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On two kinds of ellipsoidal inhomogeneities in an infinite elastic body: An application to a hybrid composite
w2o-7683/81/c6os53-11suz.00/0 PcrgamonPressLtd.
ON TWO KINDS OF ELLIPSOIDAL INHOMOGENEITIES IN AN INFINITE ELASTIC BODY: AN APPLICATION TO A HYBRID COMPOSITE? MINORU TAYAand TSU-WEICHOU Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, DE 19711,U.S.A. (Received 21 July 1980;in revised form 10 October 1980) Ahstraet-The problem of two kinds of ellipsoidal inhomogeneities embedded in an elastic body is formulated with an application to a hybrid (three-phase) composite. The analytical tool used in this study is a combination of Eshelby’s equivalent inclusion method [l] and Mori-Tanaka’s back stress analysis [2], and therefore the results are valid for large volume fraction of inhomogeneities. As a demonstration, two types of hybrid composites are examined: (i) fiber-fiber; and (ii) fiber-particulate systems. 1. INTRODUCTION
When an ellipsoidal inhomogeneity is embedded in an infinite elastic body, several related problems can be solved rather simply by Eshelby’s equivalent inclusion methodll]. For example, the overall stiffness of a two-phase composite material can be easily computed once the corresponding eigenstrain is solved by this method. However, when the volume fraction of inhomogeneity (filler) becomes large, Eshelby’s equivalent inclusion method must be modified so as to take into account the interaction among inhomogeneities as well as that between an inhomogeneity and the outer boundary of the composite. The effect of the above interactions have been known as “a back stress” to material scientists, Mori and Tanaka[2] discussed such-a problem within the framework of Eshelby’s equivalent inclusion method. In Mori-Tanaka’s paper only one kind of inclusion was treated. Recently Taya and Mura[3] have applied MO&Tanaka’s method to a penny-shaped crack at a fiber end in a short-fiber reinforced composite to compute the energy release rate of the fiber end crack and the weakened Young’s modulus of the composite. In their model one kind of inhomogeneity (fiber) was treated by Mori-Tanaka’s back stress analysis. Here we extend Mori-Tanaka’s back stress analysis to hybrid (three-phase) composites where two kinds of inhomogeneities are embedded in an infinite elastic body in order to obtain the overall stiffness of the composite. To compute the overall stiffness of a hybrid composite, one can also use a “self-consistent method”[4,5]. However, it requires a numerical computation and also gives rise to inaccurate results when the stiffness of the constituent phases differ from one another to a great extent, e.g. the case of soft matrix-rigid fiber, or fiber-crack system. On the other hand, the present formulation gives us closed form results for the overall stiffness. Hence the computation is simply a parametric one. We first describe a theoretical formulation and then apply it to two types of hybrid composites: (i) fiber-fiber; and (ii) fiber-particulate systems. 2. FORMULATION
Consider an infinite elastic body which contains infinite number of two kinds of inhomogeneities and is subjected to ‘the applied stress crf as shown in Fig. 1. For later conveniences, the domains of two kinds of ellipsoidal inhomogeneties are denoted by fi, and a2 and that of the infinite body is denoted by D. Hence the domain of the matrix is D-&-0,. Note that fl, can represent a particular inhomogeneity of type 1 or all inhomogeneitiesof type 1. This is also the case with a. Let the elastic constants of the matrix, and inhomogeneities 0, and & be C&,, C&, and C~H,respectively. The volume fractions of 0, and Q are denoted by f, and fi, respectively. We assume in this paper that all inhomogeneities are aligned in the uniaxial loading direction YIbis research was supported by IVSF under Grant No. CME7918249. 553
554
M. TAYAand TSU-WEICHOU
Fig. 1. A calculation model
(say x3axis). It should be noted that our formulation below is applicableto more general cases of geometry. Further we assume that the three-phases (matrix, 0, and a,) are linearly elastic and isotropic. Under the applied stress ui the average of the total stress in the matrix is given by ~8 t (aij)M [2] and (aij)M
=
G&Sl
(1)
where iu is the average strain disturbance due to all 0, and Q. Now introduce a single inhomogeneityof type 1 (fir) into the composite, then the equivalent inclusion method yields in D
=
Cfj& t i$t EL)
(2)
where CT:and l t are the disturbance of the stress and strain due to this single a,, respectively. E; is the corresponding eigenstrain which has non-vanishingcomponents in the domain of this single fi, and becomes zero outside this single Cl,. For the entire domain D the following relation always holds; CT;= C&E;.
(3)
With eqn (3), eqn (2) yields
u; =c&&
t e: -
e;>.
