- Email: [email protected]
Composite materials with viscoelastic interphase: creep and relaxation
Mechanics of Materials 11 (1991) 135-148 Elsevier
135
Composite materials with viscoelastic interphase: creep and relaxation Zvi Hashin 1 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA Received 8 May 1990; revised version received 16 August 1990
The effect of viscoelasticinterphase between elastic constituents of composites is investigated on the basis of a correspondence principle relating effective elastic and viscoelastic properties of composites, which has been previously established. Previously derived results for thermoelastic properties of unidirectional fiber composites and spherical particle composites with elastic interphase are utilized in this fashion to obtain correspondingviscoelasticeffectiveproperties. The analysis shows that significant viscoelastic effect in such unidirectional composites is confined to axial and transverse shear relaxation and creep and in particulate composites to any shear loading. The viscoelastic effect is negligible for all inputs which do not include such shear components.
1. I n t r o d u c t i o n
In recent years there has been considerable interest in the effect of the nature of the interface on the properties of composite materials. Imperfect interface conditions have been quantitatively defined as linear relations between interface displacement discontinuities and associated traction components. The effect of such interfaces on the elastic properties of composite materials has been investigated in Benveniste (1985), for particulate composites, Achenbach and Zhu (1990), numerically, for fiber composites modeled as periodic hexagonal arrays, Hashin (1990a), for fiber composites, and Hashin (1990b) for particulate composites. The last two papers also dealt with the effect of interface on thermal expansion. In particular it was shown in Hashin (1990a, 1990b) that the effect of an interphase, by which is meant a thin coating on the reinforcing fibers or particles, can be expressed in terms of imperfect interface conditions and the interface parameters involved can be determined in terms of interphase characteristics. In the present work we consider the case of viscoelastic interphase between two elastic components. An interphase of such nature may be a by-product of manufacture or it might be introduced by design, e.g. to provide relaxation and damping characteristics to an otherwise elastic brittle composite. It is our purpose to evaluate effective quasi-static viscoelastic properties of such composites. We do this on the basis of analytical results for elastic composites with interphase given in Hashin (1990a, 1990b), and correspondence principles for viscoelastic composites established in Hashin (1965, 1966). The case of viscoelastic interphase in unidirectional fiber composites has also been recently considered by Gosz et al. (1990) in terms of finite element analysis of periodic hexagonal arrays and also some simple applications of the correspondence principle.
1 On leave of absence from Tel Aviv University, Tel Aviv, Israel. 0167-6636/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
Z. Hashin / Viscoelastic interphase
136
2. General
Effective viscoelastic properties of composites are expressed by the effective relaxation moduli tensor C * ( t ) and the effective creep compliance tensor S * (t). The former defines the average time-dependent stress response to sudden imposition, in the Heaviside sense, of average strain. Dually, the latter defines the average time-dependent strain response to imposition of similar average stress. For details see Hashin (1965, 1972). A general correspondence principle relating effective elastic properties to effective viscoelastic properties has been established in Hashin (1965). To recapitulate, suppose that the effective elastic moduli tensor eC* of some composite material is related to the elastic phase moduli tensors eC ( ' ) by the tensor function F. Thus, c c * = F C C (1),
. . . . . e c ( " ) . . . . ).
(1)
where of course F is also a function of phase geometry. Now consider a composite with precisely the same phase geometry, but now some or all of the phases are linearly viscoelastic. This will be called the associated viscoelastic composite and its phase relaxation moduli and creep compliance tensors are denoted C* (t) and S *(t), respectively. Denoting the Laplace Transform (LT) of a function as follows ¢ ( p ) = f0~q~(t) e-m dt,
(2)
the correspondence principle expresses the LT of the effective relaxation moduli of the associated viscoelastic composite in the form = ,r(
p (2, . . . . .
.... ).
(3)
The p multiplied LT of phase relaxation moduli C(m)(t), have been named T D (transform domain) moduli in Hashin (1965), for they are the viscoelasticity analogues of elastic moduli. Note that the T D moduli of an elastic phase are precisely the elastic moduli, and it follows therefore that if some of the phases are elastic their elastic moduli are to be entered into (3). The LT of the effective creep compliance tensor can be expressed analogously to (3) and obviously T D phase compliances can be entered into any one of the relations instead of T D phase moduli, if more convenient. Accordingly, the evaluation of effective viscoelastic properties has been reduced to inversion of LT, provided that analytical expressions for effective elastic properties are known. Such inversion can be performed analytically or otherwise numerically. There are however important cases when LT inversion is not required. It has been shown in Hashin (1966) that it follows from Tauberian theorems for LT that the values of C*(t) at times t = 0, ~ are given by C * (0) = r [C (1)(0), C (2)(0) . . . . . C (m) (0) .... ], .....
(4)
.... ],
with analogous results for S * (t). The result for t = 0 merely states that the initial, elastic response of the composite is defined by its effective elastic moduli, as is certainly to be expected. The result for t = ~ is more interesting; it states that final values of effective elastic properties are given by effective property expressions in terms of final values of phase viscoelastic properties. This result often has important practical significance for if it turns out that initial values and final values of effective viscoelastic properties are close, this implies that the viscoelastic effect is negligible, Several applications to fiber composites have been given in Hashin (1972).
Z. Hashin / Viscoelasticinterphase
137
3. Viscoelastic interface conditions
We recall the elastic form of imperfect interface conditions as previously used. Referring to Fig. 1, the interface S]2 separates the two elastic media 1 and 2. Let n denote normal direction to the interface, and s, t orthogonal tangential directions. For reasons of equilibrium, traction components must be continuous across the interface. The interface imperfection is expressed by displacement jumps which are proportional to their associated traction components. Thus,
T,?) =
= D.[u.],
T~(1) = Ts(2) = D s [ u s ],
v,<') = T,<2)= D, [,,,
(5a) (5b)
on S,2,
(5c)
].
It has been shown in Hashin (1990a, 1990b) that for an isotropic thin interphase whose stiffness is much smaller than that of the reinforcement, D. = ( K i + 4 G i ) / 8 ,
(6a)
D s = D, = O y 6 ,
(6b)
where i denotes interphase property and $ is interphase thickness. It may be shown that when the above stiffness condition is not imposed, (6a) assumes the more complicated form
Ki £ 4Gi -~- E2( "]-c
[Un]/~ =
Pi) Tn
E2~ -= Pi--) (O's(s2)"1"-Ot~2))'
(7)
while (6b) remains as before. A derivation will be given elsewhere. It is seen that (7) is no longer of the form (5a). In the case when E i << E2, (7) reduces to (5a) with (6a). The formal viscoelastic counterpart of (5a) is t
0
T~(t) = D . ( t ) [ u . ( O ) ] + f o D . ( t - t ' ) - ~ - i T [ u . ( t ' ) ]
dt',
(8)
where D . ( t ) is an interface function. Similar expressions replace (5b) and (5c). The LT of (8) is given by
L=pb.,
(9) ^
and it is seen that pD~, is the normal T D interface parameter. We can similarly define tangential T D interface parameters p D s and P/)t- In the case of viscoelastic interphase we shall make the reasonable
MATERIAL 1 ~S12
Fig. 1. Interface.
138
Z. Hashin / Viscoelastic"interphase
simplifying assumption that the viscoelastic effect is restricted to shear. Then the T D domain counterparts of (6) are
p b . = (K~ + 4 p d i ) / 6 ,
(10)
pbs = p d i / ~ . The T D domain form of (7) is much more complicated. To make use of the elastic Ki restriction, u~ in (7) must first be expressed in terms of K~ and Gi, and then G~ is replaced by pG~. It follows from the correspondence principle, that in order to obtain the LT of effective relaxation moduli or creep compliances of a composite with viscoelastic interphase, the interface parameters in the corresponding results for elastic interphase are replaced by (11), in the case of interphase of initial elastic stiffness much smaller than reinforcement stiffness. If the last condition is not fulfilled, the normal interface parameter D, is replaced by the T D version of (7). We shall now proceed to use this scheme to investigate viscoelastic behavior due to interphase on the basis of the elastic results obtained in Hashin (1990a, 1990b).
