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An energy method for calculating the stiffness of aligned short-fiber composites
Mechanics of Materials 6 (1987) 197-210 North-Holland
197
AN ENERGY M E T H O D FOR CALCULATING T H E S T I F F N E S S OF ALIGNED S H O R T - F I B E R C O M P O S I T E S
Paul S. STEIF and Steven F. H O Y S A N Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. Received 23 January 1987; revised version received 24 March 1987
A procedure is presented for calculating the longitudinal elastic modulus of a composite possessing dilute concentrations of aligned cylindrical fibers. For a dilute concentration of fibers, the modulus can be written in terms of the perturbation that a single fiber induces in the displacement field of an otherwise homogeneous body subjected to remote uniaxial tension. A boundary value problem, referred to as the auxiliary problem, is formulated which yields the perturbed displacement field directly. The longitudinal modulus can then be expressed in terms of the potential energy of the auxiliary problem; thus, energy-based approximate methods (e.g. FEM) can be exploited to calculate the modulus accurately. With a small modification, this procedure is shown to be applicable to the case of fibers which are imperfectly bonded to the matrix. Moduli for a few representative cases are calculated using a finite element method. Finally, a new means of defining and computing the ineffective length of a fiber is introduced.
1. Introduction
Theoretical predictions of the effective properties of composite materials can play an important role in the design of these materials, particularly as the variety of constituent and processing options rapidly expands. Recent surveys by Christensen (1979) and Hashin (1983) indicate the extent of interest in this subject. With regard to effective elastic moduli, the theory of heterogeneous materials has been most successful in dealing with composites reinforced by very long fibers, as well as composites with spherical reinforcements. In a variety of applications where low cost is important, however, the use of chopped fibers or whiskers has often been preferred. The challenge in predicting the effective properties with these reinforcements, of course, is to account for the effect of the fiber ends, which can no longer be neglected. The most prevalent means of calculating the effective properties of so-called short-fiber composites has been to model the fibers as ellipsoids (Russel, 1973; Laws and McLaughlin, 1979; Chou, Nomura, and Taya, 1980). This derives from the possibility of exploiting Eshelby's (1957) solution for an ellipsoidal, elastic inclusion located in an infinite elastic body subject to a remotely applied stress. While the ellipsoid model does give the long fiber results when the ellipsoid aspect ratio become large, it is by no means clear that the stiffening effect of short fibers shaped as circular cylinders--apparently the most common shape --is accurately predicted by the ellipsoid model. One goal of the present work is to develop a method of calculating the stiffness of a composite made up of short cylindrical fibers. In addition to addressing the cylindrical fiber problem more accurately, the method offered here permits one to explore the effect of fiber-matrix interface character on composite stiffness. One expects that the degree of contact at the fiber-matrix interface can influence the stiffness of short-fiber composites because of the important role played by load transfer in stiffness, and the possibility of the interface markedly affecting load transfer. In fact, it has been demonstrated by Sanadi and Piggott (1985) on 0167-6636/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
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aligned short-fiber systems that different fiber surface treatments result in very different longitudinal stiffnesses. Recently, there have been efforts to assess the degree to which degradation of the interface properties has significant consequences for the effective stiffness of the composite, lncluded are: Benveniste's (1984) pointing out the gradual failure of St. Venant's principle at an interface where the interfacial integrity is substantially diminished; Benveniste's (1985) computation of the effective modulus of a composite reinforced by spherical particles which are not well bonded to the matrix: and. Steif and Hoysan's (1987) calculation of load transfer in a pullout test in which the fiber and matrix are no1 perfectly bonded to the matrix. In order to focus on the effects of fiber aspect ratio and material and interface properties, we choose to confine attention to dilute concentrations of fibers. While the dilute solution approximation by itself may have limited application, it forms the basis of a number methods of calculating the moduli of a composite with a finite volume fraction of fibers. Within the dilute solution approximation, one can view each fiber as being effectively located in an infinite body of matrix material. However, even this problem, involving a finite cylindrical fiber partially bonded to an unbounded matrix, is sufficiently complicated that we must resign ourselves to the necessity of employing numerical methods. In the present paper we develop a procedure which makes the best of this bad situation. This procedure minimizes the results that need to be extracted from the numerical calculations, and it takes advantage of the most accurate aspects of energy-based approximate methods.
2. Interface model
The interface model we employ states that the displacement components in the fiber and in the matrix are not the same across the interfaces. The jumps in the displacement are taken to be proportional to the tractions acting across the interface. While this model is simple, it is general in that it allows for a variety of interface situations. One could view this model as representing, for example, a thin fiber coating or a series of cracks along the interface. Formally, the interface model may be expressed by 7;,= o~/nj = k,i Au:
(I)
where T, is the traction acting across the interface, oij is the stress, n: is the outward normal from the fiber, k~j is the tensor of interface stiffnesses, and Auj is the jump in the displacement at the interface. Indices refer to Cartesian components, and summation on repeated indices is implied. We assume the interface stiffness tensor is symmetric. In general, one can take the interface stiffnesses to vary from point to point along the interface. By taking the coupling stiffnesses k,j to be very large, one recovers the usual condition of continuous displacements. On the other hand, vanishingly small stiffnesses correspond to complete decoupling of the fiber and the matrix, i.e., zero tractions across their interface. This interface model has been studied previously (e.g., Benveniste, 1984, 1985; Steif and Hoysan, 1987). To the extent that the model is intended to represent a fiber coating or other actual layer of material separating the fiber and the matrix, it is tacitly assumed that the interface layer is small compared with other relevant dimensions, such as the fiber diameter. The model simplifies the situation considerably in that one needs to specify only a single set of parameters (stiffnesses) in order to describe the interface; thicknesses and modufi need not be separately prescribed. This has an added advantage in that it leads to a more straightforward approach to solving boundary value problems. Without more detailed information regarding the interface, a more complex model is not warranted; this is particularly so as our present aim is only to assess qualitatively the sensitivity of composite stiffness to interface integrity.
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3. Energy-based calculation of effective modulus The present analysis is intended to derive an energy-based method of estimating the effective longitudinal Young's modulus of a composite containing a dilute concentration of aligned short fibers. Within the dilute solution approximation, one can express the composite longitudinal modulus, E~, in the form Ec/E m =
1+
(2)
cfA
where E m is the matrix Young's modulus, cf is the volume fraction of fibers, and A is the reinforcement coefficient (defined by Russel (1973)). The assumption of dilute fiber concentrations leads one to consider the problem of a single fiber located in an infinite body of matrix material. The reinforcement coefficient is then calculated by comparing the strain energy of this body with the strain energy of a similarly loaded body of pure matrix material. Clearly, the energy difference will be proportional to the fiber volume. To motivate the method we develop, it is worth considering a rather naive approach in which we employ, say, a finite element method to analyze a large but finite region containing a fiber, which is subjected to remote uniaxial tension. Using the boundary displacements or some other means, we could calculate the strain energy of the body. The problem arises in deciding how large to make the domain. The need to mimic an isolated fiber in an infinite body dictates that the finite region should be as large as possible. But, as the domain gets larger, the relative effect of the fiber becomes less and the energy difference more inaccurate. In order to develop a method that calculates only the energy difference, we take an approach similar to that of Eshelby (1956) and consider two infinite bodies: body B is homogeneous with moduli C, lkl, and body B' contains a fiber of moduli Cjjkt embedded in an otherwise homogeneous matrix of moduli Ci Jk/. Both bodies are subjected, in general, to a remote uniform stress )Zzj (see Figs. l(a) and l(b) for schema(ics of two such bodies subjected to remote uniaxial tension). A Cartesian coordinate system and a cylindrical polar coordinate system (x 3 = z), with origins coinciding with the center of the fiber, will be used t
t~
(3
r
t3
(Y
Fig. 1. (a) Schematic of a single fiber located in an infinite b o d y which is subjected to r e m o t e uniaxial tension. (b) S c h e m a t i c of an infinite b o d y which is subjected to r e m o t e uniaxial tension.
