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composite model with a combined confocal elliptical cylinder unit cell for predicting the effective longitudinal shear modulus
Int. J. SolidsStructuresVo|. 35,No 30, pp. 3977-3987,1998
Pergamon
PII: S0020--7683(97)00266--7
© 1998ElsevierScie~aceLid All rightsreserved.Printedin GreatBritain 0020-7683/98$19:00+ .00
A FIBER/MATRIX/COMPOSITE MODEL WITH A COMBINED CONFOCAL ELLIPTICAL CYLINDER UNIT CELL FOR PREDICTING THE EFFECTIVE LONGITUDINAL SHEAR MODULUS C. P. JIANG Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing ! 00083, P.R. China
and Y. K. CHEUNG* Department of Civil and Structural Engineering, The University of Hong Kong, Hong Kong (Received4 April 1997; in revised form 10 July 1997)
Abstract--A fiber/matrix/composite model with a combined confocal elliptical cylinder unit cell is proposed for predicting the effective longitudinal shear modulus of fiber reinforced composite materials from the constituent materials. The conformal mapping technique is applied to the model, resulting in closed form solutions which are applicable to extreme types of the inclusion phases (from voids to rigid inclusions), as well as can accommodate a full range of variations in fiber section aspect ratios (from circular fibers to ribbons). A comparison is made with other numerical and experimental results to demonstrate the superiority of the proposed model and the accuracy of the solutions. © 1998 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The fiber/matrix model of a combined circular cylinder is a classical and simple model for predicting the effective moduli of unidirectional fiber reinforced composite materials (Hashin, 1964 ; Whitney and Riley, 1966). The theoretical predictions from the model show reasonable agreement with experimental data for the tensile modulus, whereas very poor agreement is demonstrated for the longitudinal shear modulus (Whitney and Riley, 1966). The reason is that the combined circular cylinder cell does not account for the effect of the aspect ratio of the fiber section and the action of surrounding unit cells, which seem to be more sensitive in changing the value of the longitudinal shear modulus. In recent years considerable attention has been focused on the effective moduli of composite materials as conventional materials are continuously being replaced by a variety of composite materials, and several approaches have been developed. The Moil-Tanaka method (Mori and Tanaka, 1973; Taya and Chou, 1981; Benveniste, 1985; Zhao and Weng, 1990) has many diverse applications in that it can accommodate elliptical inclusions, thereby accounting for a wide range of variations in inclusion shapes. Another method, the inclusion/matrix/composite model (Christensen and Lo, 1979; Luo and Weng, 1987; Christensen, 1993 ; Huang and Hu, 1995) (also referred to as the generalized self-consistent method) provides accurate predictions for extreme types of inclusions (i.e. voids and rigid inclusions), and the method also gives the correct asymptotic behavior of composites as the inclusion volume fraction approaches 1 (fully packed). The early inclusion/ matrix/composite model can only accommodate cylindrical and spherical inclusions. Huang and Hu (1995) extended the model to cover the case of elliptical inclusions, which can account for a wide range of shape variations, and they addressed in detail the problem of an in-plane isotropic distribution of elliptical inclusions (the transverse moduli of the fiber reinforced composite materials). In Huang and Hu's model the unit cell is an elliptical * Author to whom correspondence should be addressed. 3977
3978
C.P. Jiang and Y. K. Cheung
inclusion and a surrounding elliptical ring matrix, which, in turn, is embedded in an infinite composite. The elliptical inclusion and ring share the coinciding major axes and the same aspect ratio (the ratio of the semiminor axis to semimajor axis), 7, which can be taken as 0 < 7 ~< 1 without loss of generality. Huang and Hu's investigation (1995) shows that the series solution for the model can guarantee the convergence for y > 0.1. However, oscillation occurs as the number of series terms increases for y < 0. I. Thus, the model has difficulty in accommodating ribbon shape inclusions. The reason is that for a small value of 7, a large amount of the matrix material in a unit cell of the model will aggregate around the two sharp ends of the inclusion, which would seem to be physically unreasonable. In this paper a fiber/matrix~composite model, in which a unit cell is a combined cylinder with confocal elliptical cross sections, is proposed and the case of longitudinal shearing is addressed in detail for predicting the effective shear modulus. The new model gives a reasonable distribution of the matrix material in a unit cell for a full range of variation in y (from ribbons to circular cylinders). Furthermore, by using the conformal mapping technique, a closed form solution for the effective longitudinal shear modulus for unidirectional fiber reinforced composite materials is obtained, which is particularly convenient for applications in engineering. Lastly a comparison with other numerical and experimental results is made to demonstrate the superiority of the proposed model and the accuracy of the solutions. 2. MODEL
The unit cell of the proposed model is a combined cylinder (a fiber embedded in a finite matrix), which, in turn, is embedded in an infinite "homogenization" composite. Figure 1 is a schematic diagram of the cross section of a unit cell. The elliptical region D~ encircled by the contour L1 represents the fiber section and the elliptical ring region D2 between LI and Z 2 represents the matrix section in the unit cell. Let 2 denote the volume fraction of the fiber, then we have
-
(1)
gll +g12
The two ellipses share the common foci O~ and 02, which is on the x-axis. Outside the L 2 is the "homogenization" composite, the moduli of which are to be determined. O102 is an imaginary cut for the convenience of analyses later. In order to determine the effective longitudinal shear modulus of the composite, a uniform longitudinal shear stress r at infinity is applied to the composite in the direction of the x'-axis with fl being the angle
®I
10
l ®1
s
I I
y
®1
J 1(9
cz
L
xl
I i0
I ',0
I ®1
I
I ~"
I0
I I
et-
Io Fig. 1. Cross section of a unit cell.
