Stiffness Predictions for Unidirectional Short-Fiber Composites: Review and Evaluation

Transcription

1 Stiffness Predictions for Unidirectional Short-Fiber Composites: Review and Evaluation Charles L. Tucker III Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign 1206 W. Green St. Urbana, IL Erwin Liang GE Corporate Research and Development Schenectady, NY September 29, 1998 To appear in Composites Science and Technology Abstract Micromechanics models for the stiffness of aligned short-fiber composites are reviewed and evaluated. These include the dilute model based on Eshelby s equivalent inclusion, the self-consistent model for finite-length fibers, Mori-Tanaka type models, bounding models, the Halpin-Tsai equation and its extensions, and shear lag models. Several models are found to be equivalent to the Mori-Tanaka approach, which is also equivalent to the generalization of the Hashin-Shtrikman-Walpole lower bound. The models are evaluated by comparison to finite element calculations using periodic arrays of fibers, and to Ingber and Papathanasiou s boundary element results for random arrays of aligned fibers. The finite element calculations provide E 11, E 22, 12, and 23 for a range of fiber aspect ratios and packing geometries, with other properties typical of injection-molded thermoplastic matrix composites. The Halpin- Tsai equations give reasonable estimates for stiffness, but the best predictions come from the Mori-Tanaka model and the bound interpolation model of Lielens et al. To whom correspondence should be addressed

2 1 Introduction This paper reviews and evaluates models that predict the stiffness of short-fiber composites. The overall goal of the research is to improve processing-property predictions for injection-molded composites. The polymer processing community has made substantial progress in modeling processinduced fiber orientation, particularly in injection molding, and these results are now routinely used to predict mechanical properties. The purpose of this paper is to review the relevant micromechanics literature, and to provide a critical evaluation of the available models. Real injection-molded composites invariably have misoriented fibers of highly variable length, but aligned-fiber properties are always calculated as a prelude to modeling the more realistic situation. Hence, we focus here on composites having aligned fibers with uniform length and mechanical properties. The modeling of composites with distributions of fiber orientation and fiber length, and the treatment of multiple types of reinforcement, will be discussed in a subsequent paper 1. In selecting models for consideration, we impose the general requirements that each model must include the effects of fiber and matrix properties and the fiber volume fraction, include the effect of fiber aspect ratio, and predict a complete set of elastic constants for the composite. Any model not meeting these criteria was excluded from consideration. All of the models use the same basic assumptions: The fibers and the matrix are linearly elastic, the matrix is isotropic, and the fibers are either isotropic or transversely isotropic. The fibers are axisymmetric, identical in shape and size, and can be characterized by an aspect ratio `=d. The fibers and matrix are well bonded at their interface, and remain that way during deformation. Thus, we do not consider interfacial slip, fiber-matrix debonding or matrix microcracking. Section 2 presents some important preliminary concepts, emphasizing strain-concentration tensors and their relationship to composite stiffness. Section 3 then reviews the various theories. Section 4 compares and evaluates the available models. We use finite element computations of periodic arrays of short fibers to provide reference propert ies, since it has not proved possible to create physical specimens with perfectly aligned fibers. A subsequent paper 1 will compare model predictions to experiments on well-characterized composites with misaligned fibers. 2 Preliminaries 2.1 Notation Vectors will be denoted by lower-case Roman letters, second-order tensors by lower-case Greek letters, and fourth-order tensors by capital Roman letters. Whenever possible, vectors and tensors are written as boldface characters; indicial notation is used where necessary. A subscript or superscript f indicates a quantity associated with the fibers, and m denotes a matrix quantity. Thus, the fibers have Young s modulus E f and Poisson ratio f, while the corresponding matrix properties are E m and m. 1

3 The symbol I represents the fourth-order unit tensor. C and S denote the stiffness and compliance tensors, respectively, and and " are the total stress and infinitesimal strain tensors. Hence, the constitutive equations for the fiber and matrix materials are 2.2 Average Stress and Strain f = C f " f (1) m = C m " m (2) Let x denote the position vector. When a composite material is loaded, the pointwise stress field (x) and the corresponding strain field "(x) will be non-uniform on the microscale. The solution of these non-uniform fields is a formidable problem. However, many useful results can be obtained in terms of the average stress and strain 2. We now define these averages. Consider a representative averaging volume V. Choose V large enough to contain many fibers, but small compared to any length scale over which the average loading or deformation of the composite varies. The volume-average stress is defined as the average of the pointwise stress (x) over the volume V : 1 Z (x)dv (3) V V The average strain " is defined similarly. It is also convenient to define volume-average stresses and strains for the fiber and matrix phases. To obtain these, first partition the averaging volume V into the volume occupied by the fibers V f and the volume occupied by the matrix V m. We consider only two-phase composites, so that V = V f + V m (4) The fiber and matrix volume fractions are simply v f = V f =V and v m = V m =V and, since only fibers and matrix are present, v m + v f = 1. The average fiber and matrix stresses are the averages over the corresponding volumes, f 1 V f Z Vf (x)dv and m 1 V m Z (x)dv (5) Vm The average strains for the fiber and matrix are defined similarly. The relationships between the fiber and matrix averages and the overall averages can be derived from the preceding definitions; they are = v f f + v m m (6) " = v f " f + v m " m (7) An important related result is the average strain theorem. Let the averaging volume V be subjected to surface displacements u 0 (x) consistent with a uniform strain " 0. Then the average strain within the region is " = " 0 (8) This theorem is proved 2 by substituting the definition of the strain tensor " in terms of the dis- 2