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interchangeably as convenient. The fiber is taken to be separated from the matrix by an interface described by (1). / r t Let primed quantities u s, ~sj, osj be associated with body B' and unprimed quantities u,. ~,~, <. wiih body B, where e~j denotes the strain. Then, the strain energy in body B' minus the strain energy in body B. Aq), is given by
AI~
ksj Aus' kuj' d S -
1 /" Oijf. , i' i d V + ~ 1[ = ,,~] a Vzc
:~ f o~ie,i dV
J SI
(3)
JV
where V~ is the infinite volume, and Sf is the surface of the fiber. Note that Au~ denotes the displacement jump at the interface in body B' and that there is no displacement jump in B. We expect the energy difference A~ to be finite; in a manner analogous to Eshelby (1956), we put Aq~ in a form which explicitly reflects that fact. To do that, we use the divergence theorem (including separate volume integrations for fiber and matrix) to derive the equality
(oij-oij)(f.ij-~-ff.ij)
(os,-oii)
dV:
Ausnj d S
f ¢
where we have made use of the fact that aij and oij both approach ~ i j a t infinity. Now, subtract the left hand side of (4) from (3) and add the right hand side to give
Adb = ½
(5)
(oOei"J - a,je,j) d V + ½ o,a Ausn j dS. '
f,Sf
'
Furthermore, since the moduli differ only in the fiber, one finds m~:
~il
(Cijkl _
C, j' k , ) e s j e 'k / d V + 7 ~
oi i Ausn I d S
(6)
f
where Vf is the fiber volume. The derivation thus far mirrors the development of equation (6.5) in Eshelby (1956), except for the surface integral term which reflects the compliant interface. A similar derivation can also be found in Eshelby (1966). We now introduce the quantity Usp, defined so that u~ = us + Usp, which represents the perturbation induced by the fiber in the uniform field u s. One can now rewrite (6) as
A ~ = ½fvf(Cijk,--Cv'jk,)e,j(%z + e~,) d V + ½fsf°iJ AuVinj dS.
(7)
We now use the above expression for the energy difference to calculate the longitudinal Young's modulus of a short fiber composite composed of isotropic fibers of circularly cylindrical cross-section. Though there is no essential necessity to restrict the values of the material parameters, the results are simplified somewhat by assuming that Poisson ratios are equal in the fiber and in the matrix. The case of different Poisson ratios is discussed below. Now, let the remote stress state be one of uniaxial tension o parallel to the axis of the fiber. Then certain terms in (7) may be calculated directly: Cijkl£ kl = (Y~i3~j3
(8)
C/jk,,~, ' = XoSi38j3
(9)
and
where )k = E f / E m and 8 0 is the Kroneker delta. Note that ~33
=
°/Em"
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In the subsequent development we concentrate on interfaces which are judged to have the most significant influence on the longitudinal stiffness of short fiber composites. Those are interfaces having weakness, or compliance, in paths of load transfer from fiber to matrix. Specifically, we will assume that a shear displacement jump is possible along the sides of the fiber, and that a normal displacement jump is possible at the fiber ends. With reference to Figure la, this means that Au 3 ¢ 0 along r = a, - L < z < L, and along z = + L , 0 < r < a. It is further assumed that the shear discontinuity induces only a shear traction and the normal discontinuity only a normal traction. Then, since the energy difference can be written as A~
=
[½o2/Ec
--
½o2/Em]Voo
(]0)
where E c is the effective modulus of the composite, one can deduce that
EmE c -l+(1-Xlc r+(1-x)cf~ Emfv,~P,d V + c f o~ffEmfz = + LA u P
dS
(11)
where cf = V f / V ~ is the fiber volume fraction. Note that 1 + (1 - X)c r is the uniform strain prediction, provided ()t - 1)cf << 1. The last two terms represent the energy associated with the fiber's perturbing the uniform strain field Eo. It is now shown that the integrals in (11) can be put into such a form that they may be evaluated in a straightforward manner with approximate energy methods. This is done by decomposing the problem of body B' subjected to uniaxial tension into two problems: one with loadings that produce the uniform strain eij, plus one with loadings that produce the perturbed displacement uP. Such a decomposition is similar to what is done in analyses of cracks and cavities. The contribution of the present paper is to show that the potential energy of the loading which produces the perturbed displacement is precisely the energy required to evaluate the reinforcement coefficient A. The particular decomposition given here requires that the Poisson's ratios be equal, though a somewhat more complicated decomposition is possible when they are different. First, we will consider the case in which the interfacial stiffness is infinite, i.e., the fiber and the matrix are perfectly bonded. Let the body B' be subjected to a remote tension o, as well as to equal and opposite concentrated body force distributions of magnitude (X - 1)o (per unit area) as shown in Fig. 2(a). This loading results in the uniform strain field q j; the stress is uniaxial tension o in the matrix and uniaxial tension Xo in the fiber. The second problem, which we refer to as the auxiliary problem, consists only of equal and opposite body force distributions at the fiber ends (see Fig. 2(b)), such that, when superposed with the first loading, is equivalent to the original problem (Fig. l(a)). Therefore, the displacement field induced in the auxiliary problem is, by definition, uP. The potential energy of the auxiliary problem //aux is defined by //aux = 1_2t f "V~
uij~ij~P~Pd V -
f
~u p dV-- - ½ f ./,uP dV
~"V~
(12)
J V~
where fi are the body forces. Again, the body forces are applied at the fiber ends, with fl =f2 = 0, f3 = - ( X - 1)o on z = L and fl =f2 = 0, f3 = ( X - 1)0 on z = - L . Using the symmetry about z = 0, one finds Hau x = ( ) t - 1)of~=Lu3P dS.
(13)
Consider now the integral over the fiber volume in (11). (The surface integral in (11) is zero for perfectly bonded fibers since hu[ = 0). Since the strain is related to the displacement according to eP3 = O u P / O z ,
(14)
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(~.-1)c~
(~,-1)(y
hll
ell (~,-1)o
G-1)o
ff Fig. 2. (a) Loading which produces a uniform strain of the fiber and the matrix when they are perfectly bonded. (b) Loading for auxiliary problem when fiber and matrix are perfectly bonded.
the term involving the volume integral becomes Em
2(1 - X)cf-~fff~=LuP dS.
(15)
By (13), this can be written as ( --Emcf/02'rra2t)l~aux
(16)
which means the reinforcement coefficient A can be expressed directly in terms of the potential energy of the auxiliary problem, i.e., A = ~ - ] + EmlIaux/~O2a2L.
(17)
Note that for fiber ends that are far apart, that is, for long fibers, Hau x is of order a ~, and thus,
A = X- 1 + O(a/L), ~mplying that the correction due to the auxiliary problem vanishes as a l L approaches 0. An analogous decomposition of the problem is possible when displacement jumps are permitted at the fiber ends. As before, the first problem consists of loadings that produce a uniform straining of the body, with no extension of the springs. This is done by applying tensile stress o at infinity, together with tensile tractions o on the matrix side of the interface at the fiber and tensile tractions Xo on the fiber side of the interface at the fiber ends. The uniform strain in the body is o/Em, the springs do not stretch, and the stress is o in the matrix and Xo in the fiber. An auxiliary problem can then be defined involving equal and opposite internal tractions (see Fig. 3) such that the superposition gives the original problem. Note that the interface does open up in the auxiliary problem, and that this auxiliary problem is equivalent to the previous one when displacement continuity is enforced at the fiber end. A careful accounting of all the energy (including the energy of the stretched interface springs) leads one to conclude that the two integrals
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Fig. 3. Details of loading near the fiber ends for auxiliary problem when fiber and matrix are not perfectly bonded. in (11) are equal to (16). Now, however, the potential energy of the auxiliary problem is not given by (13), but by Haux =
kof~=L uP as-of~=L+uP as
(181
where z = L - denotes the fiber side of the interface at the fiber end, and z = L + denotes the matrix side. As mentioned above, a decomposition is also possible when the Poisson ratios are different. The corresponding auxiliary problem involves longitudinal body forces f3 applied at the fiber ends and radial body forces fr applied at the sides of the fiber, where Pf -- Pm
]
and
fr=--~.
P f - Pm ] (l_Gf-~-(~-_l_pm) o.
In these expressions, X denotes the ratio of the shear moduli. It can be shown, as was done above, that the potential energy of the auxiliary problem is again equal to the required strain energy difference. The potential energy of the auxiliary problem was computed for several sets of material parameters using a finite element method. Because of the procedure used here, in which the energy difference is on the order of the fiber volume, the maximum size of the domain was limited only by practical considerations of solution time. For the domains chosen, the accuracy was improved by using a simple substructuring procedure, reported elsewhere (Steif and Hoysan, 1987). The accuracy of the finite element method is discussed below.