A liber matrix composite model
3979
made by the x-axis and x'-axis. For the convenience of describing the continuous conditions in the interfaces of the fiber/matrix and matrix/composite, the curvilinear orthogonal coordinates, n and s, which lie. respectively, ahmg the normal and tangent of the contours L~ or L> are set up with ~ being the angle made by the n-axis and x-axis. Let w denote the longitudinal displacement, r,_ and r,:, r,,: and r~: denote the longitudinal shear stresses in the Cartesian coordinates and the curvilinear orthogonal coordinates, respectively, then the continuous conditions of the displacements and stresses on L~ and L~ can be expressed as % -
w,,,.
w,,, = w,.
r,,~ -
r,,-,,,
r,,_,,, - r,,_,,
on L,
(2)
on L,
(3)
where the subscriptsj~ m and e refer to the fiber, matrix and effective composite, respectively. The stresses at infinity in the Cartesian coordinates can be expressed as r( -rcosfi,
r,~ =
rsinfi
(4)
Since the shear stress components are equal in pairs there are shear stresses on the cross section of the unit cell and their resultants in the Cartesian coordinates, Q, and Q,, can be calculated :
Q,:fer,:ld~+ffr ..... df~ {5)
In the case when the fibers are uniformly dispersed and the elliptical cross sections of the fibers are randomly oriented in the transverse plane, ~he composite can be regarded as transversely isotropic, and the resultants of stresses of the unit cell given by eqn (5) can be taken to be average over the azimuth ft. The resultants of shear stresses on the cross section of a unit cell are equivalent between two different views of the composite : one regards the composite as an overall effective medium, and the other goes into the detail of the unit cell. The equivalent condition of the shear stress in the direction of the x'-axis leads to the following equation:
2~
(Q, cosfi }
Q~.sinfl)dfi='c(f~+f~2)
{6}
in terms of which, the longitudinal shear modulus of the composite can be determined.
3. ANALYS1S A N D SOLUTION
It is well known that the governing equation for the longitudinal shear problem is a harmonic equation, and the longitudinal displacement w. the shear stress components r~_and r,.~ in the Cartesian coordinates and r,,_ and r£ in the curvilinear coordinates can be expressed in terms of an analytical function J*(z) (where the same symbol is used for a complex variable z = x + i y and for the axial coordinate z of a unit cell, and should not be confused with each other) :
3981)
C.P. Jiang and Y. K. Cheung
w = Ref*(z)
(7)
~ , : - - i'c~,_ = G f * ' ( z )
(8)
r,,: - - i t , : = G e ' ~ f * ' ( z )
(9)
where G is the shear modulus, the superscript prime denotes differentiation with respect to the argument, and Re denotes the real part of a function. To solve the problem, another complex variable = {+it/ = p e '°
(10)
is introduced, where p and 0 are the polar coordinates in the complex ¢-plane. Making use of the conformal transformation of the z-plane onto the C-plane (Muskhelishivili, 1975)
z = ~,~(;) = R (~+ ~)
ll)
where R is a constant to be determined, the regions fl~ and ~2 surrounded by the imaginary cut O102, the elliptical contours Lt and L2 in the z-plane (Oxy-plane, Fig. 1) are mapped onto the circular ring regions fl] and ~2 surrounded by LI), L] and L~ with radii Po = 1, Pl and P2 in the ~-plane (Fig. 2), respectively. From this transformation, it is seen that
a, = R Pk+
,
b, = R
px
(13)
where ak and bk are the length of the semimajor and semiminor axes, and the subscript k = 1, 2 refers to L~ and L> respectively. Since the geometrical parameters of the fiber section, a~ and b~, are known, R and Pl can be determined by using eqn (13). From Fig. 1, it is seen that
rl
Fig. 2, Conformal mapping.
A fiber/matrix/composite model
3981
f ~ = na~bl = n R Z ( p ~ - - l/'p~)
(14)
~~t + ~ 2 = ga2b2 = n R e ( P ~ -
l/p~)
(15)
The substitution o f e q n s (14) and (15) into eqn (1) yields -- 1/pT
2 -
(16)
in terms of which, P2 can be determined as the volume fraction of fibers 2 is prescribed. Let
.fl ) =.f*(:)
(17)
then from the chain rule for differentiation, we have .f*'(z) =.f"(~)/~'(~)
(18)
Applying eqn (18) and noting (Muskhelishivili, 1975) that Y
e '~ - ~ {~'(~) p I{'/(OI
(19)
where I'] expresses the absolute value, eqns (7)-(9) become w = Re.fl~)
(20)
r,~: - ir,.~ = G -f'(¢)/{o'(~)
(21)
~,,--- i~,- =
G"
~" f'(~) p I,-'(01
(22)
Through the above transformation, the problem under consideration (Fig. 1) has been reduced to a connection problem of the concentric circular rings in the ~-plane (Fig. 2), which is easy to solve by using the method of power series. In the region ~ ] in the ~-plane, the analytical function relevant to the fiber in a unit cell,,11(~), can be expanded into the Laurent series : ./,(O=a*ln~+
y, ],=
~tk~k
(23)
~-
Since the geometry of a unit cell is symmetric with respect to the x-axis and ),-axis, the resultant shear stress c o m p o n e n t Q., on the unit cell section ( ~ +f12) is only dependent on the shear stress at infinity r~, and Q, on r,.~. For simplicity, the two cases will be dealt with separately. First let T~z ~
r cos fi,
"el:: ~
= 0
(24)
apparently the displacement w is symmetric with respect to the x-axis. Noting that ~k
pk(coskO+isinkO)
(25)
from eqns (20) and (23), it can be seen that the undetermined coefficients a* and ak are real constants. Since O~O2 in the z-plane is only an imaginary cut, ~ = e "~ and ~ = e ~0in the ~plane are corresponding to a same point in the z-plane, which leads to
C. P. Jiang and Y. K. Cheung
3982
(26)
A (e') = fl (e "3 F r o m eqn (26), it can be inferred that a*=0.
a i~=a¢,
k=0.1,2
....