4 placement vector u into the definition of average strain ", and applying Gauss s theorem. The result is " ij = 1 Z u 0 i V n j + n i u 0 j ds (9) S where S denotes the surface of V and n is a unit vector normal to ds. The average strain within a volume V is completely determined by the displacements on the surface of the volume, so displacements consistent with a uniform strain must produce the identical value of average strain. A corollary of this principle is that, if we define a perturbation strain " C (x) as the difference between the local strain and the average, then the volume-average of " C (x) must equal zero: " C (x) "(x)? " (10) " C = 1 V Z V " C (x)dv = 0 (11) The corresponding theorem for average stress also holds. Thus, if surface tractions consistent with a 0 are exerted on S then the average stress is = 0 (12) 2.3 Average Properties and Strain Concentration The goal of micromechanics models is to predict the average elastic properties of the composite, but even these need careful definition. Here we follow the direct approach 3. Subject the representative volume V to surface displacements consistent with a uniform strain " 0 ; the average stiffness of the composite is the tensor C that maps this uniform strain to the average stress. Using eqn (8) we have = C" (13) The average compliance S is defined in the same way, applying tractions consistent with a uniform stress 0 on the surface of the averaging volume. Then, using eqn (12), " = S (14) It should be clear that S = C?1. Other authors define the average stiffness and compliance through the integral of the strain energy over V ; this is equivalent to the direct approach 2,4. An important related concept, first introduced by Hill 2, is the idea of strain- and stress-concentration tensors A and B. These are essentially the ratios between the average fiber strain (or stress) and the corresponding average in the composite. More precisely, " f = A" (15) f = B (16) 3

5 A and B are fourth-order tensors and, in general, they must be found from a solution of the microscopic stress or strain fields. Different micromechanics models provide different ways to approximate A or B. Note that A and B have both the minor symmetries of a stiffness or compliance tensor, but lack the major symmetry. That is, but in general, A ijkl = A jikl = A ijlk (17) A ijkl 6= A klij (18) For later use it will be convenient to have an alternate strain concentration tensor ^A that relates the average fiber strain to the average matrix strain, This is related to A by A = ^A " f = ^A" m (19) h (1? v f )I + v f ^Ai?1 (20) so the two forms are easily interchanged. Using equations now in hand, one can express the average composite stiffness in terms of the strain-concentration tensor A and the fiber and matrix properties 2. Combining eqns (1), (2), (6), (7), (13), and (15), one obtains The dual equation for the compliance is C = C m + v f C f? C m A (21) S = S m + v f S f? S m B (22) Equations (21) and (22) are not independent, since S = [C]?1. Hence, the strain-concentration tensor A and the stress-concentration tensor B are not independent either. The choice of which one to use in any instance is a matter of convenience. To illustrate the use of the strain-concentration and stress-concentration tensors, we note that the Voigt average corresponds to the assumption that the fiber and the matrix both experience the same, uniform strain. Then " f = ", A = I, and from eqn (21) the composite modulus is C Voigt = C m + v f C f? C m (23) = v f C f + v m C m Recall that the Voigt average is an upper bound on the composite modulus. The Reuss average assumes that the fiber and matrix both experience the same, uniform stress. This means that the stress-concentration tensor B equals the unit tensor I, and from (22) the compliance is 4

6 ε T ε C (x) (a) (b) (c) Figure 1: Eshelby s inclusion problem. Starting from the stress-free state (a), the inclusion undergoes a stress-free transformation strain " T (b). Fitting the inclusion and matrix back together (c) produces the strain state " C (x) in both the inclusion and the matrix. S Reuss = S m + v f S f? S m (24) = v f S f + v m S m This represents a lower bound on the stiffness of the composite. 3 Theories 3.1 Eshelby s Equivalent Inclusion A fundamental result used in several different models is Eshelby s equivalent inclusion 5,6. Eshelby solved for the elastic stress field in and around an ellipsoidal particle in an infinite matrix. By letting the particle be a prolate ellipsoid of revolution, one can use Eshelby s result to model the stress and strain fields around a cylindrical fiber. Eshelby first posed and solved a different problem, that of a homogeneous inclusion (Fig. 1). Consider an infinite solid body with stiffness C m that is initially stress-free. All subsequent strains will be measured from this state. A particular small region of the body will be called the inclusion, and the rest of the body will be called the matrix. Suppose that the inclusion undergoes some type of transformation such that, if it were a separate body, it would acquire a uniform strain " T with no surface traction or stress. " T is called the transformation strain, or the eigenstrain. This strain might be acquired through a phase transformation, or by a combination of a temperature change and a different thermal expansion coefficient in the inclusion. In fact the inclusion is bonded to the matrix, so when the transformation occurs the whole body develops some complicated strain field " C (x) relative to its shape before the transformation. Within the matrix the stress m is simply the stiffness times this strain, 5

7 ε A C m ε T C f σ I (a) (b) Figure 2: Eshelby s equivalent inclusion problem. The inclusion (a) with transformation strain " T has the same stress I and strain as the inhomogeneity (b) when both bodies are subject to a far-field strain " A m (x) = C m " C (x) (25) but within the inclusion the transformation strain does not contribute to the stress, so the inclusion stress is I = C m " C? " T (26) The key result of Eshelby was to show that within an ellipsoidal inclusion the strain " C is uniform, and is related to the transformation strain by " C = E" T (27) E is called Eshelby s tensor, and it depends only on the inclusion aspect ratio and the matrix elastic constants. A detailed derivation and applications are given by Mura 7, and analytical expressions for Eshelby s tensor for an ellipsoid of revolution in an isotropic matrix appear in many papers The strain field " C (x) in the matrix is highly non-uniform 13, but this more complicated part of the solution can often be ignored. The second step in Eshelby s approach is to demonstrate an equivalence between the homogeneous inclusion problem and an inhomogeneous inclusion of the same shape. Consider two infinite bodies of matrix, as shown in Fig. 2. One has a homogeneous inclusion with some transformation strain " T ; the other has an inclusion with a different stiffness C f, but no transformation strain. Subject both bodies to a uniform applied strain " A at infinity. We wish to find the transformation strain " T that gives the two problems the same stress and strain distributions. For the first problem the inclusion stress is just eqn (26) with the applied strain added, I = C m " A + " C? " T (28) 6