4. Moduli of composite with rigid fibers In the limit as the modulus of the fiber becomes large compared with that of the matrix, the auxiliary problem presented above becomes one of applying body forces of infinite magnitude to a fiber of infinite
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204
modulus. This is not possible to do in practice with a finite element method, for which the solution will often be inaccurate as one constituent becomes considerably stiffer than the other. For this reason, we offer a method intended specifically for calculating the effective longitudinal modulus of a composite reinforced by rigid fibers. Again, we compare the strain energy of bodies B' and B, both of which are subject to a remote uniaxial tension o. Now, however, body B' has a fiber which does not deform, Considering only the case of perfect bonding between the fiber and the matrix (the modification for an elastic interface being wholly analogous to that given above), the difference in the strain energies, Aq), is given by
A~):- ~
OijEij d V -
~
t
Making use of the fact that o;/ and on &, one can derive the identity
:l f v "
aij
(19)
Oij~.iSdV.
both correspond to uniaxial tension o at infinity, and that u / = 0
(ai/- oi/)(C/+ t C:) d V - ~I f+rqoijc;:d V - v,. ,*
- ~lfsC~flUiFll d S ,
( 2o )
from which it is found that AcI, can be written as
A+ = -½.sf +'+"' d S -
dS,
(2l)
!
where we have again written o ; / = o;j + og and nj still points from the fiber to the matrix. Note that the first integral can be identified as minus the strain energy in the volume Vf of body B. To develop an auxiliary problem for which the strain energy is proportional to the fiber volume, we view the straining of body B' as accomplished in three steps. First, take homogeneous body B and let it deform uniformly under uniaxial tension o. Secondly, cut out the material in volume Vr (this is represented by the first integral in (21)). Thirdly, apply displacements u2 to the exposed surface Sf that return it to its original position. These displacements are u a = - ui, where u i are the displacements brought about by the homogeneous stressing of B. The region exterior to Vf is now precisely in the configuration that B' would be in, were it subjected to a remote uniaxial tension o. If, consistent with previous notation, the stresses induced during the third step are denoted by o;p, then the tractions 7];a required to produce the displacements u~ are -njo,~. Thus, the work done in the auxiliary problem (the third step), W~ux, is given by W~.~ = ½fsrOi~u;nj ds > 0.
(22)
Using (10) and (21), one can then write the reinforcement coefficient A as A = 1 + EmWau,J'rro2a2L.
(231
A simple interpretation of (23) is that the reinforcement coefficient exceeds 1 (the rule-of-mixtures prediction based on compliances) by an amount which reflects the extent to which the fiber prevents the rest of the material from deforming.
5. Approximate solution of auxiliary problem for perfectly bonded interfaces In this section, we consider further the auxiliary problem in order to derive a useful approximate solution and to assess the influence of the fiber aspect ratio. Recall that the auxiliary problem consists of
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equal and opposite patches of body force applied to the fiber ends (when the fiber is perfectly bonded to the matrix). The magnitude of the body forces is proportional to the applied stress, and it depends on the fiber-to-matrix modulus ratio. Thus, the modulus ratio enters the problem in two ways: it is a scale factor for the body forces, and it characterizes the degree of inhomogeneity of the body that is being loaded. We begin by deriving an approximate solution which ignores this second role of the modulus ratio. That is, we imagine the energy of the auxiliary problem to be calculated by applying body forces (X - 1)o to the ends of a fiber which has the same moduli as the matrix. Applying body forces of magnitude ( X - 1)o is equivalent to applying body forces of unit magnitude and then scaling up the resulting energy appropriately. Since the fiber and the matrix now have identical moduli, this problem of unit body forces applied to the fiber ends can be solved with the use of the Kelvin point load solution. Omitting the unessential details, one finds the approximate potential energy for unit body forces, /Tapp, to be given by the triple integral
_ /Tapp
a'
[1[1[~
4/*(i -- v) ~y=0~x=0~r_0 {(3 1
-
4v)
_
(x2 + y2_ 2xy cos
1
_
¢x 2 + y2 + 4t L/a)2- 2xy cos
[x2+y2+4(L/a)2
2xycosv]3/2
}xydvdxdy,
where /, is the shear modulus. The potential energy of the auxiliary problem would then be given approximately by ()~- 1)202/-lapp. An inspection of the expression for /~app reveals that the second and third integrals vanish as L / a ---, ~. Hence, when the fiber is relatively long, Happ simplifies to the form
a3(3-4v) fl /app--
fl f'rr
xydy__d__f_.xdy__
Jx=OJy=OJy_O Cx2 + y2
2xy cos i •
The triple integral was evaluated numerically and was found to equal 1.37. Obviously, this corresponds to twice the energy of a single circular patch of unit body force in an infinite solid. Recalling equation (17), which links the energy Haux with the reinforcement coefficient A, one can see that, to first order in (a/L), the reinforcement coefficient A is given by A=?t_1_[(?t_l)21"37 (l+v)(3-4V)]L 2"~ 1-Tv
"
This simple result, which is approximately vahd for a l L << 1 and for X nearly 1 (the inhomogeneity of the fiber-matrix system having been neglected), can be extended to a somewhat larger range of )~ in the following way. In the limit of alE--+ 0, the auxiliary problem, for any value of X, consists of a single circular patch of load applied to the end of a very long fiber. By dimensional considerations, the potential energy of this problem (with unit body forces) must be of the form /-/aux = ~(~, v)(1/Em) a3
(24)
provided that the Poisson ratios are equal to v. Now, since the body force on the fiber ends is (X - 1)o, one finds that the reinforcement coefficient A can be written as
7~]
A = ( X _ 1 ) [ 1 _ ( X _ 1 ) ~ ( ~ v) a
(25)
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206
short-fiber cornposite.s
in the limit of a / L << 1. Below, we will find an approximate form for the function ~(?,., v) using the finite element results.
6. Results
Calculations described in the previous sections were carried out for a range of material parameters, including a relatively short fiber L / a = 4 and a relatively long fiber L / a = 12. As has been discussed previously, the finite element results may be subject to some error when the modulus ratio is rather large. This motivated the derivation of a method suitable for infinitely rigid fibers, which was given in Section 4. The results are now presented which show the influence of several parameters (v = 0.3) on the reinforcement coefficient A. In Fig. 4, we depict the dependence of A on ?~ for three fiber aspect ratios. The infinite fiber results, A = ?, - 1, is shown as the dashed line. Finite element results for L / a = 4 and 12 (?~ = 2, 5 and 10) are represented by symbols as indicated. The solid lines represent the approximate solution, which was derived in Section 5 and is discussed further below. As expected, and as found by Russel (1973), reducing the aspect ratio reduces the stiffening effect. Furthermore, this effect becomes more pronounced as the modulus ratio ?, increases. Since previous calculations of the stiffening effect of discontinuous fibers are based for the most part on modeling the fiber as an ellipsoid, it is interesting to compare the stiffening effect of cylindrical fibers with that of ellipsoidal fibers. But an immediate question arises: What is the ellipsoid to be compared with? Three possibilities that suggest themselves are depicted in Fig. 5. Ellipsoid I is inscribed in the cylinder and, therefore, has a minor axis equal to the fiber radius and a major axis equal to the fiber half length; its volume is 2 of the cylinder's volume. Ellipsoid II has the same minor axis as the cylinder's radius, and the same volume; its major axis is ~ times the half length of the cylinder. Ellipsoid III has the same major axis as the cylinder half length, and the same volume; its minor axis is (~)1/2 times the radius of the cylinder. A comparison of the A-values is shown in Table 1 for a cylinder having an aspect ratio of L / a = 4. In general, the degrees of reinforcement provided by Ellipsoids I and III and cylinders are comparable; Ellipsoid II provides significantly more stiffening, as one might expect from the larger aspect ratio. Some sense of the accuracy of the finite element results can be gained from considering the patch-load solution given in Section 5. Recall that the patch-load solution is precisely the auxiliary problem for ~ = 1. It was found that the patch-load solution and the finite element results differed by 2.9 percent for L / a = 4
10
/-
9
/
8
/
6
/
0
/
]t7.
A aL'~" = 12
5 4
0
/
/
L.-L-- = 12
[]
",/'ff"~l
I
I
I
I
I
I
I
I
1
3
4
5
6
7
8
9
10
2
Fig. 4. R e i n f o r c e m e n t c o e f f i c i e n t as a f u n c t i o n o f m o d u l u s r a t i o f o r p e r f e c t l y b o n d e d fibers.
P.S. Steif, S.F. Hoysan / Alignedshort-fiber composites
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c~
kJ ELLIPSOID II
ELLIPSOID I
0
ELLIPSOID III
Fig. 5. Ellipsoids for comparison with cylindrical fiber results.
Table 1 Comparison of reinforcement coefficient for cylindrical and ellipsoidal fibers
2.0 5.0 10.0 20.0 oc
Cylinder
Ellipsoid !