4. Unidirectional fiber composites A unidirectional fiber composite (UFC) consists of parallel fibers, generally of circular cross-sections, which are embedded in a matrix at random locations in the transverse plane. The fibers are assumed elastic transversely isotropic, which is a suitable description of c a r b o n / g r a p h i t e fibers, while the matrix will be assumed isotropic. In between the fibers and matrix there is a very thin viscoelastic interphase. The fiber composite is macroscopically transversely isotropic. The case of elastic interphase has been considered in Hashin (1990a), and in this case the composite is elastic. Referring a composite specimen to a Cartesian coordinate system with x 1 in fiber direction and x2, x 3 in transverse directions, the effective thermoelastic stress-strain relations can be written in the form Oil = n*~ll +
1"422 +
1"~:33 + FA*~,
0"22 = ]*~11 q- ( k * -}- G t ) ~ 2 = -{- ( k * - Gt)~33 + F-~q~,
(II)
0-33= / * g , , + ( k * - G~- )i:2 + ( k * + G~ )i33 + Fa~;
F~, = -(n*a~, + 2 l * a { ) , O12 = 2 G ~ e , 2 ,
ill = E 2
(i-23 = 2 G ~ 2 3 ,
_
(i-31 = 2 G ~ 3 1 ;
(12)
E~ 0.22 -- -~-~ O33 q- 0 / ~ , E ? o33 + a~q~,
%2 = - ~-~-~6,1 + E¢
C33 --
F~- = - ( l * a ~ + 2k*a~-);
_
E 2 ¢lll --
o22
(i% +
E{
+
Here, A indicates axial fiber direction, and T transverse, normal to fibers, direction, * indicates effective property, n and l are moduli associated with uniaxial strain in 1 direction, k is transverse bulk modulus, G is transverse shear modulus, E is Young's modulus, ~ is Poisson's ratio, a is thermal expansion coefficient, overbar denotes average, and q~ is uniform temperature change.
139
Z. Hashin / Viscoelasticinterphase
When the interphase is viscoelastic, the effective stress-strain relations of the composite also become viscoelastic. The viscoelastic counterparts of (11)-(13) can be written in terms of convolutions in the time domain or in terms of TD moduli and compliances in the transform domain, the latter being entirely analogous to (12)-(14). See Hashin (1972) for details.
5. Axisymmetric properties
Axisymmetric properties are associated with states of average strain and stress which are axisymmetric with respect to the fiber direction, together with a uniform temperature change. Such states of strain and stress are
=
£11 0 0
0 i22 0
0 1 0 ], i22
=
011
0
0 ]
0
622
0 ].
0
0
(14)
0"22
It follows from (11)-(13) that the elastic stress-strain relations have the form 611 = n *~11 at- 2l *~22 -at-/-'~, 022 = o33 = l*i n + 2k'g22 + / ~ t h , =
011
Et
2P~
_
+
~A
+
( 1
)o22+ t .
(15)
It has been shown in Hashin (1990a) that in the case of imperfect interface conditions or interphase, all of the above properties can be evaluated on the basis of the composite cylinder assemblage (CCA), or the generalized self-consistent scheme (GSCS) models, with identical results. The results were conveniently expressed in terms of equivalent properties of a fiber with imperfect interface or coated with interphase. In the axisymmetric case the only equivalent fiber property which is different from the usual fiber properties is the transverse bulk modulus. This is given by
kf ke -
(16)
1 + 2kf/D.a"
In the case of isotropic interphase of thickness 8 and elastic moduli K i and provided that K i, G i << kf,
~J<< a.
Gi,
Dn
in (16) is given by (6a) (17)
Here kf is fiber transverse bulk modulus, and a is fiber radius. The properties appearing in (15) are then given by k* = k~ +
1
Uf
v~
'
(18a)
ke__ km q" km._t_~ E ~ ~- Emvm "t- EAf Vf,
(18b)
Z. Hashin / Viscoelastic"interphase
140
(VAfP/~ = VmVm + ltAfV f Jr-
O~
=
Vm)(1/km
-
1/ke)vmV f
(18c)
Um/k e + vr/k m + 1/G m
Olm -~- (O/Af- am)[ pl,( S,*1 - s,n~) + 2P,2( S,~ - S~)] + 2(aVf- am)[P12(S,*~ - $1~) + (P22 + P23)($1~ - S ~ ) ] ,
ot~ = an, + (OtAf- am) { P,,(S,*2 - $1~) + P12 [ 32~ + 32~ - ( S ~ + S~)] } + 2(avr- am){ P12(S~'2 - S ~ ) + ½(P22 + P23) [ $2~ + S~ - ($2~ + $2~)] },
(19)
where P,, --- (AS22 + AS23)/D,
P,2 = - A S , 2 / D ,
P22 + P23 = A & , / D ,
D --- AS,~(AS22 + ZXS23) - 2AS~2,
ASll
ASI2 =
= I / / E A f -- $1~ ,
AS22 + AS23 = ½(1/k e + 4 ~ 2 f / E A r )
--
Snm ___ 1~Era '
m = $12
S,*~ = I / E 2 ,
S~2= -v~./E~,,
__ P m / E m ,
_ P A f / E A f - - Sln~,
($2~ + S2n~), Sdm2 +
$23 m=
(1
-
b,m
)/Em,
Sff2 + Sff3= ½(1/k* +4vy2/EA),
where m and f indicate isotropic matrix and transversely isotropic fiber properties, respectively. The effective moduli n* and l * can be evaluated by use of the transverse isotropy relations n* = E 2 + 4k*v2
2,
l* = 2k*v~,.
(20)
In order to evaluate the T D properties corresponding to (18)-(20), we evaluate the T D equivalent transverse bulk modulus P/~e by replacement of D, in (16) by (10). We then replace k e in (18)-(20) by p/~¢ everywhere. In order to invert the T D properties analytically into the time domain, it is necessary to describe interphase shear viscoelasticity by a differential time operator, which leads to cumbersome LT inversions. Fortunately this is not needed, from a practical point of view, for it will be shown that the viscoelastic effect for the axisymmetric properties is very small. To see this, we evaluate the initial and final values of the viscoelastic properties k*(t), E*(t), n*(t), l*(t), a,~(t), and a-~(t), at times t = 0, oo, by use of the Tauberian results (4). To do this, we first evaluate the initial and final values of the interface functions D,(t) and D,(t), equation (8), by use of (11) and the Tauberian theorems. The results are just as (6) with initial and final relaxation moduli. We assume that K~ does not change with time, and that G~(t) diminishes to 0 after infinite time. We then have
D,(O) = [Ki+ 4Gi(O)]/8,
D,(oo)-=Ki/8,
D,(O) = G i ( 0 ) / 8 ,
Ds(m ) = O.
(21)
The last result implies a condition of free tangential sliding in the final state. The normal interface parameters can be related in terms of the interphase initial Poisson's ratio vi(0 ). Thus, D.(oe) D.(0)
1 + vi(0 ) 311 - v~(0)] "
(22)
To obtain the initial and final values of the axisymmetric viscoelastic properties, the initial and final values of D n are used in evaluation of (18)-(20) with (16). We have done this for two fiber composites:
Z. Hashin / Viscoelastic interphase
141
T300 graphite fibers in epoxy matrix, and a ceramic composite consisting of Nikalon fibers and BSi matrix. Fiber and matrix elastic properties, and thermal expansion coefficients are: T300 (Transversely Isotropic ) EAr ----231 GPa, GAf =
ETf =
22.4 GPa,
kf = 17.5 GPa,
GTf =
8.3 GPa,
22.0 GPa,
PAf = 0.30,
t)Tf = 0.35,
aAf= --1.33)<10 -6 C °,
aTf = 7.04 × 10 -6 C ° ;
Nikalon ( lsotropic ) Ef = 193 GPa,
Gr = 80.4 GPa,
uf = 0.20,
a f = 3.1
Gm = 1.28 GPa,
~'m= 0.35,
a m = 6 3 × 10 -6 C ° ;
Gm = 2 5 . 7 G P a ,
~m=0"22,
a m = 3.2 ×
x 10 - 6 C ° ;
Epoxy Em=
3.45 GPa,
BSi Em=62.7GPa,
10 -6
C ° .