(27)
F r o m further analyses, it is seen that the exact solution can be obtained by taking two terms (k = - 1.1) in the expansion (23), so the function,If(() in the region f)] in the (plane can be expressed as
f,(() = A ((+ ~)
(28)
where A is a real constant. Substituting eqn (28) into eqns (20) real and imaginary parts, one obtains
(22) and
separating out the
w~= A (p+ ~)cosO
(29)
r,~ 1 = GI A
(30)
r,,~-I+~'(~)1
1-
p- /
cos0
(31)
Similarly, in the region f ~ in the i-plane, the functionj~(~) relevant to the matrix in a unit cell can be expressed
as
./;(~) = B
,
+ B~ ~
(32)
where B ~and B~ are real constants. The substitution o f e q n (32) into eqns (20) (22) yields
w,,
+ B~p cos 0
r~.,, = Gm Re
(33)
- B i/7.2+ Bi
(34)
R(1 - 1/~ 2)
r ...... - l e / ( ( ) l
Bl
cos0
(35)
In the external region o f the c o n t o u r L~ in the (-plane, the function j;((), which is relevant to the effective composite, can be expressed as
f; (() = C i + c, .
(36)
¢
where C
and C~ are real constants. In the same manner,
w,,,~:,, and
r,:,, can be written as
A fiber,matrix 'composite model
w,.=
+C~p
.... = Gc Re
-C
3983
(37)
cos0
~/(.2 +C~
R(I
(38)
1"~ 2) C
I
(39)
where G,~ is the effective longitudinal shear modulus. Substituting eqns (29), (31), (33), (35), (37) and (39) into eqns (2), (3) and (24), six equations are obtained. Since ~ = 0 leads to an identity, only five linear algebraic equations are independent, in terms of which, five real coefficients A. B t. Bl, C ~ and ("~ can be determined : /
A = G,nR~,:- / ,'
(GI+G,,,)
1+
l
-(G,-G,,,)
4-
G,
~]]<,I P~_/J
:
'~'
B,
(c.,+c_,. \ ~_c,,,
= (Gj+G,,, < 2<,
.... 2<, p~)A
(40) )
o,-c
G,-G,,,
(41)
1)
2<, "pi .4
c l= 2C;-(I+p_~)- 5~,;- p?+p;
t43)
A- C,, p~
(43)
R~B C1 -
G~.
(44)
Substituting eqns (40)-(42) into eqns (30) and (34), then substituting them into the first equation of (5), one obtains
Q, =
213 G,,, • ~(~2~ +f23) cos/3 Ii G,. +I2G,,
(45)
where
11 = (G,, + G/)
1+
- (G, - G.,,)
+
(46) P~/
(47)
3984
c. P. Jiang and Y. K. Cheung
I~. = 2G/Z+(Gm+GI.)(I--2)--(Gr--G.,
) P5 _ P_T o p~/,,, /' P2--
1
(48)
N o w consider another stress state at infinity : ,._ = 0 ,
r,,: = - r s i n f l
(49)
Apparently, in this case the displacement w is antisymmetric with respect to the x-axis. In the same manner, one can obtain 216 Gm • r ( ~ +f~2) sin fi Q' = - 14G,, + I5G,,,
(50)
where
I4=(G.,+G/)(1-~i-)+{G/-Gm)(1 KPT
-P!- t P~/
(51,
OS/
(52)
/' p ~ -
(53)
1
Is=(G"'+G"(I+~)+(G/-Gm)(;i-+~!t-
16 = 2 G / 2 + ( G m + G I } ( I - ) . ) + ( G t - G m ) ( P ~
\pT
05/
Substitution of eqns (45) and (50) into eqn (6) yields 13
10
1
ltG,,+ I2G,,, + 14G,,+ IsG,,, - G,,,
(54)
Eqn (54) can be transformed into a quadratic equation in G,, :
I, I4G~.+(I,15+1214-1314
Ijl)G,,,G,,+(I215-13Is-I21~,)G~,
= 0
(55)
For a practical composite, Ge has a positive solution and a negative one from eqn (55). Deleting the unreasonable negative solution, we obtain the closed form solution of the effective longitudinal shear modulus G,, :
G,. = ( - B + x / B 2 -4C)G,,,/2
(56)
where B = (ll Is + 12 I4 - 1314
- - 16
Ii
),/( II 14 )
C = (1215 - lals - 1216)/(1,14)
(57)
(58)
4. RESULT COMPARISON AND CONCLUSION It is of interest to see how the shape o f the fiber cross section would affect the effective longitudinal shear modulus o f a composite. First, we use the properties of b o r o n fibers and epoxy matrix (Whitney and Riley, 1966) in our calculation. The elastic constants are
A fiber/matrix/composite model
3985
Table 1. Variation of the longitudinal shear modulus vs ratio aspect 7 Longitudinal shear modulus (GPa) ),
"'= 1"
0.9
0.8
0.7
0.6
0.5
(I.4
0.2
0.1
0.000 l
0.20 0.50 0.70 (I.75
2.28 4.49 8.28 10.1
2.28 4.50 8.32 10.2
2.29 4.57 8.50 10.4
2.31 4.70 8.87 10.9
2.35 4.93 9.52 11.8
2.42 5.34 10.7 13.3
2.53 6.10 12.8 16.1
3.32 11.6 26.7 33.1
5.62 25.0 50.4 59.1
35.4 86.5 121 129
* F o r 7 - 1, the longitudinal shear modulus cannot be directly computed by using the present formula. Instead, ?' is taken as a value approaching 1, for example ? = 0.9999
El=413.69GPa,
G / = 172.37GPa,
E,,, = 4 . 1 3 6 9 G P a ,
G,,, = 1.5322GPa,
v~=0.2 r,,, = 0.35
The l o n g i t u d i n a l shear m o d u l u s from the present theoretical f o r m u l a [eqn (56)] for various fiber section aspect ratio ), (7' = bl/al) are shown in Table 1. It can be seen that the l o n g i t u d i n a l shear m o d u l u s is very sensitive to the aspect ratio. As the aspect ratio decreases (noting t h a t 0 < i' ~< 1 w i t h o u t loss o f generality), the l o n g i t u d i n a l shear m o d u l u s increases for a certain fiber v o l u m e fraction. F r o m Fig. 5 in W h i t n e y a n d Riley (1966) it is seen that the shape o f fiber section o f the specimens are irregular, so it is i m p r o p e r to be r e g a r d e d as a circle. To account for the influence o f the shape, the fiber sections are t a k e n as an ellipse with ~, = 0.4 a n d the results are listed in T a b l e 2, which shows that the theoretical predictions are in r e a s o n a b l e a g r e e m e n t with the e x p e r i m e n t a l results. T h e second e x a m p l e is the glass-epoxy composite, with the elastic c o n s t a n t s being ( Z h a o a n d Weng, 1990): EI=72.4GPa,
GI=30.2GPa,
E,,, = 2 . 7 6 G P a ,
G,,, = 1.02GPa.
