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In this paper we consider a sample of a linearly elastic heterogeneouscomposite in elastodynamic equilibrium and present universal theorems whichprovide lower bounds for the total elastic strain energy plus the kineticenergy, and the total complementary elastic energy plus the kinetic energy. Fora general heterogeneous sample which undergoes harmonic motion at a singlefrequency, we show that, among all consistent boundary data which produce thesame average strain, the uniform-stress boundary data render the total elasticstrain energy plus the kinetic energy an absolute minimum. We also show that,among all consistent boundary data which produce the same average momentum inthe sample, the uniform velocity boundary data render the total complementaryelastic energy plus the kinetic energy an absolute minimum. We do not assumestatistical homogeneity or material isotropy in our treatment, although theyare not excluded. These universal theorems are the dynamic equivalent of theuniversal theorems already known for the static case (Nemat-Nasser and Hori1995). It is envisaged that the bounds on the total energy presented in thispaper will be used to formulate computable bounds on the overall dynamicproperties of linearly elastic heterogeneous composites with arbitrarymicrostructures.

Universal Theorems for Total Energy of the Dynamics of Linearly Elastic Heterogeneous Solids