frequency scans were performed by increasing the temperature from 30 T to 200 T at several frequencies over the range from 5 to 100 Hz. Several measurements were repeated for both
FIRER DISTRIBUTION IN COMPOSITES
107
frequency and tempemtum scans in order to verify that no physical ageing of the materials had occuned during the experiments. 3 - Experimental results and discussion FIG. 2 shows a typical photomicrograph section of the top (FIG. 2-a) and the bottom (FIG. 2-b) of samples (S2). These samples can be considered as a two layered composite with a mean fiber density of 2.5 fiber/mm2 at the top and 1.7 fiber/mm2 at the bottom. For samples (Sl) the photomicrograph analysis shows a homogeneous fiber distribution with a mean fiber density of
FIG. 2 Typical photomicrograph section of the sample (S2) : (a) top and (b) bottom of the sample. FIG. 3 shows the experimental storage modulus Ek and loss factor tan&,J of the epoxy matrix, the composites (Sl) and (S2) at 5 Hz. It can be observed that the two composites which have the same mean fiber volume fraction present a significant difference in their transverse storage modulus and loss factor. It is obvious that the transverse properties are sensitive to fiber distribution.
FIG. 3 Plots of experimental storage moduli and loss factors of ( -) Matrix, () composite (Sl) and (W composite (S2).
108
A-
A. AGBOSSOU .
.
.
and M. LAGACHE . .
.
For modelling the observed fiber distribution effects in viscoelastic behavior we propose in this section a model based on two homogenization models (PHM and GSCM), which produce satisfactory results for high and low reinforcement contents respectively. The basic procedure for these models is as follows : - select a microscopic volume in the material, which could be considered as a representative elementary volume (RVE). The RVE and its boundaries are respectively noted V and &. - apply average strain (respectively stmss) to the RVE with appropriate boundary conditions that produce homogeneous fields in a homogeneous medium. For elastic bodies, the two classic ui = Eti.xj over 3V (a) (1) kinds of homogeneous boundary conditions am : i T = Z,.nj over aV (b) where Elj is a constant strain and Cij is a constant stress. - calculate the average stress (respectively strain) on the RVE. - determine, from the relation between average strain and stress, the elastic properties of the X=Q:E composite medium (2) where C and E am average stress and strain over the RVE and Q is the stiffness tensor. 1 - General Self-Consistent
Scheme (GSCS)
In the first version of this model [4], [5], it was assumed that each fiber is embedded in a homogeneous boundless medium, which had the unknown properties of the composite. The homogeneous medium was subjected to a boundary condition of the avemge strain type at infinity (Equation l-a). After computation of the boundary value problem, algebraic equations could be obtained for elastic properties of the composite. An improved model, sometimes called (GSCS) model or double-embedded scheme, was developed by Kemer and Christensen [4], [6]. In this model, the medium considered is a concentric cylindrical body (FIG. 4-a), where : - the innermost cylinder is the fiber, - the next ring models the matrix, - the outermost ring represents a homogenous material with the unknown effective properties of the composite. The fiber and the matrix volume fraction were directly taken into account. Indeed, the mtio (a/6)’ represents the fiber volume contents. In spite of this improvement, the GSCS model tends to mask the fiber correlation effects (e.g. fiber-fiber interaction effects) which can become important at high volume fraction concentrations. The algebraic formulae of the GSCS model [4] have been computed, with complex variables in FORTRAN 171. 2 - Periodic Homogeneization
Model (PHM)
This homogeneization technique, which appeared in the 1970’s, assumes that the microstructure of the composite is mechanically and geometrically periodic. This method was especially developed by Sanchez-Palencia [8], Duvaut et al. [9]. Suquet [IO], among others. It assumes periodic
FIRER DISTRIBUTION IN COMPOSITES
109
distribution of the RVE. On account of the transverse isotropy of the unidirectional composite material, it could be proved that this period has a hexagonal structure as shown in FIG. 4.b.