(4)
Following Eshelby[l], we have
e:=S:&:,
in 0,
(9
Two kinds of emiisoidalinhomogeneitiesin an infiniteelastic WY
555
where Sh,, is the Eshelby’s tensor which depends on C&+and the geometry of fzi. Since the added single inhomogeneity Qr can represent any single ai, the domain Qi in the preceding equations is meant for any i~omogeneity of type 1. Next we add another single inhomogeneity of type 2 (&) to this composite system D Then we have in I3 oft u$ = c$&
t & + E”w - e;*)
= C&(&t & + e:,.
(6)
With eqn (2), eqn (4) provides
e: is again related to BE: as EL= S~~~~
in 4
(8)
where cr is the e~ens~ain define in a, and Sk,,,,,depends on C& and the geome~y of inhomogeneity of type 2. Since the disturbed stress aij must satisfy _fMijd V = 0, we obtain
(1- fr- f&@tj)M
t fi(dj)
+ fi(c$
= 0
(9)
where ( ) denotes the volume averaged quantity. Eliminating ei and et through eqns (5) and (8), we have three unknowns, i.e. $, a: and ei*, which will be solved by the three equations (eqns (2), (6) and (9)). Once et and e$* are solved, we can compute the overall stiffness of the composite by using the equivalence of the strain energies:
where CT&-’and Cc&-i are the compliances of the matrix and the composite, respectively. The details of the derivation of eqn (10)are given in Appendix A. The compu~tion of the e~ens~a~s and the overail stiffness of the composite are described below for two types of hybrid composites: (i) fiber-fiber; and (ii) fiber-particulatesystems. Let the inhomogeneity of type 1 be fiber-l, and the homogeneity of type 2 be fiber-2(fiber-fiber system) or particulate ~fiber-particulatesystem).
Referring to Fii. 1, the non-vanishingapplied stress is a$ (denoted by o-“)and all fibers are aligned along the loading axis &-axis). Hence the non-vanishingcomponents of Efj, ~77and +$j are ij = 11, 22 and 33. It is also noted that the system of Fig. 1 gives rise to a transverse insotropy, i.e. ef, = 4, rff = a2Tand d, = izz. In setting ij = 11 and
556
where CT,
=-J--. - 1+6%-(@,22_1)+3(1-2%)& ) 2(1- vo)(
(13)
In eqn (13)A,,S are Lame constants, v is Poisson’s ratio, a is the aspect ratio of the fiber, and g is defined in Appendix B. The subscripts 0 and 1 in the above equation denote the matrix and fiber-l, respectively. Noting that ET,= - v~u”~~~and E$ = &‘/E,,where & is the matrix Young’s moduIus, we solve for CT,and & in eqns (11)and’(l2) to obtain (14)
&=
B3*_ B,“, -e A* 11+-p33+g
00 (Be* - v&3*) A*
(15)
where A” = C;9C,*,- C:,C:, B,* = 2(C:*-
D,*C,:)
Bz* = DX:2
- C:,
B3* = 2(D1*Czi-
C:,)
B,* = C:; - &*C;“l.
(16)
Two kinds of ellipsoidal inhomogeneities in an infinite elastic body
557
The stress disturbed by fir, (r;j is obtained by eqns (4), (5), (14) and (15) as
(17)
43 T=’
I
4vo (1-2~~)~
2(1-
+ I
(1-
WA* + WWL A*
I
2~0) + (H;B2*
+ H&*1
A”
2vo)
61
;33
f
+” H:dBz* - voB,*) + H;(Bd* - VOB?) A* A* Eot I
H;,=2
(19)
~(SIIII+Slln+s~3,,-I)+SIIII+S112t-IJ
k(2$,,, HTz= (l -2v,)
(18)
t $33 - 1) + 2st,3j
and S&, is the Eshelby’s tensor for fiber-l and is given in Appendix B. Next we solve for lt? and ej;* in eqns (6) and (8), and for a:, and a& in eqn (7). The solution procedure is the same as in the foregoing steps except for eqns (13) and (19), which will be obtained below, depending on the type of inhomogeneity &, (a) a2 denotes fiber phase. The solutions corresponding to eqns (14) and 05) are ** B r;,* = ---J--g AS” 11t
**
** 82 A*$433
u” -I- K
(B:” - vOB:*) A**
(20)
**
83 B., _ u” (B:* - v,B;*) t-E t4* =AIC*q, A*“’ 33 E. A**
(21)
where the superscript ** denotes the inhomogeneity & In the above equations A**, Bf*, B?, Bf* and Bb* are given by eqn (16) where C$‘and DB are replaced by C$* and Dt* respectively. The coefficients C$* and D)* are of the same forms as those given by eqn (17) except that the superscript * and the subscript 1 are replaced by ** and 2, respectively.
