Thermal Expansion of Composite Materials
`R. R. TummalaA. L. Friedberg
`
`Citation: Journal of Applied Physics 41, 5104 (1970); doi: 10.1063/1.1658618
`View online: http://dx.doi.org/10.1063/1.1658618
`View Table of Contents: http://aip.scitation.org/toc/jap/41/13
`Published by the American Institute of Physics
`
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`

`JOURNAL OF APPLIED PHYSICS
`
`VOLUME 41, NUMBER 13
`
`DECEMBER 1970
`
`Composites, Carbides
`
`Thermal Expansion of Composite Materials
`R. R. TUMMALA
`IBM Components Division, East Fishkill Facility, Hopewell Junction, New York 12533
`AND
`A. L. FRIEDBERG
`University of Illinois, Urbana, Iinois 61801
`
`An equation for predicting the thermal expansion coefficient of dilute binary composites is presented
`treating the dispersed particles as elastic spheres and taking into consideration the physical interactions
`between the dispersed phase and the matrix. Application of this particular equation to a variety of systems
`such as ceramic-glass, glass-metal, metal—metal, and organic~metal, is discussed, as well as the application
`of other equations to other material systems extant in the literature.
`
`INTRODUCTION
`
`Manyengineering materials contain multiple phases;
`among these are composites in high degree of techno-
`logical development during the last decade. The com-
`plexities of these materials such as chemical compati-
`bility, wettability, adsorption characteristics, and stress
`development resulting from differences in expansion,
`have so far restricted their complete characterization.
`Understanding the behavior of composites relative to
`the properties of individual phases is desired not only
`for the practical need of predicting the properties of
`composites but also for the fundamental knowledge
`required in developing new materials. Several theoreti-
`cal analyses! have been derived in the literature,
`which define thermal properties of composites. These
`analyses are due to Turner,! Kerner,’ Blackburn,’ and
`Thomas,‘ and their equations for predicting the ther-
`mal expansion coefficient of composite materials are
`given in the appendix. Someof these equations!'® have
`been verified experimentally on certain practical sys-
`tems, but poor agreement was found between theo-
`retical analysis and experimental results with certain
`other systems.?” Excellent review articles* and mathe-
`matical treatments based on energy principles’ can be
`found in the literature. The objective of this paper
`is to present a general equation, based on a theoretical
`model,
`to predict the thermal expansion behavior of
`a variety of binary composites containing limited con-
`centration of the dispersed phase.
`
`RAs
`PRY
`=eS48
`
`NOMENCLATURE
`
`(13)
`
`constant defined by Eq.
`density
`Young’s modulus
`shear modulus
`bulk modulus
`pressure
`radius of dispersed particle
`radius of matrix
`temperature change from reference
`value
`total volume of phase
`
`v
`x
`
`a
`
`8
`
`ub
`o
`T
`subscript ¢
`subscript m
`subscript ¢
`subscript +
`subscript ¢
`subscripts 7,7
`
`the dis-
`
`volume fraction
`restrained displacement of
`persed particle
`linear thermal expansioncoefficient per
`degree centigrade
`volumetric thermal expansion coefhi-
`cient per degree centigrade
`strain
`Poisson’s ratio
`stress
`shear stress
`pertaining to dispersed phase
`pertaining to matrix phase
`pertaining to composite
`radial stress
`tangential stress
`components of tensor
`
`THEORETICAL ANALYSIS
`
`In deriving the expression for predicting the thermal
`expansion coefficient of composites, the following as-
`sumptions are made:
`
`(1) The matrix and the dispersed phase obey
`Hooke’s law.
`(2) Each dispersed particle is treated as an elastic
`sphere embedded in an infinite elastic continuum re-
`sulting in an axially symmetrical stress distribution
`around the dispersion.
`(3) There is no chemical reaction between phases
`at the fabricating temperature of the composite.
`is
`(4) The observed thermal expansion coefficient
`the sum of the matrix thermal expansion coefficient
`modified by the effect of the dispersed phase upon
`the expansion of the matrix; and the expansion co-
`efficient of the dispersed phase modified by the effect
`of matrix on the expansion of the dispersed phase.
`
`Referring to Fig. 1, consider a spherical particle of
`radius r of the dispersed phase being embedded in a
`matrix of radius R. If the unrestrained strains of the
`particle and the matrix due to an increase in tem-
`perature AT are designed ez and ¢,, then these strains
`5104
`
`€
`

