Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions

Arch Appl Mech DOI 1.17/s419-8-246-1 ORIGINAL R. Rodríguez-Ramos R. Guinovart-Díaz J. Bravo-Castillero F. J. Sabina H. Berger S. Kari U. Gabbert Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions Received: 31 March 28 / Accepted: 9 May 28 Springer-Verlag 28 Abstract In the present work, unified formulae for the overall elastic bounds for multiphase transversely isotropic composites with different geometrical types of inclusions embedded in a matrix are calculated, including the spherical and long or short continuous cylindrical fiber cases. The influence of the different geometrical configurations of the inclusions on the composites is studied. The transversely isotropic effective bounds are obtained by applying the variational formulation for anisotropic composites developed by Willis, which relies on expressions for the static transversely isotropic Green s function. Some numerical calculations and comparisons with the effective coefficients derived from the self-consistent approach, asymptotic homogenization method, and finite element method (FEM) are shown for different aspect ratio values, exhibiting good agreement. Keywords Particle-reinforced composites Variational bounds Transversely isotropic components Green s function 1 Introduction A classical problem in solid mechanics is the determination of the effective elastic properties of a composite material made up of a statistically isotropic random distribution of isotropic and elastinclusions embedded in a continuous isotropic and elastic matrix. Various models have been developed to compute the two independent elastic constants of the composite,as in the Mori Tanakaestimates [2,25], the self-consistent scheme [16,17], and the differential method [22]. They are based on the mean-field approximation, which assumes that the stress and strain fields in the matrix and in the inclusion are adequately represented by their volume-averaged values, and differ in the way they account for the elastinteraction between the inclusions. In other cases, rigorous bounds for the elastic properties of isotropic two-phase composites were obtained from variational principles. Besides the Voigt Reuss bounds, which are usually very wide, Hashin and Shtrikman [12] derived much tighter bounds by employing variational principles which involve polarization fields. The two-point correlation bounds are the best possible ones given only the volume fraction information. They were improved R. Rodríguez-Ramos R. Guinovart-Díaz J. Bravo-Castillero Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado, CP 14 Habana 4, Cuba F. J. Sabina Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 2-726, Delegación de Alvaro Obregón, 1 Mexico D.F., Mexico H. Berger (B) S. Kari U. Gabbert Institute of Mechanics, Otto-von-Guericke-University of Magdeburg, Universitaetsplatz 2, 3916 Magdeburg, Germany Tel.: +49-391-671246 E-mail: Harald.Berger@Masch-Bau.Uni-Magdeburg.DE

R. Rodríguez-Ramos et al. by three-point bounds [4,24,35], which incorporate information about the phase arrangement through the statistical correlation parameters mentioned above. Derivation of the effective coefficients for anisotropic elastic composites using the asymptotic homogenization method (AHM) and self-consistent methods (SCM) has been studied by Pobedrya [29] and Willis [39], and more recently by Levin et al. [21], Sabina et al. [31], Guinovart-Diaz et al. [1], and Rodriguez-Ramos et al. [3]. The estimation of bounds for the overall properties of anisotropic composites has also been considered by many authors. The main overall elastic moduli of fiber composites with transversely isotropic phases are shown to be connected by simple universal relations which are independent of the geometry at given concentration. Upper and lower bounds obtained in terms of phase properties and concentrations were reported by Hill [15]. On the other hand, the plane-strain bulk modulus and the two shear moduli of multiphase transversely isotropic fiber-reinforced materials of arbitrary transverse phase geometry were bounded in terms of phase moduli and phase volume fractions by Hashin [13]. Moreover, expressions and bounds for the five effective elastic moduli of a unidirectional fiber composite, consisting of transversely isotropic fibers and matrix, were derived on the basis of analogies between isotropic and transversely isotropic elasticity equations by Hashin [14]. Bounds of Hashin Shtrikman type and self-consistent estimates for the overall properties of composites, which may be anisotropic, were developed in Willis [36]. In Nemat-Nasser and Hori [27], the Hashin-Shtrikman variational principle is applied to solids with periodic microstructure, and bounds on the overall moduli are obtained by defining energy functionals for the eigenstrain or eigenstress fields in the equivalent homogeneous solid. Finally, bounds for the overall elastic energy of the unit cell, and hence bounds for the overall elasticity parameters of elastic solids with periodic microstructure, were obtained. Moreover, improvable bounds on overall properties of heterogeneous finite solids were also studied by Nemat-Nasser and Hori [26]. In the present paper, the derivationof the upperand lower boundsfor N-phase elastic transversely isotropic composite materials using the procedure of Willis [4] is obtained. The constituents of the two- or three-phase composite studied here are transversely isotropic materials. Unified formulae for the bounds as a function of the aspect ratio of the inclusions are given. Some numerical results for each elastic coefficient are presented. Section 2 starts with the basic equations of Willis and gives the variational bounds explicitly. In Sect. 3 some numerical calculations of the bounds for different geometrical configurations of the inclusions and comparisons with other models are done. Section 4 provides some final remarks. 2 Variational bounds for the anisotropic elastic multiphase composite The problem of estimating bounds for the mean elastic energy density, W (e), in a given sample of material, when it is subjected to loads that generate within it a mean strain field e, has been addressed by Willis [4]. In that chapter, in particular, for a two-phase composite, i.e., a particle-reinforced matrix with anisotropic elastic constituents, he gave a general explicit formulae (21.38) for the upper and lower bounds of the mean energy, which relies on the evaluation of a certain P function depending on the derivatives of the Green s function G. The Willis formulae are given at the outset as follows: where and { N } 1 L U = c r [I (L o L r ) P] 1 r=1 { N } 1 L L = c r [I + (L r L o ) P] 1 r=1 1 2 el Le W (e) 1 2 el Ue, (1) c r [I (L o L r ) P] 1 L r (2) r=1 c r [I + (L r L o ) P] 1 L r (3) are the upper and lower bounds, respectively; the value r = 1 denotes the matrix and r = 2,...,N correspond to the inclusions; c r are the volume fractions of each phase, respectively, such that c 1 + +c N = 1; L r is r=1