The interphase initial Poisson's ratio was chosen as Pi ( 0 ) = 0 . 3 7 5 .
Initial and final values of relaxation moduli have been evaluated for various values of the interface constant kf/aD,(O) and volume fractions. It was found that the difference between final and initial values of axisymmetric properties was very small and it may thus be assumed that the viscoelastic effect for these properties can be neglected.
Axial shear The effective elastic axial shear modulus for fibers coated with thin interphase is given by, Hashin (1990a), Of
a~-am
6e
q-
1//(Ge -
Gf 1 + paf/6 i '
a m ) -k- V m / 2 a m '
p = 8 / a , Gf = GAl.
(23)
This result is valid for any ratio Ge/Gi, but since 6 << a the effect of interphase on the elastic shear modulus will only be significant if G f / G i is large enough. We now consider the case of viscoelastic interphase. The simplest viscoelastic material which exhibits the necessary features of initial elastic response and unlimited creep is described by the Maxwell model. We accordingly assume that the interphase material deviatoric stress and strain obey the stress-strain relation
skt
e~t = ~
i~1
(24)
+ 2G'---~'
where "//i is a viscosity coefficient, G i is initial shear modulus, i indicates interphase property, and the dot denotes time derivative. The LT of (24) is 2r/iP ^ Sk,--'~ l - - - ~ i p ( k l ,
Ti = ~i/G,,
(25)
Z. Hashin / Viscoelastic interphase
142
and therefore the TD shear modulus of the interphase is ~P 1T~p"
pdj-
(26)
Replacing G~ in (23) by (26) yields the TD effective axial shear modulus pG~. The effective TD creep compliance p ~ is then given by (27)
p~,~ = 1/pGfl,.
The LT inversion required is elementary and yields the explicit results G;(t)
= Gm[AH(t)
+ (B-A)
e-'/~r], (28)
gX(t)=l/Gm[AH(t)+(1-1)e-'/"],
where Vm A = - -
l+vf'
B=
y(l
"1"-Vf ) / U r n nt- I -1- p G f / G i
3' + (1 + p G f / G i ) ( 1 + v f ) / v m ' yVm/(1 + vf) + 1 + pGf/6i
rr = r i
pGf/Gi
'
y(1 + v f ) / v m + 1 + p G f / G i ,rc = r i
pGf/Gi
'
Y = Gf/am,
(29)
and H ( t ) is the Heaviside step function. It is seen that the initial elastic axial shear response is given by (30)
G ~ (0) = 1 / g ~ (0) = GmB,
while the final response is Vm
G~ (oc) = 1 / g ~ (o¢) = Gm 1 + vf"
(31)
The result (29) is the same as the elastic result (23). The result (30) is the axial shear modulus of an elastic matrix perforated with continuous aligned cylindrical holes, Hashin (1972). This result implies that after long time the interracial shear stresses in the interphase relax to zero and thus the coated fiber behaves in the final stage as a cylindrical void. It should be emphasized that this last conclusion is valid for any linear viscoelastic interphase whose final relaxation modulus vanishes. This is easily shown from the Tauberian results (4), applied to the present case. Other noteworthy special cases relate to the value of the quantity p G f / G i. If Gi is not very much smaller than Gf, this quantity becomes a small number, and therefore the characteristic times in (29) become very large. This implies negligible viscoelastic effect since very long time has to elapse before such effect becomes noticeable. If, on the other hand, G~ is so small that p G f / G i becomes a very large number, then the characteristic times approach the characteristic time of the interphase material and there is significant viscoelastic effect. It is thus seen that significance of viscoelastic effect is strongly dependent on the elastic shear stiffness of the interphase.
143
Z. Hashin / Viscoelastic interphase
UNIDIRECTIONALFIBER COMPOSITEWITH VISCOELASTICINTERPHASE NIKALON FIBERS BSi MATRIX vf =0.50
NIDIRECTIONALFIBER COMPOSITE WITH VISCOELASTICINTERPHASE NIKALON FIBERS BSi MATRIX 1.5
~ ~ , ~
3.0
,jDvf=0.50
GA(t) Gm
q= pGAf /Gi
2,0
1.0
1
Z *d 0.5
Y
0.1
~
CYLIvNglRIsCAL
I
I
I
I
l
1
10
102
10 3
10 4
t/ri
0.1
I
I
I
1
10
10
2
I
103
I
104
t/ri
Fig. 3. Axial shear creep compliances.
Fig. 2. Axial shear relaxation moduli.
zo~
~E1.0
UNIDIRECTIONALFIBER COMPOSITEWITH VISCOELASTICINTERPHASE NIKALON FIBERS BSi MATRIX vf =0.50
3 '0 F/~q= I
I[ /
I
CYLINDRICAL VOIDS
q=PGMIGi
I
I
I
I
2500 5000 7500 10,000 Fig. 4. Axial shear creep compliances -- real time.
t/'r;
Plots of relaxation moduli and creep compliances as functions of log(t/~i) for various values of p G f / G i are shown in Figs. 2 and 3, and a plot of relaxation moduli in real time in Fig. 4. It is seen that a minute amount of viscoelastic interphase can produce significant time dependence in axial shear for an otherwise elastic material.
Transverse response We first consider the case of effective transverse shear relaxation modulus G~-(t), and creep compliance g~ (t). The elastic transverse shear modulus for fibers coated with interphase has been evaluated in Hashin (1990a) on the basis of the GSCS. This requires the solution of a quadratic equation whose coefficients are defined in terms of 8 × 8 determinants. Following the correspondence principle we can use these results to write similar equations to evaluate the normalized effective T D modulus,
1-'*(p) =pG~(p)/Gm,
(32)
144
Z. Hashin / Viscoelastic interphase
as a root of the determinantal equation 1
-1
1
1
F*
3I"*
1
2+~m
0
1
1
~m
1
2
1
l+~m 1
1
0
0
--1
0
0
3
0
0
3
0
0
3/c2
0
-~,
3"fc l-~f
-3'
F* -3F*
3
0
0
0
1
0
0
3c l_~m
1
1 (1 +~m) c
3/c2
0
0
c
1
l/c
1/c 2
1
D(I'*, p)=
0
0
(2+~m)C 1 --~m
1
2
(1 + ~Sm)c
~m (l q- ~m)C
_l/c 2
-c
(6q+2+~f)c 1 --~f
=0,
- (2m + 1)
-(2q+
1)
(33)
where ~nl = G m / k m ,
~f = G f / k f ,
y = Gf/Gm,
c = of,
pGf ki+4pdi/3,
m=m(p) G r = GTf ,
pGf q = q ( P ) = PGi'
(34)
km = K m + Gm/3.
The determinant in (33) is evidently a quadratic equation in F*, and therefore (33) can be written as
D(F*, p ) = A ( p ) F * 2 + 2B(p)F* + C ( p ) = O ,
(35)
where the coefficients can be expressed in the form A=~[D(1,
p ) + D ( - 1 , p)] - D ( 0 , p ) ,
B=¼[D(1, p)-D(-1,
p)],
C = D(0, p ) .
(36)
The LT of the effective transverse shear creep compliance g * ( t ) , is then defined by the usual relation
p~. = 1/pG~-.
(37)
The T D effective transverse Young's modulus/3E~ is given by the elastic relation
4/pff~ = 1/pd~ + l / k * + 4v~Z/E,~,
(38)
where it has been assumed, on the basis of previous conclusions above, that k*, v~ and E 2 are elastic. Then, the LT of the associated compliance is defined by
pe.~ (p) = 1left.S. (p).
