In this section, we briefly recall the tangent second-order variational method developed in Chapter 1 for the effective constitutive behavior of (two-phase) particle-reinforced, hyperelastic composites consisting of aligned, ellipsoidal, rigid particles distributed randomly with volume fractionc in an incompressible matrix phase with energy function Wµ(1). (Note that since particles are rigid and
the matrix is incompressible, c = c(2)0 for all macroscopic deformations.) As already mentioned, the TSO method makes use of a fictitious “linear comparison composite” (LCC) with the same microstructure (i.e., same characteristic functionsχ(r)(X)) as the actual (nonlinear) composite ma-
terial (in the undeformed configuration). The moduli of the constituent phases in the LCC are identified with “tangent” linearizations of the given nonlinear phases evaluated at the macroscopic deformation gradient ¯F. This allows the use of already available methods for estimating the effec- tive behavior of linear composites to generate corresponding estimates for nonlinear composites. In Chapter 1, we made use of the generalized Hashin-Shtrikman estimates of the Willis type (Willis, 1977; Ponte Casta˜neda and Willis, 1995) for the effective behavior of the linear-elastic composite materials consisting of random distributions of aligned ellipsoidal particles with prescribed “ellip- soidal symmetry” for the particle centers (i.e., the two-point correlation functions). These estimates are exact to second-order in the heterogeneity contrast and to first order in the particle volume fraction, and are known to be quite accurate for the type of “particulate” microstructures of interest here, up to moderate concentrations of particles. In order to ensure compliance with the overall incompressibility constraint ( ¯J= det( ¯F) = 1), in Chapter 1, we made use of the expression (2.1) to split the distortional and deviatoric components of the energy and arrived at the following estimate for the effective stored-energy functionWc( ¯F) of the reinforced elastomers:
c W( ¯F) = (1−c)Wµ(1)( ¯F(1)) + 1 2 c 1−c( ¯F−R¯ (2)) ·E( ¯F−R¯(2)). (2.13) In this expression, ¯Fis the macroscopic deformation which satisfies the incompressibility condition det( ¯F) = 1, ¯R(2) is a second-order orthogonal tensor characterizing the average rotation of the
rigid particles under the macroscopic deformation gradient ¯F, and determined by the kinematical equation
Skewn( ¯R(2))T[E( ¯F−R¯(2))] + (1−c)( ¯R(2))TS(1)µ ( ¯F(1))
o
=0, (2.14) whereSkew denotes the skew-symmetric part of the quantities inside the curly brackets. Note that equation (2.14) provides in general a set of three scalar algebraic equations for the three independent components of ¯R(2). The second-order tensor ¯F(1)corresponds to the average deformation gradient
in the matrix phase of the LCC, which can be expressed in terms of the macroscopic deformation gradient ¯F, and the rotation tensor ¯R(2)as
¯
F(1)= 1
1−c( ¯F−cR¯
(2)), (2.15)
Finally,Eis a fourth-order, microstructural tensor given by1
E= lim
µ′(1)→∞(P
−1
−L(1)). (2.16) In this relation,L(1) is a fourth-order moduli tensor determined by the tangent modulus evaluated
at the macroscopic deformation, such that
L(1)= ∂2W(1)
∂F∂F( ¯F), (2.17)
while Pis an Eshelby-type (fourth-order) tensor containing information about the shape and dis- tribution of the particles in the undeformed configuration (Ponte Casta˜neda and Willis, 1995). For ellipsoidal particles, distributed in an infinite matrix with the elastic modulus tensorL(1), the com- ponents ofPread as Pijkl= 1 4π|Z0| Z |ξξξ|=1 Bik(ξξξ)ξjξl h ξξξT(ZT0Z0)−1ξξξ i−3 2 dS, (2.18) where the symmetric, second-order tensorZ0serves to characterize the “shape” and “orientation” of
the particles in the undeformed configuration, and the tensorBdenotes the inverse of the acoustic tensor K of the matrix with components Kik =L(1)ijklξjξl. In this work, we will also assume that
the initial (in the undeformed configuration) “shape” and “orientation” of the two-point correlation function for the distribution of the ellipsoidal particles are identically the same as for the particles themselves, as specified by the tensorZ0. It should be remarked that this assumption is not essential,
and the shapes and orientations of the distribution functions could, in general, be different from those of the particles (Ponte Casta˜neda and Willis, 1995) leading to the use of two different P tensors. It is also worth mentioning that all fourth-order tensorsE,P, and L(1) have major symmetry, but
not generally minor symmetry. Furthermore, in view of definitions (2.16)-(2.18), together with the
1In Chapter 1, the fourth-order tensorEwas labelledEI
, but, for simplicity, the superscriptI has been dropped
objectivity assumption forW(1)(F), it can be verified that
Eijkl( ¯F) = ¯RipR¯kqEpjql( ¯U), (2.19)
where ¯Uand ¯Rdenote the macroscopic stretch and (rigid-body) rotation tensors, respectively (note that ¯F= ¯RU¯). Making use of (2.19) along with (2.14) and (2.15), it can be shown that expression (2.13) forWc( ¯F) satisfies theobjectivity condition
c
W( ¯F) =cW( ¯U). (2.20) In summary, for a given ellipsoidal microstructure, macroscopic loading ¯F, and matrix strain energyW(1), the computation of the effective stored-energy function Wc( ¯F) in (2.13), as well as of
the associated rotation tensor ¯R(2) in (2.14), requires the calculation ofE. As defined in (2.16),
the tensorE should be calculated in the incompressibility limit of the matrix phase (µ′(1) → ∞).
To this end, in Chapter 1, we carried out a general asymptotic analysis for the computation of the tensor Ein the incompressibility limit (µ′(1) → ∞). For completeness and to maintain continuity,
the procedure for computing the tensorEis provided in Appendix B.1.
We conclude this section by noting that, in Chapter 1, we already considered the application of the above-described results for the special case of elastomers reinforced with spherical particles under triaxial loadings, and for the special case of composites with a neo-Hookean matrix, they derived closed-form expressions for the corresponding effective stored-energy function. In addition, they considered elastomers reinforced with two-dimensional fibers of elliptical cross-section under plane-strain loading. For the special case of composites with a neo-Hookean matrix and dilute concentration of fibers, they recovered exactly the results obtained by Lopez-Pamies and Ponte Casta˜neda (2006b) using the more sophisticated GSO method. In addition, for finite concentrations of particles and Gent-type matrices, the agreement of the TSO and GSO results was quite good. In the present work, we will consider for the first time applications for elastomers reinforced with three-dimensional spheroidal fibers.