An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials

We discuss the consequences of a different application of the principle the Modified Secant Method is C. R. Acad. Sci. Paris Sér. IIb 320 (1995) 563] based on. In fact, we directly compute the second-order averages of the local fields available from the linear elastic homogenization procedure exploited in order to evaluate the effective elastic moduli. This method, which can be seen either as a simplification of the Modified Secant Method or as an extension of the Secant Method J. Mech. Phys. Solids 26 (1979) 325], may be useful for any composite whose overall elastic constants need to be estimated by modeling the microstructure through Morphologically Representative Patterns J. Mech. Phys. Solids 44 (1996) 307], which is for instance the case of syntactic foams Int. J. Solids and Structures 38/40-41 (2001) 7235]. In order to show the accuracy of the proposed method, we apply it to several examples and compare its results with those obtainable by means of other analytical methods available in the literature, with numerical results of Finite Element simulations, and with experimental results. Closed-form solutions are derived for the effective yield stress of porous metals and incompressible composites reinforced with rigid spheres.