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions the anisotropic elastic tensor for r-phase and L o is the anisotropic elastic tensor of the comparison material. P is related to the integral of the Green s function which is defined as a linear operator as follows (see [4]) ( τ) ij (x) = jkl (x, x )τ kl (x )dx, (4) with jkl (x, x ) = 2 G ij x k x l. (5) (ij)(kl) The parenthesis stand for symmetrization, see Willis [4]. Expressions (2) and(3) give an upper and lower bound, respectively, for any L o such that the quadratic forms e(l o L r )e and e(l r L o )e are positive definite for every r. Now, consider that the inclusions are aligned spheroids of aspect ratio δ.letox 3 be the axis of alignment and let the elastic properties of both the inclusions and the matrix be transversely isotropic with its axis parallel to Ox 3. Then both media are characterized by five parameters and the operations indicated in (2) and(3) can be carried out. It is useful to introduce the symbolic notation described in Appendix A. Examples of this can be found in Sabina et al. [32] and Sabina and Willis [33]. In order to work out (2)and(3) it remains to obtain P in the notation of Appendix A, P = (2P 1, P 2, P 3, P 4, 2P 5, 2P 6 ), (6) where P i (i = 1, 2,..., 6) are given in Appendix B. Using the algebraic rules of Appendix A and after some algebraic manipulations, the bounds for the transversely isotropic elastic composite are obtained. They have the following general form: k L k k U, l L l l U, l L l l U, n L n n U, m L m m U, p L p p U, where the particular expressions for the bounds depend on the number of phases. The analytic expressions for the lower and upper bounds are given as follows: ( k B = 1 N a i 4 i ( l B = 1 N d i 4 i ( l B = 1 N e i 4 i ( n B = 1 N d i 4 i (b i k i e i l i ) + (b i k i e i l i ) + e i b i (a i n i 2d i l i 2 ) + N i (b i l i e in i ) + b i a i ) (a i l i 2d i k i ), ) (a i l i 2d i k i ), 2 ) (b i l i 2 e in i ), i ) (a i n i 2d i l i 2 ), i m B = 2 m i f i ( N ) 1, p B = 2 f i p i g i ( N g i ) 1, (7)

