Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 57

When assessing the validity of undamaged particulate composites, it is particularly valuable to compare predictions using formulae for the relevant effective properties with those obtained from the use of alternative methods. This procedure can provide the confidence for use of the formulae in practical situations, and this approach to validation is now followed. When considering the effective bulk modulus, thermal expansion coefficient and thermal conductivity for two-phase composites having spherical particles of the same size, the results obtained using Maxwell’s methodology are identical to the realistic bounds. They are also identical to estimates for effective properties obtained by applying the composite spheres assemblage model (see the review by Hashin [1]) for a particulate composite to a representative volume element comprising just one particle and a matrix region that is consistent with the volume fractions of the composite. In the case of shear modulus, the result (3.58) obtained using Maxwell’s methodology, corresponds exactly to one of the variational bounds whenever (kp−km)(μp−μm)≥0. For the case (kp−km)(μp−μm)≤0, it can be shown that μmin*≤μm*≤μmax* and a comparison of (3.59) and (3.67) then indicates that the result (3.58) for the effective shear modulus derived using Maxwell’s methodology must lie between the bounds (3.67). In addition, the results (3.55–3.58), for two-phase composites arising from the use of Maxwell’s methodology, are such that κeff→κp, keff→kp, μeff→μp and αeff→αp, respectively, when Vp→1, limits requiring Vp values attained only for a range of particle sizes, as for the composite spheres assembly model.

Effective properties of two-phase composites, derived using Maxwell’s methodology, may be expressed as a mixtures estimate plus a correction term, as seen from (3.55)–(3.58). The correction is always proportional to the product VpVm, and it involves the square of property differences for the case of conductivity, bulk and shear moduli, and the product of differences of the bulk compressibility and expansion coefficient for the case of thermal expansion. These results are the preferred common form for effective properties, having the advantage that conditions governing whether an extreme value is an upper or lower bound are then easily determined. In addition, such conditions determine when both upper and lower bounds coincide with each other, and with predictions based on Maxwell’s methodology, leading to exact nontrivial predictions for all volume fractions. For example, when μp=μm the bounds for bulk modulus given by (3.62) are equal to the exact solution for any values of kp, km and the volume fractions, and they are equal to the result (3.56) indicating that Maxwell’s methodology leads, in this special nontrivial case, to an exact result for all volume fractions for which the composite is isotropic. For the case of thermal expansion, it follows from (3.65) and (3.66) that exact results are also obtained for any values of kp, km, αp, αm and Vp, and they are equal to (3.57) indicating that Maxwell’s methodology again leads, in a special nontrivial case, to an exact result for all volume fractions.

Results for effective properties of two-phase composites, are such that Maxwell’s methodology, the composite spheres assemblage model when it can generate exact results, and the realistic variational bound, all lead to the same result. This suggests very strongly that the realistic bound is a much better estimate of effective properties for spherical particles than the other bound, which can be obtained simply by interchanging particle and matrix properties and the volume fractions. Further evidence of this phenomenon is provided in Figures 3.2–3.4 and Tables 3.1 and 3.2, where use has been made of the conductivity results of Sangani and Acrivos [10], based on the use of spherical harmonic expansions, and the results of Arridge [11] who used harmonics up to 11th order to estimate accurate values of bulk modulus and thermal expansion for body-centred cubic (b.c.c.) and face-centred cubic (f.c.c.) arrays of spherical particles having the same size. Whereas Sangani and Acrivos considered simple cubic, b.c.c. and f.c.c. arrays of spheres, only the f.c.c. results are shown in Figure 3.2, for the values κp/κm=0.01,10,∞ of the phase contrast, as a larger range of volume fractions can be considered. It is seen that there is excellent agreement between predictions based on Maxwell’s result (3.55) and the results of Sangani and Acrivos for a wide range of particulate volume fractions. Results (not shown) indicate that the agreement is less good at large volume fractions of particulate, when comparing Maxwell’s result with the simple-cubic and b.c.c. results of Sangani and Acrivos.

Figure 3.2 Dependence of ratio of effective and matrix thermal conductivities for a two-phase composite on particulate volume fraction for a face-centred cubic array of spherical particles, at various phase contrasts.

Figure 3.4 Dependence of the effective thermal expansion coefficient for a two-phase composite on particulate volume fraction (see Table 3.2 for numerical values).

Table 3.1 Estimates of effective bulk modulus (GPa) for a two-phase particulate composite.