(39)
Analytical inversion into the time domain does not appear feasible, but numerical inversion can be performed on the basis of the results listed above. Here, we shall confine ourselves to exploitation of the Tauberian theorems to obtain initial and final values of properties of interest. To do this we merely have to
145
Z. Hashin / Viscoelastic interphase UNIDIRECTIONAL FIBER COMPOSITE WITH VISCOELASTIC INTERPHASE NIKALON FIBERS
BSi MATRIX
vf = 0 4 0
1.50
E~E~ (0)/ E m
100
050
O.OJ
I 0.1
I I
I I0
l 10 2
I 10 3
q(O)
Fig. 5. Initial and final values of transverse Young's relaxation modulus.
perform an elastic analysis with initial and final values of the interface parameters as given by (21)-(22). Figure 5 shows plots of E~ (0)/E m and E~ (oo)/E m versus the initial interface shear constant pGf q(0) = ai(0 ) .
(40)
In the final state there is free tangential sliding at the interface and the normal bond interface parameter is magnified by the factor (22). The plots show the extent of the viscoelastic effect of the interphase and it is seen that the effect diminishes with weakening of normal bond until initial and final values both converge to the elastic result for cylindrical voids.
6. Particulate composites Elastic properties of spherical particle composites with imperfect interface or interphase have been analyzed in Hashin (1990b) and the results can be transcribed to the case of viscoelastic interphase exactly as was done above for UFC. Labeling matrix by 1 and particles by 2, the effective bulk modulus K * for the case of thin elastic interphase is given by /;2
K* = K 1+
1
- - + Ke - K1
K2
3v I
,
Ke - 1 +
3K2/aD
. '
(41)
3K 1 -I- 4G 1
where D, is expressed by (6a). Then, the effective thermal expansion coefficient is given by a2--al 0/*=0~1"}- 1
Ke
!
( 1 K*
1) K1
"
(42)
K1
Explicit results in the time domain for the corresponding viscoelastic properties can be obtained by assuming a simple viscoelastic model for the interphase and using the correspondence principle as was done above in the case of axial shear. This, however, is hardly worthwhile, for evaluation of final and
Z. Hashin / Viscoelastic interphase
146
initial values K * (0), K * (m), a* (0), and a * ( m ) , by introduction of (21) into the elastic results (41)-(42), shows that there is very little difference between initial and final values, and thus the effect of viscoelastic interphase on these properties is very small. The situation is of course quite different for shear. In this case, analytical treatment is entirely analogous to the method employed above for transverse shear of a unidirectional fiber composite. The effective elastic shear modulus for the case of imperfect interface or elastic interphase has been evaluated on the basis of the GSCS in Hashin (1990b), requiring again the solution of an 8 X 8 determinantal equation. Employing again the correspondence principle, and defining in analogy to (32) the normalized effective T D modulus,
F* =pG*(p)/Gm,
(43)
we find that this quantity is the root of the equation
D(F*)=
2
3Ns
12Vl~, 2
2
2(5 -- 4 v 1 )/)~t
- 3/x ~
0
0
1
-1/X 5
(7-- 4Vl)h 2
1
2(1 - 2 v 1 )/,k ~
1/x~
0
0
2F*
- 12F*/X 5
- 6v~X2
2
-4(5 -
12/x'
0
0
2F*
4F*/X 5
(7 + 2 v l ) ~ 2
1
2(1 + v~ ) / X 3
- 4/)~ 5
0
0
0
0
-6v t
2
- 4 ( 5 - v1
12
6v2y
-- 2"),
0
0
7 + 2v 1
1
2(1 + v I )
-4
0
0
12v 1
2
2(5 - 4 v I )
0
0
1
2(1
7
4v 1
Ul)/X ~
2vl)
(7 + 2v2Y ) 3
1
=0,
--y
12u2(m - 1)
- 2(1 + 2 m )
--[2(7 + 2v2)q+ 7-- 4v:]
--(1 + 2 q )
(44) where v1 and v2 are matrix and particle Poisson's ratios, respectively, and
y = G2/G1,
X3 = v2,
oG2 Ki+4pGi/3,
m=m(p)
q = q ( p ) = pG: pG i '
(45)
where a in p, equation (23), is now particle radius. Equation (44) can again be put into the form (35), (36). The LT of the effective shear creep compliance g*(t) is evaluated by the usual relation
p~,*(p) = 1/pG*(p),
(46)
and the LT of effective Young's relaxation modulus E * (t), and creep compliance e* (t) are defined by
pff~*(p)
9K *pG* ( p ) =
3K* + p G * ( p ) '
(47)
p~*(p) = 1/p£*(p), where it has been assumed that K * is elastic. Inversion into the time domain can again only be done numerically and we again confine ourselves to evaluation of initial and final values of relaxation moduli. This is done exactly as in the case of transverse shear modulus as outlined above. Figure 6 shows such results for normalized effective shear relaxation moduli G* ( 0 ) / G m and G * ( m ) / G m for the case of glass spheres embedded in epoxy matrix, plotted versus the initial value of shear interface constant pG2 q(O) = Gi(O ) .
Z. Hashin / Viscoelastic interphase
147
SPHERICAL PARTICLE COMPOSITE WITH VISCOELASTIC INTERPHASE GLASS PARTICLES
E P O X Y MATRIX
v2 = 0 4 0
25 O)IG m 2.0 15
G*(m)IGm
10VOIDS
i
05
001
I 0.1
I I
I I0
I I0 z
I 103
q (0)
Fig. 6. Initial and final values of shear relaxation modulus -- particulate composite. The elastic properties of glass are: E 2 =
72.4 GPa,
G 2 =
30.2 GPa,
1"2= 0.20,
and the epoxy properties the same as used before. The final state is again characterized by freely sliding interface, and with weakening normal bond both properties converge to the elastic effective shear modulus for spherical voids.
7. Discussion and conclusion Effective relaxation moduli, and creep compliances of fiber composites, and particulate composites with viscoelastic interphase have been derived on the basis of previous results for elastic interphase and a correspondence principle for viscoelastic composites. The results obtained reveal that time dependence of effective properties produced by viscoelastic interphase is significant only for shear inputs such as axial and transverse shear for UFC, and for any shear for a particulate composite. Inputs such as uniaxial stress which include such shear components will of course also produce such time effects. Axisymmetric inputs for UFC and spherically symmetric inputs for particulate composites, thermal expansion being included in these, will produce negligible time effects. Note that the insignificance of time effects for thermal expansion also remains for the case when interphase viscoelastic properties are temperature dependent, a case which has not been considered in this work.
Acknowledgement This work has been supported by the Air Force Office of Scientific Research (AFOSR), Lt. Col. G. Haritos, and by the Office of Naval Research (ONR), Dr. R. Jones, through contract between the National Center for Composite Materials Research at the University of Illinois, Urbana, Dr. S.S. Wang, Director, and the University of Pennsylvania. I am indebted to the aforementioned for support and encouragement. Helpful discussions with Professor J.D. Achenbach and Professor B. Moran are also acknowledged.
148
Z. Hashin / Viscoelastic interphase
References Achenbach, J.D. and H. Zhu (1989), Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites, J. Mech. Phys. Solids 37, 381-393. Achenbach, J.D. and H. Zhu (1990), Effect of interphases on micro- and macromechanical behavior of hexagonal array fiber composites, J. Appl. Mech. 57, 956-963. Benveniste, Y. (1985), The effective mechanical behavior of composite materials with imperfect contact between the constituents, Mech. Mater. 4, 197-208. Gosz, M., B. Moran and J.D. Achenbach (1990), Effect of viscoelastic interface on the transverse behavior of fiber reinforced composites, to be published.
Hashin, Z. (1965), Viscoelastic behavior of heterogenous media, J. Appl. Mech. 32, 630-636. Hashin, Z. (1966), Viscoelastic fiber reinforced materials, AIAA J. 4, 1411-1417. Hashin, Z. (1972), Theoo' of Fiber Reinforced Materials, NASA CR, 1974. Hashin, Z. (1990a), Thermoelastic properties of fiber composites with imperfect interface, Mech. Mater. 8, 333-348. Hashin, Z. (1990b), Thermoelastic properties of fiber composites with imperfect interface, J. Mech. Phys. Solids, in press.