R. Rodríguez-Ramos et al. where a i = 1 + 4(k i k )P 1 + 2(l i l )P 2, b i = 1 + 2(l i l )P 3 + (n i n )P 4, d i = 2(l i l )P 1 + (n i n )P 2, e i = (l i l )P 4 + 2(k i k )P 3, f i = 1 + 4(m i m )P 5, g i = 1 + 4(p i p )P 6, = 1 2 a ib i d i e i, ( = 1 N a i b i 4 2 d i ) e i. (8) B = L, U denotes the lower and upper bounds, respectively, related to the choice of the comparison body L o. In the calculations that follow the comparison moduli of L o is taken as L o = min(l r ) for the lower bound and L o = max(l r ) for the upper bound, r = 1, 2,...,N. It is not difficult to verify that the bounds in Eq. (7) are identical to those found originally by Hashin and Shtrikman [12] and Hashin [13]. 3 Numerical results In this section, the general expressions for the upper and lower bounds (7) are applied to two- and three-phase composites with different types of inclusions (Fig. 1) for the prediction of their effective elastic properties. 3.1 Two-phase composites (N = 2) It is to be noted that the numerical values of l L, l U, l L,andl U are such that l L = l L and l U = l U, although the given expressions look different. These formulae are provided for completeness and can be used as a further check on the final results. It is worthwhile to note that the upper and lower bounds of m are both equal to the Voigt average when m 1 = m 2. Moreover, the upper and lower role is interchanged when m 2 m 1 changes sign, i.e., for m 2 < m 1, (7) is a lower bound, while for m 1 < m 2,(7) is an upper bound. The same considerations can be applied to p. Fig. 1 Different configuration of composites: a short fibers, b spherical inclusions, c long fibers, and d three-phase fibrous composite

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions In the calculations that follow the comparison moduli of L o is taken as that of the matrix L 2 for the lower bound and as that of the inclusion L 1 for the upper bound. The presentation of the results is divided according to different shapes of inclusions: long cylindrical fibers, spherical inclusions, and short cylindrical fibers (see Fig. 1). The inclusions can be considered in general as ellipsoidal inclusions with different aspect ratios of length to width denoted by δ. Long cylindrical fibers The first example involves ellipsoidal inclusions of the piezoelectric material BaTiO 3 embedded in the piezomagnetic medium CoFe 2 O 4. Only the elastic constants are involved in the calculations of this paper, i.e., the bounds relate only to the elastic properties of both materials. The elastic constants (in the Voigt notation) are given in Table 1 and were taken from Aboudi [1]. Figure 2 displays the five stiffnesses k, l, n, m, and p in GPa plotted versus the inclusion volume fraction c 2 for an aspect ratio δ = 1 and δ =.1. Since the inclusion is softer than the matrix, it shows a characteristic pattern of a weakly reinforced composite. The properties are monotone decreasing functions of c 2. Moreover, the upper and lower bounds calculated using (7) are also shown. Although it is not shown in the figure, it is convenient to comment that, for δ = 1, the effective coefficients k, l, n, m, and p were computed using the asymptotic homogenization method (AHM) [1] and the self-consistent effective field method (SCEF) [21]. All curves agree with the lower and upper bounds (7). These results can be compared with other calculations by Aboudi [1] using the Mori Tanaka method, the generalized method of cells, and a numerical implementation of the asymptotic homogenization method, which also show curves lying on top of each other for each stiffness. The comparison is reasonably good except for a slight maximum near c 2 = 1for m in Aboudi s approach (Fig. 2). This may be due to the numerical implementation of the methods for a rectangular shaped inclusion, which differs from the very long ellipsoidal one used here. The bounds do not overlap for δ =.1, except for m and p. Another combination of materials is now considered: inclusions of PZT-5 embedded in a polymer matrix (see the material data in Table 1). The results (7) are also valid for an isotropic matrix such as the epoxy considered here. Figure 3 shows k, l, m,and n (in GPa) versus c 2, respectively, for a long ellipsoidal inclusion [i.e., δ = 1 in (7)]. This time, the upper and lower bounds are wide but not as wide as the Voigt and Reuss bounds (not shown here). All curves are monotone increasing. The self-consistent effective field (SCEF) solution and finite-element method (FEM) [5 9] lie on top of the lower bound. The asymptotic homogenization method (AHM) also lies on top of the lower bound. Therefore, the lower bound can be considered as an effective value for that composite. In order to validate the obtained formulae (7) an ideal material (stiffer) embedded in a BaTiO 3 matrix (softer) is considered. It was chosen so that m 1 = m 2, i.e., a Hill s solid [15]. All the curves for the effective coefficients k, l, n, and p agree with each other, that is to say, the upper and lower bounds, self-consistent effective field (SCEF), asymptotic homogenization (AHM), and Hill s method curves coincide. 3 25 n UB, δ=1 LB, δ=1 UB, δ=.1 LB, δ=.1 2 15 k 1 5 l m p.1.2.3.4.5.6.7.8.9 1 Volume fraction c 2 Fig. 2 Plot of the upper and lower bound (7) of the five effective elastic constants k, l, m, n,and p as a function of the inclusion volume fraction c 2 for an aspect ratio δ = 1 and δ =.1. The materials are inclusion of BaTiO 3 embedded in a matrix of CoFe 2 O 4