Ellipsoid II
Ellipsoid III
0.869 2.573 4.095 5.484 8.167
0.898 2.746 4.438 5.992 8.748
0.935 3.128 5.530 8.173 14.34
0.875 2.547 3.939 5.117 7.000
a n d 1.3 percent for L / a = 12. O n e should b e a r in mind, however, that the finite e l e m e n t m e t h o d is p r o b a b l y most accurate when )~ = 1 (see Steif a n d H o y s a n , 1987), a l t h o u g h p r e v i o u s experience with such p r o b l e m s suggests that the finite element m e t h o d is reliable at least up to )~ = 10. This is also b o r n e out b y c o n s i d e r a t i o n of the r e i n f o r c e m e n t coefficient for rigid fibers. The p a t c h - l o a d solution also p r o v i d e s a m e a s u r e of the degree to which the ends of a fiber ' i n t e r a c t ' with one another, at least for the case X = 1. This m a y b e seen in Fig. 6, in which we have p l o t t e d /~app at L / a , n o r m a l i z e d b y /~app for L / a = oo, as a f u n c t i o n of L / a (v = 0.3). A s L / a a p p r o a c h e s zero, /-/app vanishes since the equal a n d o p p o s i t e p a t c h e s of b o d y force cancel out one another. Since l o a d transfer
1
0.8
FI(L/a)
0.6
I-l(oo)
0.4 0.2 0 0
1
I
I
5
10
15
L/a Fig. 6. Effect of fiber aspect ratio on potential energy of auxiliary problem.
I
20
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P.S. Steif S.F. Hoysan / Aligned short-fiber composites
occurs more slowly when X is larger, one expects the interaction of the ends 1o be greater when ~ ~ greater than one, i.e., the corresponding curves would lie below the one shown in Fig. 6 Consider now the function ~(~, v) which was part of the approximate solution derived in Section 5. Recall that the approximate solution was intended to be valid for relatively long fibers, i.e., for L / a :,-- t. Hence, we chose to evaluate ~(h, v) approximately by fitting it to the finite element results for L / a = 12.0, the largest fiber aspect ratio considered. In particular, the form ~(X, ~,) - ('./~'~ was fit with ~ least squares m e t h o d using H ~ for X = 1, 2, 5, 10 and 20 (v = 0.3), with the result: ~ ( ~ - , P ) = 2 . 3 9 / ~ . 0289.
We have evaluated the reinforcement coefficient A using (25), and the results are included in Fig. 4. As expected, this approximate solution is most accurate when L / a is relatively large. Note also that the approximate solution becomes progressively worse for shorter fibers as h increases. This is because the interaction of the fiber ends, which is ignored in the approximate solution, becomes increasingly important as the fiber becomes stiffer. We turn now to the effect of interfacial compliance on the reinforcement coefficient. It is useful to define dimensionless interface compliances which represent the stiffnesses of the matrix relative to that of the interfaces. Let k~ and k~ be the shear stiffness on the sides and the extensional stiffness on the end, respectively; then, we define dimensionless compliances C, and C; by C s = E m / k Sa,
C¢ = E m / k ~ a.
N o t e that a compliance value of 0 corresponds to a rigid interface, i.e., a perfect bond, and infinite compliance corresponds to zero stiffness. The influences of shear and extensional interface compliance on A are shown in Tables 2 and 3. respectively, for the case of L / a = 4. Shear compliance along the sides has relatively little effect when X is near one, while it can have more influence when ;k is large. However, even with zero transfer of load through the sides (C~ = oc), there is still reinforcement. This means that a substantial part of the load enters through the fiber ends. Extensional compliance at the ends has a strong effect when 2, is near one: for larger h, such compliance has an effect similar to that of shear compliance on the sides.
Table 2 Effect of interfacial compliance along the fiber sides ( L / a = 4)
x
C, = o.o
c, = 1.o
G = o¢
2.0 5.0 10.0
0.869 2.573 4.095
0.856 2.426 3.690
0.811 2.080 2.936
Table 3 Effect of interfacial compliance along the fiber ends ( L / a = 4)
2.0 5.0 10.0
G = o.o
Ce = 1.0
G = o~
0.869 2.573 4.095
0.608 2.054 3.350
0.311 1.634 2.818
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7. Calculation of ineffective length Finally, we show it is possible to exploit the ideas contained here to provide an alternative means of calculating the ineffective length. The ineffective length, a concept used in calculations of composite strength, refers to the portion of the fiber near its end or near a break which is not fully loaded. Estimates of the ineffective length which are most often used are based on shear-lag analyses. Aside from the highly approximate nature of such analyses, there is a degree of arbitrariness in the usual definition of the ineffective length: the length over which less than, say, 90% of the load is transferred to the fiber. Here, we propose an alternative definition of ineffective length which is based on the effective stiffness of the composite. Recall the rule-of-mixtures approximation, E . . . . of the longitudinal stiffness of an aligned fiber composite with n fibers, all of radius a and length 2 L, in a volume V: Ero m =
(2"rra2Ln/V)Ef + (1 - 2,rra2Ln/V)Em
= E m -1- ( E f -
Em)cf,
or E r o m / E m = 1 q- (~k -- 1 ) O f
The rule-of-mixtures approximation is based on the assumption that the strain in the fiber and in the matrix is equal to the applied strain. This means that the fiber carries the 'infinite fiber load' over the whole of its finite length. As seen earlier, a real short-fiber composite will actually have an effective modulus which is less than that given by the rule-of-mixtures prediction (Hau x is always less than zero). Based on the notion that the ineffective length is that part of the fiber which does not carry the 'infinite fiber load', we can find an imaginary fiber length (less than the true fiber length) which, when incorporated into an imaginary composite having the same number of fibers as the real composite, gives a rule-of-mixtures stiffness that is equal to the true stiffness of the actual short-fiber composite. The ineffective length is then defined to be half the difference between the true length and the imaginary length. Consider a composite with n fibers of length 2 L ' and radius a in volume V. According to the rule of mixtures, its stiffness E . . . . is given by E r o m / E m = 1 + (X - 1)2,~a2L'n/V.
If the actual composite has the same number of fibers, but of length 2L, and it has a reinforcement coefficient A, then the true composite stiffness is
Ec/E m = 1 + 2 A ~ a 2 L n / V . Equating the stiffnesses, one finds that
L ' / L = A / ( X - 1). With the ineffective length, Lineff
a
L 1 a
Lineff ,
- 1
defined to be L - L', one obtains
'rro2a3(~ - 1)
Since for long fibers /-/aux scales like ( X - 1)2o2a 3, the ineffective length depends only on the modulus ratio. With the finite element results for L / a = 12.0, one finds that Lineef/a ranges from 0.3 for X = 2 to 2.7 for ?, = 20. If these calculations had been carried out using a larger L / a , the ineffective length would be slightly larger.
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Acknowledgements The authors thank Mr. R.W. Dill for carrying out some of the computations. This work has been supported by the National Science Foundation under grant MSM-8451080, by the General Electric Engine Business Group, by the Alcoa Technical Center, and by the Department of Mechanical Engineering, Carnegie-Mellon University.
References Benveniste, Y. (1984), " O n the effect of debonding on the overall behavior of composite materials", Mechanics of Materials 3, 349. Benveniste, Y. (1985), "The effective mechanical behavior of composite materials with imperfect contact between the constituents", Mechanics of Materials 4, 197. Christensen, R.M. (1979), Mechanics of Composite Materials, Wiley-Interscience, New York. Chou, T.-W., Nomura, S., and Taya, M. (1980), "A self-consistent approach to the elastic stiffness of short-fiber composites", J. Composite Mater. 14, 178. Eshelby, J.D. (1956), "Continuum theory of lattice defects". in: F. Seitz and D. Turnball, eds., Progress in Solid State Physics Vol. 3, Academic Press, New York. Eshelby, J.D. (1957), "The determination of the elastic field of an ellipsoidal inclusion, and related problems", Proc. Roy. Soc. London 241A, 376.
Eshelby, J.D. (1966), " O n the interaction between inclusions", Acta Metallurgica 14, 1306. Hashin, Z. (1983), "Analysis of composite materials--a survey", A S M E J. Appl. Mech. 50, 481. Laws, N. and McLaughlin, R. (1979), "The effect of fibre length on the overall moduli of composite materials", J Mech. Phys. Solids 27, 1. Russel, W.B. (1973), "On the effective moduli of composite materials: effect of fiber length and geometry, at dilute concentrations", J. Appl. Math. Phys. (ZAMP) 24, 581. Sanadi, A.R. and Piggott, M.R. (1985), "Interfacial effects in carbon-epoxies. Part 1: strength and modulus with short aligned fibers", J. Mater. Sci. 20, 421. Steif, P.S. and Hoysan, S.F. (1987), "On load transfer between imperfectly bonded constituents", Mechanics of Materials 5, 375.
