Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I: Theory.

*(English)*Zbl 1116.74412Summary: This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method [Ponte Castañeda, J. Mech. Phys. Solids 39, 45-71 (1991; Zbl 0734.73052)] makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method [Ponte Castañeda, ibid. 44, No. 6, 827-862 (1966; Zbl 1054.74708)] makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work (Zbl 1116.74411).

##### MSC:

74Q20 | Bounds on effective properties in solid mechanics |

74A40 | Random materials and composite materials |

74E30 | Composite and mixture properties |

##### Keywords:

Voids and inclusions; Anisotropic material; Porous material; Viscoplastic material; Variational calculus
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\textit{P. Ponte Castañeda}, J. Mech. Phys. Solids 50, No. 4, 737--757 (2002; Zbl 1116.74412)

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