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3.1 SIMPLE MICROMECHANICAL MODELS
he simplest method of estimating the stiffness of a composite in which all of the fibres are aligned in the direction of the applied load (a unidirectional composite) is to assume that the structure is a simple beam, as in Figure 3.1, in which the two components are perfectly bonded together so that they deform together. We shall ignore the
possibility that the polymer matrix can exhibit time-dependent deformation. The elastic (Young) moduli of the matrix and reinforcement are Em and Ef, respectively. We let the cross-sectional area of the fibre
‘component’ be Af and that of the matrix component be Am. If the length of the beam is L, then we can represent the quantities of the two components in terms of their volume fractions, Vf and Vm, which is more usual, and we know that their sum Vf + Vm = 1. The fibre volume fraction, Vf, is the critical material parameter for most purposes. The subscript ‘c’ refers to the composite.
The load on the composite, Pc, is shared between the two phases, so that Pc = Pf + Pm, and the strain in the two phases is the same as that in the composite, εc = εf= εm (ie. this is an ‘iso-strain’ condition).
Since stress = load/area, we can write:
σcAc = σfAf + σmAm
and from the iso-strain condition, dividing through by the relevant strains, we have:
σ σ σ
= +
ε ε ε
c c f f m m
c f m
A A A
or Ec = Ef Vf + Em(1 - Vf ) ... [3.1]
This equation is referred to as the Voigt estimate, but is more familiarly known as the rule of mixtures.
It makes the implicit assumption that the Poisson ratios of the two components are equal (νf = νm), thus ignoring elastic constraints caused by differential lateral contractions. More sophisticated models have been developed which allow for such effects, the most familiar being that of Hill (1964) which shows that the true stiffness of a unidirectional composite beam would be greater than the prediction of equation 3.1 by an amount which is proportional to the square of the difference in Poisson ratios, (νf – νm)2, but for most practical purposes this difference is so small as to be negligible. For example, for high-performance reinforced plastics, νm ≈ 0.35 and νf ≈ 0.25, and for a fibre volume fraction, Vf, of about 0.6, the correction needed to account for the Poisson constraints is only about 2%. A good indication of the validity of the mixture rule for the longitudinal moduli of two
fibre matrix
longitudinal modulus, Ec
load, Pc
Figure 3.1. Simplified parallel model of a unidirectional composite.
T
0 0.2 0.4 0.6 0.8 1.0
0 100 200 300 400
W/Al-4%Cu glass/epoxy
Young modulus, E c , GPa
Fibre volume fraction, Vf
Figure 3.2. Confirmation of the rule-of-mixtures relationship for the Young moduli, Ec, of undirectional composites consisting
of tungsten wires in Al-4%Cu alloy and glass rods in epoxy resin.
composites of quite different kinds is given in Figure 3.2.
To estimate the transverse modulus, Et, we use a similar approach with a block model such as that shown in Figure 3.3, with the same constraints as before, ie. well-bonded components with similar Poisson ratios, and no
visco-elastic response from the matrix. This is now an ‘iso-stress’ model, so that σc = σf = σm. The total extension of the model is the sum of the extensions of the two components:
εcLc = εf Lf + εm Lm
If the cross-sections of both phases are the same, L ≡ V, so dividing through by the stress (and remembering that Vf+ Vm = 1) we have:
This is referred to as the Reuss estimate, sometimes called the inverse rule of mixtures: the transverse modulus is therefore:
and the relationship between the Voigt and Reuss models can be seen in Figure 3.4. The Reuss estimate is sometimes modified to account for the Poisson effect in the matrix by introducing a ‘constrained’
matrix modulus, effectively by dividing Em by (1 – ν2m) , so that:
The relatively small effect of this correction is also shown in Figure 3.4.
The problem with the Reuss model for the transverse case is that the geometry shown in Figure 3.3 in no way resembles that of a fibre composite perpendicular to the fibres. And in assuming that the Poisson ratios of the phases are the same, it ignores constraints due to strain concentrations in the matrix between the fibres. As a consequence, although it appears to give the correct form of the variation of Et with Vf, the values predicted seldom agree with experimental measurements. The model also implicitly assumes that the transverse stiffness of the fibre is the same as its longitudinal stiffness, and while this is true of isotropic fibres like
fibre matrix
transverse modulus, Et
load, Pc
Figure 3.3. Simple series model of a composite.