197
AN ENERGY M E T H O D FOR CALCULATING T H E S T I F F N E S S OF ALIGNED S H O R T - F I B E R C O M P O S I T E S
Paul S. STEIF and Steven F. H O Y S A N Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. Received 23 January 1987; revised version received 24 March 1987
A procedure is presented for calculating the longitudinal elastic modulus of a composite possessing dilute concentrations of aligned cylindrical fibers. For a dilute concentration of fibers, the modulus can be written in terms of the perturbation that a single fiber induces in the displacement field of an otherwise homogeneous body subjected to remote uniaxial tension. A boundary value problem, referred to as the auxiliary problem, is formulated which yields the perturbed displacement field directly. The longitudinal modulus can then be expressed in terms of the potential energy of the auxiliary problem; thus, energy-based approximate methods (e.g. FEM) can be exploited to calculate the modulus accurately. With a small modification, this procedure is shown to be applicable to the case of fibers which are imperfectly bonded to the matrix. Moduli for a few representative cases are calculated using a finite element method. Finally, a new means of defining and computing the ineffective length of a fiber is introduced.
1. Introduction
Theoretical predictions of the effective properties of composite materials can play an important role in the design of these materials, particularly as the variety of constituent and processing options rapidly expands. Recent surveys by Christensen (1979) and Hashin (1983) indicate the extent of interest in this subject. With regard to effective elastic moduli, the theory of heterogeneous materials has been most successful in dealing with composites reinforced by very long fibers, as well as composites with spherical reinforcements. In a variety of applications where low cost is important, however, the use of chopped fibers or whiskers has often been preferred. The challenge in predicting the effective properties with these reinforcements, of course, is to account for the effect of the fiber ends, which can no longer be neglected. The most prevalent means of calculating the effective properties of so-called short-fiber composites has been to model the fibers as ellipsoids (Russel, 1973; Laws and McLaughlin, 1979; Chou, Nomura, and Taya, 1980). This derives from the possibility of exploiting Eshelby's (1957) solution for an ellipsoidal, elastic inclusion located in an infinite elastic body subject to a remotely applied stress. While the ellipsoid model does give the long fiber results when the ellipsoid aspect ratio become large, it is by no means clear that the stiffening effect of short fibers shaped as circular cylinders--apparently the most common shape --is accurately predicted by the ellipsoid model. One goal of the present work is to develop a method of calculating the stiffness of a composite made up of short cylindrical fibers. In addition to addressing the cylindrical fiber problem more accurately, the method offered here permits one to explore the effect of fiber-matrix interface character on composite stiffness. One expects that the degree of contact at the fiber-matrix interface can influence the stiffness of short-fiber composites because of the important role played by load transfer in stiffness, and the possibility of the interface markedly affecting load transfer. In fact, it has been demonstrated by Sanadi and Piggott (1985) on 0167-6636/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
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aligned short-fiber systems that different fiber surface treatments result in very different longitudinal stiffnesses. Recently, there have been efforts to assess the degree to which degradation of the interface properties has significant consequences for the effective stiffness of the composite, lncluded are: Benveniste's (1984) pointing out the gradual failure of St. Venant's principle at an interface where the interfacial integrity is substantially diminished; Benveniste's (1985) computation of the effective modulus of a composite reinforced by spherical particles which are not well bonded to the matrix: and. Steif and Hoysan's (1987) calculation of load transfer in a pullout test in which the fiber and matrix are no1 perfectly bonded to the matrix. In order to focus on the effects of fiber aspect ratio and material and interface properties, we choose to confine attention to dilute concentrations of fibers. While the dilute solution approximation by itself may have limited application, it forms the basis of a number methods of calculating the moduli of a composite with a finite volume fraction of fibers. Within the dilute solution approximation, one can view each fiber as being effectively located in an infinite body of matrix material. However, even this problem, involving a finite cylindrical fiber partially bonded to an unbounded matrix, is sufficiently complicated that we must resign ourselves to the necessity of employing numerical methods. In the present paper we develop a procedure which makes the best of this bad situation. This procedure minimizes the results that need to be extracted from the numerical calculations, and it takes advantage of the most accurate aspects of energy-based approximate methods.
2. Interface model
The interface model we employ states that the displacement components in the fiber and in the matrix are not the same across the interfaces. The jumps in the displacement are taken to be proportional to the tractions acting across the interface. While this model is simple, it is general in that it allows for a variety of interface situations. One could view this model as representing, for example, a thin fiber coating or a series of cracks along the interface. Formally, the interface model may be expressed by 7;,= o~/nj = k,i Au:
(I)
where T, is the traction acting across the interface, oij is the stress, n: is the outward normal from the fiber, k~j is the tensor of interface stiffnesses, and Auj is the jump in the displacement at the interface. Indices refer to Cartesian components, and summation on repeated indices is implied. We assume the interface stiffness tensor is symmetric. In general, one can take the interface stiffnesses to vary from point to point along the interface. By taking the coupling stiffnesses k,j to be very large, one recovers the usual condition of continuous displacements. On the other hand, vanishingly small stiffnesses correspond to complete decoupling of the fiber and the matrix, i.e., zero tractions across their interface. This interface model has been studied previously (e.g., Benveniste, 1984, 1985; Steif and Hoysan, 1987). To the extent that the model is intended to represent a fiber coating or other actual layer of material separating the fiber and the matrix, it is tacitly assumed that the interface layer is small compared with other relevant dimensions, such as the fiber diameter. The model simplifies the situation considerably in that one needs to specify only a single set of parameters (stiffnesses) in order to describe the interface; thicknesses and modufi need not be separately prescribed. This has an added advantage in that it leads to a more straightforward approach to solving boundary value problems. Without more detailed information regarding the interface, a more complex model is not warranted; this is particularly so as our present aim is only to assess qualitatively the sensitivity of composite stiffness to interface integrity.
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3. Energy-based calculation of effective modulus The present analysis is intended to derive an energy-based method of estimating the effective longitudinal Young's modulus of a composite containing a dilute concentration of aligned short fibers. Within the dilute solution approximation, one can express the composite longitudinal modulus, E~, in the form Ec/E m =
1+
(2)
cfA
where E m is the matrix Young's modulus, cf is the volume fraction of fibers, and A is the reinforcement coefficient (defined by Russel (1973)). The assumption of dilute fiber concentrations leads one to consider the problem of a single fiber located in an infinite body of matrix material. The reinforcement coefficient is then calculated by comparing the strain energy of this body with the strain energy of a similarly loaded body of pure matrix material. Clearly, the energy difference will be proportional to the fiber volume. To motivate the method we develop, it is worth considering a rather naive approach in which we employ, say, a finite element method to analyze a large but finite region containing a fiber, which is subjected to remote uniaxial tension. Using the boundary displacements or some other means, we could calculate the strain energy of the body. The problem arises in deciding how large to make the domain. The need to mimic an isolated fiber in an infinite body dictates that the finite region should be as large as possible. But, as the domain gets larger, the relative effect of the fiber becomes less and the energy difference more inaccurate. In order to develop a method that calculates only the energy difference, we take an approach similar to that of Eshelby (1956) and consider two infinite bodies: body B is homogeneous with moduli C, lkl, and body B' contains a fiber of moduli Cjjkt embedded in an otherwise homogeneous matrix of moduli Ci Jk/. Both bodies are subjected, in general, to a remote uniform stress )Zzj (see Figs. l(a) and l(b) for schema(ics of two such bodies subjected to remote uniaxial tension). A Cartesian coordinate system and a cylindrical polar coordinate system (x 3 = z), with origins coinciding with the center of the fiber, will be used t
t~
(3
r
t3
(Y
Fig. 1. (a) Schematic of a single fiber located in an infinite b o d y which is subjected to r e m o t e uniaxial tension. (b) S c h e m a t i c of an infinite b o d y which is subjected to r e m o t e uniaxial tension.
200
P.S. Steif S.F. Hoysan /Alignedshortzfiber composites
interchangeably as convenient. The fiber is taken to be separated from the matrix by an interface described by (1). / r t Let primed quantities u s, ~sj, osj be associated with body B' and unprimed quantities u,. ~,~, <. wiih body B, where e~j denotes the strain. Then, the strain energy in body B' minus the strain energy in body B. Aq), is given by
AI~
ksj Aus' kuj' d S -
1 /" Oijf. , i' i d V + ~ 1[ = ,,~] a Vzc
:~ f o~ie,i dV
J SI
(3)
JV
where V~ is the infinite volume, and Sf is the surface of the fiber. Note that Au~ denotes the displacement jump at the interface in body B' and that there is no displacement jump in B. We expect the energy difference A~ to be finite; in a manner analogous to Eshelby (1956), we put Aq~ in a form which explicitly reflects that fact. To do that, we use the divergence theorem (including separate volume integrations for fiber and matrix) to derive the equality
(oij-oij)(f.ij-~-ff.ij)
(os,-oii)
dV:
Ausnj d S
f ¢
where we have made use of the fact that aij and oij both approach ~ i j a t infinity. Now, subtract the left hand side of (4) from (3) and add the right hand side to give
Adb = ½
(5)
(oOei"J - a,je,j) d V + ½ o,a Ausn j dS. '
f,Sf
'
Furthermore, since the moduli differ only in the fiber, one finds m~:
~il
(Cijkl _
C, j' k , ) e s j e 'k / d V + 7 ~
oi i Ausn I d S
(6)
f
where Vf is the fiber volume. The derivation thus far mirrors the development of equation (6.5) in Eshelby (1956), except for the surface integral term which reflects the compliant interface. A similar derivation can also be found in Eshelby (1966). We now introduce the quantity Usp, defined so that u~ = us + Usp, which represents the perturbation induced by the fiber in the uniform field u s. One can now rewrite (6) as
A ~ = ½fvf(Cijk,--Cv'jk,)e,j(%z + e~,) d V + ½fsf°iJ AuVinj dS.
