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We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing thin infinite elastic rods with material coefficients that rapidly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the rod thickness and the period of material oscillations, which are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under the assumption on material symmetries as well as in the general case when these two types of displacements are intertwined. [ABSTRACT FROM AUTHOR]

For arbitrarily small values of ε > 0 , we formulate and analyse the Maxwell system of equations of electromagnetism on ε -periodic sets S ε ⊂ R 3. Assuming that a family of Borel measures μ ε , such that supp (μ ε) = S ε , is obtained by ε -contraction of a fixed 1-periodic measure μ , and for right-hand sides f ε ∈ L 2 (R 3 , d μ ε) , we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic "singular structures", when μ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for L 2 functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method. [ABSTRACT FROM AUTHOR]

We show that nonlinearly elastic plates of thickness $$h\rightarrow 0$$ with an $$\varepsilon $$ -periodic structure such that $$\varepsilon ^{-2}h\rightarrow 0$$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is 'discontinuously anisotropic' in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale. [ABSTRACT FROM AUTHOR]