135
Composite materials with viscoelastic interphase: creep and relaxation Zvi Hashin 1 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA Received 8 May 1990; revised version received 16 August 1990
The effect of viscoelasticinterphase between elastic constituents of composites is investigated on the basis of a correspondence principle relating effective elastic and viscoelastic properties of composites, which has been previously established. Previously derived results for thermoelastic properties of unidirectional fiber composites and spherical particle composites with elastic interphase are utilized in this fashion to obtain correspondingviscoelasticeffectiveproperties. The analysis shows that significant viscoelastic effect in such unidirectional composites is confined to axial and transverse shear relaxation and creep and in particulate composites to any shear loading. The viscoelastic effect is negligible for all inputs which do not include such shear components.
1. I n t r o d u c t i o n
In recent years there has been considerable interest in the effect of the nature of the interface on the properties of composite materials. Imperfect interface conditions have been quantitatively defined as linear relations between interface displacement discontinuities and associated traction components. The effect of such interfaces on the elastic properties of composite materials has been investigated in Benveniste (1985), for particulate composites, Achenbach and Zhu (1990), numerically, for fiber composites modeled as periodic hexagonal arrays, Hashin (1990a), for fiber composites, and Hashin (1990b) for particulate composites. The last two papers also dealt with the effect of interface on thermal expansion. In particular it was shown in Hashin (1990a, 1990b) that the effect of an interphase, by which is meant a thin coating on the reinforcing fibers or particles, can be expressed in terms of imperfect interface conditions and the interface parameters involved can be determined in terms of interphase characteristics. In the present work we consider the case of viscoelastic interphase between two elastic components. An interphase of such nature may be a by-product of manufacture or it might be introduced by design, e.g. to provide relaxation and damping characteristics to an otherwise elastic brittle composite. It is our purpose to evaluate effective quasi-static viscoelastic properties of such composites. We do this on the basis of analytical results for elastic composites with interphase given in Hashin (1990a, 1990b), and correspondence principles for viscoelastic composites established in Hashin (1965, 1966). The case of viscoelastic interphase in unidirectional fiber composites has also been recently considered by Gosz et al. (1990) in terms of finite element analysis of periodic hexagonal arrays and also some simple applications of the correspondence principle.
1 On leave of absence from Tel Aviv University, Tel Aviv, Israel. 0167-6636/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
Z. Hashin / Viscoelastic interphase
136
2. General
Effective viscoelastic properties of composites are expressed by the effective relaxation moduli tensor C * ( t ) and the effective creep compliance tensor S * (t). The former defines the average time-dependent stress response to sudden imposition, in the Heaviside sense, of average strain. Dually, the latter defines the average time-dependent strain response to imposition of similar average stress. For details see Hashin (1965, 1972). A general correspondence principle relating effective elastic properties to effective viscoelastic properties has been established in Hashin (1965). To recapitulate, suppose that the effective elastic moduli tensor eC* of some composite material is related to the elastic phase moduli tensors eC ( ' ) by the tensor function F. Thus, c c * = F C C (1),
. . . . . e c ( " ) . . . . ).
(1)
where of course F is also a function of phase geometry. Now consider a composite with precisely the same phase geometry, but now some or all of the phases are linearly viscoelastic. This will be called the associated viscoelastic composite and its phase relaxation moduli and creep compliance tensors are denoted C* (t) and S *(t), respectively. Denoting the Laplace Transform (LT) of a function as follows ¢ ( p ) = f0~q~(t) e-m dt,
(2)
the correspondence principle expresses the LT of the effective relaxation moduli of the associated viscoelastic composite in the form = ,r(
p (2, . . . . .
.... ).
(3)
The p multiplied LT of phase relaxation moduli C(m)(t), have been named T D (transform domain) moduli in Hashin (1965), for they are the viscoelasticity analogues of elastic moduli. Note that the T D moduli of an elastic phase are precisely the elastic moduli, and it follows therefore that if some of the phases are elastic their elastic moduli are to be entered into (3). The LT of the effective creep compliance tensor can be expressed analogously to (3) and obviously T D phase compliances can be entered into any one of the relations instead of T D phase moduli, if more convenient. Accordingly, the evaluation of effective viscoelastic properties has been reduced to inversion of LT, provided that analytical expressions for effective elastic properties are known. Such inversion can be performed analytically or otherwise numerically. There are however important cases when LT inversion is not required. It has been shown in Hashin (1966) that it follows from Tauberian theorems for LT that the values of C*(t) at times t = 0, ~ are given by C * (0) = r [C (1)(0), C (2)(0) . . . . . C (m) (0) .... ], .....
(4)
.... ],
with analogous results for S * (t). The result for t = 0 merely states that the initial, elastic response of the composite is defined by its effective elastic moduli, as is certainly to be expected. The result for t = ~ is more interesting; it states that final values of effective elastic properties are given by effective property expressions in terms of final values of phase viscoelastic properties. This result often has important practical significance for if it turns out that initial values and final values of effective viscoelastic properties are close, this implies that the viscoelastic effect is negligible, Several applications to fiber composites have been given in Hashin (1972).
Z. Hashin / Viscoelasticinterphase
137
3. Viscoelastic interface conditions
We recall the elastic form of imperfect interface conditions as previously used. Referring to Fig. 1, the interface S]2 separates the two elastic media 1 and 2. Let n denote normal direction to the interface, and s, t orthogonal tangential directions. For reasons of equilibrium, traction components must be continuous across the interface. The interface imperfection is expressed by displacement jumps which are proportional to their associated traction components. Thus,
T,?) =
= D.[u.],
T~(1) = Ts(2) = D s [ u s ],
v,<') = T,<2)= D, [,,,
(5a) (5b)
on S,2,
(5c)
].
It has been shown in Hashin (1990a, 1990b) that for an isotropic thin interphase whose stiffness is much smaller than that of the reinforcement, D. = ( K i + 4 G i ) / 8 ,
(6a)
D s = D, = O y 6 ,
(6b)
where i denotes interphase property and $ is interphase thickness. It may be shown that when the above stiffness condition is not imposed, (6a) assumes the more complicated form
Ki £ 4Gi -~- E2( "]-c
[Un]/~ =
Pi) Tn
E2~ -= Pi--) (O's(s2)"1"-Ot~2))'
(7)
while (6b) remains as before. A derivation will be given elsewhere. It is seen that (7) is no longer of the form (5a). In the case when E i << E2, (7) reduces to (5a) with (6a). The formal viscoelastic counterpart of (5a) is t
0
T~(t) = D . ( t ) [ u . ( O ) ] + f o D . ( t - t ' ) - ~ - i T [ u . ( t ' ) ]
dt',
(8)
where D . ( t ) is an interface function. Similar expressions replace (5b) and (5c). The LT of (8) is given by
L=pb.,
(9) ^
and it is seen that pD~, is the normal T D interface parameter. We can similarly define tangential T D interface parameters p D s and P/)t- In the case of viscoelastic interphase we shall make the reasonable
MATERIAL 1 ~S12
Fig. 1. Interface.
138
Z. Hashin / Viscoelastic"interphase
simplifying assumption that the viscoelastic effect is restricted to shear. Then the T D domain counterparts of (6) are
p b . = (K~ + 4 p d i ) / 6 ,
(10)
pbs = p d i / ~ . The T D domain form of (7) is much more complicated. To make use of the elastic Ki restriction, u~ in (7) must first be expressed in terms of K~ and Gi, and then G~ is replaced by pG~. It follows from the correspondence principle, that in order to obtain the LT of effective relaxation moduli or creep compliances of a composite with viscoelastic interphase, the interface parameters in the corresponding results for elastic interphase are replaced by (11), in the case of interphase of initial elastic stiffness much smaller than reinforcement stiffness. If the last condition is not fulfilled, the normal interface parameter D, is replaced by the T D version of (7). We shall now proceed to use this scheme to investigate viscoelastic behavior due to interphase on the basis of the elastic results obtained in Hashin (1990a, 1990b).