FIG. 4 (a) General Self Consistent Scheme GSCM (b) RVE of a unidirectional composite material PHM. The set of boundary conditions on the RVE have to preserve the perk&city of the structure :
Iui = iii + Eij.xj
for every point M(x ,,xZ,xJ) of V
17~ is periodic on dv, i.e. ri has the same value at opposite points of aV bi.nj is antiperiodic on
(3)
i.e. the stress vector has opposite value at opposites points of
The conditions (3) imply that in the periodic medium the mean value of a is exactly E ( (Eij>= E, ) . In the same manner as the strain, the macroscopic stress tensor is the mean value of 6, i.e. ( oi >” = 2,. i! is assumed continuous and with bounded first derivatives, i.e. iii E H,( V ) in terms of Sobolev Space. Under these conditions i; satisfies the divergence theorem. In addition, it can be shown [lo] that conditions (1) and (3) give a single (a, u) solution in the RVE. Except near the boundary of the composite structure, this solution could be extended to all periodic media. In this case the filler interaction effects arc totally taken into account especially for high fiber volume fraction. For further detail, we recommend Sanchez-Palancia’s and Suquet’s lectures [8], [ 101. The material studied in this paper is a unidirectional fiber composite. Thus its propertics along fiber direction y3 arc constant. Consequently, the RVE (a, u) solution does not depend on the length of the RVE along the direction y9 To take that into account, we can assume the following relations in theRVE: l;ri=ii(Yl*YZ) (4) and
cii = & + Ei ; 2, = iti (fi) ; & = 0 ; $, = +
;
In relations (4) and (5). comma derivation notation is used. At every point of the RVE, with homogeneous and isotropic constituents, the local stress-strain elastic relations become :
A. AGBOSSOU
110
and M. LAGACHE
G, +&I 222 +E22 43
0
0 0
2.Z,2 + 2.E,
Yc O O 0 ’ x ’ 0 0 ’ %
0
O
0 0
(6)
SE,, 3 h+E,, ?h
In the case of a circular fiber in an RVE with two planes of symmetry,
the resulting elastic material
is orthotropic. So, the relations between mean values Z and E hold :
YE,
-5,
-v&
-“I/E, )/E, -v$,
-vs2
fi3
0
0
0
0
0
0
0
0
0
’ 41 E22 E33 =
0
0
0
Y G2
0
0
0
0
0
0
’
’
’
Y G 23
’
0
0
YG
2.43 .2’E31 31
The matrix of equation (6) can be divided in two disconnected matrixconcerns O23
9 c31
and&
~ll,~22,033,c~12
variables.
analogous sub-systems.
andat
Furthermore
(7)
2q2
sub-matrices
: the first (4x4) sub-
variables,
while the second is related only to
the macroscopic
law (7) can also be divided in two
,&
Finally, the general problem (7) can be decomposed
problems : (i) the generalized plane strain problem noted Pl in which i.l3 , E,
in two different 2-D , E3t ,623 and 631
are eliminated, and (ii) the antiplane problem noted P2 in which only ii, , E2, , E3t ,623 and 631 do not vanish. If, for instance, we impose only El2 = E.o. the elastic problem Pl under condition (3) in the RVE solution gives Et2 = (0 t2)” from the (CJ , u) elastic solution. E equation (7) leads to G,, ( G ,2 = <
The introduction
of the El2
in
). Moreover, inverting (7), we can note that the solution of
four problems Pl and two problems P2 with only one non-zero term in the macroscopic strain E, leads to the five independent components of the stiffness matrix of the transverse isotropic media. Following a Desbordes’s [ 111 suggestion in 3-D problem, the periodic homogenization problems have been solved by finite element formulation [ 121 in FORTRAN using complex variables [ 131.
111
FIRER DISTRIBUTION IN COMPOSITES 3 - Combination model
As shown in the previous sections and as observed by many authors in the literature, actual composites present inhomogeneous fiber distributions (aggregate, ateas of high fibers, ateas of low fibers, ...) which affect experimental results. A simple and practical method to take into account this effect is to assume that the representative volume of the composite is as shown in FIG. 5, with areas of high fiber (I-IF) volume fraction (more than 50 96) and atea of low fiber (LF) volume fraction. These areas can be grouped, giving a composite consisting of two major parts, one exclusively I-IFand the other exclusively LF, as shown in FIG. 5-b. lbe composite can now be cons&ted to be either a region HF and a region LF in series, together in parallel with the remaining portion LF (FIG. 5-c). or a region HF and a region LF in parallel, together in series with the remaining portion LF (FIG. 5-d).
Cc)
-It (4 (a)
FIG. 5 (a) actual composite ; (b) combination models ; (c) parallel-series model ; (d) series-parallel model. In the (I-IF) and (LF) portions PHM and GSCM models were applied respectively. The proposed combination model consists of applying the GSCM model to the region LF and the PHM model to the region HF. Therefore the complex modulus E* of the composite is expressed by 1141,[I51 : 0-4d E*= [ (1-h)E;A+kEk+G where E,‘(,
l
(8)
r
and E&,,., am the complex moduli of LF region and HF region respectively. The
parameter k expressing the degree of parallelism and the parameter $ expressing the degree of series connection are defined as shown in FIG. 5 with h $ = v(w) where v~p, represents the proportion of I-IFcomposite in the sample.
