Experimental Values for the Elastic Constants of a Particulate-Filled Glassy Polymer

Young’s modulus and Poisson’s ratio have been measured simultaneously on a series of particulate composites containing volume fractions of filler up to 0.50. The composites consisted of small glass spheres imbedded in a rigid epoxy polymer matrix. The measured values were compared with theoretical values calculated from current theories. A recently generalized and simplified version of van der Poel’s theory provided the best agreement. It predicted values of Young’s modulus for composites with filler volume fractions up to 0.35. Measured values of Poisson’s ratio exhibited scattering, but were consistent with values calculated from van der Poel’s theory.


Introduction
It has bee n known for so me tim e that th e additi on of a partic ulate filler to a polymeric material can greatly affect the elastic properties of the res ulting co mposite. The effect is a mechani cal one co mpli cated by several superimposed effects of a c he mical and physical nature . The pure mechanical effect is best und er stood. It is accounted for in terms of the local strain distortions due to dissimilarities be tween the matrix and filler material. Variou s th eori es have been propounded to explain it, and revie ws of these th eories are available [1-3)1.
In order to test these theories good data are needed for a series of macroscopically homogeneous and isotropi c composites containin g various known amounts of fille r. The filler and matrix materials should be isotropi c, and the two elastic co nstants necessary to characterize each of the m should be known. These two elastic cons tants s hould preferably be obtained by simultaneous meas ure me nts on the same specimen in orde r to min imize effects due to viscoelasticity and s pecime n variation.
There are a numbe r of physical a nd c he mi cal effec ts that mu s t be considered in c hoosing an appropri ate co mposite sys te m for testin g. Th e rubbercarbon blac k sys te m for instance is com plicated by the following effects [4][5][6]: The small carbon particles I Figu res in bracke ts indica te th e li te rature refe rence s at th e e nd of thi s paper. 45 present in large numbers bond to the rubber molec ules and signifi cantly increase th e degree of cr osslin king of th e rubber matrix ; th e carbon particles associate together in lon g c hain s thus producing a stru c ture effect ; and th e filled rubbe r co mposite softens after a previou s deformation. For th ese reasons thi s sys te m does not provide a satisfactory tes t of theories d ealin g with th e mechanical effect.
In some cases the presence of the filler particle may induce crystallization, crosslin kin g or other s truc tural changes in the s urrounding s hell of polymer matrix , thus introducing an additional phase with conseque nt changes in mechanical prope rties [7][8][9][10][11]. This effect should be avoided by using a filler whic h adheres firmly to the matrix, but doe~ not exercise a long-ran ge influence on the matrix structure. In addition the filler particle dimensions should be large enough that the total fill er surface area is comparatively small.
The room temperature be havior of a composite form ed at elevated temperatures may be affected by froz en-in stresses due to differences in th e thermal expansion properties of the m atrix and filler material [12]. In cas t co mposites thi s effect can be reduced by gelling at room te mperature, and coolin g slowly after an y post c ure at elevated te mperatures.
Th e s hape and orie ntation of the filler particles may cau se the co mposition to be ani sotropic, but this effec t is easily avoid ed by usin g s pherical filler particles.
Previous checks of the theori es have met so me of the above criteria in various ways. For those com· posites consisting of a rigid filler imbedded in a rubbery matrix of Poisson's ratio 0.5, the shear modulus of the composite increases as the volume fraction of the filler is increased. This effect is analogous to the increase in viscosity of a suspension of particles in a liquid [13], so that viscosity data such as those provided by Eilers [14] can be used in a partial check of the theories. Data are also available on composites with a rubbery matrix, but these data are usually incomplete in that only one elastic constant is measured as a function of the volume fraction of filler. In these cases the dependence on filler content of the other constant, that is needed to characterize an isotropic material, cannot be checked. Schwarzl et a1.
[15] and Waterman [16] however, have provided complete data on a composite system consisting of NaCl crystals imbedded in a rubbery polyurethane matrix, and have compared them with the predictions of van der Poel's theory [17]. Additional data on this material have been provided by Nicholas and Freudenthal [18]. Payne [6] has measured the shear moduli of composites consisting of natural rubber filled with glass spheres, whiting and a "nonstructure" carbon black, and has given a valuable discussion of this system.
When the polymer matrix is in the leathery or glassy state, one can no longer assume as in the rubbery case that Poisson's ratio of the matrix is close to 0.5. Yet, unless two elastic constants are known for the matrix material, and reasonable values of the constants can be assumed for the filler, theoretical predictions for the composite's elastic behavior can only be approximate. In much of the data for glassy-matrix composites the constants of the components are not completely specified, and in most cases only one constant characterizing the composite (i.e. Young's modulus or shear modulus) is measured as a function of filler content. Some useful but usually incomplete data are available [17,[19][20][21][22]. More data on glassy-matrix composites are desirable, and for this reason the work reported here was undertaken.