`

`THERMAL EXPANSION OF COMPOSITE MATERIALS
`
`5105
`
`
`
`can be written as
`
`eg=aaAT?,
`
`Em= OmATY,
`
`(€m— €a) = TAT (im— aa).
`
`(1a)
`
`(1b)
`
`(1c)
`
`Because of the restraints developed, the actual bound-
`ary between the dispersed particle and the matrix
`will be as shown by the dashed line in Fig. 1. The
`restrained displacement of the particle is x, and the
`restrained displacement of
`the matrix is then é.—
`eg~ x. Since thermal expansion is a second-order ten-
`sor, we can write the strain tensor in termsof principal
`strains: €)j= en-++ eet ¢3. From Assumption (4), it fol-
`lows that
`
`Be=Bm+Ba= (AT) [J (et eet €¢3) dv/ f dvIm
`+ (AT)[f (ert erot eas) dv/fdvja.
`
`(2)
`
`Fic. 1. Model showing the development of restraints due to
`differences in thermal expansion of phases; thermal expansion
`coefficient of matrix is assumed to be higher than that of the
`dispersed phaseforillustration purposes.
`
`The tangential strains in the particle and the matrix
`are
`
`matrix and the particle at the matrix-particle inter-
`face. That is, at r=R
`
`(en+ ef €33) m= ( 1- 2m) (out O29-+ 033) /Em
`
`t{agAT+ (01/Ea) —((oyMa)/Ea]— (ota/Ea) }
`
`(eat eet €33) ¢ = (1 — 2a) (out oat 933) /Ea
`
`or
`
`+3a,A4T,
`
`(3a)
`
`= tLamAT+-o1Em— (ott/Em) |
`
`(9)
`
`+3agAT.
`
`(3b)
`
`[(1—2pe) /Ea](—P)+agAT
`
`Applying Assumption (2), we can use Timoshenko’s®
`analysis relating the principal stresses to the pressure
`P, on the matrix due to the spherical particle:
`
`G1 6y= — (r/R) 3Pa
`
`620= 033= 0:=4(7/R)?°Prm.
`
`(4a)
`
`(4b)
`
`Again applying Assumption (2), no shear stresses act
`on the dispersed particle. That is
`
`Tmax > 4 (ou— 033) =0.
`
`(5)
`
`Therefore, o11=033, and since the pressure exerted by
`the matrix on the particle is — Pu, the stresses inside
`the particles are
`
`on oy= —P,
`
`02= 033=01= — Pa.
`
`(6a)
`
`(6b)
`
`Equations (3a) and (3b) can now be written as
`
`(ent €22+ €53) m= 3amAT
`(et €22t 633) a= (1— 2a) /Ea(—3Pa)+3agAT.
`
`(7a)
`(7b)
`
`=[(1-un)/2Em|(P) +omAT.
`
`Solving for P, we get
`
`pH
`(a@g— Om) AT
`[1+bm) /2Em|+C(1— Que) /Ea]
`
`(10)
`
`Equation (10) is identical to the expression derived
`by Selsing® and Lundin" for calculation of stresses
`in composites resulting from differences in thermal
`expansion coefficients of the individual components.
`Selsing proved the validity of this equation for Al,O,—
`glass system by comparing the experimentally deter-
`mined stresses from x-ray measurements with thecal-
`culated values. Since vg+2, we can write
`
`[ dy= 1—w
`
`(11a)
`
`f dv=q.
`Substituting (10) and (11) into Eq. (8), we have
`
`a
`
`(11b)
`
` f=30m(1—Vs) —3(
`(1—2Qpa)/Ea
`
`C(At-im)/2EmJ+[(1— 2a) /Ea]
`
`)
`
`X Llaa— em) Vat 3agVa].
`
`(12)
`
`Substituting (7a) and (7b), in Eq. (2) and defining
`a= (1/L)(8L/dT), we get
`wwe fd So (GA) JPadedefae
`3
`/1—2u,g
`Be=
`300m J dy
`aT \ ; Pydvt3ag ; dv.
`(8)
`To evaluate P, we will equate the strains in the
`
`Writing @.=3a, for isotropic composites, and rear-
`ranging Eq. (12). we get
`
`O1p= Om— Vad (am— era),
`
`(13)
`
`