4. Unidirectional fiber composites A unidirectional fiber composite (UFC) consists of parallel fibers, generally of circular cross-sections, which are embedded in a matrix at random locations in the transverse plane. The fibers are assumed elastic transversely isotropic, which is a suitable description of c a r b o n / g r a p h i t e fibers, while the matrix will be assumed isotropic. In between the fibers and matrix there is a very thin viscoelastic interphase. The fiber composite is macroscopically transversely isotropic. The case of elastic interphase has been considered in Hashin (1990a), and in this case the composite is elastic. Referring a composite specimen to a Cartesian coordinate system with x 1 in fiber direction and x2, x 3 in transverse directions, the effective thermoelastic stress-strain relations can be written in the form Oil = n*~ll +
1"422 +
1"~:33 + FA*~,
0"22 = ]*~11 q- ( k * -}- G t ) ~ 2 = -{- ( k * - Gt)~33 + F-~q~,
(II)
0-33= / * g , , + ( k * - G~- )i:2 + ( k * + G~ )i33 + Fa~;
F~, = -(n*a~, + 2 l * a { ) , O12 = 2 G ~ e , 2 ,
ill = E 2
(i-23 = 2 G ~ 2 3 ,
_
(i-31 = 2 G ~ 3 1 ;
(12)
E~ 0.22 -- -~-~ O33 q- 0 / ~ , E ? o33 + a~q~,
%2 = - ~-~-~6,1 + E¢
C33 --
F~- = - ( l * a ~ + 2k*a~-);
_
E 2 ¢lll --
o22
(i% +
E{
+
Here, A indicates axial fiber direction, and T transverse, normal to fibers, direction, * indicates effective property, n and l are moduli associated with uniaxial strain in 1 direction, k is transverse bulk modulus, G is transverse shear modulus, E is Young's modulus, ~ is Poisson's ratio, a is thermal expansion coefficient, overbar denotes average, and q~ is uniform temperature change.
139
Z. Hashin / Viscoelasticinterphase
When the interphase is viscoelastic, the effective stress-strain relations of the composite also become viscoelastic. The viscoelastic counterparts of (11)-(13) can be written in terms of convolutions in the time domain or in terms of TD moduli and compliances in the transform domain, the latter being entirely analogous to (12)-(14). See Hashin (1972) for details.
5. Axisymmetric properties
Axisymmetric properties are associated with states of average strain and stress which are axisymmetric with respect to the fiber direction, together with a uniform temperature change. Such states of strain and stress are
=
£11 0 0
0 i22 0
0 1 0 ], i22
=
011
0
0 ]
0
622
0 ].
0
0
(14)
0"22
It follows from (11)-(13) that the elastic stress-strain relations have the form 611 = n *~11 at- 2l *~22 -at-/-'~, 022 = o33 = l*i n + 2k'g22 + / ~ t h , =
011
Et
2P~
_
+
~A
+
( 1
)o22+ t .
(15)
It has been shown in Hashin (1990a) that in the case of imperfect interface conditions or interphase, all of the above properties can be evaluated on the basis of the composite cylinder assemblage (CCA), or the generalized self-consistent scheme (GSCS) models, with identical results. The results were conveniently expressed in terms of equivalent properties of a fiber with imperfect interface or coated with interphase. In the axisymmetric case the only equivalent fiber property which is different from the usual fiber properties is the transverse bulk modulus. This is given by
kf ke -
(16)
1 + 2kf/D.a"
In the case of isotropic interphase of thickness 8 and elastic moduli K i and provided that K i, G i << kf,
~J<< a.
Gi,
Dn
in (16) is given by (6a) (17)
Here kf is fiber transverse bulk modulus, and a is fiber radius. The properties appearing in (15) are then given by k* = k~ +
1
Uf
v~
'
(18a)
ke__ km q" km._t_~ E ~ ~- Emvm "t- EAf Vf,
(18b)
Z. Hashin / Viscoelastic"interphase
140
(VAfP/~ = VmVm + ltAfV f Jr-
O~
=
Vm)(1/km
-
1/ke)vmV f
(18c)
Um/k e + vr/k m + 1/G m
Olm -~- (O/Af- am)[ pl,( S,*1 - s,n~) + 2P,2( S,~ - S~)] + 2(aVf- am)[P12(S,*~ - $1~) + (P22 + P23)($1~ - S ~ ) ] ,
ot~ = an, + (OtAf- am) { P,,(S,*2 - $1~) + P12 [ 32~ + 32~ - ( S ~ + S~)] } + 2(avr- am){ P12(S~'2 - S ~ ) + ½(P22 + P23) [ $2~ + S~ - ($2~ + $2~)] },
(19)
where P,, --- (AS22 + AS23)/D,
P,2 = - A S , 2 / D ,
P22 + P23 = A & , / D ,
D --- AS,~(AS22 + ZXS23) - 2AS~2,
ASll
ASI2 =
= I / / E A f -- $1~ ,
AS22 + AS23 = ½(1/k e + 4 ~ 2 f / E A r )
--
Snm ___ 1~Era '
m = $12
S,*~ = I / E 2 ,
S~2= -v~./E~,,
__ P m / E m ,
_ P A f / E A f - - Sln~,
($2~ + S2n~), Sdm2 +
$23 m=
(1
-
b,m
)/Em,
Sff2 + Sff3= ½(1/k* +4vy2/EA),
where m and f indicate isotropic matrix and transversely isotropic fiber properties, respectively. The effective moduli n* and l * can be evaluated by use of the transverse isotropy relations n* = E 2 + 4k*v2
2,
l* = 2k*v~,.
(20)
In order to evaluate the T D properties corresponding to (18)-(20), we evaluate the T D equivalent transverse bulk modulus P/~e by replacement of D, in (16) by (10). We then replace k e in (18)-(20) by p/~¢ everywhere. In order to invert the T D properties analytically into the time domain, it is necessary to describe interphase shear viscoelasticity by a differential time operator, which leads to cumbersome LT inversions. Fortunately this is not needed, from a practical point of view, for it will be shown that the viscoelastic effect for the axisymmetric properties is very small. To see this, we evaluate the initial and final values of the viscoelastic properties k*(t), E*(t), n*(t), l*(t), a,~(t), and a-~(t), at times t = 0, oo, by use of the Tauberian results (4). To do this, we first evaluate the initial and final values of the interface functions D,(t) and D,(t), equation (8), by use of (11) and the Tauberian theorems. The results are just as (6) with initial and final relaxation moduli. We assume that K~ does not change with time, and that G~(t) diminishes to 0 after infinite time. We then have
D,(O) = [Ki+ 4Gi(O)]/8,
D,(oo)-=Ki/8,
D,(O) = G i ( 0 ) / 8 ,
Ds(m ) = O.
(21)
The last result implies a condition of free tangential sliding in the final state. The normal interface parameters can be related in terms of the interphase initial Poisson's ratio vi(0 ). Thus, D.(oe) D.(0)
1 + vi(0 ) 311 - v~(0)] "
(22)
To obtain the initial and final values of the axisymmetric viscoelastic properties, the initial and final values of D n are used in evaluation of (18)-(20) with (16). We have done this for two fiber composites:
Z. Hashin / Viscoelastic interphase
141
T300 graphite fibers in epoxy matrix, and a ceramic composite consisting of Nikalon fibers and BSi matrix. Fiber and matrix elastic properties, and thermal expansion coefficients are: T300 (Transversely Isotropic ) EAr ----231 GPa, GAf =
ETf =
22.4 GPa,
kf = 17.5 GPa,
GTf =
8.3 GPa,
22.0 GPa,
PAf = 0.30,
t)Tf = 0.35,
aAf= --1.33)<10 -6 C °,
aTf = 7.04 × 10 -6 C ° ;
Nikalon ( lsotropic ) Ef = 193 GPa,
Gr = 80.4 GPa,
uf = 0.20,
a f = 3.1
Gm = 1.28 GPa,
~'m= 0.35,
a m = 6 3 × 10 -6 C ° ;
Gm = 2 5 . 7 G P a ,
~m=0"22,
a m = 3.2 ×
x 10 - 6 C ° ;
Epoxy Em=
3.45 GPa,
BSi Em=62.7GPa,
10 -6
C ° .