R. Rodríguez-Ramos et al. Table 1 Elastic material properties in GPa Material C 11 = k + m C 12 = k m C 13 = l C 33 = n C 44 = p BaTiO 3 166 77 78 162 43 CoFe 2 O 4 286 173 17 269.5 45.3 PZT-5 12.1 7.54 7.52 11.1 2.11 Polymer.386.257.257.386.64 Isotropinclusion 483.68 99.7 99.7 483.68 192.31 Epoxy matrix I 94.23 4.38 4.38 94.23 26.92 PZT-7A 157 85.4 73 175 47.2 Epoxy matrix II 8 4.4 4.4 8 1.8 Yttria coating 191.53 47.88 47.88 191.53 71.83 fiber 7.91 1.59 2.85 691.21 15.17 SiC matrix 536.28 134.7 134.7 536.28 21.1 ATH filler 82.51 26.5 26.5 82.51 28.23 PMMA matrix 4.85 2.18 2.18 4.85 1.34 Effective Coefficient k 1 8 6 4 2 Upper Bound AHM SCEF FEM Lower Bound Effective Coefficient l 8 6 4 2.2.4.6.8 1.2.4.6.8 1 Effective Coefficient m 2.5 2 1.5 1.5 Effective Coefficient n 12 1 8 6 4 2.2.4.6.8 1 Volume fraction fiber.2.4.6.8 1 Volume fraction fiber Fig. 3 Plots of the four effective elastic constants k, l, m,and n versus c 2 for long cylindrical fibers. The materials here are PZT-5 inclusions in an polymer matrix. The upper and lower bound (7) and the results of the self-consistent effective field (SCEF), finite element method (FEM), and asymptotic homogenization methods (AHM) areshown Spherical inclusions As another example, two isotropic materials of Young s moduli E 1 = 3GPa and E 2 = 76 GPa and Poisson s ratios ν 1 =.4 andν 2 =.23 are studied since there are some experimental data available [34] for spherical inclusions (δ = 1). The upper and lower bounds of the axial Young s modulus Ē a (in GPa) and the dimensionless transverse shear modulus m/m 1 computed from (7)usingδ = 1 and finite element method (FEM) [19] are plotted against the inclusion volume fraction c 2 in Fig. 4. The experimental data and FEM results lie between the calculated bounds. One relevant fact is that the FEM results and the experimental data coincide very well. Short cylindrical fibers Figure 5 shows plots the four elastic constants k, m, n, and p (in GPa) versus c 2. The composite is made of isotropinclusions embedded in an epoxy matrix I. The material parameters are in Table 1. The upper and

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions 8 6 UB FEM Exp LB 3 25 2 E a 4 m/m 1 15 1 2 5.2.4.6.8 1 Volume fraction c 2.2.4.6.8 Volume fraction c 2 1 Fig. 4 Plots of the bounds of effective axial Young s modulus and effective property m/m 1 versus c 2 for spherical inclusions. A comparison between the bounds, the FEM results, and the experimental data of Smith for two isotropic materials is shown lower bounds (7) and finite element method (FEM) [2] are shown for aspect ratio δ = 3 (short fibers). In this case, the lower bound can be considered as an effective property for the parameters m and p, respectively. The next example in Fig. 6 considers four elastic constants k, l, n, and p (in GPa) for the materials combination of PZT-7A inclusions in an epoxy matrix II for two fixed values of the inclusion volume fraction c 2 =.3and.8 as a function of the aspect ratio δ. For relatively large values of δ the upper bounds are almost constant for all the elastic constants. The lower bound is likewise almost constant except for n. Near δ =, all curves change very rapidly. 14 25 13 Upper Bound FEM Lower Bound 225 12 2 Effective Coefficient (GPa) 11 1 9 8 7 6 k m Effective Coefficient (GPa) 175 15 125 1 75 n p 5 4 5 3 25 2.1.2.3.4 Volume fraction fiber.1.2.3.4 Volume fraction fiber Fig. 5 Plots of the four effective elastic constants k, m, n,and p versus c 2. The materials here are isotropinclusions in an epoxy matrix. The upper and lower bound (7) and FEM results are shown for aspect ratio δ = 3(short fibers)