hf. TAYAand Tsu-WEICHOU
558
Likewise, the disturbed stress oi is given by 2
_d* =
; wT,*G*
+ fc**BT*)
A**
( (1-2vo)
PO
41
I
(H:;rB:* + HyB:*) A*”
+ t (1 fz,+
& I
v&*) + H:;(B:* - I$&*) A** I
&_
z-
** **
IAt
W21Bl
+H22&
(22)
** **
A**
I
+ 20 - vo)+ (H;&* + H;%*) A** 1(1 - 2vo)
61 & I
+ jf H;;(B:* - vOBT*)+ Hg(B4** - V&*) A”” A** I Eo1
(23)
where Hz* is given by eqn (19) and Sg is replaced by S& which is given in Appendix B. (b) 02 denotes partuculate phase. A particulate filler is assumed to be of spherical shape. The expressions for eff and ui are again given by eqns (20)-(23) except that the coefficients Cff and HTf are now given by c** = 2(1 •t Vo)+ -(““> 4 ” 3(1-v,) 5(1--V,) &-I\0 c;“,“=
--(l+vo)
w-5~0)
t 2(E)
p2-PO
+
3(1- Vg) 15(1- Vo)( &-A0 >
ho
AZ-Ao
c,“: = c;:
**
H,, =-
(24)
2(9 t 5 vg) 15(1- vo)
H::,=_2(l+5vo) 15(1- v(j)
(25)
Hz: = 2H:: H;+_
l6
15(1- VI)) From eqn (l), we have
bh --PO
_
2
&I+ (1- 2vo)
-((r33)M_ - --i,,4vo Pa
(1 - 2vo)
(12;J33
t ;;~-p&,.
(26) (27)
After having expressed (aii)M,(a&) and (0;) in terms of Zi,we substitute them into eqn (9) to solve for i,, and Q3:
(28)
(29)
Two kinds of ellip~~al ~homogeneit~sin an infinite elasticbody
559
where S, S, and S, are given in Appendix C. With eqns (28)and (29),we can obtain eigenstrains E; and P:* as
(&*sl+ B**s*)+ @2*- vc&*)
W)
A*S E$c3 =
@3”%
+
B4*s2)
+ @4*
-
VOB3*)
(31) (32)
ey =
(33) 2.2 C~~put~ti~~ of ~~~gi~udi~ulYou~g’~ modulus EL of a ~yb~d co~posire When all fibers are aligned in the uniaxial loading direction, the equation of the eq~valence of strain energy (eqn {lo)) is reduced to
u@a@U0$3fl+ u0G;“f2 --
-.--C---f
2&
2&
2
2
’
(34)
With eqns (31)and (33),eqn (34)provides us with the lon~tudinal Young’s modulus EL of the hybrid composite:
!L-- 1
(35)
Eo l+rl where rt
=
ft
WSSI
+
MS21
+
A*S + fi
fBT*s,+ Bf’S2)
@a
-
Yom
A* + wf*
-
A**S
v&*)
136)
It should be noted that S, S, and S2 are also functions of fl and f2 {see Appendix C). When the two kinds of fillers are identical, i.e. a, = & and the volume fraction of the filler is small, one can easily obtain the results based on Eshelby’s equivalent incIusion method (without back stress analysis). 3. RESULTS AND DISCUSSION
Since a hybrid composite consists of three phases, a number of parameters characteristic of the three phases must be specified in order to compute EL. Here we set Poisson’s ratio of the matrix as 0.35 and those of two kinds of fillers as 0.3 throu~out our ~ompu~tion, As a demonstration of our results, the following cases are computed: (i) Case-l: Compu~ EJEo for given stiffnesses of the fillers, ~~~Eo)~= 50 and (B/E& = 100, and given ratio of the volume fraction of the fillers, filfi = 3 for various aspect ratios of the fibers. (ii) Case-2: Compute EJEo of a fi~r-particulate composite for (~/Eo)* = 50, (~Eok = 50 and ft = f2. The results of Case-l are shown as solid curves in Fii. 2 where EJE, computed by a “rule of mixtures” which can be obtained asymptotic~iy by increasing the aspect ratios of fibers, is shown as a dotted line. It is noted in Fig. 2 that the aspect ratio lid being 1 corresponds to the case of a particulate filler. The results of Case-2 are shown in Fii. 3 where two extreme cases are also inves~t~, i.e. two kinds of fillers are identical and they are either of the fiber type or of the particulate type. It is concluded from Fig. 3 that “a volume average approach” to combine the results of ssVol. 17. No. 6-B
hf. TAYAand Tsu.WH Crrou
5
Eo 40
-
30
-
20 -
IO -
I
0.1
0.2
0.3
0.4
O.3
f=f,+f*
Fig. 2. LongitudinalYoung’smodulusof a hybridcompositevs. volumefractionof tillerswith jz = 3fi. 40 EL G
1’ 30
20
IO
PofflcuMo
I
0.1
02
0.3
0.4
0.5
only
,
Fig. 3. LongitudinalYoung’s modulus of a fiber-particulatecomposite vs volume fraction of tilers with
h = h.