0 0.2 0.4 0.6 0.8 1.0
Volume fraction of reinforcement, Vf
Figure 3.4. Predicted variations of the longitudinal elastic modulus, Ec, and the transverse modulus, Et, of composites of
glass and epoxy resin connected in parallel and in series. The solid curves give the Voigt (parallel) and Reuss (series) estimates. The dashed line represents the Reuss estimate modified to include the effect of matrix Poisson constraint.
glass, it is not true of reinforcements with a textile origin like carbon and Kevlar.
The need for a more refined procedure is apparent if we look at even an idealised model of the structure of a composite loaded transverse to the fibres, as shown in Figure 3.5. A model based on hexagonal (close-packing) geometry would clearly give different results from this square-packing model, and both would be different from the true packing geometry of
any real composite (see Figure 3.6, for example) which will almost always tend to be much less regular than either of these idealised geometries. In practice the true transverse modulus lies some way above the lower bound given by the Reuss estimate.
In discussing the behaviour of anisotropic materials, it becomes inconvenient to continue using subscripts like c and t and we resort to using a numerical index system associated with orthogonal axes, x1, x2 , and x3. We choose the x1 direction to coincide with the fibre axis and the x2 direction to coincide with the transverse in-plane direction (assuming that we are dealing, as is usually the case, with a thin plate of material). The longitudinal stiffness of a unidirectional composite is then referred to as E1 and the transverse stiffness is E2. Since in most practical reinforced plastics composites, Ef » Em the Reuss and Voigt estimates of stiffness given in equations 3.1 and 3.2 can be usefully approximated (in the new notation) by the relationships:
E1 ≈ Ef Vf... [3.4]a E2 ≈ Em (1 – Vf )-1... [3.4]b from which it is clear that the stiffness in the fibre direction is dominated by the fibre modulus, while that in the transverse direction is dominated by the matrix modulus.
The Poisson ratio, ν, of an isotropic material is defined as the (negative) ratio of the lateral strain, ε2, when a stress is applied in the longitudinal (x1) direction, divided by the longitudinal strain, ε1,
ie. ν = –ε2/ε1. Consideration of equations 3.1 and 3.2 shows that in a unidirectional composite lamina there will be two in-plane Poisson ratios, not one as in isotropic materials, and it is convenient to label these ν12, called the major Poisson ratio (relating to the lateral strain, ε2, when a stress is applied in the longitudinal (x1) direction) and ν21, the minor Poisson ratio (relating to the strain in the x1 direction when a stress is applied in the x2 direction). The two are not the same, since it is obvious that ν12 must be much larger than ν21. By means of arguments similar to those above for the determination of E1, it can be shown that for a stress applied in the x1 direction only, the major Poisson ratio is given by:
ν12 = νf Vf +νm(1 - Vf)... [3.5]
ie. it also obeys the rule of mixtures. Further thought will show that the minor Poisson ratio must be related to ν12 by the equation:
ν21 / E2 = ν12 / E1... [3.6]
Figure 3.5. Idealised square packing geometry in a transverse section of a unidirectional composite.
Figure 3.6. Real packing geometry: the cross-section of a unidirectional SiC/CAS laminate of Vf
≈ 0. 4. The magnification can be judged from the fibre diameter, 15μm.
Values of the fibre and matrix Poisson ratios rarely differ by a great deal, so that neither matrix nor fibre characteristics dominate these two elastic constants.
When a unidirectional fibre composite is loaded by an in-plane (x1x2) shear force, it distorts to a parallelogram, as shown in Figure 3.7. The shear stress in the fibre direction, τ
12, is matched by its complementary shear stress, τ21. The simplest model assumes that the fibres and matrix carry the same stress:
τ12 = G12 γ12 = Gfγf = Gmγm
and the (apparent) in-plane shear modulus, G12, is given by:
= f + − f
12 f m
1 V (1 V )
G G G
... [3.7]
By analogy with equation 3.2 for the transverse Young's modulus, it can be seen that the matrix shear stiffness again dominates the composite shear modulus unless the fibre volume fraction is very large.