The interphase initial Poisson's ratio was chosen as Pi ( 0 ) = 0 . 3 7 5 .
Initial and final values of relaxation moduli have been evaluated for various values of the interface constant kf/aD,(O) and volume fractions. It was found that the difference between final and initial values of axisymmetric properties was very small and it may thus be assumed that the viscoelastic effect for these properties can be neglected.
Axial shear The effective elastic axial shear modulus for fibers coated with thin interphase is given by, Hashin (1990a), Of
a~-am
6e
q-
1//(Ge -
Gf 1 + paf/6 i '
a m ) -k- V m / 2 a m '
p = 8 / a , Gf = GAl.
(23)
This result is valid for any ratio Ge/Gi, but since 6 << a the effect of interphase on the elastic shear modulus will only be significant if G f / G i is large enough. We now consider the case of viscoelastic interphase. The simplest viscoelastic material which exhibits the necessary features of initial elastic response and unlimited creep is described by the Maxwell model. We accordingly assume that the interphase material deviatoric stress and strain obey the stress-strain relation
skt
e~t = ~
i~1
(24)
+ 2G'---~'
where "//i is a viscosity coefficient, G i is initial shear modulus, i indicates interphase property, and the dot denotes time derivative. The LT of (24) is 2r/iP ^ Sk,--'~ l - - - ~ i p ( k l ,
Ti = ~i/G,,
(25)
Z. Hashin / Viscoelastic interphase
142
and therefore the TD shear modulus of the interphase is ~P 1T~p"
pdj-
(26)
Replacing G~ in (23) by (26) yields the TD effective axial shear modulus pG~. The effective TD creep compliance p ~ is then given by (27)
p~,~ = 1/pGfl,.
The LT inversion required is elementary and yields the explicit results G;(t)
= Gm[AH(t)
+ (B-A)
e-'/~r], (28)
gX(t)=l/Gm[AH(t)+(1-1)e-'/"],
where Vm A = - -
l+vf'
B=
y(l
"1"-Vf ) / U r n nt- I -1- p G f / G i
3' + (1 + p G f / G i ) ( 1 + v f ) / v m ' yVm/(1 + vf) + 1 + pGf/6i
rr = r i
pGf/Gi
'
y(1 + v f ) / v m + 1 + p G f / G i ,rc = r i
pGf/Gi
'
Y = Gf/am,
(29)
and H ( t ) is the Heaviside step function. It is seen that the initial elastic axial shear response is given by (30)
G ~ (0) = 1 / g ~ (0) = GmB,
while the final response is Vm
G~ (oc) = 1 / g ~ (o¢) = Gm 1 + vf"
(31)
The result (29) is the same as the elastic result (23). The result (30) is the axial shear modulus of an elastic matrix perforated with continuous aligned cylindrical holes, Hashin (1972). This result implies that after long time the interracial shear stresses in the interphase relax to zero and thus the coated fiber behaves in the final stage as a cylindrical void. It should be emphasized that this last conclusion is valid for any linear viscoelastic interphase whose final relaxation modulus vanishes. This is easily shown from the Tauberian results (4), applied to the present case. Other noteworthy special cases relate to the value of the quantity p G f / G i. If Gi is not very much smaller than Gf, this quantity becomes a small number, and therefore the characteristic times in (29) become very large. This implies negligible viscoelastic effect since very long time has to elapse before such effect becomes noticeable. If, on the other hand, G~ is so small that p G f / G i becomes a very large number, then the characteristic times approach the characteristic time of the interphase material and there is significant viscoelastic effect. It is thus seen that significance of viscoelastic effect is strongly dependent on the elastic shear stiffness of the interphase.
143
Z. Hashin / Viscoelastic interphase
UNIDIRECTIONALFIBER COMPOSITEWITH VISCOELASTICINTERPHASE NIKALON FIBERS BSi MATRIX vf =0.50
NIDIRECTIONALFIBER COMPOSITE WITH VISCOELASTICINTERPHASE NIKALON FIBERS BSi MATRIX 1.5
~ ~ , ~
3.0
,jDvf=0.50
GA(t) Gm
q= pGAf /Gi
2,0
1.0
1
Z *d 0.5
Y
0.1
~
CYLIvNglRIsCAL
I
I
I
I
l
1
10
102
10 3
10 4
t/ri
0.1
I
I
I
1
10
10
2
I
103
I
104
t/ri
Fig. 3. Axial shear creep compliances.
Fig. 2. Axial shear relaxation moduli.
zo~
~E1.0
UNIDIRECTIONALFIBER COMPOSITEWITH VISCOELASTICINTERPHASE NIKALON FIBERS BSi MATRIX vf =0.50
3 '0 F/~q= I
I[ /
I
CYLINDRICAL VOIDS
q=PGMIGi
I
I
I
I
2500 5000 7500 10,000 Fig. 4. Axial shear creep compliances -- real time.
t/'r;
Plots of relaxation moduli and creep compliances as functions of log(t/~i) for various values of p G f / G i are shown in Figs. 2 and 3, and a plot of relaxation moduli in real time in Fig. 4. It is seen that a minute amount of viscoelastic interphase can produce significant time dependence in axial shear for an otherwise elastic material.
Transverse response We first consider the case of effective transverse shear relaxation modulus G~-(t), and creep compliance g~ (t). The elastic transverse shear modulus for fibers coated with interphase has been evaluated in Hashin (1990a) on the basis of the GSCS. This requires the solution of a quadratic equation whose coefficients are defined in terms of 8 × 8 determinants. Following the correspondence principle we can use these results to write similar equations to evaluate the normalized effective T D modulus,
1-'*(p) =pG~(p)/Gm,
(32)
144
Z. Hashin / Viscoelastic interphase
as a root of the determinantal equation 1
-1
1
1
F*
3I"*
1
2+~m
0
1
1
~m
1
2
1
l+~m 1
1
0
0
--1
0
0
3
0
0
3
0
0
3/c2
0
-~,
3"fc l-~f
-3'
F* -3F*
3
0
0
0
1
0
0
3c l_~m
1
1 (1 +~m) c
3/c2
0
0
c
1
l/c
1/c 2
1
D(I'*, p)=
0
0
(2+~m)C 1 --~m
1
2
(1 + ~Sm)c
~m (l q- ~m)C
_l/c 2
-c
(6q+2+~f)c 1 --~f
=0,
- (2m + 1)
-(2q+
1)
(33)
where ~nl = G m / k m ,
~f = G f / k f ,
y = Gf/Gm,
c = of,
pGf ki+4pdi/3,
m=m(p) G r = GTf ,
pGf q = q ( P ) = PGi'
(34)
km = K m + Gm/3.
The determinant in (33) is evidently a quadratic equation in F*, and therefore (33) can be written as
D(F*, p ) = A ( p ) F * 2 + 2B(p)F* + C ( p ) = O ,
(35)
where the coefficients can be expressed in the form A=~[D(1,
p ) + D ( - 1 , p)] - D ( 0 , p ) ,
B=¼[D(1, p)-D(-1,
p)],
C = D(0, p ) .
(36)
The LT of the effective transverse shear creep compliance g * ( t ) , is then defined by the usual relation
p~. = 1/pG~-.
(37)
The T D effective transverse Young's modulus/3E~ is given by the elastic relation
4/pff~ = 1/pd~ + l / k * + 4v~Z/E,~,
(38)
where it has been assumed, on the basis of previous conclusions above, that k*, v~ and E 2 are elastic. Then, the LT of the associated compliance is defined by
pe.~ (p) = 1left.S. (p).
