Asymptotic homogenisation via the method of multiple scales is considered for problems in which the microstructure comprises inclusions of one material embedded in a matrix formed from another. In particular, problems are considered in which the interface conditions include a global balance law in the form of an integral constraint; this may be zero net charge on the inclusion, for example. It is shown that for such problems care must be taken in determining the precise location of the interface; a naive approach leads to an incorrect homogenised model. The method is applied to the problems of perfectly dielectric inclusions in an insulator, and acoustic wave propagation through a bubbly fluid in which the gas density is taken to be negligible.

A semi-analytic method is proposed for two problem settings for a Kirchhoff plate containing an absolutely rigid circular inclusion and undergoing a transverse point force. The settings differ by the location (within and out of inclusion) of the force application point. In both cases, the plate's stress-strain state is simulated with a boundary value problem for the biharmonic equation stated over a doubly connected region whose inner contour represents the edge of the inclusion. Boundary conditions imposed on the inner contour bring some parameters which are found via the equations of static equilibrium of the inclusion. A modification of the Kupradze's method of functional equations is proposed for obtaining influence functions of a point force for such plates. Green's functions of the biharmonic equation for appropriately shaped simply connected regions are employed. Numerical differentiation is never required in the computing of stress components and the latter are subsequently found with accuracy level comparable with that attained for the deflection function.