Effective properties of nonlinear composites

These lectures describe several procedures commonly used or recently developed to predict the overall behavior of nonlinear composites from the behavior of their individual constituents and from statistical information about their microstructure. Secant methods are discussed in section 2. A modified method based on the second-order moment of the strain field is proposed and compared with the classical secant method in specific situations, composites with large or small contrast and power-law materials. The incremental method is presented in section 3. It appears much stiffer than the two secant methods. Its predictions for isotropic two-phase power-law composites even violate a rigorous upper bound when the nonlinearity is strong. A variational procedure leading to rigorous upper bounds for the effective potential of the composite is presented in section 4. Specific forms for voided or rigidly reinforced power-law composites are given first. Then a general upper bound applying to a general class of nonlinear composites is derived. The variational procedure coincides with the secant approach based on second-order moments and with the variational procedure of Ponte Castnneda. These different schemes are applied in section 5 to predict the overall behavior of metal-matrix composites. A simplified model based on the variational procedure is proposed. Its predictions compared well with simulations performed by the Finite Element Method.

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