8 while the second problem has no " T but a different stiffness, giving a stress of I = C f " A + " C (29) Equating these two expressions gives the transformation strain that makes the two problems equivalent. Using eqn (27) and some rearrangement, the result is? h C m + C f? C m i E " T = C f? C m " A (30) Note that " T is proportional to " A, which makes the stress in the equivalent inhomogeneity proportional to the applied strain. 3.2 Dilute Eshelby Model One can use Eshelby s result to find the stiffness of a composite with ellipsoidal fibers at dilute concentrations. Recall from eqn (21) that to find the stiffness one only has to find the strainconcentration tensor A. To do this, first note that for a dilute composite the average strain is identical to the applied strain, " = " A (31) since this is the strain at infinity. Also, from Eshelby, the fiber strain is uniform, and is given by " f = " A + " C (32) where the right-hand side is evaluated within the fiber. Now write the equivalence between the stresses in the homogeneous and the inhomogeneous inclusions, eqns (28) and (29), C f " A + " C = C m " A + " C? " T (33) then use eqns (27), (31) and (32) to eliminate " T, " A and " C from this equation, giving h I + ES m C f? C mi " f = " (34) Comparing this to eqn (15) shows that the strain-concentration tensor for Eshelby s equivalent inclusion is A Eshelby = hi + ES m C f? C mi?1 This can be used in eqn (21) to predict the moduli of aligned-fiber composites, a result first developed by Russel 14. Calculations using this model to explore the effects of particle aspect ratio on stiffness are presented by Chow 15. While Eshelby s solution treats only ellipsoidal fibers, the fibers in most short-fiber composites are much better approximated as right circular cylinders. The relationship between ellipsoidal and cylindrical particles was considered by Steif and Hoysan 16, who developed a very accurate finite element technique for determining the stiffening effect of a single fiber of given shape. For very short particles, `=d = 4, they found reasonable agreement for E 11 by letting the cylinder and the ellipsoid have the same `=d. The ellipsoidal particle gave a slightly stiffer composite, with the 7 (35)

9 difference between the two results increasing as the modulus ratio E f =E m increased. Henceforth we will use the cylinder aspect ratio in place of the ellipsoid aspect ratio in Eshelby-type models. Because Eshelby s solution only applies to a single particle surrounded by an infinite matrix, A Eshelby is independent of fiber volume fraction and the stiffness predicted by this model increases linearly with fiber volume fraction. Modulus predictions based on eqns (35) and (21) should be accurate only at low volume fractions, say up to v f of 1%. The more difficult problem is to find some way to include interactions between fibers in the model, and so produce accurate results at higher volume fractions. We next consider approaches for doing that. 3.3 Mori-Tanaka Model A family of models for non-dilute composite materials has evolved from a proposal originally made by Mori and Tanaka 17. Benveniste 18 has provided a particularly simple and clear explanation of the Mori-Tanaka approach, which we use here to introduce the approach. We have already introduced the strain-concentration tensor in eqn (15). Suppose that a composite is to be made of a certain type of reinforcing particle, and that, for a single particle in an infinite matrix, we know the dilute strain-concentration tensor A Eshelby, " f = A Eshelby " (36) The Mori-Tanaka assumption is that, when many identical particles are introduced in the composite, the average fiber strain is given by " f = A Eshelby " m (37) That is, within a concentrated composite each particle sees a far-field strain equal to the average strain in the matrix. Using the alternate strain concentrator defined in eqn (19), the Mori-Tanaka assumption can be re-stated as ^A MT = A Eshelby (38) Equation (20) then gives the Mori-Tanaka strain concentrator as A MT = A Eshelby h (1? v f )I + v f A Eshelbyi?1 This is the basic equation for implementing a Mori-Tanaka model. The Mori-Tanaka approach for modeling composites was first introduced by Wakashima, Otsuka and Umekawa 19 for modeling thermal expansions of composites with aligned ellipsoidal inclusions. (Mori and Tanaka s paper 17 treats only the homogeneous inclusion problem, and says nothing about composites). Mori-Tanaka predictions for the longitudinal modulus of a short-fiber composite were first developed by Taya and Mura 8 and Taya and Chou 9, whose work also included the effects of cracks and of a second type of reinforcement. Weng 20 generalized their method, and Tandon and Weng 11 used the Mori-Tanaka approach to develop equations for the complete set of elastic constants of a short-fiber composite. Tandon and Weng s equations for the plane-strain bulk modulus k 23 and the major Poisson ratio 12 must be solved iteratively. However, this iteration can be avoided by using an alterate formula for 12 ; details are given in Appendix A. The usual development of the Mori-Tanaka model 8,9,11 differs somewhat from Benveniste s (39) 8

