Relationships Between Derivations of the Overall Properties of Composites by Perturbation Expansions and Variational Principles

摘要: ABSTRACT The overall properties of composites may be defined from the solution some standard boundary value problem. For example, for elasticity, displacements consistent with a uniform strain throughout composite prescribed on boundary. actual fluctuates but its mean takes and object is to deduce stress. random composites, an ergodic hypothesis invoked expectation stress at chosen point sought instead. problem so generated requires integral equation. medium containing dilute distribution particles or fibres, expressions moduli as power series in concentration c sought, by ensemble averaging equation keeping one more fixed. An infinite hierarchy equations this way, which solved making appropriate closure assumption, valid limit low concentrations, stage: closing nth stage correctly estimates up terms order cn. If such assumptions are made high they become no than ad hoc approximations yet, even simplest, “quasicrystalline approximation,” has been found give agreement those obtained “cell” approximation believed generate reasonable results any concentration. This demonstrated via formulation polarization associated variational principle, due Hashin Shtrikman. Allowance higher interactions shown many cases yield bounds coefficients c2 example worked out detail.