l R. Rodríguez-Ramos et al. k 8 6 4 2 UB, c =.8 2 PZT 7A/ LB, cepoxy =.8 2 UB, c 2 =.3 LB, c 2 =.3 5 1 15 2 5 4 3 2 1 5 1 15 2 14 35 12 3 1 25 8 2 n 6 p 15 4 1 2 5 5 1 15 2 Aspect ratio δ 5 1 15 2 Aspect ratio δ Fig. 6 Plots of the upper and lower bounds of effective moduli k, l, n,and p against the aspect ratio δ for two values of c 2 =.3 and.8. The materials are PZT-7A inclusions in an epoxy matrix 3.2 Three-phase composites (N = 3) In the calculations that follow the comparison moduli of L o are taken as L o = min(l 1, L 2 ) for the lower bound and L o = max(l 1, L 2 ) for the upper bound. The presentation of the results is divided according to the different shapes of the inclusions as before, i.e, long cylindrical fibers, spherical inclusions, and short cylindrical fibers (Fig. 1). Long cylindrical fibers As an example of the results obtained for a three-phase composite, in Fig. 7a, b, we display the normalized shear moduli (transversal and longitudinal) for a coated-fiber composite versus the fiber volume fraction c 3. They have been calculated from the analytical expressions for square and hexagonal symmetries reported by Guinovart-Díaz et al. [11] andcompared with the correspondingupperand lower boundsmoduliobtainedfrom (7), for an aspect ratio of δ = 1 and interphase volume fraction of c 2 =.5. The properties of the constituent materials (SiC matrix/yttria coating/carbon fiber) were taken from Benveniste et al. [3](see Table 1). In Fig.7a we observe that the effective shear modulus m/m 1 is decreasing and stays within the bounds when the volume fraction of fiber increases. For small values of c 3, the effective coefficient m/m 1 lies on the upper bound for both symmetries, while for high concentrations of fiber it lies on the lower bound for square symmetry. As illustrated in Fig. 7b, the other effective properties lie within the bounds and are very close to the corresponding upper bound for all values of the fiber volume fraction. The effective properties and the bound coefficients of the coated-fiber composites have been calculated using the present models for other systems of materials that appear in Benveniste et al. [3] and agree quite well with the effective properties calculated by using the Mori and Tanaka approach in Benveniste et al. [3]. Due to the very small thickness of the coating in the systems considered, the effective properties are almost equal to the effective properties obtained in the lower bound and in the uncoated fiber case. The bound formulae are functions of the aspect ratio δ and they are used for the estimations of the effective properties of particulate reinforced composites. Spherical inclusions Figure 8 presents the effective Young s modulus for spherical inclusions. A comparison between the models reported in Nie and Basaran [28] and Ju and Chen [18] with the lower bound (7) for the three-phase composite

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions 1.9.8 Upper Bound AHM Hex AHM Square Lower Bound 1.9.8.7.7.6.6 m / m 1.5 p /p 1.5.4.4.3.3.2.2.1.1.2.4.6.8 1.2.4.6.8 1 Fiber volume fraction c 3 Fiber volume fraction c 3 Fig. 7 Bounds and AHM results for the normalized effective shear moduli (transversal and longitudinal) for a three-phase composite (SiC matrix/yttria coating/carbon fiber) as a function of volume fraction of filler c 3 is shown. The interphase thickness is c 2 =.1 and the fiber volume fraction is c 3 =.48. The composite is made of Poli-metil metacrilato (PMMA) matrix/alumina trihydrate (ATH) filler. The material parameters are listed in Table 1. The effective Young s modulus and the lower bound can be calculated from the well-known expression E = 9km/(3k + m) for an isotropic composite. Both models are very close to the lower bound, which gives a good estimation for this property. The bounds and the effective properties of the bulk and shear normalized properties k/k 1 and µ/µ 1, respectively, for transversely isotropic elastic composite as a function of interphase region elastic modulus q = E 3 /E 2 and ν 2 = ν 3 were studied as well. The theoretical models 12 1 Nie Basaran (25) Ju and Chen (1994) Lower Bound SCS 8 E (GPa) 6 4 2 1 2 3 4 5 6 Interface parameter log(q) Fig. 8 Behavior of the effective Young s modulus for spherical inclusions in a PMMA matrix/ath filler composite as a function of the interphase region elastic modulus q = E 3 /E 2