v1=0.2 r,,, = 0.35
By using the present formula, the n o r m a l i z e d l o n g i t u d i n a l shear m o d u l u s as a function o f )~ is depicted in Fig. 3. A c o m p a r i s o n with the numerical results ( Z h a o a n d W e n g , 1990) by using E s h e l b y - M o r i T a n a k a t h e o r y shows that the results o f the two m e t h o d s are in g o o d agreement. The present f o r m u l a is a p p l i c a b l e to the whole spectrum o f inclusion types, i.e. from voids to rigid inclusions. W h e n the elastic m o d u l u s o f fibers are equal to zero, the present f o r m u l a gives the results for m a t e r i a l s with voids. The n o r m a l i z e d shear m o d u l u s p r e d i c t i o n s for the m a t e r i a l with voids are depicted in Fig. 4, which shows that the shear m o d u l u s is also very sensitive to the aspect ratio o f voids. Based on the a b o v e results and c o m p a r i s o n we can m a k e the following c o n c l u s i o n s : 1. The p r o p o s e d i n c l u s i o n / m a t r i x / c o m p o s i t e m o d e l o f a c o m b i n e d confocal elliptical cylinder unit cell is a p p l i c a b l e to the whole spectrum o f inclusion types o f inclusions ( f r o m
Table 2. A comparison of the theoretical predictions of the longitudinal shear modulus with experimental results Longitudinal shear modulus (GPa) ). 0.70 0.75
Classical model ~
Present model ~
Experiment 3
8.27 10.10
12.82
12.21 16.75
16.06
Referring to Hashin and Rosen, 1964: Whitney and Riley. 1966. : Taking ~, = 0.4. 3Referring to Whitney and Riley. 1966.
3986
C. P. Jiang and Y. K. Cheung 30 25 2O
E "~
15
j"
~o.ooooo~/
0
//
10
0.2
I 0.4
t 0.6
I 0.8
Fig. 3. Variation of the longitudinal shear modulus.
1 0.9 0.8 0.7 0.6 (.9
0.5 0.4 0.3 0.2 0.1 0 0.2
0.4
0.6
0.8
Fig. 4. Shear modulus for the material with ~oids.
voids to rigid inclusions), and can also a c c o m m o d a t e a full range o f aspect ratios ( from circular fibers to r i b b o n ) . 2. By using the c o n f o r m a l m a p p i n g technique integrated with the Laurent e x p a n s i o n m e t h o d , the l o n g i t u d i n a l shear m o d u l u s is derived in a closed form, which is p a r t i c u l a r l y convenient for a p p l i c a t i o n in engineering. 3. A c o m p a r i s o n with experimental d a t a and numerical solutions shows the effectiveness and accuracy o f the present formula. The i n c l u s i o n / m a t r i x / c o m p o s i t e m o d e l of a c o m b i n e d confocal elliptical cylinder unit cell is also a p p l i c a b l e to predict the effective tensile m o d u l u s in terms o f the other two c o m p l e x stress functions ~ ( z ) and ~P(z) (Muskhelishivili, 1975): however, it is difficult to o b t a i n the closed form solution. The p r o b l e m will be dealt with elsewhere. REFERENCES Benveniste, Y. (1985) The effective mechanical behavior of composite materials with imperfect contact between the constituents. Mechanics (~fMaterials 4, 197 208. Christensen, R. M. (1993) Effective properties o1 composite materials containing voids. Proceedings O/the Royal Society ~[London A440, 461 473. Christensen, R. M. and Lo, K. M. (1979) Solutions lot effective shear properties m three phase sphere and cylinder models. Journal of the Mechanics and Physics of'Solids 27, 315 330. Hashin, Z. and Rosen, B. W. (1964) The elastic moduli of fiber reinforced materials. Journal (~fApplied Mechanics 31,223 232. Huang, Y. and Hu, K. X. (1995) A generalized sell-consistent mechanics method for solids containing elliptical inclusions. ASME Journal Of Applied Mechanics 62, 566 572.
A liber matrix composite medel
3987
Luo. H. A. and Weng, G. J. (1987) On gshelby's inclusion problem in a three-phase spherically concentric solid, and a modification of Mori T a n a k a ' s method. Mechamcs o/Materials 8, 347 361. Mori, T. and Tanaka. K (1973) Average stress in the matrix and average elastic energy of'materials with misfitting inclusions. Acta Metallurgica 21,571 574. M uskhelishivili, N. 1. (1975) Some basic problems of mathematical theory o1"elasticity. N oordhoff, Leyden. Taya. M. and Chou. T. W. {1981) On two kinds of ellipsoidal mhomogeneities in an infinite elastic body: an application to a h~brid composite. InlernationalJom'md o/Solids atutStruclure.s 17, 553 563. Whitne>. J. M. and Riley. M. B. (1966) Elastic properties of fiber reintk)rced composite materials..4 IAA .lomwal 4(9), 1537 1542. Zhao. Y. H. and Weng, G. H. (1990) Effective elastic moduli of ribbon-reinforced composites. Ah'ME ,Iomwa/~!! Ii)l)/icd M~'chanics 57. 158 167.
