Second-order homogenization estimates incorporating field fluctuations in finite elasticity

This paper presents the application of a recently proposed "second-order" homogenization method (Ponte Castañeda 2002; J. Mech. Phys. Solids 50, 737) to the estimation of the effective behavior of hyperelastic composites subjected to finite deformations. The key idea is to introduce an optimally chosen "linear comparison composite" which can then be used to convert available homogenization estimates for linear composites directly into new estimates for the nonlinear hyperelastic composites. More precisely, the method makes use of "generalized" secant moduli that are intermediate between the standard "secant" and "tangent" moduli of the nonlinear phases, and that depend not only on the averages, or first moments of the fields in the phases, but also on the second moments of the field fluctuations, or phase covariance tensors. The use of the method is illustrated in the context of carbon-black-filled, and fiber-reinforced elastomers, and estimates analogous to the well-known Hashin=-Shtrikman and self-consistent estimates for linear-elastic composites are generated. The new estimates are compared with corresponding estimates using an earlier version of the method (Ponte Castañeda and Tiberio 2000; J. Mech. Phys. Solids 48, 1389) neglecting the use of fluctuations, and the new results are found to be superior. In particular, the new estimates, unlike the earlier ones, are found to satisfy a rigorous bound, and to give more realistic predictions in the important limit of incompressible behavior.