R. Rodríguez-Ramos et al. for the calculations of the above properties were taken from Nie and Basaran [28] and Ju and Chen [18]. The comparison between these two models and the bounds (not shown) illustrates that both models are consistent with the bounds and are near to the lower bound. The bulk modulus in the model of Ju and Chen for the lower values of the parameter q is not consistent with the bounds (it falls outside of the bounds). This indicates that the model presented by Nie Basaran gives a better approximation of the overall bulk modulus, as was mentioned recently in the work reported by Nie and Basaran [28]. However, the shear modulus in both mentioned models falls within the bounds. A comparison for three-phase spherical inclusions with the numerical results by FEM [19,2]isshownin Table 2. The numerical results show good concordance between the FEM results and the variational bounds. In Fig. 9, the behavior of the axial shear modulus p of the composite as a function of the aspect ratio δ and the volume fraction c 3 is illustrated. The composite has the constituents SiC matrix/yttria coating/carbon inclusion (see Table 1). The volume fraction of the coating is c 2 =.5. The aspect ratio δ influences the behavior of the bounds of the composite. The shaded and the mesh surfaces indicated the lower and upper bound, respectively, for the value of the axial shear effective coefficient p with respect to δ (aspect ratio) and c 3 (volume fraction). Notice that for values of δ 2 (short fibers) the bounds are tighter than for values of δ 2. Table 2 Comparison between effective properties using FEM and variational bounds [lower bound (LB) and upper bound (UB)] in GPa Sphere Interphase C 11 C 12 volume fraction volume fraction LB FEM UB LB FEM UB.6.5 248.61 451.91 455.52 82.51 113.36 113.53.15.65 134.91 366.3 376.32 48.27 92.4 93.35.25.72 84.97 3.34 34.87 3.63 74.73 75.22 C 13 C 14.6.5 66.92 114.99 114.7 482.39 514.41 516.59.15.65 37.77 93.32 94.27 472.79 51.59 517.45.25.72 24.44 76.4 76.4 476.39 56.26 524.61 C 44 C 66.6.5 83.5 167.88 171. 13.65 17.22 173.43.15.65 43.32 134.31 141.48 87.56 14.92 146.53.25.72 27.17 18.74 114.83 62.99 117.86 121.9 p (GPa) 2 18 16 14 12 1 8 6 4 2.2.4 c 3.6.8 2 4 6 δ 8 1 Fig. 9 Surface of the effective axial shear modulus p of the three-phase composite SiC matrix/yttria coating/carbon inclusion as a function of the aspect ratio δ and volume fraction of inner inclusion c 3

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions 4 Conclusions Explicit formulae for the variational upper and lower bounds for transversely isotropic ellipsoidal inclusions embedded in a transversely isotropic matrix are obtained. The inclusions are all aligned in the x 3 direction. The geometrical configuration of the inclusions exerts a significant contribution on the effective properties of the composite and therefore on their bounds. The method used is based on that of Willis [4], who derived a tensor formula for the bounds. Using the symbolic notation and the algebra of Hill [15], it was possible to obtain explicit formulae for the bounds of the effective elastic constants k, l, m, n and p. Several examples for different two- and three-phase composites are shown. Acknowledgments This work was sponsored by CoNACYT project no. 47218-F and by DFG Graduiertenkolleg 828 Micro Macro Interactions in Structured Media and Particle Systems. The provisions of the Basic Sciences Program Project CITMA no. 9/24 and the Department of Basic Science of the Monterrey Institute of Technology, Campus of México State are also acknowledged. The authors gratefully acknowledge the suggestions and comments given by Prof. John Willis. Thanks are due to Ms. Ana Pérez Arteaga and Fernando Rodríguez for computational support. Appendix A: Transversely isotropic symmetry considerations In the case that the fourth-order tensor L of elastic moduli has transversely isotropic symmetry relative, say, to the Ox 3 axis, then, it can be represented symbolically using the notation employed by Hill [16] in the form L = ( 2k, l, l, n, 2m, 2p ), where the appropriate elastic constants 2k, l, l, n,2m,and 2p appear in the constitutive relation between stress σ and strain e, viz.σ = Le. It is written explicitly as follows 1 2 (σ 11 + σ 22 ) = k(e 11 + e 22 ) + le 33, σ 33 = l (e 11 + e 22 ) + ne 33, σ 11 σ 22 = 2m(e 11 e 22 ), σ 12 = 2me 12, σ 13 = 2pe 13, σ 23 = 2pe 23. It is assumed that l = l since the diagonal symmetry is lost, in general, in a triple product of transversely isotropic tensors. Moreover, let L 1 = (2k, l, l, n, 2m, 2p) and L 2 = ( 2k, l, l, n, 2m, 2p ) be two fourthorder transversely isotropic tensors; then the following algebraic rules hold L 1 + L 2 = ( 2(k 1 + k 2 ), l 1 + l 2, l 1 + l 2, n 1 + n 2, 2(m 1 + m 2 ), 2(p 1 + p 2 ) ), L 1 L 2 = ( 4k 1 k 2 + 2l 1 l 2, 2l 1 k 2 + n 1 l 2, l 1 n 2 + 2k 1 l 2, 2l 1l 2 + n ) 1n 2, 4m 1 m 2, 4p 1 p 2. Then, it follows that the inverse of L, the compliance tensor M, isgivenby ( n M = L 1 = 2, l 2, l 2, k ), 1 2m, 1, 2p where = kn ll. The identity tensor I in this notation is I = (1,,, 1, 1, 1). In particular, if L is isotropic, with Lamé moduli λ, µ, then it can also be represented in this way as L = (2(λ + µ),λ,λ,λ+ 2µ, 2µ, 2µ).

