A Mixture Theory for Response Fields in Complex Structures

A formal procedure is developed for the calculation of fields when two (or more) component subsystems are thoroughly interlocked, so that their surface of contact extends through the total structure. Such a structure can properly be termed a structural mixture. An example is a system of frames and ribs which is completely covered by and connected to an outer skin (or shell) at a large number of points. The coupling between subsystems is accounted for in a global fashion, using Green’s functions for each of the subsystems, together with matching conditions at their interface. This leads in general to a pair of coupled integral equations, each giving the response in one of the two interpenetrating subsystems. For a disparate structural mixture comprised of “weakly-coupled” subsystems, the Green’s functions used are obtained for complementary (i.e., non-equivalent) homogeneous interface conditions. The equations can then be solved by an alternating perturbation procedure, which gives rise to a pair of coupled series. The procedure is applied to the calculation of waves in a beam stiffened at various points along its length by contact with a second subsystem. Numerical results are presented and their convergence is discussed.