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Many biological and engineering materials have nonperiodic microstructuresfor which classical periodic homogenization results do not apply. Certainnonperiodic microstructures may be approximated by locally periodicmicrostructures for which homogenization techniques are available. Motivated bythe consideration that such materials are often anisotropic and can possesresidual stresses, a broad class of locally periodic microstructures isconsidered and the resulting effective macroscopic equations are derived. Theeffective residual stress and effective elasticity tensor are determined bysolving unit cell problems at each point in the domain. However, it is foundthat for a certain class of locally periodic microstructures, solving the unitcell problems at only one point in the domain completely determines theeffective elasticity tensor.

On the homogenization of a new class of locally periodic microstructures in linear elasticity with residual stress