Pergamon
PII: S0020--7683(97)00266--7
© 1998ElsevierScie~aceLid All rightsreserved.Printedin GreatBritain 0020-7683/98$19:00+ .00
A FIBER/MATRIX/COMPOSITE MODEL WITH A COMBINED CONFOCAL ELLIPTICAL CYLINDER UNIT CELL FOR PREDICTING THE EFFECTIVE LONGITUDINAL SHEAR MODULUS C. P. JIANG Department of Flight Vehicle Design and Applied Mechanics, Beijing University of Aeronautics and Astronautics, Beijing ! 00083, P.R. China
and Y. K. CHEUNG* Department of Civil and Structural Engineering, The University of Hong Kong, Hong Kong (Received4 April 1997; in revised form 10 July 1997)
Abstract--A fiber/matrix/composite model with a combined confocal elliptical cylinder unit cell is proposed for predicting the effective longitudinal shear modulus of fiber reinforced composite materials from the constituent materials. The conformal mapping technique is applied to the model, resulting in closed form solutions which are applicable to extreme types of the inclusion phases (from voids to rigid inclusions), as well as can accommodate a full range of variations in fiber section aspect ratios (from circular fibers to ribbons). A comparison is made with other numerical and experimental results to demonstrate the superiority of the proposed model and the accuracy of the solutions. © 1998 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The fiber/matrix model of a combined circular cylinder is a classical and simple model for predicting the effective moduli of unidirectional fiber reinforced composite materials (Hashin, 1964 ; Whitney and Riley, 1966). The theoretical predictions from the model show reasonable agreement with experimental data for the tensile modulus, whereas very poor agreement is demonstrated for the longitudinal shear modulus (Whitney and Riley, 1966). The reason is that the combined circular cylinder cell does not account for the effect of the aspect ratio of the fiber section and the action of surrounding unit cells, which seem to be more sensitive in changing the value of the longitudinal shear modulus. In recent years considerable attention has been focused on the effective moduli of composite materials as conventional materials are continuously being replaced by a variety of composite materials, and several approaches have been developed. The Moil-Tanaka method (Mori and Tanaka, 1973; Taya and Chou, 1981; Benveniste, 1985; Zhao and Weng, 1990) has many diverse applications in that it can accommodate elliptical inclusions, thereby accounting for a wide range of variations in inclusion shapes. Another method, the inclusion/matrix/composite model (Christensen and Lo, 1979; Luo and Weng, 1987; Christensen, 1993 ; Huang and Hu, 1995) (also referred to as the generalized self-consistent method) provides accurate predictions for extreme types of inclusions (i.e. voids and rigid inclusions), and the method also gives the correct asymptotic behavior of composites as the inclusion volume fraction approaches 1 (fully packed). The early inclusion/ matrix/composite model can only accommodate cylindrical and spherical inclusions. Huang and Hu (1995) extended the model to cover the case of elliptical inclusions, which can account for a wide range of shape variations, and they addressed in detail the problem of an in-plane isotropic distribution of elliptical inclusions (the transverse moduli of the fiber reinforced composite materials). In Huang and Hu's model the unit cell is an elliptical * Author to whom correspondence should be addressed. 3977
3978
C.P. Jiang and Y. K. Cheung
inclusion and a surrounding elliptical ring matrix, which, in turn, is embedded in an infinite composite. The elliptical inclusion and ring share the coinciding major axes and the same aspect ratio (the ratio of the semiminor axis to semimajor axis), 7, which can be taken as 0 < 7 ~< 1 without loss of generality. Huang and Hu's investigation (1995) shows that the series solution for the model can guarantee the convergence for y > 0.1. However, oscillation occurs as the number of series terms increases for y < 0. I. Thus, the model has difficulty in accommodating ribbon shape inclusions. The reason is that for a small value of 7, a large amount of the matrix material in a unit cell of the model will aggregate around the two sharp ends of the inclusion, which would seem to be physically unreasonable. In this paper a fiber/matrix~composite model, in which a unit cell is a combined cylinder with confocal elliptical cross sections, is proposed and the case of longitudinal shearing is addressed in detail for predicting the effective shear modulus. The new model gives a reasonable distribution of the matrix material in a unit cell for a full range of variation in y (from ribbons to circular cylinders). Furthermore, by using the conformal mapping technique, a closed form solution for the effective longitudinal shear modulus for unidirectional fiber reinforced composite materials is obtained, which is particularly convenient for applications in engineering. Lastly a comparison with other numerical and experimental results is made to demonstrate the superiority of the proposed model and the accuracy of the solutions. 2. MODEL
The unit cell of the proposed model is a combined cylinder (a fiber embedded in a finite matrix), which, in turn, is embedded in an infinite "homogenization" composite. Figure 1 is a schematic diagram of the cross section of a unit cell. The elliptical region D~ encircled by the contour L1 represents the fiber section and the elliptical ring region D2 between LI and Z 2 represents the matrix section in the unit cell. Let 2 denote the volume fraction of the fiber, then we have
-
(1)
gll +g12
The two ellipses share the common foci O~ and 02, which is on the x-axis. Outside the L 2 is the "homogenization" composite, the moduli of which are to be determined. O102 is an imaginary cut for the convenience of analyses later. In order to determine the effective longitudinal shear modulus of the composite, a uniform longitudinal shear stress r at infinity is applied to the composite in the direction of the x'-axis with fl being the angle
®I
10
l ®1
s
I I
y
®1
J 1(9
cz
L
xl
I i0
I ',0
I ®1
I
I ~"
I0
I I
et-
Io Fig. 1. Cross section of a unit cell.