Two kinds of ellipsoidalinhomogeneitiesin ad:inthtiteelasticbody
561
only and partic~ate only does not provide the results of fi~r-~~~~ate system. In other words, if it does, then the curve of fi~r-particulate system would have been located exactly in the middle between those cases of fiber only and particulate only since fr = fi. Also the results based on a “rule of mixtures” are plotted as a dotted line in Fii. 3. Finally it should be noted that the present formulation can be easily extended to the case of more than three-phase materials since all the interactions among various hinds of inhomogeneity are carried by iii (see eqns (2) and (6)). Hence in the eq~tion of the equivalence of the strain energies (eqn (lo)), an addition of another kind of inhomogeneity simply leads to that of another term carrying the corresponding eigenstrain to the r.h.s. of eqn (10).
fiber
REFERENCES 1. 3, D. Eshelby, The determi~~n of the elastic geld of an el~p~id~ inclusion, and related problems.Proc. R. Sot.
(~ndon), Vol. Awl, pp. 376-3%(1957). 2. T. Moriand K. Tanaka, Averagestress in matrixand averageelasticenergyof materialswith misfittinginclusions.Acta Mett7lhffgicaZ&571-574(May 1973). 3. hi. Taya and T. Mura, On stiffnessand strength of an aliied short-fiberreinforcedcompositeunder uniaxial applied stress when the compositecontains fiber-endcracks,in press. J. App. Mech. 4. R. Hill, A self-consistentmechanicsof compositematerialsJ. Me& Physics and solids 13,213-222(l%S). 5. 3. Budiansky, On the elastic moduli of some heterogeneousmaterials. I Mech., Physics and SMds, 13, 223-227 fl%5).
APPENDIXA A strain energy W of a hybrid compositecontainingtwo khtdsof inhomogeneities(fir and t&) is given by 1 I
W=z ~(~~+~~~a~+~+~)dV
(Al)
wherethe domainof integrationis D (entire body) includingthe matrix,0, and h, u,, and uu are the disturbancesof the stress and strain (displacementgradient)due to inhomogeneitiesand G,is the averagedisplacementin the matrix and the symmetriccomponentof its gradientis iU (see eqn (1)). We expand the integrandin eqn (Al) as
Note that
since OijJ= 0 in D and g$r = 0 on IDI,where 101is the boundaryof D. In the derivationof (A3) Gauss’divergencetheorem has been used. Next considerthe second term on the right hand side of eqn (AZ).This term can be rewrittenas in termsof eigenstrain ef dfinedin ft. fl$(dUt 4) = Ufjj(liij+ Uu
-
e;)
+ ei}
= cr&
t crje;
(A4)
where the Rshelby’sequivalentinclusionmethodis applied; (A9
uij= C&(~~~+ Q - e$) in D,
The first term on the righthand side of eqn (A4) vanishes upon integrationover D by the same token leadingto eqn (A3). Thus, a strainenergy W (eqn (Al)) is simplifiedas
l
Since we have two kinds of eigenstrainsc): and J*“,e: is given by in n, in & in D-i&-&.
(A71
With eqn (A% eqn (A@ is furtherreducedto
&v,* d V.
(A8)
M. TAYAand Tsu-WSICHOU
562
A strain energy W can be interpretedas per a unit volume.Then, we obtain from (A8) w = ;+;
++)e$
+&$*.
(A91
Under the applied stress I& a compositehas a strain energy 1/2f&%~o~ where C&-i is the complianceof the composite.Thus we can obtain the equationof the equivalenceof strainenergy (eqn (10)).
APPENDIX B SHELBY’S TENSORS For a fiber-lie inclusion, S& are given by
(Bl)
where vOis Poisson’sratio of a matrix,II is aspect ratio of a fiber ( = I/d), and g is given by g = ,&+(a2
(B2)
- l)‘n - cash-‘a}.