Vp	Maxwell’s Methodology	Arridge [11] (f.c.c.)	Arridge [11] (b.c.c.)	Torquato [13] Lower bound
0	80.56	80.56	80.56	80.56
0.1	88.06	88.01	88.09	88.09
0.2	96.61	96.47	96.67	96.71
0.3	106.42	106.28	106.56	106.66
0.4	117.80	117.76	118.12	118.23
0.5	131.16	131.56	131.90	131.83
0.6	147.07	148.79	148.76	—
0.6802 (max.)	162.20	—	165.39	—
0.7	166.33	171.67	—	—
0.7405 (max.)	175.33	183.54	—	—
0.8	190.12	—	—	—
0.9	220.26	—	—	—

Table 3.2 Estimates of thermal expansion coefficient (×10⁶ K ^–1) of a two-phase particulate composite.

Vp	Maxwell’s Methodology	Arridge [11] (f.c.c.)	Arridge [11] (b.c.c.)	Torquato [13] Upper bound
0	22.5	22.5	22.5	22.5
0.1	20.13	20.1	20.1	20.12
0.2	17.87	17.9	17.9	17.85
0.3	15.73	15.8	15.7	15.69
0.4	13.70	13.7	13.7	13.63
0.5	11.76	11.7	11.7	11.67
0.6	9.91	9.7	9.7	—
0.6802 (max.)	8.49	—	8.1	—
0.7	8.15	7.7	—	—
0.7405 (max.)	7.45	6.9	—	—
0.8	6.46	—	—	—
0.9	4.85	—	—	—

It is worth noting from Bonnecaze and Brady [12, Tables 2–4], who used a multipole method to estimate the conductivity of cubic arrays of spherical particles, that their results for the case that retains only dipole–dipole interactions correspond almost exactly (to three significant figures) with the results shown in Figure 3.2 obtained using Maxwell’s result (3.55). They did not make this comparison, considering only the results of Sangani and Acrivos [10]. This result suggests that Maxwell’s result, which was derived assuming particles do not interact, is in fact valid also for the case when particle interactions are represented by dipole–dipole interactions, and this might explain why Maxwell’s result is found to be a good approximation for a wide range of volume fractions. Further discussion of this issue is beyond the scope of this chapter. We note that for composites used in practice, the difference in the values of the thermomechanical properties (e.g. bulk modulus, thermal expansion coefficient) of the reinforcement and matrix, seldom lead to values of phase contrast that are greater than 10 or so. The phase contrast of the transport properties (such as electrical or thermal conductivity) can be very much greater. It follows that in practical situations, greater confidence may be placed in the Maxwell formulation being accurate at relatively large volume fractions for the thermomechanical properties when compared with the case of transport properties.

The results of Arridge [11] are based on the following properties for silicon carbide spheres in an aluminium matrix:

The corresponding values of bulk and shear moduli are kp=​​​259.68GPa, km=​​​80.56GPa, μp=​​​202.94GPa, μm=​​​26.85GPa, and are such that kp>km and μp>μm. It should be noted that an array of spheres in a b.c.c. or in an f.c.c. arrangement possesses cubic symmetry. The thermal expansion coefficient of such an array is, therefore, isotropic. Arridge’s results are given as mean values implying that the expansion coefficients differ slightly in various directions, a situation that could arise because an insufficient number of harmonics has been included in the representation.

Additional evidence concerning the accuracy of realistic bounds is given by Torquato [13] who considered the effect on bounds of geometrical factors relating to the reinforcement, and developed three-point bounds that are more restrictive than the conventional two-point bounds (equivalent to the Hashin–Shtrikman [6] bounds) for the case of bulk modulus and thermal expansion of suspensions of spheres. The definition, (Torquato [13], Equation (25)), has in fact been replaced by ζ2=(512−316ln3)Vp. Torquato’s [13] three-point bounds are compared in Figures 3.3 and 3.4 with the almost exact results of Arridge [11], and results obtained from Maxwell’s methodology and the two-point variational bounds. From (3.56) and (3.62), bulk modulus results using the Hashin–Shtrikman [6] lower bound and Maxwell’s methodology are identical as μp>μmfor Arridge’s properties. These results are very close to those obtained using the Arridge model for both f.c.c. and b.c.c. particle arrangements, and to the three-point lower bound estimate of Torquato [13]. The results of Arridge are shown for all volume fractions up to the closest packing value for f.c.c. and b.c.c. configurations of spherical particles. The f.c.c. and b.c.c. packing configurations lead to bulk moduli that are very close together for particulate volume fractions in the range 0 < Vp < 0.6. Furthermore, the results obtained using Maxwell’s methodology lie between the f.c.c. and b.c.c. estimates for volume fractions in the range 0 < Vp < 0.4. For a significant range of volume fractions, the Hashin–Shtrikman upper bound is seen in Figure 3.3 to be significantly different to the corresponding lower bound, and to the three-point upper bound of Torquato.