`

`5106
`
`R. R.
`
`TUMMALA AND A. L. FRIEDBERG
`
`
`
`
`
`
`
`THERMALEXPANSIONCOEFFICIENTx107%
`
`
`
` EQUATION (13) AND
`
`THOMAS 47
`
`BLACKBURN
`
`2¢
`
`¢ EXPERIMENTAL
`RESULTS
`
`
`1
`20
`0
`10
`30
`VOLUME PERCENT OF ZIRCONIA
`
`
`
`Fic. 2. Thermal expansion coefficient of glass-zirconia com-
`posites as predicted by various equations; experimental results
`are also shown.
`
`where
`
`
`( +b) [2Em
`[1+em) /2EmJ+ £(1— 2a) /Ea)”
`
`The size and shape of dispersed particles, as well as
`other factors, would influence only the composite ex-
`pansion through the term A. Equation (13)
`then
`predicts the thermal expansion coefficient of isotropic,
`dilute binary composites formed of
`two chemically
`insoluble phases for which accurate data is available
`on elastic properties. Further,
`the equation can be
`applied only if
`the temperature variation of elastic
`constants of the components is known and that the
`linear thermal expansion coefficient of the composite
`can be calculated in the same temperature range as
`are the linear thermal expansion coefficients of
`the
`two phases. The Al,O;-glass system studied by Hunter
`and Brownell? can be considered a typical example
`for which the equation cannot be applied, for:
`
`(1) It contained anisotropic corundum crystals;
`(2) The temperature used to form the borosilicate-
`corundum composites was so high (up to 1500°C)
`that chemical reaction between phases was more than
`likely.
`
`VERIFICATION OF THE EQUATION
`
`Three systems were chosen to prove the validity
`of Eq. (13).
`
`Ceramic~Glass Composites
`
`One of the systems selected for verification of the
`foregoing theory consisted of a low-melting glass ma-
`trix (80% B2Os, 10% BaO, and 10% Al.Os3) dispersed
`with spherical particles of yttria-stabilized zirconia.
`Glass-zirconia compacts were produced by intimately
`
`mixing desired proportions of the glass powder (average
`particle size 20 «) with 10-20-u diameter zirconia and
`hot pressing at as low as 480°C for 3 min (to mini-
`mize chemical reactions). Rods $-in. diameter by ap-
`proximately3-in, long were prepared, and linear thermal
`expansion coefficients of the composites measured in
`the temperature range 20°-300°C. Calculated thermal
`expansion coefficients are, therefore, applicable in the
`same temperature range. Volumefraction of zirconia
`ranged from 0.05 to 0.40. Densities approaching the
`theoretical density were obtained in all composites
`containing less than 30-vol% zirconia. Figure 2 shows
`the coefficient of thermal expansion of glass—zirconia
`composites as predicted by various equations from
`the following properties of components:
`
`Zirconia
`
`Glass
`
`aa=9.4X 10-4 in./in./PC
`Fa=28X 108 psi
`Ky=15.6X 108 psi
`Ga=12.2X 10° psi
`pa= 0.23
`Da= 5.2 g/cc.
`
`Om = 6.45X 107in./in./PC
`Em = 6.3% 10° psi
`K,, = 2.84X 10° psi
`Gn = 2.78 X 10° psi
`Hm = 0.13
`Dm = 2.45 g/cc.
`
`It is clear from Fig. 2 that (13) predicts very ac-
`curately the thermal expansion coefficient of glass-
`zirconia composites containing less
`than 25-vol%
`zirconia. It is interesting to note that Thomas’ equa-
`tion predicts within 1% of (13).
`
`Metal-Metal Composites
`
`Toillustrate the applicability of his equation, ‘Turner
`studied two metallic systems: antimony—lead and alu-
`minum-beryllium. The experimental thermal expansion
`
`:
`—
`
` 30
`
`20+
`|
`
`2~oO
`2x
`b
`Zz
`iu
`5kLu
`5Oo
`z
`oO
`
`az<aoxw
`4
`S
`ao
`i
`rz
`F
`
`
`
`
`KERNER
`
`®
`
`. EXPERIMENTAL
`VALUES
`
`(po
`
`(=)
`
`aa.
`
`C EQUATION (13)
`a od
`-
`-
`aol
`40
`60
`VOLUME PERCENT LEAD
`
`-
`
`4
`80
`
`100
`
`|
`ge
`AF
`10 L_ 1
`6
`20
`
`Fic. 3.
`
`Experimental! and calculated thermal expansion co-
`efficients of antimony—lead composites.
`
`