(7)
We now use the above expression for the energy difference to calculate the longitudinal Young's modulus of a short fiber composite composed of isotropic fibers of circularly cylindrical cross-section. Though there is no essential necessity to restrict the values of the material parameters, the results are simplified somewhat by assuming that Poisson ratios are equal in the fiber and in the matrix. The case of different Poisson ratios is discussed below. Now, let the remote stress state be one of uniaxial tension o parallel to the axis of the fiber. Then certain terms in (7) may be calculated directly: Cijkl£ kl = (Y~i3~j3
(8)
C/jk,,~, ' = XoSi38j3
(9)
and
where )k = E f / E m and 8 0 is the Kroneker delta. Note that ~33
=
°/Em"
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201
In the subsequent development we concentrate on interfaces which are judged to have the most significant influence on the longitudinal stiffness of short fiber composites. Those are interfaces having weakness, or compliance, in paths of load transfer from fiber to matrix. Specifically, we will assume that a shear displacement jump is possible along the sides of the fiber, and that a normal displacement jump is possible at the fiber ends. With reference to Figure la, this means that Au 3 ¢ 0 along r = a, - L < z < L, and along z = + L , 0 < r < a. It is further assumed that the shear discontinuity induces only a shear traction and the normal discontinuity only a normal traction. Then, since the energy difference can be written as A~
=
[½o2/Ec
--
½o2/Em]Voo
(]0)
where E c is the effective modulus of the composite, one can deduce that
EmE c -l+(1-Xlc r+(1-x)cf~ Emfv,~P,d V + c f o~ffEmfz = + LA u P
dS
(11)
where cf = V f / V ~ is the fiber volume fraction. Note that 1 + (1 - X)c r is the uniform strain prediction, provided ()t - 1)cf << 1. The last two terms represent the energy associated with the fiber's perturbing the uniform strain field Eo. It is now shown that the integrals in (11) can be put into such a form that they may be evaluated in a straightforward manner with approximate energy methods. This is done by decomposing the problem of body B' subjected to uniaxial tension into two problems: one with loadings that produce the uniform strain eij, plus one with loadings that produce the perturbed displacement uP. Such a decomposition is similar to what is done in analyses of cracks and cavities. The contribution of the present paper is to show that the potential energy of the loading which produces the perturbed displacement is precisely the energy required to evaluate the reinforcement coefficient A. The particular decomposition given here requires that the Poisson's ratios be equal, though a somewhat more complicated decomposition is possible when they are different. First, we will consider the case in which the interfacial stiffness is infinite, i.e., the fiber and the matrix are perfectly bonded. Let the body B' be subjected to a remote tension o, as well as to equal and opposite concentrated body force distributions of magnitude (X - 1)o (per unit area) as shown in Fig. 2(a). This loading results in the uniform strain field q j; the stress is uniaxial tension o in the matrix and uniaxial tension Xo in the fiber. The second problem, which we refer to as the auxiliary problem, consists only of equal and opposite body force distributions at the fiber ends (see Fig. 2(b)), such that, when superposed with the first loading, is equivalent to the original problem (Fig. l(a)). Therefore, the displacement field induced in the auxiliary problem is, by definition, uP. The potential energy of the auxiliary problem //aux is defined by //aux = 1_2t f "V~
uij~ij~P~Pd V -
f
~u p dV-- - ½ f ./,uP dV
~"V~
(12)
J V~
where fi are the body forces. Again, the body forces are applied at the fiber ends, with fl =f2 = 0, f3 = - ( X - 1)o on z = L and fl =f2 = 0, f3 = ( X - 1)0 on z = - L . Using the symmetry about z = 0, one finds Hau x = ( ) t - 1)of~=Lu3P dS.
(13)
Consider now the integral over the fiber volume in (11). (The surface integral in (11) is zero for perfectly bonded fibers since hu[ = 0). Since the strain is related to the displacement according to eP3 = O u P / O z ,
(14)
202
P.S. Steif S.F. Hoysan / Aligned short-fiber composites (Y
(~.-1)c~
(~,-1)(y
hll
ell (~,-1)o
G-1)o
ff Fig. 2. (a) Loading which produces a uniform strain of the fiber and the matrix when they are perfectly bonded. (b) Loading for auxiliary problem when fiber and matrix are perfectly bonded.
the term involving the volume integral becomes Em
2(1 - X)cf-~fff~=LuP dS.
(15)
By (13), this can be written as ( --Emcf/02'rra2t)l~aux
(16)
which means the reinforcement coefficient A can be expressed directly in terms of the potential energy of the auxiliary problem, i.e., A = ~ - ] + EmlIaux/~O2a2L.
(17)
Note that for fiber ends that are far apart, that is, for long fibers, Hau x is of order a ~, and thus,
A = X- 1 + O(a/L), ~mplying that the correction due to the auxiliary problem vanishes as a l L approaches 0. An analogous decomposition of the problem is possible when displacement jumps are permitted at the fiber ends. As before, the first problem consists of loadings that produce a uniform straining of the body, with no extension of the springs. This is done by applying tensile stress o at infinity, together with tensile tractions o on the matrix side of the interface at the fiber and tensile tractions Xo on the fiber side of the interface at the fiber ends. The uniform strain in the body is o/Em, the springs do not stretch, and the stress is o in the matrix and Xo in the fiber. An auxiliary problem can then be defined involving equal and opposite internal tractions (see Fig. 3) such that the superposition gives the original problem. Note that the interface does open up in the auxiliary problem, and that this auxiliary problem is equivalent to the previous one when displacement continuity is enforced at the fiber end. A careful accounting of all the energy (including the energy of the stretched interface springs) leads one to conclude that the two integrals
P.S. Steif S.F. Hoysan / Aligned short-fiber composites
203
Fig. 3. Details of loading near the fiber ends for auxiliary problem when fiber and matrix are not perfectly bonded. in (11) are equal to (16). Now, however, the potential energy of the auxiliary problem is not given by (13), but by Haux =
kof~=L uP as-of~=L+uP as
(181
where z = L - denotes the fiber side of the interface at the fiber end, and z = L + denotes the matrix side. As mentioned above, a decomposition is also possible when the Poisson ratios are different. The corresponding auxiliary problem involves longitudinal body forces f3 applied at the fiber ends and radial body forces fr applied at the sides of the fiber, where Pf -- Pm
]
and
fr=--~.
P f - Pm ] (l_Gf-~-(~-_l_pm) o.
In these expressions, X denotes the ratio of the shear moduli. It can be shown, as was done above, that the potential energy of the auxiliary problem is again equal to the required strain energy difference. The potential energy of the auxiliary problem was computed for several sets of material parameters using a finite element method. Because of the procedure used here, in which the energy difference is on the order of the fiber volume, the maximum size of the domain was limited only by practical considerations of solution time. For the domains chosen, the accuracy was improved by using a simple substructuring procedure, reported elsewhere (Steif and Hoysan, 1987). The accuracy of the finite element method is discussed below.