A. AGBOSSOU
112
and M. LAGACHE
A specific program which includes GSCS, PHM and the combination with complex variables in FORTRAN. . . . of fiber dtstnbutton 1 - Computation
. . m vtsco&&c
models has been developed
behavtor
and experimental comparison
FIG. 6 shows the predicted and experimental results. One can note that for a fixed homogeneous fiber volume fraction the GSCM model gives a curve slightly higher than that predicted by the PHM model, a fact that has been observed by other authors. Both curves for samples (Sl) are within experimental error, showing that all the models work well.
Computation modelling and experimental comparison. Experimental Predicted (-
) matrix, ( I
(-
, composite (Sl),
) composite (Sl) by GSCM, tand (-
(-
) composite (S2),
1 composite (Sl) by PHM
) composite (S2) by combination
model.
For samples (S2) we can observe that the GSCM and PHM models am not sensitive to the fiber distribution. Only the combination model fits well the experimental data of these samples because it takes into account their particular fiber distribution. This two-layer composite can be modelled as a combination of a region HF (vf = 64 %) and a region LF (vf = 31 %) in parallel, with the HF region dominate (with 2/3 of the thickness of the samples). This explains the increase in storage modulus and the decrease in loss factor in comparison to the samples (Sl). 2 - Numerical analysis of the fiber distribution effect in viscoelastic behavior FIG. 7 shows in 3-D diagrams the evolution of the storage modulus and the loss factor of composite ( v,,, = 20 96 ; region HF vr = 64 % ; region LF vt = 3 1 %I) with the increasing of h in the range of 0.2 to 1. For a fixed v,,, one can note that as the value of h approaches one, the nature of the (LF)-(HF) series is intensified and the viscoelastic behavior of low fiber composite predominates. In the reverse case of tj~approaching one, the character of the (LF)-(HF) parallelism is intensified and a large fall in loss factor with increasing temperature appears. Therefore, for a fixed proportion of
FIRER DISTRIBUTION IN COMPOSITES
113
HF, it is necessaq to take the distribution into account because them is a remarkable fall in storage modulus and loss factor with a change in the value of parameter k.
FIG. 7 3-D diagram of the composite (a) storage modulus and (b) loss factor as function of parameter A and tempemture for a tixed aggregate volume fraction v_ = 20% at 5 Hz. FIG. 8 shows, in 3-D diagrams, a typical dependence of the temperature on storage modulus and loss factor at 5 Hz for composite with (VW,= 1- v& in the range of 5% to 25% (region HF vf = 64 96 ; region LF vr = 31%). From this figure it follows that the discrepancy between experimental and theoretical data for high fiber composites is not necessarily due to a change in matrix properties or the existence of mesophase as proposed by Theocaris[l6]. The fiber distribution in and of itself could cause a significant modification in viscoelastic behavior.
(a)
(b)
FIG. 8 3-D diagram of the composite (a) storage modulus, (b) loss factor as function of aggregate volume fraction (0.05 < vcHFj<20%) and temperature for a different values of parameter Aand 41at 5 Hz.
114
A. AGBOSSOU
and M. LAGACHE
The aim of the present work is to experimentally point out the effects of fiber distributions in transversal viscoelastic behavior of two-phase composites. By changing the winding velocity during manufacturing, two specific fiber composites were made and tested in transverse direction. The experimental results show that the transverse properties ate sensitive to fiber distribution. To account for the effect of fiber distribution in viscoelastic behavior of two-phase composites we proposed a model based on the periodical homogenization model (PHM) and the general selfconsistent model (GSCM). ‘Ihe method presented fits the observed experimentaI data well. Applying this model, it has been shown that them are particular fiber distributions which could modify the storage modulus with no change of in-situ matrix properties. The proposed model could be used to analyse the fiber aggregate effect in viscoelastic behavior of two-phase composite.
Acknowledaements The authors acknowledge Kendall Isaac for his help in preparing this manuscript. We would also like to thank Vetrotex International (Mrs D. Muller, J. Abba) and Composi’tec (Mrs J.L Favre, Y. Gardet) for putting the experimental equipment at our disposal. The authors ate also indebted to Pr. J. Pastor for his scientific help in this work.
