Optimized functional perturbation method and morphology based effective properties of randomly heterogeneous beams

A Functional Perturbation Method (FPM) has been recently developed for the analysis of stochastically heterogeneous structures, for which the heterogeneity scale is not negligible relative to the macro dimensions. The FPM is based on considering the target function (here, the buckling load P) as a functional of the stochastic morphology. The target function is written as a functional perturbation series near a convenient homogeneous property, usually stiffness (K) or compliance (S). Thus, the accuracy depends on the choice of the property around which the perturbation is carried out. An Optimized FPM (OFPM) is presented here, which concentrates on finding a property h(K), which is a function of K or S, such that the target function converges faster. This is accomplished by looking for h(K) which minimizes (or nulls, if possible) the second term in the functional perturbation series. Besides its improved accuracy, h has also a dual meaning, which is related to the notion of ‘‘effective’’ property. However, the ‘‘effectiveness’’ is weak, since the property is not ‘‘purely material’’, but depends on external loading shapes. An example of a buckling problem is examined in detail, for which h is found analytically as a simple power of K, which directly depends on morphology. Comparing the new OFPM with previous FPM and numerical Monte Carlo—Finite Element results shows the desired improved accuracy. The advantages of the OFPM are then shown and discussed.