4. Moduli of composite with rigid fibers In the limit as the modulus of the fiber becomes large compared with that of the matrix, the auxiliary problem presented above becomes one of applying body forces of infinite magnitude to a fiber of infinite
P.S. Steif S.F Hovsan /Alignedshort-fiher composite,s
204
modulus. This is not possible to do in practice with a finite element method, for which the solution will often be inaccurate as one constituent becomes considerably stiffer than the other. For this reason, we offer a method intended specifically for calculating the effective longitudinal modulus of a composite reinforced by rigid fibers. Again, we compare the strain energy of bodies B' and B, both of which are subject to a remote uniaxial tension o. Now, however, body B' has a fiber which does not deform, Considering only the case of perfect bonding between the fiber and the matrix (the modification for an elastic interface being wholly analogous to that given above), the difference in the strain energies, Aq), is given by
A~):- ~
OijEij d V -
~
t
Making use of the fact that o;/ and on &, one can derive the identity
:l f v "
aij
(19)
Oij~.iSdV.
both correspond to uniaxial tension o at infinity, and that u / = 0
(ai/- oi/)(C/+ t C:) d V - ~I f+rqoijc;:d V - v,. ,*
- ~lfsC~flUiFll d S ,
( 2o )
from which it is found that AcI, can be written as
A+ = -½.sf +'+"' d S -
dS,
(2l)
!
where we have again written o ; / = o;j + og and nj still points from the fiber to the matrix. Note that the first integral can be identified as minus the strain energy in the volume Vf of body B. To develop an auxiliary problem for which the strain energy is proportional to the fiber volume, we view the straining of body B' as accomplished in three steps. First, take homogeneous body B and let it deform uniformly under uniaxial tension o. Secondly, cut out the material in volume Vr (this is represented by the first integral in (21)). Thirdly, apply displacements u2 to the exposed surface Sf that return it to its original position. These displacements are u a = - ui, where u i are the displacements brought about by the homogeneous stressing of B. The region exterior to Vf is now precisely in the configuration that B' would be in, were it subjected to a remote uniaxial tension o. If, consistent with previous notation, the stresses induced during the third step are denoted by o;p, then the tractions 7];a required to produce the displacements u~ are -njo,~. Thus, the work done in the auxiliary problem (the third step), W~ux, is given by W~.~ = ½fsrOi~u;nj ds > 0.
(22)
Using (10) and (21), one can then write the reinforcement coefficient A as A = 1 + EmWau,J'rro2a2L.
(231
A simple interpretation of (23) is that the reinforcement coefficient exceeds 1 (the rule-of-mixtures prediction based on compliances) by an amount which reflects the extent to which the fiber prevents the rest of the material from deforming.
5. Approximate solution of auxiliary problem for perfectly bonded interfaces In this section, we consider further the auxiliary problem in order to derive a useful approximate solution and to assess the influence of the fiber aspect ratio. Recall that the auxiliary problem consists of
205
P.S. Steif S.F. Hoysan /Aligned short-fiber composites
equal and opposite patches of body force applied to the fiber ends (when the fiber is perfectly bonded to the matrix). The magnitude of the body forces is proportional to the applied stress, and it depends on the fiber-to-matrix modulus ratio. Thus, the modulus ratio enters the problem in two ways: it is a scale factor for the body forces, and it characterizes the degree of inhomogeneity of the body that is being loaded. We begin by deriving an approximate solution which ignores this second role of the modulus ratio. That is, we imagine the energy of the auxiliary problem to be calculated by applying body forces (X - 1)o to the ends of a fiber which has the same moduli as the matrix. Applying body forces of magnitude ( X - 1)o is equivalent to applying body forces of unit magnitude and then scaling up the resulting energy appropriately. Since the fiber and the matrix now have identical moduli, this problem of unit body forces applied to the fiber ends can be solved with the use of the Kelvin point load solution. Omitting the unessential details, one finds the approximate potential energy for unit body forces, /Tapp, to be given by the triple integral
_ /Tapp
a'
[1[1[~
4/*(i -- v) ~y=0~x=0~r_0 {(3 1
-
4v)
_
(x2 + y2_ 2xy cos
1
_
¢x 2 + y2 + 4t L/a)2- 2xy cos
[x2+y2+4(L/a)2
2xycosv]3/2
}xydvdxdy,
where /, is the shear modulus. The potential energy of the auxiliary problem would then be given approximately by ()~- 1)202/-lapp. An inspection of the expression for /~app reveals that the second and third integrals vanish as L / a ---, ~. Hence, when the fiber is relatively long, Happ simplifies to the form
a3(3-4v) fl /app--
fl f'rr
xydy__d__f_.xdy__
Jx=OJy=OJy_O Cx2 + y2
2xy cos i •
The triple integral was evaluated numerically and was found to equal 1.37. Obviously, this corresponds to twice the energy of a single circular patch of unit body force in an infinite solid. Recalling equation (17), which links the energy Haux with the reinforcement coefficient A, one can see that, to first order in (a/L), the reinforcement coefficient A is given by A=?t_1_[(?t_l)21"37 (l+v)(3-4V)]L 2"~ 1-Tv
"
This simple result, which is approximately vahd for a l L << 1 and for X nearly 1 (the inhomogeneity of the fiber-matrix system having been neglected), can be extended to a somewhat larger range of )~ in the following way. In the limit of alE--+ 0, the auxiliary problem, for any value of X, consists of a single circular patch of load applied to the end of a very long fiber. By dimensional considerations, the potential energy of this problem (with unit body forces) must be of the form /-/aux = ~(~, v)(1/Em) a3
(24)
provided that the Poisson ratios are equal to v. Now, since the body force on the fiber ends is (X - 1)o, one finds that the reinforcement coefficient A can be written as
7~]
A = ( X _ 1 ) [ 1 _ ( X _ 1 ) ~ ( ~ v) a
(25)
P.S. Steif S.F. Hovsan /Aligned
206
short-fiber cornposite.s
in the limit of a / L << 1. Below, we will find an approximate form for the function ~(?,., v) using the finite element results.
6. Results
Calculations described in the previous sections were carried out for a range of material parameters, including a relatively short fiber L / a = 4 and a relatively long fiber L / a = 12. As has been discussed previously, the finite element results may be subject to some error when the modulus ratio is rather large. This motivated the derivation of a method suitable for infinitely rigid fibers, which was given in Section 4. The results are now presented which show the influence of several parameters (v = 0.3) on the reinforcement coefficient A. In Fig. 4, we depict the dependence of A on ?~ for three fiber aspect ratios. The infinite fiber results, A = ?, - 1, is shown as the dashed line. Finite element results for L / a = 4 and 12 (?~ = 2, 5 and 10) are represented by symbols as indicated. The solid lines represent the approximate solution, which was derived in Section 5 and is discussed further below. As expected, and as found by Russel (1973), reducing the aspect ratio reduces the stiffening effect. Furthermore, this effect becomes more pronounced as the modulus ratio ?, increases. Since previous calculations of the stiffening effect of discontinuous fibers are based for the most part on modeling the fiber as an ellipsoid, it is interesting to compare the stiffening effect of cylindrical fibers with that of ellipsoidal fibers. But an immediate question arises: What is the ellipsoid to be compared with? Three possibilities that suggest themselves are depicted in Fig. 5. Ellipsoid I is inscribed in the cylinder and, therefore, has a minor axis equal to the fiber radius and a major axis equal to the fiber half length; its volume is 2 of the cylinder's volume. Ellipsoid II has the same minor axis as the cylinder's radius, and the same volume; its major axis is ~ times the half length of the cylinder. Ellipsoid III has the same major axis as the cylinder half length, and the same volume; its minor axis is (~)1/2 times the radius of the cylinder. A comparison of the A-values is shown in Table 1 for a cylinder having an aspect ratio of L / a = 4. In general, the degrees of reinforcement provided by Ellipsoids I and III and cylinders are comparable; Ellipsoid II provides significantly more stiffening, as one might expect from the larger aspect ratio. Some sense of the accuracy of the finite element results can be gained from considering the patch-load solution given in Section 5. Recall that the patch-load solution is precisely the auxiliary problem for ~ = 1. It was found that the patch-load solution and the finite element results differed by 2.9 percent for L / a = 4
10
/-
9
/
8
/
6
/
0
/
]t7.
A aL'~" = 12
5 4
0
/
/
L.-L-- = 12
[]
",/'ff"~l
I
I
I
I
I
I
I
I
1
3
4
5
6
7
8
9
10
2
Fig. 4. R e i n f o r c e m e n t c o e f f i c i e n t as a f u n c t i o n o f m o d u l u s r a t i o f o r p e r f e c t l y b o n d e d fibers.
P.S. Steif, S.F. Hoysan / Alignedshort-fiber composites
207
c~
kJ ELLIPSOID II
ELLIPSOID I
0
ELLIPSOID III
Fig. 5. Ellipsoids for comparison with cylindrical fiber results.
Table 1 Comparison of reinforcement coefficient for cylindrical and ellipsoidal fibers
2.0 5.0 10.0 20.0 oc
Cylinder
Ellipsoid !