FIBER DISTRIBUTION IN COMPOSITES
115
1. SAFFRE P., BERGERET A., LIGIER J.L. and PASTOR J., A four-layered self-consistent scheme applied to two-phase composites, Mech. res. comm., Vol. 23, N’4, p. 359-366 (1996) 2. DE LARRARD F., LE ROY R., Un mod&legkom&rique dhomog&u%ation pour les composites bi-phasiques a inclusion gram&tire de large &endue, C. R. Acad. Sci., Vol. 314, SBrie II, p. 1253 (1992) 3. HERVE E. and ZAOUI A., N-layered inclusion-based micromechanical modelling, Int. J. Eng. Sci., Vol. 31, No 1, p. 10 (1993) 4. CHRISTENSEN R.M. ; Mechanics of composite materials, New-York, Wiley Intersc. (1979) 5. HILL R., A self-consistent mechanics of composite materials, J. of Mech. and Phys. of Solids, Vol. 13, p. 213-222 (1965) 6. CHRISTENSEN R.M. and LO, R. H., Solutions for effective shear properties in three phase sphere and cylinder models, J. of Mech. and Phys. of solids, Vol. 27, p. 315-330 (1979) 7. AGBOSSOU A., An approximate methode for the determination of Poisson’s complex ratio in harmonic viscoelastic behavior, Polymer composites, Vol. 16, No 2, p. 135-143 (1995) 8. SANCHEZPALENCIA E., Non homogenous media and vibration theory, Lecture notes in Physics, n0127, Springer-Verlag. Berlin (1980) 9. DUVAUT G., Analyse fonctionnclle et mdcanique des milieux continus. Application a l’&ude des mat&iaux composites B structure p&iodique. Homogk?isation, Theoretical and Applied Mechanics, WT Koiter Ed. North Holland Publishing Compagny, 119-134 (1976) 10. SUQUET, P.M., Elements of homogeneization for inelastic solids mechanics, Homogeneization techniques for composite media, CISM Lectures, Udine, Ed. Springer-Verlag (1987) 11. DESBORDES, 0.. LICHT C., MARIGO J.J., MIALON P. MICHEL J.C. & SUQUET P., Analyse limite dc structures fottement heterogencs, 3e Colloque tendances actuclles en calcul des structures, Ed. Pluralis, Batia (1985) 12. LAGACHE M., AGBOSSOU A. and PASTOR J., Role of interphase on the elastic behavior of composite materails : theoretical and experimental analysis, J. camp. mat., Vol. 28, no 12, p. 1140-l 157 (1994) 13. NGUYEN VIET H., PASTOR J. and MULLER D., A method for predicting linear viscoelastic mechanical behavior of composites, a comparison with other methods and experimental validation, Eur. J. Mech., A/Solids, Vol. 14, No 6, p. 939-960 (1995) 14. WARD I. M. ; Mechanical properties of solid polymers, John Wiley & Sons, N-Y (1971) 15. TAKAYANAGI M., IMADA K. and KAJIYAMA T., Mechanical properties and fine structure of drawn polymers, J. Polym. SC. Part C, No 15, p. 263-281 (1966) 16. THEOCARIS P.S.; The Mesophase Concept in Composites, Ed. Springer-Verlag, (1987).
PII SOO!I3-6413(98)00013-5
FIBER DISTRIBUTION EFFECTS IN LINEAR VISCOELASTIC BEHAVIOR OF POLYMER
MATRIX
EXPERIMENTAL
COMPOSITES
: COMPUTATION
MODELLING
AND
COMPARISON
A. Agbossou & M. Lagache Laboratoim Mat&iaux Composites (LaMaCo), ESIGEC - Universitd de Savoie, 73376 Le Bourget du Lac - Chambdry, France. (Received 19 May 1997;acceptedforprint
IS September 1997)
Viilastic properties of composites with polymeric constituents am strongly temperature dependent. This behavior manifests itself in different ways : creep under constant load, stress relaxation under constant deformation, damping of dynamic response, etc. Recently it has been pointed out that the tune-dependent properties may be used to explain and chamcte& the interphase in polymer composite. The efficient design and utilisation of such composites in structurally related applications requites a good understanding of viscoelasticity. Property analysis of viscoelastic composites is closely mlated to the analysis of elastic composites via Lee-Man&l’s or Biot’s correspondence principle. Somewhat arbitrarily, all available predictions of mechanical properties am classiii into thme large classes : micm mechanical, analogical and statistical. The micro mechanical models have as an objective the establishment of a functional dependence between the random, heterogeneous microstructure and the macro-response. The statistical methods examine in their turn the validity of certain assumptions introduced into the mictomechanical models to enhance their tractability. The analogical methods provide simple and practice models. Relevant hterature which refers to viscoelastic behavior uses the micromechanical models. Among these models, several propose a particular space distribution of components. The two mictomechanial methods recommended for modelling composite elastic behavior are the periodic homogenization model and the general self-consistent model which produce satisfactory results for high and low reinforcement contents respectively. In actual composites them is local variation of fiber density, which could gives rise to a considerable discrepancy between theoretical and experimental results. Recently, Saffre et al. [ 1] have proposed using De Larrard and Le Roy’s [2] geometric model to along with Herve and Zaoui’s [3] n+l layered models in order to better define the morphological characteristics of a twophase composite. Although the proposed model improves the abiity to predict the mechanical characteristics of the high particulate composites with different distributions of particle sizes, the approximation still requires improvement, especially for fiber composites. The aim of the present work is to experimentally point out the effects of fiber distribution in the transversal viscoelastic behavior of two-phase composites and to propose a model based on
105
106
A. AGBOSSOU and M. LAGACHE
combination of periodical homogenization model (PHM) and the general self-consistent approach @SCM) to account for these effects. In section 1 we present the experimental results. Section 2 is devoted to present the PHM and GSCM micromechanical models and to propose a model which combines the two to take into account the effects of fiber distribution. Finally the paper concludes by applying this model to the specific composites studied, and by extending it to other fiber composites.
