In the context of long-wavelength linear elastic waves, a well-posed strain-gradient model is built using the two-scale asymptotic homogenisation method, pushed to the second-order.

We combine (i) classical formal asymptotic expansions in terms of the periodicity length-to-wavelength ratio and (ii) original reciprocity identities between the so-called cell problems at various orders to obtain new relations between the higher-order homogenised tensors entering the model. The latter results allow to highlight the symmetry properties of those tensors and reduce the overall cost of their computation for a given periodicity cell.

A “Boussinesq trick” then allows to ensure the positivity and (respectively) coercivity and ellipticity of the stiffness and inertial bilinear forms featured in the obtained strain-gradient wave equation. Those properties, in turn, are used to establish the well-posedness of initial-value problems in the free space featuring that equation. Finally, numerical simulations show the expected second-order asymptotic accuracy of the model.

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