For sphericalinclusion,non vanishingS’, are
s:,,,=sha=S:u,=~~~~~~ (B3)
S&n = s&s,,= s:,,, = -gg$
APPENDIX C
A substitutionof (a&, (u$) and (~3) into eqn (9) yields the value of S, S, and & definedin eqns (281and (29)as
s=Qd2a-Q21Qn (Cl)
SI = Qd2 - Qvh sz=Q21R,-
Q,2=w
Qn=
tf,
At (1- 2vol
Q&
(H:*B:*
Wf,&*+Hf2&*)
(WfiB,*+ &B,*l) A*
t H::B:*) A”
A+
(
4vo
+f2 (I_2vo)+
(H:,*B:* t H;*B:*) A”
a)
Two kinds of ellipsoidalinbomogeneitiesin an inhite elastic body
563
ON TWO KINDS OF ELLIPSOIDAL INHOMOGENEITIES IN AN INFINITE ELASTIC BODY: AN APPLICATION TO A HYBRID COMPOSITE? MINORU TAYAand TSU-WEICHOU Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, DE 19711,U.S.A. (Received 21 July 1980;in revised form 10 October 1980) Ahstraet-The problem of two kinds of ellipsoidal inhomogeneities embedded in an elastic body is formulated with an application to a hybrid (three-phase) composite. The analytical tool used in this study is a combination of Eshelby’s equivalent inclusion method [l] and Mori-Tanaka’s back stress analysis [2], and therefore the results are valid for large volume fraction of inhomogeneities. As a demonstration, two types of hybrid composites are examined: (i) fiber-fiber; and (ii) fiber-particulate systems. 1. INTRODUCTION
When an ellipsoidal inhomogeneity is embedded in an infinite elastic body, several related problems can be solved rather simply by Eshelby’s equivalent inclusion methodll]. For example, the overall stiffness of a two-phase composite material can be easily computed once the corresponding eigenstrain is solved by this method. However, when the volume fraction of inhomogeneity (filler) becomes large, Eshelby’s equivalent inclusion method must be modified so as to take into account the interaction among inhomogeneities as well as that between an inhomogeneity and the outer boundary of the composite. The effect of the above interactions have been known as “a back stress” to material scientists, Mori and Tanaka[2] discussed such-a problem within the framework of Eshelby’s equivalent inclusion method. In Mori-Tanaka’s paper only one kind of inclusion was treated. Recently Taya and Mura[3] have applied MO&Tanaka’s method to a penny-shaped crack at a fiber end in a short-fiber reinforced composite to compute the energy release rate of the fiber end crack and the weakened Young’s modulus of the composite. In their model one kind of inhomogeneity (fiber) was treated by Mori-Tanaka’s back stress analysis. Here we extend Mori-Tanaka’s back stress analysis to hybrid (three-phase) composites where two kinds of inhomogeneities are embedded in an infinite elastic body in order to obtain the overall stiffness of the composite. To compute the overall stiffness of a hybrid composite, one can also use a “self-consistent method”[4,5]. However, it requires a numerical computation and also gives rise to inaccurate results when the stiffness of the constituent phases differ from one another to a great extent, e.g. the case of soft matrix-rigid fiber, or fiber-crack system. On the other hand, the present formulation gives us closed form results for the overall stiffness. Hence the computation is simply a parametric one. We first describe a theoretical formulation and then apply it to two types of hybrid composites: (i) fiber-fiber; and (ii) fiber-particulate systems. 2. FORMULATION
Consider an infinite elastic body which contains infinite number of two kinds of inhomogeneities and is subjected to ‘the applied stress crf as shown in Fig. 1. For later conveniences, the domains of two kinds of ellipsoidal inhomogeneties are denoted by fi, and a2 and that of the infinite body is denoted by D. Hence the domain of the matrix is D-&-0,. Note that fl, can represent a particular inhomogeneity of type 1 or all inhomogeneitiesof type 1. This is also the case with a. Let the elastic constants of the matrix, and inhomogeneities 0, and & be C&,, C&, and C~H,respectively. The volume fractions of 0, and Q are denoted by f, and fi, respectively. We assume in this paper that all inhomogeneities are aligned in the uniaxial loading direction YIbis research was supported by IVSF under Grant No. CME7918249. 553
554
M. TAYAand TSU-WEICHOU
Fig. 1. A calculation model
(say x3axis). It should be noted that our formulation below is applicableto more general cases of geometry. Further we assume that the three-phases (matrix, 0, and a,) are linearly elastic and isotropic. Under the applied stress ui the average of the total stress in the matrix is given by ~8 t (aij)M [2] and (aij)M
=
G&Sl
(1)
where iu is the average strain disturbance due to all 0, and Q. Now introduce a single inhomogeneityof type 1 (fir) into the composite, then the equivalent inclusion method yields in D
=
Cfj& t i$t EL)
(2)
where CT:and l t are the disturbance of the stress and strain due to this single a,, respectively. E; is the corresponding eigenstrain which has non-vanishingcomponents in the domain of this single fi, and becomes zero outside this single Cl,. For the entire domain D the following relation always holds; CT;= C&E;.
(3)
With eqn (3), eqn (2) yields
u; =c&&
t e: -
e;>.