A liber matrix composite model
3979
made by the x-axis and x'-axis. For the convenience of describing the continuous conditions in the interfaces of the fiber/matrix and matrix/composite, the curvilinear orthogonal coordinates, n and s, which lie. respectively, ahmg the normal and tangent of the contours L~ or L> are set up with ~ being the angle made by the n-axis and x-axis. Let w denote the longitudinal displacement, r,_ and r,:, r,,: and r~: denote the longitudinal shear stresses in the Cartesian coordinates and the curvilinear orthogonal coordinates, respectively, then the continuous conditions of the displacements and stresses on L~ and L~ can be expressed as % -
w,,,.
w,,, = w,.
r,,~ -
r,,-,,,
r,,_,,, - r,,_,,
on L,
(2)
on L,
(3)
where the subscriptsj~ m and e refer to the fiber, matrix and effective composite, respectively. The stresses at infinity in the Cartesian coordinates can be expressed as r( -rcosfi,
r,~ =
rsinfi
(4)
Since the shear stress components are equal in pairs there are shear stresses on the cross section of the unit cell and their resultants in the Cartesian coordinates, Q, and Q,, can be calculated :
Q,:fer,:ld~+ffr ..... df~ {5)
In the case when the fibers are uniformly dispersed and the elliptical cross sections of the fibers are randomly oriented in the transverse plane, ~he composite can be regarded as transversely isotropic, and the resultants of stresses of the unit cell given by eqn (5) can be taken to be average over the azimuth ft. The resultants of shear stresses on the cross section of a unit cell are equivalent between two different views of the composite : one regards the composite as an overall effective medium, and the other goes into the detail of the unit cell. The equivalent condition of the shear stress in the direction of the x'-axis leads to the following equation:
2~
(Q, cosfi }
Q~.sinfl)dfi='c(f~+f~2)
{6}
in terms of which, the longitudinal shear modulus of the composite can be determined.
3. ANALYS1S A N D SOLUTION
It is well known that the governing equation for the longitudinal shear problem is a harmonic equation, and the longitudinal displacement w. the shear stress components r~_and r,.~ in the Cartesian coordinates and r,,_ and r£ in the curvilinear coordinates can be expressed in terms of an analytical function J*(z) (where the same symbol is used for a complex variable z = x + i y and for the axial coordinate z of a unit cell, and should not be confused with each other) :
3981)
C.P. Jiang and Y. K. Cheung
w = Ref*(z)
(7)
~ , : - - i'c~,_ = G f * ' ( z )
(8)
r,,: - - i t , : = G e ' ~ f * ' ( z )
(9)
where G is the shear modulus, the superscript prime denotes differentiation with respect to the argument, and Re denotes the real part of a function. To solve the problem, another complex variable = {+it/ = p e '°
(10)
is introduced, where p and 0 are the polar coordinates in the complex ¢-plane. Making use of the conformal transformation of the z-plane onto the C-plane (Muskhelishivili, 1975)
z = ~,~(;) = R (~+ ~)
ll)
where R is a constant to be determined, the regions fl~ and ~2 surrounded by the imaginary cut O102, the elliptical contours Lt and L2 in the z-plane (Oxy-plane, Fig. 1) are mapped onto the circular ring regions fl] and ~2 surrounded by LI), L] and L~ with radii Po = 1, Pl and P2 in the ~-plane (Fig. 2), respectively. From this transformation, it is seen that
a, = R Pk+
,
b, = R
px
(13)
where ak and bk are the length of the semimajor and semiminor axes, and the subscript k = 1, 2 refers to L~ and L> respectively. Since the geometrical parameters of the fiber section, a~ and b~, are known, R and Pl can be determined by using eqn (13). From Fig. 1, it is seen that
rl
Fig. 2, Conformal mapping.
A fiber/matrix/composite model
3981
f ~ = na~bl = n R Z ( p ~ - - l/'p~)
(14)
~~t + ~ 2 = ga2b2 = n R e ( P ~ -
l/p~)
(15)
The substitution o f e q n s (14) and (15) into eqn (1) yields -- 1/pT
2 -
(16)
in terms of which, P2 can be determined as the volume fraction of fibers 2 is prescribed. Let
.fl ) =.f*(:)
(17)
then from the chain rule for differentiation, we have .f*'(z) =.f"(~)/~'(~)
(18)
Applying eqn (18) and noting (Muskhelishivili, 1975) that Y
e '~ - ~ {~'(~) p I{'/(OI
(19)
where I'] expresses the absolute value, eqns (7)-(9) become w = Re.fl~)
(20)
r,~: - ir,.~ = G -f'(¢)/{o'(~)
(21)
~,,--- i~,- =
G"
~" f'(~) p I,-'(01
(22)
Through the above transformation, the problem under consideration (Fig. 1) has been reduced to a connection problem of the concentric circular rings in the ~-plane (Fig. 2), which is easy to solve by using the method of power series. In the region ~ ] in the ~-plane, the analytical function relevant to the fiber in a unit cell,,11(~), can be expanded into the Laurent series : ./,(O=a*ln~+
y, ],=
~tk~k
(23)
~-
Since the geometry of a unit cell is symmetric with respect to the x-axis and ),-axis, the resultant shear stress c o m p o n e n t Q., on the unit cell section ( ~ +f12) is only dependent on the shear stress at infinity r~, and Q, on r,.~. For simplicity, the two cases will be dealt with separately. First let T~z ~
r cos fi,
"el:: ~
= 0
(24)
apparently the displacement w is symmetric with respect to the x-axis. Noting that ~k
pk(coskO+isinkO)
(25)
from eqns (20) and (23), it can be seen that the undetermined coefficients a* and ak are real constants. Since O~O2 in the z-plane is only an imaginary cut, ~ = e "~ and ~ = e ~0in the ~plane are corresponding to a same point in the z-plane, which leads to
C. P. Jiang and Y. K. Cheung
3982
(26)
A (e') = fl (e "3 F r o m eqn (26), it can be inferred that a*=0.
a i~=a¢,
k=0.1,2
....