1 - Materials
In order to demonstrate the change in viscoelasctic behavior of a composite due to fiber distribution, two composites samples of viscoelastic epoxy reinforced by elastic continuous glass fibers were provided by Vetrotex International. These samples were made by filament winding, and were composed of approximately 50 % in volume fraction of (DY 026) matrix and 50 8 of E glass fiber (Et = 73 GPa and Vt = 0.22). ‘Ihe first group of samples (Sl) had a homogeneous fiber distribution (FIG. l-a). The second group (S2), which also contained the same fiber volume fraction, presented heterogeneous fiber distribution (FIG. l-b).
m
o”& 0 o”oooo
0~0000000 000 o(yoooo 0000 0 000
~~&~“~-f~o
000000
00
Composite (S I)
(a)
Composite (S2) lb)
FIG. 1 Fiber distribution in the analysed composite ; (a) hwpneous fiber distribution (b) heterogeneous fiber distribution. This
distribution, obtained by changing the winding velocity during manufacturing, was approximately 64 % at the top and 31 % at the bottom of the samples. All samples were cured at 100° C for 1 hr followed by post curing at 160’ C for 5 hr. After curing, the composite were cooled at room temperature in the oven. The winding samples were furally cut perpendicular to the fiber direction. The fiber’ volume fraction was determined by burning the composite samples at 600’ C for 2 hr. 2 - Dynamic mechanical spectrometxy Dynamic mechanical spectrometry was done on a Viscoanalyser (Metravib Company, France). This set-up provides the real (E’) and imaginary (E”) parts of the dynamic stress modulus and the internal friction tan $I( = E/E’) as a function of the temperature or as a function of the frequency under isothermal conditions. Frequency scans were performed by increasing the temperature from 30 T to 200 T at several frequencies over the range from 5 to 100 Hz. Several measurements were repeated for both
FIRER DISTRIBUTION IN COMPOSITES
107
frequency and tempemtum scans in order to verify that no physical ageing of the materials had occuned during the experiments. 3 - Experimental results and discussion FIG. 2 shows a typical photomicrograph section of the top (FIG. 2-a) and the bottom (FIG. 2-b) of samples (S2). These samples can be considered as a two layered composite with a mean fiber density of 2.5 fiber/mm2 at the top and 1.7 fiber/mm2 at the bottom. For samples (Sl) the photomicrograph analysis shows a homogeneous fiber distribution with a mean fiber density of
FIG. 2 Typical photomicrograph section of the sample (S2) : (a) top and (b) bottom of the sample. FIG. 3 shows the experimental storage modulus Ek and loss factor tan&,J of the epoxy matrix, the composites (Sl) and (S2) at 5 Hz. It can be observed that the two composites which have the same mean fiber volume fraction present a significant difference in their transverse storage modulus and loss factor. It is obvious that the transverse properties are sensitive to fiber distribution.
FIG. 3 Plots of experimental storage moduli and loss factors of ( -) Matrix, () composite (Sl) and (W composite (S2).
108
A-
A. AGBOSSOU .
.
.
and M. LAGACHE . .
.