R. Rodríguez-Ramos et al. Appendix B This appendix is concerned with the definition of the components of P in (7) and the corresponding Green s function. A suitable representation of the Green s function for the time-reduced elastodynamic equation developed by Willis [37,38] appears slightly modified in Sabina et al. [32]. In the formulae below, two terms are given. The first one represents the static Green function, which is used here; the second term, proportional to ω is the term that includes the contribution to dynamics. Thus the complete formulae are given, so that the contribution of the static term is obtained by setting the circular frequency to zero. G ij (x) = 1 8π 2 3 n=1 ξ =1 ds U i n(ξ)u n j (ξ) ( cn 2 δ(ξ x) + iϖ ) exp (iϖ ξ x c n ). (1) 2c n Here, Ui n and c n are the polarization and wave speed of a plane wave propagating in the direction of the unit vector ξ in the reference medium with properties L o and ρ o. Thus, [ (Lo ) ijkl ξ j ξ l cn 2 (ρ ] o) ik U n k =. (2) Note that here the vectors Ui n are orthonormalized relative to the tensor ρ o. The associated wavenumber k n = w/c n. In the case of transverse isotropy, the three solutions of (2) correspond to the three different wave types. The eigenvectors and eigenvalues of (2)aregiveninSabinaetal.[32]. The integrations of the derivatives of the Green s function to obtain P are carried out over the spheroidal inclusions x1 2 a 2 + x2 2 a 2 + x2 3 c 2 = 1 (3) of aspect ratio δ = c/a. The components of P are given below as: 2P 1 2k = 1 2ρ I 1 P 2 P 3 l = l = (1 u 2 ) [ m 2 1 H 1(ζ(u)) + m 2 3 H 3(ζ(u)) ] du, 1 2(ρ I ρ III ) 1/2 P 4 n = 1 1 u 2 [ m 2 3 ρ H 1(ζ(u)) + m 2 1 H 3(ζ(u)) ] du, III 2P 5 2m = 1 4ρ I 2P 6 2p = 1 4 1 1 1 (1 u 2 )um 1 m 3 [H 1 (ζ(u)) H 3 (ζ(u))] du, (1 u 2 ) [ m 2 1 H 1(ζ(u)) + H 2 (ζ(u)) + m 2 3 H 3(ζ(u)) ] du, {[ u 2 m2 1 + (1 u 2 ) m2 3 ρ 1 ρ III ] +2(1 u 2 ) 1/2 m 1 m 3 u (ρ I ρ III ) 1/2 H 1 (ζ(u)) + u2 H 2 (ζ(u)) ρ I [ ] + u 2 m2 3 + (1 u 2 ) m2 1 2(1 u 2 ) 1/2 m 1 m 3 u ρ 1 ρ III (ρ I ρ III ) 1/2 } H 3 (ζ(u)) du. (4)

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