10 explanation. For an average applied stress, the reference strain " 0 is defined as the strain in a homogeneous body of matrix at this stress, = C m " 0 (40) Within the composite the average matrix strain differs from the reference strain by some perturbation ~" m, " m = " 0 + ~" m (41) A fiber in the composite will have an additional strain perturbation ~" f, such that " f = " 0 + ~" m + ~" f (42) while the equivalent inclusion will have this strain plus the transformation strain " T. The stress equivalence between the inclusion and the fiber then becomes C f " 0 + ~" m + ~" f = C m " 0 + ~" m + ~" f? " T (43) Compare this to the dilute version, eqn (33), noting that " A in the dilute problem is equivalent to (" 0 + ~" m ) here. The development is completed by assuming that the extra fiber perturbation is related to the transformation strain by Eshelby s tensor, ~" f = E" T (44) Combining this with eqns (41) and (42) reveals that eqn (44) contains the essential Mori-Tanaka assumption: the fiber in a concentrated composite sees the average strain of the matrix. Some other micromechanics models are equivalent to the Mori-Tanaka approach, though this equivalence has not always been recognized. Chow 21 considered Eshelby s inclusion problem and conjectured that in a concentrated composite the inclusion strain would be the sum of two terms: the dilute result given by Eshelby (27) and the average strain in the matrix. (" C ) f = E" T + (" C ) m (45) This can be combined with the definition of the average strain from eqn (7) to relate the inclusion strain (" C ) f to the transformation strain " T : (" C ) f = (1? v f )E" T (46) Chow then extended this result to an inhomogeneity following the usual arguments, eqns (28) to (35). This produces a strain-concentration tensor A Chow = hi + (1? v f )ES m C f? C mi?1 which is equivalent to the Mori-Tanaka result (39). Chow was apparently unaware of the connection between his approach and the Mori-Tanaka scheme, but he seems to have been the first to apply the Mori-Tanaka approach to predict the stiffness of short-fiber composites. A more recent development is the equivalent poly-inclusion model of Ferrari 22. Rather than (47) 9

11 use the strain-concentration tensor A, Ferrari used an effective Eshelby tensor ^E, defined as the tensor that relates inclusion strain to transformation strain at finite volume fraction: (" C ) f = ^E" T (48) Once ^E has been defined, it is straightforward to derive a strain-concentration tensor A and a composite modulus. Ferrari considered admissible forms for ^E, given the requirements that ^E must (a) produce a symmetric stiffness tensor C, (b) approach Eshelby s tensor E as volume fraction approaches zero, and (c) give a composite stiffness that is independent of the matrix stiffness as volume fraction approaches unity. He proposed a simple form that satisfies these criteria, ^E = (1? v f )E (49) The combination of eqns (48) and (49) is identical to Chow s assumption (46) and, for aligned fibers of uniform length, Ferrari s equivalent poly-inclusion model, Chow s model, and the Mori- Tanaka model are identical. Important differences between the equivalent poly-inclusion model and the Mori-Tanaka model arise when the fibers are misoriented or have different lengths, a topic that will be addressed in a subsequent paper Self-Consistent Models A second approach to account for finite fiber volume fraction is the self-consistent method. This approach is generally credited to Hill 23 and Budiansky 24, whose original work focused on spherical particles and continuous, aligned fibers. The application to short-fiber composites was developed by Laws and McLaughlin 25 and by Chou, Nomura and Taya 26. In the self-consistent scheme one finds the properties of a composite in which a single particle is embedded in an infinite matrix that has the average properties of the composite. For this reason, self-consistent models are also called embedding models. Again building on Eshelby s result for a ellipsoidal particle, we can create a self-consistent version of eqn (35) by replacing the matrix stiffness and compliance tensors by the corresponding properties of the composite. This gives the self-consistent strain-concentration tensor as A SC = h?1 I + ES C f? Ci (50) Of course the properties C and S of the embedding matrix are initially unknown. When the reinforcing particle is a sphere or an infinite cylinder, the equations can be manipulated algebraically to find explicit expressions for the overall properties 23,24. For short fibers this has not proved possible, but numerical solutions are easily obtained by an iterative scheme. One starts with an initial guess at the composite properties, evaluates E and then A SC from eqn (50), and substitutes the result into eqn (21) to get an improved value for the composite stiffness. The procedure is repeated using this new value, and the iterations continue until the results for C converge. An additional, but less obvious, change is that Eshelby s tensor E depends on the matrix properties, which are now transversely isotropic. Expressions for Eshelby s tensor for an ellipsoid of revolution in a transversely isotropic matrix are given by Chou, Nomura and Taya 27 and by Lin 10

12 and Mura 28. With these expressions in hand one can use eqn (50) together with (21) to find the stiffness of the composite. This is the self-consistent approach used for short-fiber composites 25,26. A closely-related approach, called the generalized self-consistent model, also uses an embedding approach. However, in these models the embedded object comprises both fiber and matrix material. When the composite has spherical reinforcing particles, the embedded object is a sphere of the reinforcement encased in a concentric spherical shell of matrix; this is in turn surrounded by an infinite body with the average composite properties. The generalized self-consistent model is sometimes referred to as a double embedding approach. For continuous fibers the embedded object is a cylindrical fiber surrounded by a cylindrical shell of matrix. The first generalized selfconsistent models were developed for spherical particles by Kerner 29, and for cylindrical fibers by Hermans 30. Both of these papers contain an error, which is discussed and corrected by Christensen and Lo 31. While the generalized self-consistent model is widely regarded as superior to the original self-consistent approach, no such model has been developed for short fibers. 3.5 Bounding Models A rather different approach to modeling stiffness is based on finding upper and lower bounds for the composite moduli. All bounding methods are based on assuming an approximate field for either the stress or the strain in the composite. The unknown field is then found through a variational principle, by minimizing or maximizing some functional of the stress and strain. The resulting composite stiffness is not exact, but it can be guaranteed to be either greater than or less than the actual stiffness, depending on the variational principle. This rigorous bounding property is the attraction of bounding methods. Historically, the Voigt and Reuss averages were the first models to be recognized as providing rigorous upper and lower bounds 32. To derive the Voigt model, eqn (23), one assumes that the fiber and matrix have the same uniform strain, and then minimizes the potential energy. Since the potential energy will have an absolute minimum when the entire composite is in equilibrium, the potential energy under the uniform strain assumption must be greater than or equal to the exact result, and the calculated stiffness will be an upper bound on the actual stiffness. The Reuss model, eqn (24), is derived by assuming that the fiber and matrix have the same uniform stress, and then maximizing the complementary energy. Since the complementary energy must be maximum at equilibrium, the model provides a lower bound on the composite stiffness. Detailed derivations of these bounds are provided by Wu and McCullough 33. The Voigt and Reuss bounds provide isotropic results (provided the fiber and matrix are themselves isotropic), when in fact we expect aligned-fiber composites to be highly anisotropic. More importantly, when the fiber and matrix have substantially different stiffnesses then the Voigt and Reuss bounds are quite far apart, and provide little useful information about the actual composite stiffness. This latter point motivated Hashin and Shtrikman to develop a way to construct tighter bounds. Hashin and Shtrikman developed an alternate variational principle for heterogeneous materials 34,35. Their method introduces a reference material, and bases the subsequent development on the differences between this reference material and the actual composite. Rather than requiring two variational principles, like the Voigt and Reuss bounds, their single variational principle gives both the upper and lower bounds by making appropriate choices of the reference material. For an upper 11