For modelling the observed fiber distribution effects in viscoelastic behavior we propose in this section a model based on two homogenization models (PHM and GSCM), which produce satisfactory results for high and low reinforcement contents respectively. The basic procedure for these models is as follows : - select a microscopic volume in the material, which could be considered as a representative elementary volume (RVE). The RVE and its boundaries are respectively noted V and &. - apply average strain (respectively stmss) to the RVE with appropriate boundary conditions that produce homogeneous fields in a homogeneous medium. For elastic bodies, the two classic ui = Eti.xj over 3V (a) (1) kinds of homogeneous boundary conditions am : i T = Z,.nj over aV (b) where Elj is a constant strain and Cij is a constant stress. - calculate the average stress (respectively strain) on the RVE. - determine, from the relation between average strain and stress, the elastic properties of the X=Q:E composite medium (2) where C and E am average stress and strain over the RVE and Q is the stiffness tensor. 1 - General Self-Consistent
Scheme (GSCS)
In the first version of this model [4], [5], it was assumed that each fiber is embedded in a homogeneous boundless medium, which had the unknown properties of the composite. The homogeneous medium was subjected to a boundary condition of the avemge strain type at infinity (Equation l-a). After computation of the boundary value problem, algebraic equations could be obtained for elastic properties of the composite. An improved model, sometimes called (GSCS) model or double-embedded scheme, was developed by Kemer and Christensen [4], [6]. In this model, the medium considered is a concentric cylindrical body (FIG. 4-a), where : - the innermost cylinder is the fiber, - the next ring models the matrix, - the outermost ring represents a homogenous material with the unknown effective properties of the composite. The fiber and the matrix volume fraction were directly taken into account. Indeed, the mtio (a/6)’ represents the fiber volume contents. In spite of this improvement, the GSCS model tends to mask the fiber correlation effects (e.g. fiber-fiber interaction effects) which can become important at high volume fraction concentrations. The algebraic formulae of the GSCS model [4] have been computed, with complex variables in FORTRAN 171. 2 - Periodic Homogeneization
Model (PHM)
This homogeneization technique, which appeared in the 1970’s, assumes that the microstructure of the composite is mechanically and geometrically periodic. This method was especially developed by Sanchez-Palencia [8], Duvaut et al. [9]. Suquet [IO], among others. It assumes periodic
FIRER DISTRIBUTION IN COMPOSITES
109
distribution of the RVE. On account of the transverse isotropy of the unidirectional composite material, it could be proved that this period has a hexagonal structure as shown in FIG. 4.b.
FIG. 4 (a) General Self Consistent Scheme GSCM (b) RVE of a unidirectional composite material PHM. The set of boundary conditions on the RVE have to preserve the perk&city of the structure :
Iui = iii + Eij.xj
for every point M(x ,,xZ,xJ) of V
17~ is periodic on dv, i.e. ri has the same value at opposite points of aV bi.nj is antiperiodic on
(3)
i.e. the stress vector has opposite value at opposites points of
The conditions (3) imply that in the periodic medium the mean value of a is exactly E ( (Eij>= E, ) . In the same manner as the strain, the macroscopic stress tensor is the mean value of 6, i.e. ( oi >” = 2,. i! is assumed continuous and with bounded first derivatives, i.e. iii E H,( V ) in terms of Sobolev Space. Under these conditions i; satisfies the divergence theorem. In addition, it can be shown [lo] that conditions (1) and (3) give a single (a, u) solution in the RVE. Except near the boundary of the composite structure, this solution could be extended to all periodic media. In this case the filler interaction effects arc totally taken into account especially for high fiber volume fraction. For further detail, we recommend Sanchez-Palancia’s and Suquet’s lectures [8], [ 101. The material studied in this paper is a unidirectional fiber composite. Thus its propertics along fiber direction y3 arc constant. Consequently, the RVE (a, u) solution does not depend on the length of the RVE along the direction y9 To take that into account, we can assume the following relations in theRVE: l;ri=ii(Yl*YZ) (4) and
cii = & + Ei ; 2, = iti (fi) ; & = 0 ; $, = +
;
In relations (4) and (5). comma derivation notation is used. At every point of the RVE, with homogeneous and isotropic constituents, the local stress-strain elastic relations become :
A. AGBOSSOU
110
and M. LAGACHE
G, +&I 222 +E22 43
0
0 0
2.Z,2 + 2.E,
Yc O O 0 ’ x ’ 0 0 ’ %
0
O
0 0
(6)
SE,, 3 h+E,, ?h
In the case of a circular fiber in an RVE with two planes of symmetry,
the resulting elastic material
is orthotropic. So, the relations between mean values Z and E hold :
YE,
-5,
-v&
-“I/E, )/E, -v$,
-vs2
fi3
0
0
0
0
0
0
0
0
0
’ 41 E22 E33 =
0
0
0
Y G2
0
0
0
0
0
0
’
’
’
Y G 23
’
0
0
YG
2.43 .2’E31 31
The matrix of equation (6) can be divided in two disconnected matrixconcerns O23
9 c31
and&
~ll,~22,033,c~12
variables.
analogous sub-systems.
andat
Furthermore
(7)
2q2
sub-matrices
: the first (4x4) sub-
variables,
while the second is related only to
the macroscopic
law (7) can also be divided in two
,&
Finally, the general problem (7) can be decomposed
problems : (i) the generalized plane strain problem noted Pl in which i.l3 , E,
in two different 2-D , E3t ,623 and 631
are eliminated, and (ii) the antiplane problem noted P2 in which only ii, , E2, , E3t ,623 and 631 do not vanish. If, for instance, we impose only El2 = E.o. the elastic problem Pl under condition (3) in the RVE solution gives Et2 = (0 t2)” from the (CJ , u) elastic solution. E equation (7) leads to G,, ( G ,2 = <
The introduction
of the El2
in
). Moreover, inverting (7), we can note that the solution of
four problems Pl and two problems P2 with only one non-zero term in the macroscopic strain E, leads to the five independent components of the stiffness matrix of the transverse isotropic media. Following a Desbordes’s [ 111 suggestion in 3-D problem, the periodic homogenization problems have been solved by finite element formulation [ 121 in FORTRAN using complex variables [ 131.