Materials
The system of composites studied consisted of glqsS spheres imbedded in and adhering to a glassy epoxy matrix. A series of suc h materials was fabricated, each member containing a different volume fraction of glass spheres.
The matrix material was formed from the diglycidyl ether of bisphenol-A (DGEBA) hardened with a stoichiometric amount of triethylenetetramine (TET A). The DGEBA resin used (DER-332, Dow Chemical Co.)2 was almost pure monomer with an epoxy equivalent weight 172-178. The glass transition temperature 2Certain commerc ial equipm ent , in strum e nts or material s a re ide ntifi ed in this paper in ord er to provide an ade quat e desc ription. In no case does suc h ide ntification impl y' recomme ndation or e ndorse ment by the Na ti onal Bureau of Stand ard s . .

46
of the hardened polymer as determined by dilatometry was 120°C. The density determined by hydrostatic weighing was 1.187 g/cm 3 for material stored 185 days at 24°C and 54 percent R.H.
The filler material consisted of glass spheres with a distribution of diameters in the range 1 to 30 /.Lm. The glass was an optical crown glass, soda lime type, with a silica content not less than 60 percent (Standard Class IV Unispheres No. 4000, Cataphote Corp.). The manufacturer specifies an approximate Young's modulus for this glass of 7.6 X 1010 Pa (11 X 10 6 Ib/in 2 ) as measured on a bulk sample. Poisson's ratio for soda lime glass, given in the literature [23], is 0.23. The density of this glass as determined with a pycnometer is 2.392 g/cm 3 • Before use the glass spheres were cleaned by passing them near the poles of a powerful permanent magnet a number of times to remove iron impurities. They were washed twice in boiling distilled water and twice in boiling isopropanol, and dried overnight under vacuum at 130°C. The dried spheres were coated with the coupling agent y-aminopropyltriethoxysilane (A1100, Union Carbide Corp.) as follows: A given weight of spheres was added to an equal weight of freshly prepared 1 percent-solution of A1100 in water. The slurry was stirred for 15 min, filtered, and washed with an equal amount of water. The spheres were then dried overnight under vacuum at 130°C, and then lightly ground in a mortar to break up agglomerations.
Before use the DGEBA monomer was de aerated in a vacuum oven for at least 1 hour at 60°C. A weighed amount was transferred to a flask, the glass spheres added, and the slurry stirred under vacuum for an hour to remove air introduced with the spheres. The TET A hardener was added, and the mixture rapidly stirred under vacuum for 5 minutes. The mixture then was carefully poured into a mold formed from clamped glass plates separated by spacers and sealed with rubber tubing. The plates had previously been treated with a mold release a ' It.
The mold was sealed and the COlllCULS allowed to cure for 16 hours at room temperature. Before gelling occurred the mold was rotated at 1 rpm in order to prevent the glass spheres from settling. The material was post cured in an un clamped mold for 24 hours at 75°C followed by 8 hours at 150°C. The oven was then turned off and the cured composite allowed to come to room temperature overnight in the oven. According to reported research on similar materials [24,25] this should produce an almost completely cured resin with only minimal strains introduced by the molding process.
Several series of composites containing volume fractions of glass spheres up to 0.50 were prepared by this process. The content of glass was determined by hydrostatic weighing and found to be uniform throughout each of the cast samples. Test specimens of dumbbell shape [26] with 5-cm gage length, 1.3·cm gage width and 0.5-cm thickness were prepared from TABLE 1. Results for samples tested 1 to 2 weeks after preparation a the castings, using a high-s peed router. Four specimens were obtained from e ach cast sample.

Procedure
The specim ens were tested on a tensile testing machine at 0.05-c m/min rate of extension and 12.5-c m grip separation. Th e sta te of strain in the uniform narrow portion of the s pecime n was monitored with a strain gage extenso me ter of nominal 5-cm gage length' l The exact value of th e initial gage le ngth was determined in situ with a cathetom e te r. A trans verse extenso meter e mploying a s mall linear variable differential transform er was used to obtain simultaneous measurements of the width of the speci me n durin g test. The amplified outputs of the two extenso me ters and the load-exte nsion c urve of the te nsile tes ter were recorded se parately. Sim ultaneous data values were obtained by puttin g s mall simultaneo us pip marks on the recorder traces by means of a pushbutton operated pulse circuit.
The data were used to plot stress-s train c urves and c urves of transverse strain ve rsus longitudinal strain for the specimens _ The initial slope of the stress-strain c urve was taken as Young's modulus of the s pecime n, and the negative of the initial slope of the transverse s train-longitudinal strain c urve was take n as Poisson's ratio. As all of the specime ns were subj ec ted to the same low rate of straining during data acquisition , it was beli eved that the relative effects of viscoelasticity would be small. Therefore viscoelastic effec ts were not considered in the s ubsequent analysis.