(4)
Following Eshelby[l], we have
e:=S:&:,
in 0,
(9
Two kinds of emiisoidalinhomogeneitiesin an infiniteelastic WY
555
where Sh,, is the Eshelby’s tensor which depends on C&+and the geometry of fzi. Since the added single inhomogeneity Qr can represent any single ai, the domain Qi in the preceding equations is meant for any i~omogeneity of type 1. Next we add another single inhomogeneity of type 2 (&) to this composite system D Then we have in I3 oft u$ = c$&
t & + E”w - e;*)
= C&(&t & + e:,.
(6)
With eqn (2), eqn (4) provides
e: is again related to BE: as EL= S~~~~
in 4
(8)
where cr is the e~ens~ain define in a, and Sk,,,,,depends on C& and the geome~y of inhomogeneity of type 2. Since the disturbed stress aij must satisfy _fMijd V = 0, we obtain
(1- fr- f&@tj)M
t fi(dj)
+ fi(c$
= 0
(9)
where ( ) denotes the volume averaged quantity. Eliminating ei and et through eqns (5) and (8), we have three unknowns, i.e. $, a: and ei*, which will be solved by the three equations (eqns (2), (6) and (9)). Once et and e$* are solved, we can compute the overall stiffness of the composite by using the equivalence of the strain energies:
where CT&-’and Cc&-i are the compliances of the matrix and the composite, respectively. The details of the derivation of eqn (10)are given in Appendix A. The compu~tion of the e~ens~a~s and the overail stiffness of the composite are described below for two types of hybrid composites: (i) fiber-fiber; and (ii) fiber-particulatesystems. Let the inhomogeneity of type 1 be fiber-l, and the homogeneity of type 2 be fiber-2(fiber-fiber system) or particulate ~fiber-particulatesystem).
Referring to Fii. 1, the non-vanishingapplied stress is a$ (denoted by o-“)and all fibers are aligned along the loading axis &-axis). Hence the non-vanishingcomponents of Efj, ~77and +$j are ij = 11, 22 and 33. It is also noted that the system of Fig. 1 gives rise to a transverse insotropy, i.e. ef, = 4, rff = a2Tand d, = izz. In setting ij = 11 and
556
where CT,
=-J--. - 1+6%-(@,22_1)+3(1-2%)& ) 2(1- vo)(
(13)
In eqn (13)A,,S are Lame constants, v is Poisson’s ratio, a is the aspect ratio of the fiber, and g is defined in Appendix B. The subscripts 0 and 1 in the above equation denote the matrix and fiber-l, respectively. Noting that ET,= - v~u”~~~and E$ = &‘/E,,where & is the matrix Young’s moduIus, we solve for CT,and & in eqns (11)and’(l2) to obtain (14)
&=
B3*_ B,“, -e A* 11+-p33+g
00 (Be* - v&3*) A*
(15)
where A” = C;9C,*,- C:,C:, B,* = 2(C:*-
D,*C,:)
Bz* = DX:2
- C:,
B3* = 2(D1*Czi-
C:,)
B,* = C:; - &*C;“l.
(16)
Two kinds of ellipsoidal inhomogeneities in an infinite elastic body
557
The stress disturbed by fir, (r;j is obtained by eqns (4), (5), (14) and (15) as
(17)
43 T=’
I
4vo (1-2~~)~
2(1-
+ I
(1-
WA* + WWL A*
I
2~0) + (H;B2*
+ H&*1
A”
2vo)
61
;33
f
+” H:dBz* - voB,*) + H;(Bd* - VOB?) A* A* Eot I
H;,=2
(19)
~(SIIII+Slln+s~3,,-I)+SIIII+S112t-IJ
k(2$,,, HTz= (l -2v,)
(18)
t $33 - 1) + 2st,3j
and S&, is the Eshelby’s tensor for fiber-l and is given in Appendix B. Next we solve for lt? and ej;* in eqns (6) and (8), and for a:, and a& in eqn (7). The solution procedure is the same as in the foregoing steps except for eqns (13) and (19), which will be obtained below, depending on the type of inhomogeneity &, (a) a2 denotes fiber phase. The solutions corresponding to eqns (14) and 05) are ** B r;,* = ---J--g AS” 11t
**
** 82 A*$433
u” -I- K
(B:” - vOB:*) A**
(20)
**
83 B., _ u” (B:* - v,B;*) t-E t4* =AIC*q, A*“’ 33 E. A**
(21)
where the superscript ** denotes the inhomogeneity & In the above equations A**, Bf*, B?, Bf* and Bb* are given by eqn (16) where C$‘and DB are replaced by C$* and Dt* respectively. The coefficients C$* and D)* are of the same forms as those given by eqn (17) except that the superscript * and the subscript 1 are replaced by ** and 2, respectively.