Ellipsoid II
Ellipsoid III
0.869 2.573 4.095 5.484 8.167
0.898 2.746 4.438 5.992 8.748
0.935 3.128 5.530 8.173 14.34
0.875 2.547 3.939 5.117 7.000
a n d 1.3 percent for L / a = 12. O n e should b e a r in mind, however, that the finite e l e m e n t m e t h o d is p r o b a b l y most accurate when )~ = 1 (see Steif a n d H o y s a n , 1987), a l t h o u g h p r e v i o u s experience with such p r o b l e m s suggests that the finite element m e t h o d is reliable at least up to )~ = 10. This is also b o r n e out b y c o n s i d e r a t i o n of the r e i n f o r c e m e n t coefficient for rigid fibers. The p a t c h - l o a d solution also p r o v i d e s a m e a s u r e of the degree to which the ends of a fiber ' i n t e r a c t ' with one another, at least for the case X = 1. This m a y b e seen in Fig. 6, in which we have p l o t t e d /~app at L / a , n o r m a l i z e d b y /~app for L / a = oo, as a f u n c t i o n of L / a (v = 0.3). A s L / a a p p r o a c h e s zero, /-/app vanishes since the equal a n d o p p o s i t e p a t c h e s of b o d y force cancel out one another. Since l o a d transfer
1
0.8
FI(L/a)
0.6
I-l(oo)
0.4 0.2 0 0
1
I
I
5
10
15
L/a Fig. 6. Effect of fiber aspect ratio on potential energy of auxiliary problem.
I
20
208
P.S. Steif S.F. Hoysan / Aligned short-fiber composites
occurs more slowly when X is larger, one expects the interaction of the ends 1o be greater when ~ ~ greater than one, i.e., the corresponding curves would lie below the one shown in Fig. 6 Consider now the function ~(~, v) which was part of the approximate solution derived in Section 5. Recall that the approximate solution was intended to be valid for relatively long fibers, i.e., for L / a :,-- t. Hence, we chose to evaluate ~(h, v) approximately by fitting it to the finite element results for L / a = 12.0, the largest fiber aspect ratio considered. In particular, the form ~(X, ~,) - ('./~'~ was fit with ~ least squares m e t h o d using H ~ for X = 1, 2, 5, 10 and 20 (v = 0.3), with the result: ~ ( ~ - , P ) = 2 . 3 9 / ~ . 0289.
We have evaluated the reinforcement coefficient A using (25), and the results are included in Fig. 4. As expected, this approximate solution is most accurate when L / a is relatively large. Note also that the approximate solution becomes progressively worse for shorter fibers as h increases. This is because the interaction of the fiber ends, which is ignored in the approximate solution, becomes increasingly important as the fiber becomes stiffer. We turn now to the effect of interfacial compliance on the reinforcement coefficient. It is useful to define dimensionless interface compliances which represent the stiffnesses of the matrix relative to that of the interfaces. Let k~ and k~ be the shear stiffness on the sides and the extensional stiffness on the end, respectively; then, we define dimensionless compliances C, and C; by C s = E m / k Sa,
C¢ = E m / k ~ a.
N o t e that a compliance value of 0 corresponds to a rigid interface, i.e., a perfect bond, and infinite compliance corresponds to zero stiffness. The influences of shear and extensional interface compliance on A are shown in Tables 2 and 3. respectively, for the case of L / a = 4. Shear compliance along the sides has relatively little effect when X is near one, while it can have more influence when ;k is large. However, even with zero transfer of load through the sides (C~ = oc), there is still reinforcement. This means that a substantial part of the load enters through the fiber ends. Extensional compliance at the ends has a strong effect when 2, is near one: for larger h, such compliance has an effect similar to that of shear compliance on the sides.
Table 2 Effect of interfacial compliance along the fiber sides ( L / a = 4)
x
C, = o.o
c, = 1.o
G = o¢
2.0 5.0 10.0
0.869 2.573 4.095
0.856 2.426 3.690
0.811 2.080 2.936
Table 3 Effect of interfacial compliance along the fiber ends ( L / a = 4)
2.0 5.0 10.0
G = o.o
Ce = 1.0
G = o~
0.869 2.573 4.095
0.608 2.054 3.350
0.311 1.634 2.818
P.S. Steif, S.F. Hoysan /Alignedshort-fiber composites,
209
7. Calculation of ineffective length Finally, we show it is possible to exploit the ideas contained here to provide an alternative means of calculating the ineffective length. The ineffective length, a concept used in calculations of composite strength, refers to the portion of the fiber near its end or near a break which is not fully loaded. Estimates of the ineffective length which are most often used are based on shear-lag analyses. Aside from the highly approximate nature of such analyses, there is a degree of arbitrariness in the usual definition of the ineffective length: the length over which less than, say, 90% of the load is transferred to the fiber. Here, we propose an alternative definition of ineffective length which is based on the effective stiffness of the composite. Recall the rule-of-mixtures approximation, E . . . . of the longitudinal stiffness of an aligned fiber composite with n fibers, all of radius a and length 2 L, in a volume V: Ero m =
(2"rra2Ln/V)Ef + (1 - 2,rra2Ln/V)Em
= E m -1- ( E f -
Em)cf,
or E r o m / E m = 1 q- (~k -- 1 ) O f
The rule-of-mixtures approximation is based on the assumption that the strain in the fiber and in the matrix is equal to the applied strain. This means that the fiber carries the 'infinite fiber load' over the whole of its finite length. As seen earlier, a real short-fiber composite will actually have an effective modulus which is less than that given by the rule-of-mixtures prediction (Hau x is always less than zero). Based on the notion that the ineffective length is that part of the fiber which does not carry the 'infinite fiber load', we can find an imaginary fiber length (less than the true fiber length) which, when incorporated into an imaginary composite having the same number of fibers as the real composite, gives a rule-of-mixtures stiffness that is equal to the true stiffness of the actual short-fiber composite. The ineffective length is then defined to be half the difference between the true length and the imaginary length. Consider a composite with n fibers of length 2 L ' and radius a in volume V. According to the rule of mixtures, its stiffness E . . . . is given by E r o m / E m = 1 + (X - 1)2,~a2L'n/V.
If the actual composite has the same number of fibers, but of length 2L, and it has a reinforcement coefficient A, then the true composite stiffness is
Ec/E m = 1 + 2 A ~ a 2 L n / V . Equating the stiffnesses, one finds that
L ' / L = A / ( X - 1). With the ineffective length, Lineff
a
L 1 a
Lineff ,
- 1
defined to be L - L', one obtains
'rro2a3(~ - 1)
Since for long fibers /-/aux scales like ( X - 1)2o2a 3, the ineffective length depends only on the modulus ratio. With the finite element results for L / a = 12.0, one finds that Lineef/a ranges from 0.3 for X = 2 to 2.7 for ?, = 20. If these calculations had been carried out using a larger L / a , the ineffective length would be slightly larger.
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P.S. Steif S.F. Hoysan / Aligned short-fiber composites
Acknowledgements The authors thank Mr. R.W. Dill for carrying out some of the computations. This work has been supported by the National Science Foundation under grant MSM-8451080, by the General Electric Engine Business Group, by the Alcoa Technical Center, and by the Department of Mechanical Engineering, Carnegie-Mellon University.
References Benveniste, Y. (1984), " O n the effect of debonding on the overall behavior of composite materials", Mechanics of Materials 3, 349. Benveniste, Y. (1985), "The effective mechanical behavior of composite materials with imperfect contact between the constituents", Mechanics of Materials 4, 197. Christensen, R.M. (1979), Mechanics of Composite Materials, Wiley-Interscience, New York. Chou, T.-W., Nomura, S., and Taya, M. (1980), "A self-consistent approach to the elastic stiffness of short-fiber composites", J. Composite Mater. 14, 178. Eshelby, J.D. (1956), "Continuum theory of lattice defects". in: F. Seitz and D. Turnball, eds., Progress in Solid State Physics Vol. 3, Academic Press, New York. Eshelby, J.D. (1957), "The determination of the elastic field of an ellipsoidal inclusion, and related problems", Proc. Roy. Soc. London 241A, 376.
Eshelby, J.D. (1966), " O n the interaction between inclusions", Acta Metallurgica 14, 1306. Hashin, Z. (1983), "Analysis of composite materials--a survey", A S M E J. Appl. Mech. 50, 481. Laws, N. and McLaughlin, R. (1979), "The effect of fibre length on the overall moduli of composite materials", J Mech. Phys. Solids 27, 1. Russel, W.B. (1973), "On the effective moduli of composite materials: effect of fiber length and geometry, at dilute concentrations", J. Appl. Math. Phys. (ZAMP) 24, 581. Sanadi, A.R. and Piggott, M.R. (1985), "Interfacial effects in carbon-epoxies. Part 1: strength and modulus with short aligned fibers", J. Mater. Sci. 20, 421. Steif, P.S. and Hoysan, S.F. (1987), "On load transfer between imperfectly bonded constituents", Mechanics of Materials 5, 375.