Results
During early s tages of the research it was assumed that the samples were fully c ured, so that tensile propertie s co uld be meas ured 1 or 2 weeks after preparation. The res ults of these measurements are given in table 1. Eac h value lis ted is the average of measurements on the four specime ns obtained from each cast sample, and the variation betwee n specimens is expressed as the probable e rror of a single observation. The probable errors are of the order of 3 perce nt of the measured value for both Young's modulus and Pois son's ratio. The variation from sample to sample however was noticeably gr eate r, eve n though the extra samples of same filler content were made and measured by the sa me s tandardized procedure.
A late r series of samples made and meas ured is lis ted in table 2. These samples were tested 200 days after preparation to see if the re was an aging effect. During this time the samples were stored over a saturated soluti on of Mg(N0 3h' 6H zO maintained at the laboratory te mpe rature of 24°C. This provided a n e nvironm e nt of 54 percent relative humidity. As a II Expressed as valu e ± pro bab le e rror of a sin gle ohse rv ati on. Eac h va lu e is th e ave rage of meas ure ment s on four specim ens, exce pt as noted.
hAverage of three spec imens. l' Ave ra ge of two specim e ns. check on the effect of moisture two extra samples of epoxy matrix material were prepared. One was stored over water, and the other was stored over anhyd-rous CaS04 in a vacuum desiccator.
Comparison of the values given in tables 1 and 2 shows that there is an age hardening effect. The Young's moduli tabulated in table 2 are slightly higher than the corresponding moduli in table 1. However the measurement values obtained do not seem to have been affected by the relative humidity of the environment. The Young's moduli for the three samples stored in different environments are not significantly different.

Discussion
The data just presented will now be analyzed with the help of a theory due to van der Poel [1 7], and compared with the predictions of other theories. Let G represent the shear modulus, K bulk modulus, E Young's modulus, and v Poisson's ratio of the composite; let the subscripts f and m refer to the filler and matrix respectively, and let cp represent the volume fraction of filler. These elastic constants are interrelated by the equations,

3K-2G v= 6K+2G
(1 ) (2) so that only two constants are required to characterize the initial elastic behavior of an isotropic material. Van der Poel's theory provides two relationships,
Relation (3) is also used in the theory of Kerner [27], and is the same as Hashin and Shtrikman's equation for the highest lower bound of the bulk modulus [28]. However van der Poel's relation (4) for the shear modulus, as given in the original presentation, was very complicated. A table of values was provided, but this table was limited to materials for which Poisson's ratio of the matrix was 0.5. In addition there was an error in the derivation. Recently van der Poel's method has been reexamined, the error corrected, and the method extended for use with matrix materials having any value for Poisson's ratio [29]. Subsequently the calculation of G has been simplified [30].
If the elastic constants Em, v"', Ef, Vf of the matrix and filler materials are known, the corresponding values Km , Gm , K f , G f can be calculated by means 48 of relations (1) and (2). K and G for the composite can be calculated as a function of cp, using equations (3) and (4), and the corresponding values of E and v then can be calculated from relations (1) and (2). It is these calculated values of E and v that are compa' red with the experimental data.
In order to determine how well the theories agree with the experimental data it is necessary to have accurate values of E In and Vm for use in the theoretical calculations. The average of the measured values of Em and VIII was not regarded as sufficiently accurate, so more accurate values were obtained by the following extrapolation procedure: The averages of EII/ and VII! were used as a first approximation , and curves of E versus cp and V versus cp were calculated using van der Poel's theory. These curves were fitted to the experimental data for values of cp up to 0.15. This was done by shifting the curves mathematically along the E or v axis to obtain a least squ ares fit with the experimental data. The shape and slope of the curves was maintained constant during the shifting process.  It is apparent from figur e 1 that th e values of relative modulu s E/E", calculated usin g van de r Poel's theory are in good agree me nt with the measured values for filler volume fraction s up to 0.35 , and give slightly better predictions than value s calc ulated us ing the Kerner or Hashin and Shtrikman lowe r bound th eory. At higher filler volume fractions up to 0.50 measured values exceed the van der Poel predictions, but are less than tho se of Budiansky.
Poisson's ratios are plotted in figure 2. Values for the fresh samples are plotted as ope n circles, and valu es for the age hard e ned sa mples as solid circles.  in the order of these curves is a consequence of th e decrease in th e value of JJ with increasing <po Although there is considerable scatter in the data, the c urve of values calculated from van der Poel's th eory see ms to provide the best fit. These results indicate that the generalized van der Poel theory is be tter than othe r current theories in predicting the mec hani cal effect of filler on the elastic properties of a partic ula te composite. As the calculations involved in the application of this theory have now been simplifi ed [29 , 30], it is hoped that the theory will be more frequ e ntly used in the interpre tation of experimental data.