hf. TAYAand Tsu-WEICHOU
558
Likewise, the disturbed stress oi is given by 2
_d* =
; wT,*G*
+ fc**BT*)
A**
( (1-2vo)
PO
41
I
(H:;rB:* + HyB:*) A*”
+ t (1 fz,+
& I
v&*) + H:;(B:* - I$&*) A** I
&_
z-
** **
IAt
W21Bl
+H22&
(22)
** **
A**
I
+ 20 - vo)+ (H;&* + H;%*) A** 1(1 - 2vo)
61 & I
+ jf H;;(B:* - vOBT*)+ Hg(B4** - V&*) A”” A** I Eo1
(23)
where Hz* is given by eqn (19) and Sg is replaced by S& which is given in Appendix B. (b) 02 denotes partuculate phase. A particulate filler is assumed to be of spherical shape. The expressions for eff and ui are again given by eqns (20)-(23) except that the coefficients Cff and HTf are now given by c** = 2(1 •t Vo)+ -(““> 4 ” 3(1-v,) 5(1--V,) &-I\0 c;“,“=
--(l+vo)
w-5~0)
t 2(E)
p2-PO
+
3(1- Vg) 15(1- Vo)( &-A0 >
ho
AZ-Ao
c,“: = c;:
**
H,, =-
(24)
2(9 t 5 vg) 15(1- vo)
H::,=_2(l+5vo) 15(1- v(j)
(25)
Hz: = 2H:: H;+_
l6
15(1- VI)) From eqn (l), we have
bh --PO
_
2
&I+ (1- 2vo)
-((r33)M_ - --i,,4vo Pa
(1 - 2vo)
(12;J33
t ;;~-p&,.
(26) (27)
After having expressed (aii)M,(a&) and (0;) in terms of Zi,we substitute them into eqn (9) to solve for i,, and Q3:
(28)
(29)
Two kinds of ellip~~al ~homogeneit~sin an infinite elasticbody
559
where S, S, and S, are given in Appendix C. With eqns (28)and (29),we can obtain eigenstrains E; and P:* as
(&*sl+ B**s*)+ @2*- vc&*)
W)
A*S E$c3 =
@3”%
+
B4*s2)
+ @4*
-
VOB3*)
(31) (32)
ey =
(33) 2.2 C~~put~ti~~ of ~~~gi~udi~ulYou~g’~ modulus EL of a ~yb~d co~posire When all fibers are aligned in the uniaxial loading direction, the equation of the eq~valence of strain energy (eqn {lo)) is reduced to
u@a@U0$3fl+ u0G;“f2 --
-.--C---f
2&
2&
2
2
’
(34)
With eqns (31)and (33),eqn (34)provides us with the lon~tudinal Young’s modulus EL of the hybrid composite:
!L-- 1
(35)
Eo l+rl where rt
=
ft
WSSI
+
MS21
+
A*S + fi
fBT*s,+ Bf’S2)
@a
-
Yom
A* + wf*
-
A**S
v&*)
136)
It should be noted that S, S, and S2 are also functions of fl and f2 {see Appendix C). When the two kinds of fillers are identical, i.e. a, = & and the volume fraction of the filler is small, one can easily obtain the results based on Eshelby’s equivalent incIusion method (without back stress analysis). 3. RESULTS AND DISCUSSION
Since a hybrid composite consists of three phases, a number of parameters characteristic of the three phases must be specified in order to compute EL. Here we set Poisson’s ratio of the matrix as 0.35 and those of two kinds of fillers as 0.3 throu~out our ~ompu~tion, As a demonstration of our results, the following cases are computed: (i) Case-l: Compu~ EJEo for given stiffnesses of the fillers, ~~~Eo)~= 50 and (B/E& = 100, and given ratio of the volume fraction of the fillers, filfi = 3 for various aspect ratios of the fibers. (ii) Case-2: Compute EJEo of a fi~r-particulate composite for (~/Eo)* = 50, (~Eok = 50 and ft = f2. The results of Case-l are shown as solid curves in Fii. 2 where EJE, computed by a “rule of mixtures” which can be obtained asymptotic~iy by increasing the aspect ratios of fibers, is shown as a dotted line. It is noted in Fig. 2 that the aspect ratio lid being 1 corresponds to the case of a particulate filler. The results of Case-2 are shown in Fii. 3 where two extreme cases are also inves~t~, i.e. two kinds of fillers are identical and they are either of the fiber type or of the particulate type. It is concluded from Fig. 3 that “a volume average approach” to combine the results of ssVol. 17. No. 6-B
hf. TAYAand Tsu.WH Crrou
5
Eo 40
-
30
-
20 -
IO -
I
0.1
0.2
0.3
0.4
O.3
f=f,+f*
Fig. 2. LongitudinalYoung’smodulusof a hybridcompositevs. volumefractionof tillerswith jz = 3fi. 40 EL G
1’ 30
20
IO
PofflcuMo
I
0.1
02
0.3
0.4
0.5
only
,
Fig. 3. LongitudinalYoung’s modulus of a fiber-particulatecomposite vs volume fraction of tilers with
h = h.