3.2 THE HALPIN-TSAI EQUATIONS
The models in the last section are simple models, although, if treated with care, they can give useful approximations to the behaviour of many composites. There have been many more formal treatments, however, based on more realistic models of the transverse fibre distribution, which yield results of varying degrees of complexity. These more rigorous approaches may give predictions of elastic properties that are closer to experimentally observed values than the simple models of section 3.1, but they are seldom easy to use in practice. For design purposes it is more useful to have simple and rapid computational procedures for estimating ply properties rather than more exact but intractable solutions.
Convenient interpolation procedures have been developed by Halpin and Tsai (1968; see also Halpin, 1992) who showed that many of the more rigorous mathematical models could be reduced to a group of approximate relationships of the form:
E1 ≈ EfVf + Em(1 - Vf) ... [3.8]
ν12 ≈ νfVf + νm(1 - Vf)... [3.9]
2
1 1
= + ζη
− η
f m f
( V )
E E
( V )
... [3.10]
12
1 1
= + ζη
− η ff m
( V )
G G
( V ) ... [3.11]
The first two of these equations are the same rules of mixtures that we have already discussed. In the second two equations, ζ is a factor, specific to a given material, that is determined by the shape and distribution of the reinforcement (ie. whether they are fibres, plates, particles, etc. and what kind of packing geometry), and by the geometry of loading. The parameter η is a function of the ratio of the relevant fibre and matrix moduli (Ef/Em in equation 3.9 and Gf/Gm in equation 3.10) and of the reinforcement factor ζ, thus:
Figure 3.7. Definition of a shear relative to the x1 x2 Cartesian axes.
⎛ 1⎞
The parameter ζ is the only unknown, and values must be obtained empirically for a given composite material, although they are also sometimes derived by a circular argument involving comparison of equations 3.9 and 3.10 with one of the exact numerical solutions mentioned previously. ζ may vary from zero to infinity, and the Reuss and Voigt models are actually special cases of equation 3.9 for ζ = 0 and ζ
= ∞, respectively. A number of analyses have been carried out to compare the predictions of equations 3.9 and 3.10 with elasticity-theory calculations, often with a great degree of success, and it is frequently quoted from the early work of Halpin and Tsai that for practical materials reasonable values of ζ are 1 for predictions of G12 and 2 for calculations of E22. It is dangerous, however, to accept these values uncritically for any given composite. This can be illustrated by taking a specific example.
Sensitive measurements have been made of the elastic properties of a sample of unidirectional carbon-fibre-reinforced plastic (CFRP) consisting of Toray T300 carbon fibres in Ciba-Geigy 914 epoxide resin with a fibre volume fraction of 0.56 (Pierron & Vautrin, 1994). Measurements were also made of the properties of the unreinforced matrix resin. Table 3.1 gives the properties of the plain matrix and fibres.
Table 3.1. Properties of T300 carbon fibres and 914 epoxy resin.
Property Fibres Matrix
Young's modulus, E, GPa 220 3.3 Shear modulus, G, GPa 25 1.2
Poisson ratio, ν 0.15 0.37
The experimental values are first compared in Table 3.2 with the predictions of the simple micromechanical models of section 3.1. It can be seen that the fibre-dominated modulus, E1, is well predicted by the simple rule of mixtures, while the other properties, E12, G12 and ν12, are less well predicted. It is to some extent accidental that the prediction of E2 in Table 3.2 is as good as it is. The very rigid carbon-carbon bonds in carbon fibres are well-aligned with the fibre length, but in the transverse direction, the fibre is far less rigid, ie. it is highly anisotropic, with an anisotropy ratio (the ratio of the axial to transverse moduli) of about 10. Inserting a more appropriate value of Ef (which is difficult to measure) into equation 3.2 would actually result in a value of E2 for the composite that was even lower than that predicted in Table 3.2.