13 bound the reference material must be as stiff or stiffer than any phase in the composite (fiber or matrix), and for a lower bound the reference material must have a stiffness less than or equal to any phase. In most composites the fiber is stiffer than the matrix, so choosing the fiber as the reference material gives an upper bound and choosing the matrix as the reference material gives a lower bound. If the matrix is stiffer than the fiber, the bounds are reversed. The resulting bounds are tighter than the Voigt and Reuss bounds, which can be obtained from the Hashin-Shtrikman theory by giving the reference material infinite or zero stiffness, respectively. Hashin and Shtrikman s original bounds 35 apply to isotropic composites with isotropic constituents. Frequently the bounds are regarded as applying to composites with spherical particles, though a fiber composite with 3-D random fiber orientation must also obey the bounds. Walpole re-derived the Hashin-Shtrikman bounds using classical energy principles 36, and extended them to anisotropic materials 37. Walpole also derived results for infinitely long fibers and infinitely thin disks in both aligned and 3-D random orientations 38. The Hashin-Shtrikman-Walpole bounds were extended to short-fiber composites by Willis 39 and by Wu and McCullough 33. These workers introduced a two-point correlation function into the bounding scheme, allowing aligned ellipsoidal particles to be treated. Based on these extensions, explicit formulae for aligned ellipsoids were developed by Weng 40 and by Eduljee et al. 41,42. The general bounding formula, shown here in the format developed by Weng, gives the composite stiffness C as C = hv f C f Q f + v m C m Q mi h v f Q f + v m Q mi?1 (51) where the tensors Q f and Q m are defined as Q f = h I + E 0 S 0 (C f? C )i 0?1 h and Q m = I + E 0 S 0 (C m? C )i 0?1 (52) Here E 0 is Eshelby s tensor associated with the properties of the reference material, which has stiffness C 0 and compliance S 0. When the matrix is chosen as the reference material, eqn (51) gives a strain concentrator of ^A lower = h I + E m S m (C f? C m )i?1 (53) This result is labeled here as the lower bound, on the presumption that the fiber is stiffer than the matrix. The composite stiffness is found by substituting ^A lower into eqns (20) and (21). Eduljee and McCullough 41,42 argue that the lower bound provides the most accurate estimate of composite properties, and recommend it as a model. Note that this lower bound prediction is identical to the Mori-Tanaka model, eqn (39) 20,40. This correspondence lends theoretical support to the Mori- Tanaka approach, and guarantees that it will always obey the bounds. The other bound, found by using eqn (51) with the fiber as the reference material, has a strain concentrator of h i ^A upper = I + E f S f (C m? C f ) (54) Note that the Eshelby tensor E f is now computed for inclusions of matrix material surrounded by the fiber material. Equation (54) is labeled as the upper bound, presuming that the fiber is stiffer than the matrix. An identical result can be obtained from the Mori-Tanaka theory by assuming 12

14 that ellipsoidal particles of the matrix material are embedded in a continuous phase of the fiber material. If the matrix is stiffer than the fibers, then the right-hand sides of eqns (53) and (54) are unchanged but eqn (53) becomes the upper bound and eqn (54) becomes the lower bound. All of the preceding bounding formulae have been given for two-component composites, but the theory readily accommodates multiple reinforcements. At fiber volume fractions close to unity, the matrix stiffness strongly influences the composite stiffness for the lower bound/mori-tanaka models, despite the tiny amount of it that is present. Packing considerations suggest that the only way to approach such high volume fractions is for the fiber phase to become continuous, and Lielens et al. 43 suggest that at very high fiber volume fractions the composite stiffness should be much closer to the upper bound, or equivalently to the Mori-Tanaka prediction using the fiber as the continuous phase. This insight prompted Lielens and co-workers to propose a model that interpolates between the upper and lower bounds, such that the lower bound dominates at low volume fractions and the upper bound dominates at high volume fractions (again presuming the fiber is the stiffer phase). They perform this interpolation on the inverse of the strain-concentration tensor ^A, producing the predictive equation 43 ^A Lielens = n(1? f )[ ^A lower ]?1 + f [ ^A upper ]?1o?1 (55) The interpolating factor f depends on fiber volume fraction, and they propose f = v f + v 2 f 2 (56) This theory reproduces the lower bound and Mori-Tanaka results at low volume fractions, but is said to give improved results at reinforcement volume fractions in the 40 to 60% range. 3.6 Halpin-Tsai Equations The Halpin-Tsai equations 44,45 have long been popular for predicting the properties of short-fiber composites. A detailed review and derivation is provided by Halpin and Kardos 46, from which we summarize the main points. The Halpin-Tsai equations were originally developed with continuous-fiber composites in mind, and were derived from the work of Hermans 30 and Hill 47. Hermans developed the first generalized self-consistent model for a composite with continuous aligned fibers (see Section 3.4). Halpin and Tsai found that three of Hermans equations for stiffness could be expressed in a common form: P = 1 + v f with = (P f =P m )? 1 P m 1? v f (P f =P m ) + 1 (57) Here P represents any one of the composite moduli listed in Table 1, and P f and P m are the corresponding moduli of the fibers and matrix, while is a parameter that depends on the matrix Poisson ratio and on the particular elastic property being considered. Hermans derived expressions for the plane-strain bulk modulus k 23, and for the longitudinal and transverse shear moduli G 12 and G 23. The parameters for these properties are given in Table 1. Note that for an isotropic matrix 13