(27)
F r o m further analyses, it is seen that the exact solution can be obtained by taking two terms (k = - 1.1) in the expansion (23), so the function,If(() in the region f)] in the (plane can be expressed as
f,(() = A ((+ ~)
(28)
where A is a real constant. Substituting eqn (28) into eqns (20) real and imaginary parts, one obtains
(22) and
separating out the
w~= A (p+ ~)cosO
(29)
r,~ 1 = GI A
(30)
r,,~-I+~'(~)1
1-
p- /
cos0
(31)
Similarly, in the region f ~ in the i-plane, the functionj~(~) relevant to the matrix in a unit cell can be expressed
as
./;(~) = B
,
+ B~ ~
(32)
where B ~and B~ are real constants. The substitution o f e q n (32) into eqns (20) (22) yields
w,,
+ B~p cos 0
r~.,, = Gm Re
(33)
- B i/7.2+ Bi
(34)
R(1 - 1/~ 2)
r ...... - l e / ( ( ) l
Bl
cos0
(35)
In the external region o f the c o n t o u r L~ in the (-plane, the function j;((), which is relevant to the effective composite, can be expressed as
f; (() = C i + c, .
(36)
¢
where C
and C~ are real constants. In the same manner,
w,,,~:,, and
r,:,, can be written as
A fiber,matrix 'composite model
w,.=
+C~p
.... = Gc Re
-C
3983
(37)
cos0
~/(.2 +C~
R(I
(38)
1"~ 2) C
I
(39)
where G,~ is the effective longitudinal shear modulus. Substituting eqns (29), (31), (33), (35), (37) and (39) into eqns (2), (3) and (24), six equations are obtained. Since ~ = 0 leads to an identity, only five linear algebraic equations are independent, in terms of which, five real coefficients A. B t. Bl, C ~ and ("~ can be determined : /
A = G,nR~,:- / ,'
(GI+G,,,)
1+
l
-(G,-G,,,)
4-
G,
~]]<,I P~_/J
:
'~'
B,
(c.,+c_,. \ ~_c,,,
= (Gj+G,,, < 2<,
.... 2<, p~)A
(40) )
o,-c
G,-G,,,
(41)
1)
2<, "pi .4
c l= 2C;-(I+p_~)- 5~,;- p?+p;
t43)
A- C,, p~
(43)
R~B C1 -
G~.
(44)
Substituting eqns (40)-(42) into eqns (30) and (34), then substituting them into the first equation of (5), one obtains
Q, =
213 G,,, • ~(~2~ +f23) cos/3 Ii G,. +I2G,,
(45)
where
11 = (G,, + G/)
1+
- (G, - G.,,)
+
(46) P~/
(47)
3984
c. P. Jiang and Y. K. Cheung
I~. = 2G/Z+(Gm+GI.)(I--2)--(Gr--G.,
) P5 _ P_T o p~/,,, /' P2--
1
(48)
N o w consider another stress state at infinity : ,._ = 0 ,
r,,: = - r s i n f l
(49)
Apparently, in this case the displacement w is antisymmetric with respect to the x-axis. In the same manner, one can obtain 216 Gm • r ( ~ +f~2) sin fi Q' = - 14G,, + I5G,,,
(50)
where
I4=(G.,+G/)(1-~i-)+{G/-Gm)(1 KPT
-P!- t P~/
(51,
OS/
(52)
/' p ~ -
(53)
1
Is=(G"'+G"(I+~)+(G/-Gm)(;i-+~!t-
16 = 2 G / 2 + ( G m + G I } ( I - ) . ) + ( G t - G m ) ( P ~
\pT
05/
Substitution of eqns (45) and (50) into eqn (6) yields 13
10
1
ltG,,+ I2G,,, + 14G,,+ IsG,,, - G,,,
(54)
Eqn (54) can be transformed into a quadratic equation in G,, :
I, I4G~.+(I,15+1214-1314
Ijl)G,,,G,,+(I215-13Is-I21~,)G~,
= 0
(55)
For a practical composite, Ge has a positive solution and a negative one from eqn (55). Deleting the unreasonable negative solution, we obtain the closed form solution of the effective longitudinal shear modulus G,, :
G,. = ( - B + x / B 2 -4C)G,,,/2
(56)
where B = (ll Is + 12 I4 - 1314
- - 16
Ii
),/( II 14 )
C = (1215 - lals - 1216)/(1,14)
(57)
(58)
4. RESULT COMPARISON AND CONCLUSION It is of interest to see how the shape o f the fiber cross section would affect the effective longitudinal shear modulus o f a composite. First, we use the properties of b o r o n fibers and epoxy matrix (Whitney and Riley, 1966) in our calculation. The elastic constants are
A fiber/matrix/composite model
3985
Table 1. Variation of the longitudinal shear modulus vs ratio aspect 7 Longitudinal shear modulus (GPa) ),
"'= 1"
0.9
0.8
0.7
0.6
0.5
(I.4
0.2
0.1
0.000 l
0.20 0.50 0.70 (I.75
2.28 4.49 8.28 10.1
2.28 4.50 8.32 10.2
2.29 4.57 8.50 10.4
2.31 4.70 8.87 10.9
2.35 4.93 9.52 11.8
2.42 5.34 10.7 13.3
2.53 6.10 12.8 16.1
3.32 11.6 26.7 33.1
5.62 25.0 50.4 59.1
35.4 86.5 121 129
* F o r 7 - 1, the longitudinal shear modulus cannot be directly computed by using the present formula. Instead, ?' is taken as a value approaching 1, for example ? = 0.9999
El=413.69GPa,
G / = 172.37GPa,
E,,, = 4 . 1 3 6 9 G P a ,
G,,, = 1.5322GPa,
v~=0.2 r,,, = 0.35
The l o n g i t u d i n a l shear m o d u l u s from the present theoretical f o r m u l a [eqn (56)] for various fiber section aspect ratio ), (7' = bl/al) are shown in Table 1. It can be seen that the l o n g i t u d i n a l shear m o d u l u s is very sensitive to the aspect ratio. As the aspect ratio decreases (noting t h a t 0 < i' ~< 1 w i t h o u t loss o f generality), the l o n g i t u d i n a l shear m o d u l u s increases for a certain fiber v o l u m e fraction. F r o m Fig. 5 in W h i t n e y a n d Riley (1966) it is seen that the shape o f fiber section o f the specimens are irregular, so it is i m p r o p e r to be r e g a r d e d as a circle. To account for the influence o f the shape, the fiber sections are t a k e n as an ellipse with ~, = 0.4 a n d the results are listed in T a b l e 2, which shows that the theoretical predictions are in r e a s o n a b l e a g r e e m e n t with the e x p e r i m e n t a l results. T h e second e x a m p l e is the glass-epoxy composite, with the elastic c o n s t a n t s being ( Z h a o a n d Weng, 1990): EI=72.4GPa,
GI=30.2GPa,
E,,, = 2 . 7 6 G P a ,
G,,, = 1.02GPa.