Two kinds of ellipsoidalinhomogeneitiesin ad:inthtiteelasticbody
561
only and partic~ate only does not provide the results of fi~r-~~~~ate system. In other words, if it does, then the curve of fi~r-particulate system would have been located exactly in the middle between those cases of fiber only and particulate only since fr = fi. Also the results based on a “rule of mixtures” are plotted as a dotted line in Fii. 3. Finally it should be noted that the present formulation can be easily extended to the case of more than three-phase materials since all the interactions among various hinds of inhomogeneity are carried by iii (see eqns (2) and (6)). Hence in the eq~tion of the equivalence of the strain energies (eqn (lo)), an addition of another kind of inhomogeneity simply leads to that of another term carrying the corresponding eigenstrain to the r.h.s. of eqn (10).
fiber
REFERENCES 1. 3, D. Eshelby, The determi~~n of the elastic geld of an el~p~id~ inclusion, and related problems.Proc. R. Sot.
(~ndon), Vol. Awl, pp. 376-3%(1957). 2. T. Moriand K. Tanaka, Averagestress in matrixand averageelasticenergyof materialswith misfittinginclusions.Acta Mett7lhffgicaZ&571-574(May 1973). 3. hi. Taya and T. Mura, On stiffnessand strength of an aliied short-fiberreinforcedcompositeunder uniaxial applied stress when the compositecontains fiber-endcracks,in press. J. App. Mech. 4. R. Hill, A self-consistentmechanicsof compositematerialsJ. Me& Physics and solids 13,213-222(l%S). 5. 3. Budiansky, On the elastic moduli of some heterogeneousmaterials. I Mech., Physics and SMds, 13, 223-227 fl%5).
APPENDIXA A strain energy W of a hybrid compositecontainingtwo khtdsof inhomogeneities(fir and t&) is given by 1 I
W=z ~(~~+~~~a~+~+~)dV
(Al)
wherethe domainof integrationis D (entire body) includingthe matrix,0, and h, u,, and uu are the disturbancesof the stress and strain (displacementgradient)due to inhomogeneitiesand G,is the averagedisplacementin the matrix and the symmetriccomponentof its gradientis iU (see eqn (1)). We expand the integrandin eqn (Al) as
Note that
since OijJ= 0 in D and g$r = 0 on IDI,where 101is the boundaryof D. In the derivationof (A3) Gauss’divergencetheorem has been used. Next considerthe second term on the right hand side of eqn (AZ).This term can be rewrittenas in termsof eigenstrain ef dfinedin ft. fl$(dUt 4) = Ufjj(liij+ Uu
-
e;)
+ ei}
= cr&
t crje;
(A4)
where the Rshelby’sequivalentinclusionmethodis applied; (A9
uij= C&(~~~+ Q - e$) in D,
The first term on the righthand side of eqn (A4) vanishes upon integrationover D by the same token leadingto eqn (A3). Thus, a strainenergy W (eqn (Al)) is simplifiedas
l
Since we have two kinds of eigenstrainsc): and J*“,e: is given by in n, in & in D-i&-&.
(A71
With eqn (A% eqn (A@ is furtherreducedto
&v,* d V.
(A8)
M. TAYAand Tsu-WSICHOU
562
A strain energy W can be interpretedas per a unit volume.Then, we obtain from (A8) w = ;+;
++)e$
+&$*.
(A91
Under the applied stress I& a compositehas a strain energy 1/2f&%~o~ where C&-i is the complianceof the composite.Thus we can obtain the equationof the equivalenceof strainenergy (eqn (10)).
APPENDIX B SHELBY’S TENSORS For a fiber-lie inclusion, S& are given by
(Bl)
where vOis Poisson’sratio of a matrix,II is aspect ratio of a fiber ( = I/d), and g is given by g = ,&+(a2
(B2)
- l)‘n - cash-‘a}.
For sphericalinclusion,non vanishingS’, are
s:,,,=sha=S:u,=~~~~~~ (B3)
S&n = s&s,,= s:,,, = -gg$
APPENDIX C
A substitutionof (a&, (u$) and (~3) into eqn (9) yields the value of S, S, and & definedin eqns (281and (29)as
s=Qd2a-Q21Qn (Cl)
SI = Qd2 - Qvh sz=Q21R,-
Q,2=w
Qn=
tf,
At (1- 2vol
Q&
(H:*B:*
Wf,&*+Hf2&*)
(WfiB,*+ &B,*l) A*
t H::B:*) A”
A+
(
4vo
+f2 (I_2vo)+
(H:,*B:* t H;*B:*) A”
a)
Two kinds of ellipsoidalinbomogeneitiesin an inhite elastic body
563