111
FIRER DISTRIBUTION IN COMPOSITES 3 - Combination model
As shown in the previous sections and as observed by many authors in the literature, actual composites present inhomogeneous fiber distributions (aggregate, ateas of high fibers, ateas of low fibers, ...) which affect experimental results. A simple and practical method to take into account this effect is to assume that the representative volume of the composite is as shown in FIG. 5, with areas of high fiber (I-IF) volume fraction (more than 50 96) and atea of low fiber (LF) volume fraction. These areas can be grouped, giving a composite consisting of two major parts, one exclusively I-IFand the other exclusively LF, as shown in FIG. 5-b. lbe composite can now be cons&ted to be either a region HF and a region LF in series, together in parallel with the remaining portion LF (FIG. 5-c). or a region HF and a region LF in parallel, together in series with the remaining portion LF (FIG. 5-d).
Cc)
-It (4 (a)
FIG. 5 (a) actual composite ; (b) combination models ; (c) parallel-series model ; (d) series-parallel model. In the (I-IF) and (LF) portions PHM and GSCM models were applied respectively. The proposed combination model consists of applying the GSCM model to the region LF and the PHM model to the region HF. Therefore the complex modulus E* of the composite is expressed by 1141,[I51 : 0-4d E*= [ (1-h)E;A+kEk+G where E,‘(,
l
(8)
r
and E&,,., am the complex moduli of LF region and HF region respectively. The
parameter k expressing the degree of parallelism and the parameter $ expressing the degree of series connection are defined as shown in FIG. 5 with h $ = v(w) where v~p, represents the proportion of I-IFcomposite in the sample.
A. AGBOSSOU
112
and M. LAGACHE
A specific program which includes GSCS, PHM and the combination with complex variables in FORTRAN. . . . of fiber dtstnbutton 1 - Computation
. . m vtsco&&c
models has been developed
behavtor
and experimental comparison
FIG. 6 shows the predicted and experimental results. One can note that for a fixed homogeneous fiber volume fraction the GSCM model gives a curve slightly higher than that predicted by the PHM model, a fact that has been observed by other authors. Both curves for samples (Sl) are within experimental error, showing that all the models work well.
Computation modelling and experimental comparison. Experimental Predicted (-
) matrix, ( I
(-
, composite (Sl),
) composite (Sl) by GSCM, tand (-
(-
) composite (S2),
1 composite (Sl) by PHM
) composite (S2) by combination
model.
For samples (S2) we can observe that the GSCM and PHM models am not sensitive to the fiber distribution. Only the combination model fits well the experimental data of these samples because it takes into account their particular fiber distribution. This two-layer composite can be modelled as a combination of a region HF (vf = 64 %) and a region LF (vf = 31 %) in parallel, with the HF region dominate (with 2/3 of the thickness of the samples). This explains the increase in storage modulus and the decrease in loss factor in comparison to the samples (Sl). 2 - Numerical analysis of the fiber distribution effect in viscoelastic behavior FIG. 7 shows in 3-D diagrams the evolution of the storage modulus and the loss factor of composite ( v,,, = 20 96 ; region HF vr = 64 % ; region LF vt = 3 1 %I) with the increasing of h in the range of 0.2 to 1. For a fixed v,,, one can note that as the value of h approaches one, the nature of the (LF)-(HF) series is intensified and the viscoelastic behavior of low fiber composite predominates. In the reverse case of tj~approaching one, the character of the (LF)-(HF) parallelism is intensified and a large fall in loss factor with increasing temperature appears. Therefore, for a fixed proportion of
FIRER DISTRIBUTION IN COMPOSITES
113
HF, it is necessaq to take the distribution into account because them is a remarkable fall in storage modulus and loss factor with a change in the value of parameter k.
FIG. 7 3-D diagram of the composite (a) storage modulus and (b) loss factor as function of parameter A and tempemture for a tixed aggregate volume fraction v_ = 20% at 5 Hz. FIG. 8 shows, in 3-D diagrams, a typical dependence of the temperature on storage modulus and loss factor at 5 Hz for composite with (VW,= 1- v& in the range of 5% to 25% (region HF vf = 64 96 ; region LF vr = 31%). From this figure it follows that the discrepancy between experimental and theoretical data for high fiber composites is not necessarily due to a change in matrix properties or the existence of mesophase as proposed by Theocaris[l6]. The fiber distribution in and of itself could cause a significant modification in viscoelastic behavior.