(39)
Analytical inversion into the time domain does not appear feasible, but numerical inversion can be performed on the basis of the results listed above. Here, we shall confine ourselves to exploitation of the Tauberian theorems to obtain initial and final values of properties of interest. To do this we merely have to
145
Z. Hashin / Viscoelastic interphase UNIDIRECTIONAL FIBER COMPOSITE WITH VISCOELASTIC INTERPHASE NIKALON FIBERS
BSi MATRIX
vf = 0 4 0
1.50
E~E~ (0)/ E m
100
050
O.OJ
I 0.1
I I
I I0
l 10 2
I 10 3
q(O)
Fig. 5. Initial and final values of transverse Young's relaxation modulus.
perform an elastic analysis with initial and final values of the interface parameters as given by (21)-(22). Figure 5 shows plots of E~ (0)/E m and E~ (oo)/E m versus the initial interface shear constant pGf q(0) = ai(0 ) .
(40)
In the final state there is free tangential sliding at the interface and the normal bond interface parameter is magnified by the factor (22). The plots show the extent of the viscoelastic effect of the interphase and it is seen that the effect diminishes with weakening of normal bond until initial and final values both converge to the elastic result for cylindrical voids.
6. Particulate composites Elastic properties of spherical particle composites with imperfect interface or interphase have been analyzed in Hashin (1990b) and the results can be transcribed to the case of viscoelastic interphase exactly as was done above for UFC. Labeling matrix by 1 and particles by 2, the effective bulk modulus K * for the case of thin elastic interphase is given by /;2
K* = K 1+
1
- - + Ke - K1
K2
3v I
,
Ke - 1 +
3K2/aD
. '
(41)
3K 1 -I- 4G 1
where D, is expressed by (6a). Then, the effective thermal expansion coefficient is given by a2--al 0/*=0~1"}- 1
Ke
!
( 1 K*
1) K1
"
(42)
K1
Explicit results in the time domain for the corresponding viscoelastic properties can be obtained by assuming a simple viscoelastic model for the interphase and using the correspondence principle as was done above in the case of axial shear. This, however, is hardly worthwhile, for evaluation of final and
Z. Hashin / Viscoelastic interphase
146
initial values K * (0), K * (m), a* (0), and a * ( m ) , by introduction of (21) into the elastic results (41)-(42), shows that there is very little difference between initial and final values, and thus the effect of viscoelastic interphase on these properties is very small. The situation is of course quite different for shear. In this case, analytical treatment is entirely analogous to the method employed above for transverse shear of a unidirectional fiber composite. The effective elastic shear modulus for the case of imperfect interface or elastic interphase has been evaluated on the basis of the GSCS in Hashin (1990b), requiring again the solution of an 8 X 8 determinantal equation. Employing again the correspondence principle, and defining in analogy to (32) the normalized effective T D modulus,
F* =pG*(p)/Gm,
(43)
we find that this quantity is the root of the equation
D(F*)=
2
3Ns
12Vl~, 2
2
2(5 -- 4 v 1 )/)~t
- 3/x ~
0
0
1
-1/X 5
(7-- 4Vl)h 2
1
2(1 - 2 v 1 )/,k ~
1/x~
0
0
2F*
- 12F*/X 5
- 6v~X2
2
-4(5 -
12/x'
0
0
2F*
4F*/X 5
(7 + 2 v l ) ~ 2
1
2(1 + v~ ) / X 3
- 4/)~ 5
0
0
0
0
-6v t
2
- 4 ( 5 - v1
12
6v2y
-- 2"),
0
0
7 + 2v 1
1
2(1 + v I )
-4
0
0
12v 1
2
2(5 - 4 v I )
0
0
1
2(1
7
4v 1
Ul)/X ~
2vl)
(7 + 2v2Y ) 3
1
=0,
--y
12u2(m - 1)
- 2(1 + 2 m )
--[2(7 + 2v2)q+ 7-- 4v:]
--(1 + 2 q )
(44) where v1 and v2 are matrix and particle Poisson's ratios, respectively, and
y = G2/G1,
X3 = v2,
oG2 Ki+4pGi/3,
m=m(p)
q = q ( p ) = pG: pG i '
(45)
where a in p, equation (23), is now particle radius. Equation (44) can again be put into the form (35), (36). The LT of the effective shear creep compliance g*(t) is evaluated by the usual relation
p~,*(p) = 1/pG*(p),
(46)
and the LT of effective Young's relaxation modulus E * (t), and creep compliance e* (t) are defined by
pff~*(p)
9K *pG* ( p ) =
3K* + p G * ( p ) '
(47)
p~*(p) = 1/p£*(p), where it has been assumed that K * is elastic. Inversion into the time domain can again only be done numerically and we again confine ourselves to evaluation of initial and final values of relaxation moduli. This is done exactly as in the case of transverse shear modulus as outlined above. Figure 6 shows such results for normalized effective shear relaxation moduli G* ( 0 ) / G m and G * ( m ) / G m for the case of glass spheres embedded in epoxy matrix, plotted versus the initial value of shear interface constant pG2 q(O) = Gi(O ) .
Z. Hashin / Viscoelastic interphase
147
SPHERICAL PARTICLE COMPOSITE WITH VISCOELASTIC INTERPHASE GLASS PARTICLES
E P O X Y MATRIX
v2 = 0 4 0
25 O)IG m 2.0 15
G*(m)IGm
10VOIDS
i
05
001
I 0.1
I I
I I0
I I0 z
I 103
q (0)
Fig. 6. Initial and final values of shear relaxation modulus -- particulate composite. The elastic properties of glass are: E 2 =
72.4 GPa,
G 2 =
30.2 GPa,
1"2= 0.20,
and the epoxy properties the same as used before. The final state is again characterized by freely sliding interface, and with weakening normal bond both properties converge to the elastic effective shear modulus for spherical voids.
7. Discussion and conclusion Effective relaxation moduli, and creep compliances of fiber composites, and particulate composites with viscoelastic interphase have been derived on the basis of previous results for elastic interphase and a correspondence principle for viscoelastic composites. The results obtained reveal that time dependence of effective properties produced by viscoelastic interphase is significant only for shear inputs such as axial and transverse shear for UFC, and for any shear for a particulate composite. Inputs such as uniaxial stress which include such shear components will of course also produce such time effects. Axisymmetric inputs for UFC and spherically symmetric inputs for particulate composites, thermal expansion being included in these, will produce negligible time effects. Note that the insignificance of time effects for thermal expansion also remains for the case when interphase viscoelastic properties are temperature dependent, a case which has not been considered in this work.
Acknowledgement This work has been supported by the Air Force Office of Scientific Research (AFOSR), Lt. Col. G. Haritos, and by the Office of Naval Research (ONR), Dr. R. Jones, through contract between the National Center for Composite Materials Research at the University of Illinois, Urbana, Dr. S.S. Wang, Director, and the University of Pennsylvania. I am indebted to the aforementioned for support and encouragement. Helpful discussions with Professor J.D. Achenbach and Professor B. Moran are also acknowledged.
148
Z. Hashin / Viscoelastic interphase
References Achenbach, J.D. and H. Zhu (1989), Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites, J. Mech. Phys. Solids 37, 381-393. Achenbach, J.D. and H. Zhu (1990), Effect of interphases on micro- and macromechanical behavior of hexagonal array fiber composites, J. Appl. Mech. 57, 956-963. Benveniste, Y. (1985), The effective mechanical behavior of composite materials with imperfect contact between the constituents, Mech. Mater. 4, 197-208. Gosz, M., B. Moran and J.D. Achenbach (1990), Effect of viscoelastic interface on the transverse behavior of fiber reinforced composites, to be published.
Hashin, Z. (1965), Viscoelastic behavior of heterogenous media, J. Appl. Mech. 32, 630-636. Hashin, Z. (1966), Viscoelastic fiber reinforced materials, AIAA J. 4, 1411-1417. Hashin, Z. (1972), Theoo' of Fiber Reinforced Materials, NASA CR, 1974. Hashin, Z. (1990a), Thermoelastic properties of fiber composites with imperfect interface, Mech. Mater. 8, 333-348. Hashin, Z. (1990b), Thermoelastic properties of fiber composites with imperfect interface, J. Mech. Phys. Solids, in press.