15 Table 1: Correspondence between Halpin-Tsai equation (57) and generalized self-consistent predictions of Hermans 30 and Kerner 29. After Halpin and Kardos 46. P P f P m Comments k 23 k f k m 1?m?2m 2 1+m plane strain bulk modulus, aligned fibers G 23 G f G m 1+m 3?m?4m 2 transverse shear modulus, aligned fibers G 12 G f G m 1 longitudinal shear modulus, aligned fibers K K f K m 2(1?2m) 1+m G G f G m 7?5m 8?10m bulk modulus, particulates shear modulus, particulates k m = Em. 2(1+m)(1?2m) Hill 47 showed that for a continuous, aligned-fiber composite the remaining stiffness parameters are given by E 11 = v f E f + v m E m?4 4 f? m 1 kf? 5 km = v f f + v m m + 4 f? m 1 kf? 5 km ? v f? v! m k 23 k f k m 1? v f? v! m k 23 k f k m This completes Hermans model for aligned-fiber composites; note that one must know k 23 to find E 11 and 12. We now know that Hermans result for G 23 is incorrect, in that it does not satisfy all of the fiber/matrix continuity conditions 3. It is, however, identical to a lower bound on G 23 derived by Hashin 48. Hermans remaining results are identical to Hashin and Rosen s composite cylinders assemblage model 49, so Hermans k 23, and thus his E 11 and 12, are identical to the self-consistent results of Hill 23. The Halpin-Tsai form (57) can also be used to express equations for particulate composites derived by Kerner 29, who also used a generalized self-consistent model. Table 1 gives the details. Kerner s result for shear modulus G is also known to be incorrect, but reproduces the Hashin- Shtrikman-Walpole lower bound for isotropic composites, while Kerner s result for bulk modulus K is identical to Hashin s composite spheres assemblage model 50. See Christensen and Lo 31 and Hashin 3 for further discussion of Kerner s and Hermans results. To transform these results into convenient forms for continuous-fiber composites, Halpin and Tsai made three additional ad hoc approximations: Equation (57) can be used directly to calculate selected engineering constants, with E 11 or E 22 replacing P. The parameters in Table 1 are insensitive to m, and can be approximated by constant values. The underlined terms in eqns (58) and (59) can be neglected. 14 (58) (59)

16 Table 2: Traditional Halpin-Tsai parameters for short-fiber composites, used in eqn (57). For G 23 see Table 1. P P f P m Comments E 11 E f E m 2(`=d) longitudinal modulus E 22 E f E m 2 transverse modulus G 12 G f G m 1 longitudinal shear modulus 12 Poisson ratio, = v f f + v m m In eqn (58) the underlined term is typically negligible, and dropping it gives the familiar rule of mixtures for E 11 of a continuous-fiber composite. However, dropping the underlined term in eqn (59) and using a rule of mixtures for 12 is not necessarily accurate if the fiber and matrix Poisson ratios differ. Halpin and Tsai argue for this latter approximation on the grounds that laminate stiffnesses are insensitive to 12. In adapting their approach to short-fiber composites, Halpin and Tsai noted that must lie between 0 and 1. If = 0 then eqn (57) reduces to the inverse rule of mixtures 46, while for = 1 the Halpin-Tsai form becomes the rule of mixtures, 1 P = v f P f + v m P m (60) P = v f P f + v m P m (61) Halpin and Tsai suggested that was correlated with the geometry of the reinforcement and, when calculating E 11, it should vary from some small value to infinity as a function of the fiber aspect ratio `=d. By comparing model predictions with available 2-D finite element results, they found that = 2(`=d) gave good predictions for E 11 of short-fiber systems. Also, they suggested that other engineering constants of short-fiber composites were only weakly dependent on fiber aspect ratio, and could be approximated using the continuous-fiber formulae 45. The resulting equations are summarized in Table 2. The early references 44,45 do not mention G 23. When this property is needed the usual approach is to use the value given in Table 1. While the Halpin-Tsai equations have been widely used for isotropic fiber materials, the underlying results of Hermans and Hill apply to transversely isotropic fibers, so the Halpin-Tsai equations can also be used in this case. The Halpin-Tsai equations are known to fit some data very well at low volume fractions, but to under-predict some stiffnesses at high volume fractions. This has prompted some modifications to their model. Hewitt and de Malherbe 51 proposed making a function of v f, and by curve fitting found that = v 10 f (62) gave good agreement with 2-D finite element results for G 12 of continuous fiber composites. Nielsen and Lewis 52,53 focused on the analogy between the stiffness G of a composite and the viscosity of a suspension of rigid particles in a Newtonian fluid, noting that one should find 15