Figure 3.3 Dependence of effective bulk modulus for a two-phase composite on particulate volume fraction (see Table 3.1 for numerical values).

For the case of thermal expansion, the Hashin–Shtrikman [6] upper bound and Maxwell’s methodology result are identical as seen from (3.57) and (3.66), because for Arridge’s properties (kp−km)(μp−μm)(αp−αm)≤0. These results are seen in Figure 3.4 to be very close to those obtained using the Arridge model for both f.c.c. and b.c.c. particle arrangements, and to the three-point upper bound estimate of Torquato. The results of Arridge are again shown for all volume fractions up to the closest packing value for f.c.c. and b.c.c. configurations of spherical particles. The f.c.c. and b.c.c. packing configurations lead to expansion coefficients that are very close together, and very close to results obtained using Maxwell’s methodology, for particulate volume fractions in the range 0 < Vp < 0.5. For a significant range of volume fractions, the Hashin–Shtrikman lower bound is seen in Figure 3.4 to be significantly different to the corresponding upper bound, and to the three-point lower bound of Torquato. In view of the almost exact results of Arridge, and the observation that the three-point bounds for bulk modulus and thermal expansion derived by Torquato are reasonably close, it is deduced that Maxwell’s methodology provides accurate estimates of bulk modulus and thermal expansion coefficient for a wide range of volume fractions.

For the case of a simple cubic array of spherical particles with volume fractions in the range 0 < Vp < 0.4, Cohen and Bergman [14] (see Figure 3.4) have shown that bounds for shear modulus, obtained using a Fourier representation of an integrodifferential equation for the displacement field, are very close to the Hashin–Shtrikman lower bound when using properties for a glass–epoxy composite. The results of this chapter indicate that their bounds will also be very close to the result obtained using Maxwell’s methodology, showing again that its validity is not restricted to low volume fractions, as might be expected from the approximations made.

For the cases of bulk modulus and thermal expansion, Maxwell’s methodology is based on a stress distribution (3.24) in the matrix outside the sphere having radius b of effective medium, which is exact everywhere in the matrix (i.e. b<r<∞) and involves an r-dependence only through terms proportional to r−3. It follows from (3.23) that, for the discrete particle model (see Figure 3.1(a)), the asymptotic form for the stress field in the matrix as r→∞ has the same form as the exact solution for the equivalent effective medium model (see Figure 3.1(b)). The matching of the discrete and effective medium models at large distances, leading to an exact solution in the matrix (b<r<∞) of the effective medium model, is thought to be one reason why estimates for bulk modulus and thermal expansion coefficient of two-phase composites are accurate for a wide range of volume fractions. When estimating thermal conductivity using Maxwell’s methodology, relations (3.8) and (3.9) show that a similar situation arises. The r-dependence of the temperature gradient in the r-direction is through a term again proportional to r−3, and as discussed previously, estimates of thermal conductivity based on Maxwell’s methodology are again accurate for a wide range of volume fractions.

For the case of shear modulus, the exact solution for the stress field (3.44) in the matrix lying outside the sphere having radius b of effective medium (see Figure 3.1(b)) involves terms proportional to r−3 and r−5, but only terms involving r−3 are used when applying Maxwell’s methodology, as seen from (3.41). This means that, in contrast to the cases for the effective bulk modulus, thermal expansion coefficient and thermal conductivity, the resulting estimate for the effective shear modulus does not lead to an exact matrix stress distribution in the region b<r<∞ outside the sphere of effective medium, and consequently estimates for effective shear modulus are likely to be less accurate than those for other effective properties.

The results, discussed previously for various effective properties, are remarkable as one might expect Maxwell’s methodology to be accurate only for sufficiently low volume fractions of reinforcement. The reason is that the methodology involves the examination of the stress, displacement or temperature fields in the matrix at large distances from the cluster of particles, and assumes that the perturbing effect of each particle can be approximated by locating them at the same point. The nature of this approximation is such that interactions between particles are negligible, and it would be expected that resulting effective properties will be accurate only for low volume fractions, as originally suggested by Maxwell [3]. In view of compelling evidence presented in this chapter, based on a wide variety of considerations, a major conclusion is that results for two-phase composites derived using Maxwell’s methodology are not limited to small particulate volume fractions and can be used with confidence using typical volume fractions often encountered in practice.