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We study the existence of solutions arising from the modelling of elastic materials using generalized theories of continua. In view of some evidence from physics of meta-materials we focus our effort on two recent nonstandard relaxed micromorphic models including novel micro-inertia terms. These novel microinertia terms are needed to better capture the band-gap response. The existence proof is based on the Banach fixed point theorem.Mathematics Subject Classification: 35M33, 35Q74, 74H20, 74M25, 74B99

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