(a)
(b)
FIG. 8 3-D diagram of the composite (a) storage modulus, (b) loss factor as function of aggregate volume fraction (0.05 < vcHFj<20%) and temperature for a different values of parameter Aand 41at 5 Hz.
114
A. AGBOSSOU
and M. LAGACHE
The aim of the present work is to experimentally point out the effects of fiber distributions in transversal viscoelastic behavior of two-phase composites. By changing the winding velocity during manufacturing, two specific fiber composites were made and tested in transverse direction. The experimental results show that the transverse properties ate sensitive to fiber distribution. To account for the effect of fiber distribution in viscoelastic behavior of two-phase composites we proposed a model based on the periodical homogenization model (PHM) and the general selfconsistent model (GSCM). ‘Ihe method presented fits the observed experimentaI data well. Applying this model, it has been shown that them are particular fiber distributions which could modify the storage modulus with no change of in-situ matrix properties. The proposed model could be used to analyse the fiber aggregate effect in viscoelastic behavior of two-phase composite.
Acknowledaements The authors acknowledge Kendall Isaac for his help in preparing this manuscript. We would also like to thank Vetrotex International (Mrs D. Muller, J. Abba) and Composi’tec (Mrs J.L Favre, Y. Gardet) for putting the experimental equipment at our disposal. The authors ate also indebted to Pr. J. Pastor for his scientific help in this work.
FIBER DISTRIBUTION IN COMPOSITES
115
1. SAFFRE P., BERGERET A., LIGIER J.L. and PASTOR J., A four-layered self-consistent scheme applied to two-phase composites, Mech. res. comm., Vol. 23, N’4, p. 359-366 (1996) 2. DE LARRARD F., LE ROY R., Un mod&legkom&rique dhomog&u%ation pour les composites bi-phasiques a inclusion gram&tire de large &endue, C. R. Acad. Sci., Vol. 314, SBrie II, p. 1253 (1992) 3. HERVE E. and ZAOUI A., N-layered inclusion-based micromechanical modelling, Int. J. Eng. Sci., Vol. 31, No 1, p. 10 (1993) 4. CHRISTENSEN R.M. ; Mechanics of composite materials, New-York, Wiley Intersc. (1979) 5. HILL R., A self-consistent mechanics of composite materials, J. of Mech. and Phys. of Solids, Vol. 13, p. 213-222 (1965) 6. CHRISTENSEN R.M. and LO, R. H., Solutions for effective shear properties in three phase sphere and cylinder models, J. of Mech. and Phys. of solids, Vol. 27, p. 315-330 (1979) 7. AGBOSSOU A., An approximate methode for the determination of Poisson’s complex ratio in harmonic viscoelastic behavior, Polymer composites, Vol. 16, No 2, p. 135-143 (1995) 8. SANCHEZPALENCIA E., Non homogenous media and vibration theory, Lecture notes in Physics, n0127, Springer-Verlag. Berlin (1980) 9. DUVAUT G., Analyse fonctionnclle et mdcanique des milieux continus. Application a l’&ude des mat&iaux composites B structure p&iodique. Homogk?isation, Theoretical and Applied Mechanics, WT Koiter Ed. North Holland Publishing Compagny, 119-134 (1976) 10. SUQUET, P.M., Elements of homogeneization for inelastic solids mechanics, Homogeneization techniques for composite media, CISM Lectures, Udine, Ed. Springer-Verlag (1987) 11. DESBORDES, 0.. LICHT C., MARIGO J.J., MIALON P. MICHEL J.C. & SUQUET P., Analyse limite dc structures fottement heterogencs, 3e Colloque tendances actuclles en calcul des structures, Ed. Pluralis, Batia (1985) 12. LAGACHE M., AGBOSSOU A. and PASTOR J., Role of interphase on the elastic behavior of composite materails : theoretical and experimental analysis, J. camp. mat., Vol. 28, no 12, p. 1140-l 157 (1994) 13. NGUYEN VIET H., PASTOR J. and MULLER D., A method for predicting linear viscoelastic mechanical behavior of composites, a comparison with other methods and experimental validation, Eur. J. Mech., A/Solids, Vol. 14, No 6, p. 939-960 (1995) 14. WARD I. M. ; Mechanical properties of solid polymers, John Wiley & Sons, N-Y (1971) 15. TAKAYANAGI M., IMADA K. and KAJIYAMA T., Mechanical properties and fine structure of drawn polymers, J. Polym. SC. Part C, No 15, p. 263-281 (1966) 16. THEOCARIS P.S.; The Mesophase Concept in Composites, Ed. Springer-Verlag, (1987).