v1=0.2 r,,, = 0.35
By using the present formula, the n o r m a l i z e d l o n g i t u d i n a l shear m o d u l u s as a function o f )~ is depicted in Fig. 3. A c o m p a r i s o n with the numerical results ( Z h a o a n d W e n g , 1990) by using E s h e l b y - M o r i T a n a k a t h e o r y shows that the results o f the two m e t h o d s are in g o o d agreement. The present f o r m u l a is a p p l i c a b l e to the whole spectrum o f inclusion types, i.e. from voids to rigid inclusions. W h e n the elastic m o d u l u s o f fibers are equal to zero, the present f o r m u l a gives the results for m a t e r i a l s with voids. The n o r m a l i z e d shear m o d u l u s p r e d i c t i o n s for the m a t e r i a l with voids are depicted in Fig. 4, which shows that the shear m o d u l u s is also very sensitive to the aspect ratio o f voids. Based on the a b o v e results and c o m p a r i s o n we can m a k e the following c o n c l u s i o n s : 1. The p r o p o s e d i n c l u s i o n / m a t r i x / c o m p o s i t e m o d e l o f a c o m b i n e d confocal elliptical cylinder unit cell is a p p l i c a b l e to the whole spectrum o f inclusion types o f inclusions ( f r o m
Table 2. A comparison of the theoretical predictions of the longitudinal shear modulus with experimental results Longitudinal shear modulus (GPa) ). 0.70 0.75
Classical model ~
Present model ~
Experiment 3
8.27 10.10
12.82
12.21 16.75
16.06
Referring to Hashin and Rosen, 1964: Whitney and Riley. 1966. : Taking ~, = 0.4. 3Referring to Whitney and Riley. 1966.
3986
C. P. Jiang and Y. K. Cheung 30 25 2O
E "~
15
j"
~o.ooooo~/
0
//
10
0.2
I 0.4
t 0.6
I 0.8
Fig. 3. Variation of the longitudinal shear modulus.
1 0.9 0.8 0.7 0.6 (.9
0.5 0.4 0.3 0.2 0.1 0 0.2
0.4
0.6
0.8
Fig. 4. Shear modulus for the material with ~oids.
voids to rigid inclusions), and can also a c c o m m o d a t e a full range o f aspect ratios ( from circular fibers to r i b b o n ) . 2. By using the c o n f o r m a l m a p p i n g technique integrated with the Laurent e x p a n s i o n m e t h o d , the l o n g i t u d i n a l shear m o d u l u s is derived in a closed form, which is p a r t i c u l a r l y convenient for a p p l i c a t i o n in engineering. 3. A c o m p a r i s o n with experimental d a t a and numerical solutions shows the effectiveness and accuracy o f the present formula. The i n c l u s i o n / m a t r i x / c o m p o s i t e m o d e l of a c o m b i n e d confocal elliptical cylinder unit cell is also a p p l i c a b l e to predict the effective tensile m o d u l u s in terms o f the other two c o m p l e x stress functions ~ ( z ) and ~P(z) (Muskhelishivili, 1975): however, it is difficult to o b t a i n the closed form solution. The p r o b l e m will be dealt with elsewhere. REFERENCES Benveniste, Y. (1985) The effective mechanical behavior of composite materials with imperfect contact between the constituents. Mechanics (~fMaterials 4, 197 208. Christensen, R. M. (1993) Effective properties o1 composite materials containing voids. Proceedings O/the Royal Society ~[London A440, 461 473. Christensen, R. M. and Lo, K. M. (1979) Solutions lot effective shear properties m three phase sphere and cylinder models. Journal of the Mechanics and Physics of'Solids 27, 315 330. Hashin, Z. and Rosen, B. W. (1964) The elastic moduli of fiber reinforced materials. Journal (~fApplied Mechanics 31,223 232. Huang, Y. and Hu, K. X. (1995) A generalized sell-consistent mechanics method for solids containing elliptical inclusions. ASME Journal Of Applied Mechanics 62, 566 572.
A liber matrix composite medel
3987
Luo. H. A. and Weng, G. J. (1987) On gshelby's inclusion problem in a three-phase spherically concentric solid, and a modification of Mori T a n a k a ' s method. Mechamcs o/Materials 8, 347 361. Mori, T. and Tanaka. K (1973) Average stress in the matrix and average elastic energy of'materials with misfitting inclusions. Acta Metallurgica 21,571 574. M uskhelishivili, N. 1. (1975) Some basic problems of mathematical theory o1"elasticity. N oordhoff, Leyden. Taya. M. and Chou. T. W. {1981) On two kinds of ellipsoidal mhomogeneities in an infinite elastic body: an application to a h~brid composite. InlernationalJom'md o/Solids atutStruclure.s 17, 553 563. Whitne>. J. M. and Riley. M. B. (1966) Elastic properties of fiber reintk)rced composite materials..4 IAA .lomwal 4(9), 1537 1542. Zhao. Y. H. and Weng, G. H. (1990) Effective elastic moduli of ribbon-reinforced composites. Ah'ME ,Iomwa/~!! Ii)l)/icd M~'chanics 57. 158 167.