17 = m = G=G m when the reinforcement is rigid (G f =G m! 1) and the matrix is incompressible. They developed an equation in which the stiffness not only matches dilute theory at low volume fractions, but also displays G=G m! 1 as v f approaches a packing limit v fmax. This leads to a modified Halpin-Tsai form P P m = 1 + v f 1? (v f ) v f (63) with retaining its definition from eqn (57). Here the function (v f ) contains the maximum volume fraction v fmax as a parameter. is chosen to give the proper behavior at the upper and lower volume fraction limits, which leads to forms such as (v f ) = 1 + 1? v! fmax vfmax 2 v f (64)!# (v f ) = 1?v f "1? exp (65) v f 1? (v f =v fmax ) The Nielsen and Lewis model improves on the Halpin-Tsai predictions, compared to experimental data for G of particle-reinforced polymers 52 and to finite element calculations for G 12 of continuous-fiber composites 53, using v fmax values from 0.40 to Recently Ingber and Papathanasiou 54 tested the Halpin-Tsai equation and its modifications against boundary element calculations of E 11 for aligned short fibers. They found the Nielsen modification to be better than the original Halpin-Tsai form. Hewitt and de Malherbe s form could be adjusted to fit data for any single `=d, but was not useful for predictions over a range of aspect ratios. These results are discussed further in Section Shear Lag Models Historically, shear lag models were the first micromechanics models for short-fiber composites 55, as well as the first to examine behavior near the ends of broken fibers in a continuous-fiber composite 56,57. Despite some serious theoretical flaws, shear lag models have enjoyed enduring popularity, perhaps due to their algebraic simplicity and their physical appeal. Classical shear lag models only predict the longitudinal modulus E 11, so they do not meet our criterion of predicting a complete set of elastic constants. However, we include them here because of their historical importance and their widespread use. One could obtain a complete stiffness model by using the shear lag prediction for E 11 and some continuous-fiber model (such as Hermans ) for the remaining elastic constants. If the fiber is anisotropic then its axial modulus should be used in the shear lag equations. Following Cox 55, the shear lag analysis focuses on a single fiber of length ` and radius r f, which is encased in a concentric cylindrical shell of matrix having radius R. The fiber is aligned parallel to the z axis, as shown in Fig. 3. Only the axial stress 11 and axial strain " 11 are of interest, and Poisson effects are neglected so that f 11 = E f " f 11. The outer cylindrical surface of the matrix is subjected to displacement boundary conditions consistent with an average axial strain " 11, and one solves for the fiber stress f 11 (z). (More rigorously, f 11 (z) is the average stress over the fiber 16

18 z l r f R Figure 3: Idealized fiber and matrix geometry used in shear lag models. cross-section at z.) Axial equilibrium of the fiber requires that d f 11 dz =? 2 rz r f (66) where rz is the axial shear stress at the fiber surface. The key assumption of shear lag theory is that rz is proportional to the difference in displacement w between the fiber surface and the outer matrix surface: H rz (z) = [w(r; z)? w(r f ; z)] (67) 2r f where H is a constant that depends on matrix properties and fiber volume fraction. Solving eqn (66) for f 11 (z) and applying boundary conditions of zero stress at the fiber ends gives an average fiber stress of " # f 11 = E f " 11 1? tanh(`=2) (68) (`=2) with 2 H = rf 2E f It is convenient to rewrite this as an expression for the average fiber strain, (69) " f 11 = `" 11 (70) where ` is a length-dependent efficiency factor, ` = " 1? tanh(`=2) (`=2) # (71) Note that ` is a scalar analog of the strain-concentration tensor A defined in eqn (15), and (1=) is a characteristic length for stress transfer between the fiber and the matrix. 17

19 Table 3: Values for K R used in eqn (74) for shear lag models. Fiber packing K R Cox p 2= 3 = Composite Cylinders 1 = p Hexagonal =2 3 = Square =4 = Cox 55 found the coefficient H by solving a second idealized problem. The concentric cylinder geometry is maintained, but the outer cylindrical surface of the matrix is held stationary and the inner cylinder, which is now rigid, is subjected to a uniform axial displacement. An elasticity solution for the matrix layer then gives H = 2G m ln(r=r f ) (72) Rosen 56,57 simplified this part of the problem by assuming that the matrix shell was thin compared to the fiber radius, (R? r f ) r f, obtaining H = 2G m (R=r f )? 1 (73) Rosen s approximation gives an error in H of 10% at v f = 0:60, with much larger errors at lower volume fractions, and we will not consider it further. It remains to choose the radius R of the matrix cylinder, and the exact choice is important. Several choices have been used, all of which can be written in the form s R KR = (74) r f v f where K R is a constant that depends on the assumption used to find R. Table 3 summarizes the choices for K R. Cox 55 assumed a hexagonal packing, and chose R as the distance between centers of nearest-neighbor fibers (Fig. 4a). It seems more realistic to let R equal half of the distance between nearest neighbors (Fig. 4b), a choice labeled hexagonal in Table 3. Rosen 56,57, and later Carman and Reifsnider 58, chose r 2 f =R2 = v f so that the concentric cylinder model in Fig. 3 would have the same fiber volume fraction as the composite. This is the same R as the composite cylinders model of Hashin and Rosen 49. More recently, Robinson and Robinson 59,60 assumed a square array of fibers, and chose R as half the distance between centers of nearest neighbors (Fig. 4c) 61. Each of these choices gives a somewhat different dependence of ` on fiber volume fraction, with larger values of K R producing lower values of E 11. Shear lag models are usually completed by combining the average fiber stress in eqn (68) with an average matrix stress to produce a modified rule of mixtures for the axial modulus: E 11 = `v f E f + (1? v f )E m (75) 18

20 R R R (a) (b) (c) Figure 4: Fiber packing arrangements used to find R in shear lag models. (a) Cox. (b) Hexagonal. (c) Square. However, the matrix stress in this formula is not consistent with the basic concepts of average stress and average strain. Note that eqn (7) must hold for " 11, as for any other component of strain. Combining this with eqn (70) to find the average matrix strain, and following through to find the composite stiffness (with Poisson effects neglected), gives a result that is consistent with both the assumptions of shear lag theory and the basic concepts of average stress and strain: E 11 = `v f E f + (1? `v f )E m (76) = E m + v f (E f? E m )` This equation is an exact scalar analog of the general tensorial stiffness formula, eqn (21). For the cases in this paper, the difference between eqns (75) and (76) is small, and we will use the classical shear lag result (75) when testing the models. A model by Fukuda and Kawata 62 for the axial stiffness of aligned short-fiber composites is closely related to shear lag theory. They begin with a 2-D elasticity solution for the shear stress around a single slender fiber in an infinite matrix. The usual shear lag relation, eqn (66), is used to transform this into an equation for the fiber stress distribution, which is then approximated by a Fourier series. The coefficients of a truncated series are evaluated analytically using Galerkin s method. This is a dilute theory, in which modulus varies linearly with fiber volume fraction. Like any shear lag theory, Fukuda and Kawata s theory predicts that E 11 approaches the rule of mixtures result as the fiber aspect ratio approaches infinity. But for short fibers Fukuda and Kawata s theory gives much lower E 11 values than shear lag theory. In Fukuda and Kawata s theory, the ratio of fiber strain to matrix strain is governed by the parameter (`=d)(e m =E f ). In contrast, q for shear lag theory, eqn (71), the governing parameter is `=2, which is proportional to (`=d) E m =E f. Thus, for high modulus ratio and low aspect ratio, Fukuda and Kawata s theory tends to underpredict E 11. For this reason we do not pursue their theory further. 19