Table 3.2. Predictions of composite properties by simple micromechanics models
Equation Relationship Predicted values
(moduli in GPa)
Calculations of E2and G12 by the Halpin-Tsai approximate model (equations 3.9 and 3.10) depend on knowing ζ. By solving these equations for a range of ζ we can then see what values of ζ must be used in order to obtain predicted moduli that agree with the experimental values. Taking the data from table 2, but substituting a more appropriate transverse fibre stiffness of about 22GPa for the axial stiffness given in the table we find that the appropriate values of ζ are about 2 for E2 and 2.5 for G12, for this particular material. Use of the usually recommended values of 2 and 1, respectively, would thus be accurate for E2 but would result in a 25% underestimate for G12. This emphasises the importance of calibrating the model against experimental data rather than against predictions of other models if the method is to be valid for design purposes.
As a second example, we consider results obtained from ultrasonic pulse-velocity measurements on model composites consisting of 3mm diameter glass rods arranged in regular hexagonal arrays in cast blocks of epoxy resin. The velocity, v, of a compressional pulse travelling through a solid is given by:
v = √(C/ρ)
where C is an elastic constant and ρ is the density of the material. C is not exactly equal to the conventional engineering Young modulus, E, although it has been shown that for ultrasonic rod waves C and E are practically identical for a wide range of GRP materials (Harris and Phillips, 1972). Figure 3.8 shows some experimental values for the longitudinal and transverse moduli of the model composites, determined by this method, as a function of
volume fraction of the glass rods (Shiel, 1995). The E1 values are well represented by the rule-of-mixtures line joining the experimental moduli of the plain resin and the glass rods, but the E2 values lie well above the line representing the Reuss estimate (equation 3.2) and significantly above the predictions of the Halpin-Tsai equation (3.9) with the recommended value of ζ of 2. In order to obtain a good fit to the data points, a value of ζ of 5.37 is required.
It is interesting to note that the stiffnesses of model ‘particulate’ composites consisting of 3mm diameter glass beads randomly dispersed in the same resin (Birch, 1995) are almost indistinguishable from the transverse stiffnesses of the rod-reinforced resin composites, as might be expected. Nielsen (1978) has discussed at length the applicability of the Halpin-Tsai equations to particulate composites.
3.3 THE COMPRESSION MODULUS
We might expect that the compression stiffness of a unidirectional composite would be similar to its tensile stiffness, until we remember that the fine-scale structures of certain fibres —particularly those derived from textiles —are known often to result in unpredictable behaviour of these fibres under shear and bending conditions. We might take as an analogy the behaviour of a piece of rope: tensile forces draw the fibres into tighter bundles, but any attempt to squeeze a length of rope axially will force the fibres apart. Piggott and Harris (1980) illustrated the problem by measuring the compression moduli, Ec, of polyester-based composites reinforced with glass, Kevlar-49 and two varieties of carbon fibres. Their results, illustrated in Figure 3.9, show that the variation with Vf is linear up to only about 50 vol% of reinforcement, and that beyond this the rate of increase of stiffness falls or even becomes negative, depending on the fibre. The change in behaviour is evidently related to the diminishing ability of the
0 0.2 0.4 0.6 0.8 1.0
Reinforcement volume fraction, Vf
Figure 3.8. Comparison of experimental data for the elastic moduli of model glass/resin composites with various theoretical
predictions. The upper and lower bounds (rule-of-mixtures and inverse RoM) correspond to equations 3.1 and 3.2. The Halpin-Tsai curve for ζ = 2 represents the commonly used function for the transverse modulus, E2 (equation 3.9) but a value of ζ = 5.37
is necessary to obtain the best fit to the data.
matrix to constrain the fibre bundles to deform axially as Vm falls. As the inset figure plotting the initial slopes of these curves against the fibre tensile modulus shows, the rule of mixtures is not properly obeyed except by the homogeneous and isotropic glass fibre. The slopes, dEc/dVf, of the curves for the two carbon-fibre composites fall slightly below the 1:1 rule-of-mixtures values, while that for the aramid fibre composite falls well below the RoM.
These results confirm what we know of the structures of these reinforcing fibres. Some recent elegant experiments by Young et al
These results confirm what we know of the structures of these reinforcing fibres. Some recent elegant experiments by Young et al