`

`THERMAL EXPANSION OF COMPOSITE MATERIALS
`
`5107
`
`coefficients of antimony—lead composites were taken
`from his paper! and were compared in Fig. 3 with the
`predictions of (13). Theoretical predictions using Ker-
`ner and Thomas equations are also marked. Again,
`(13) predicts the thermal expansion coefficients with
`considerable accuracy.
`
`Organic-Metal Composites
`
`in
`Figure 4 is plotted from the data of Ref. 8,
`which Nielsen discusses the applicability of various
`equations for an organic~metal system. This figure is
`different from his only to the extent that the curves
`representing the Blackburn equation and (13) are
`added, Also, Turner’s curve in Fig. 4 is calculated
`using bulk moduli of
`the components, rather than
`Young’s moduli. According to Nielsen,
`‘‘the scatter
`in experimental results is so great that it is impossible
`to say which of the equations is best. However, the
`Kerner or Thomas equations are probably more ac-
`
`curate in general than Turner’s equation.” Assuming
`that
`the experimental data can be represented by
`Kerner’s equation, the agreement between (13) (shown
`by dashed line) and experimental results can be con-
`sidered to be fairly good.
`
`ACKNOWLEDGMENT
`
`The authors wish to thank J. W. Meacham for
`his help with the calculations and figures.
`
`FOR PREDICTING
`APPENDIX: EQUATIONS
`THERMAL EXPANSION OF COMPOSITES
`
`Turner’s equation:
`
`O,= (agtahat OmtmPm) | (Vakat ImRm) -
`
`(Al)
`
`Kerner’s equation:
`a= [aavaka/ (Skat4Gin) J+ [omtnkm/(3m+4Gmn) ]
`
`[vaka/ (3ka+4Gm) T+ [Onken/ (3kmt 4Gn) J”
`*
`(A2)
`
`Blackburn’s equation:
`Leettnlcineu) 8 (Ln)
`Es
`
`$1 wa) tn (1— Qya) + 1—n) (1— 2ttm) Em’
`
`Thomas’ equation:
`
`Oe = Ag”40m”.
`
`(A3)
`
`(A4)
`
`1P_S. Turner, J. Res. NBS 37, 239 (1946).
`? E. H. Kerner, Proc. Phys. Soc. 69B, 808 (1956).
`3G. Arthur and J. A. Coulson, J. Nucl. Mater. 13, 242 (1964).
`v 4 J P. Thomas, AD 287-826 (General Dynamics, Fort Worth,
`ex.),
`5R. A. Schapery, J. Comp. Mater. 2, 380 (1968).
`®W. D. Kingery, J. Amer. Ceram. Soc. 40, 351 (1957).
`(ges) Warshaw and R.Seider, J. Amer. Ceram. Soc. 50, 337
`8 L. E. Nielsen, J. Comp. Mat. 1, 100 (1967).
`9S. Timoshenko and J. N. Goodier, Theory of Elasticity
`(McGraw-Hill, New York, 1951), p. 358.
`10J. Selsing, J. Amer. Ceram. Soc. 44, 419 (1961).
`US. T. Lundin, Studies on Triaxial Whiteware Bodies (Alm-
`quist and Wiksell, Stockholm, 1960).
`20, Hunter and W. E. Brownell, J. Amer. Ceram. Soc. 50,
`19 (1967).
`
`
`
` T v ~T T
`
`
`
`
`
` Se EQUATION (13)
`
`THERMALEXPANSIONCOEFFICIENTx106/¢c
`
`
`
`
`
`BLACKBURN
`
`4
`
`
`100
`
`0
`
`80
`60
`40
`30
`VOLUME PERCENT OF FILLER
`
`Fic. 4. Calculated thermal expansion coefficients of organic—
`metal composites from the data of Ref. 8; Turner’s curve was
`recalculated using Turner’s original equation in terms of bulk
`moduli, since the Poisson ratios of the two components are not
`the same.
`
`