21 Model Table 4: Models selected for comparison. Comments Halpin-Tsai eqn (57) and Table 2 Nielsen eqns (63), (64), (57b) and Table 2 Mori-Tanaka eqns (39), (35), and (21) Lielens eqns (55), (56), (53), (54), (20), and (21) Self-Consistent eqns (50) and (21) Shear Lag eqns (75), (71), (69), (72), (74), and Table 3 4 Tests and Comparisons Obtaining reference data for unidirectional short-fiber composites presents a problem. Accurate experimental data is not available, since it has not proved possible to produce physical samples with perfectly aligned fibers. The best that can be done experimentally is to make samples with partially aligned fibers, though even in those samples the fibers may be clustered or bundled together in some unspecified way 42. Any comparison between the properties of such samples and predictions necessarily includes both the model for aligned-fiber composites and the model for fiber orientation effects. In this paper we avoid this complication by using three-dimensional finite element models of aligned short-fiber composites, rather than experimental results, as the reference data. This necessitates the assumption of a periodic arrangement of the fibers, but all of the micromechanics models are sufficiently vague about the geometric arrangement of the fibers that they admit periodic geometries. We also compare the theories to some boundary element results for random arrays of aligned fibers 54. For clarity we limit our comparisons to the models listed in Table 4. For the shear lag model we show results only for the square array, noting that this choice for R gives the highest stiffness. The models which are not shown are: the dilute Eshelby model, which is limited to small volume fractions; the Hashin-Shtrikman-Walpole lower bound, which is identical to the Mori-Tanaka model; and the upper bound, which is not claimed to be useful by itself. 4.1 Finite Element Modeling Using the finite element method we analyzed two types of periodic, three-dimensional arrays of fibers, which we call regular and staggered arrays. The representative volume elements (RVE s) are shown in Fig. 5. The unit cell dimensions were chosen with b = p a, where is a constant. We used both = 1 to obtain square packing, and = 3 which gives hexagonal packing. For the regular fibers the distance between neighboring fiber ends (equal to 2c? ` in Fig. 5a) was set to 0:538` for square packing and 0:136` for hexagonal packing. For the staggered arrays the distance along each fiber that is overlapped by neighboring fibers was set at a fixed percentage of the fiber length: 65% for square packing and 76% for hexagonal packing. These conditions, together with the fiber diameter and volume fraction, suffice to determine the dimensions a, b and c for each 20

22 Table 5: Material properties used in finite element calculations. Property Fiber Matrix E v f 0.20 `=d 1, 2, 4, 8, 16, 24, 48 RVE. Note that a new RVE and its corresponding 3-D mesh are generated for each fiber aspect ratio. Stiffnesses of these RVE s were calculated using ABAQUS 63. Twenty-node isoparametric elements were used, and a sample mesh is shown in Fig. 6. The analysis was geometrically nonlinear but the applied strain was 0.5%, so the results are in the region of linear behavior. For axial or transverse loading, symmetry requires all faces of the RVE to remain plane. To determine E 11 and 12 we fixed the normal displacements of the back, left, and bottom faces of the RVE; required the right and top faces to remain plane and parallel to the coordinate axes (using multi-point constraints); and displaced the front face uniformly in the x 1 direction. The tangential displacements on all faces were unconstrained. The average stress was computed from the reaction force in the loading direction, divided by the cross-sectional area of the RVE. Average strains were computed from the initial and deformed dimensions of the RVE. Analogous conditions were use to load the RVE in the x 2 direction to determine E 22 and 23. The longitudinal shear modulus G 12 could in principle be determined using these same RVE s, but that calculation requires a complicated application of periodic boundary conditions and we did not undertake it. All of the micromechanics theories reviewed here predict transversely isotropic properties. Transverse isotropy about the x 1 axis implies that the tensile modulus is the same for any loading direction in the 2 3 plane. This not only requires that E 22 = E 33, but also that 1 G 23 = 2 E 22 ( ) (77) RVE s with hexagonal packing should also be transversely isotropic and obey these same relationships. However, for square packing the properties are only guaranteed to be orthotropic. That is, calculations for square packing will always give E 22 = E 33, but the results will not necessarily obey eqn (77) nor will the transverse modulus necessarily be the same for other loading directions in the 2 3 plane. Here we simply report 23 and E 22 for loading in the x 2 direction, and do not explore the other orthotropic constants for square packing. The material properties used in the finite element calculations (Table 5) are typical of fiberreinforced engineering thermoplastics. All of the moduli are scaled by the matrix modulus. 4.2 Results and Discussion Figure 7 compares the theoretical and finite element results for longitudinal modulus E 11. The strong influence of fiber aspect ratio on E 11 is apparent, and all of the theories exhibit a similar 21

23 3 1 2 a l/2 c a l/2 c b b (a) (b) Figure 5: Representative volume elements used in the finite element calculations. (a) Regular array; the bold lines show the RVE. (b) Staggered array. Figure 6: Example finite element mesh for a staggered, hexagonal array. 22