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            <title xml:lang="en">Atypical dynamics of materials with periodic microstructure and local resonance</title>
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                <title xml:lang="en">Atypical dynamics of materials with periodic microstructure and local resonance</title>
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              <p>This work investigates the dynamic behavior of periodic unbraced frame structures made up of interconnected beams. Two types of microstructures are especially studied: non-orthogonal unbraced frame and honeycombs. The microstructure being much stiffer in compression than in shear, a great variety of behaviors can occur. Assuming the condition of scale separation is respected, the dynamical behaviors at the leading order are approached by the homogenisation method of periodic discrete media. In the studied ranges, the local elements behave ever in quasi-statics, ever in dynamics. For studied materials, the elastic laws are given in function of the elements properties. These laws correspond to upgraded materials as double gradient media or meta-material. To illustrate their atypical properties, propagations of 'shear' and 'compression' waves are studied. In the presence of the local resonance, the form of the equations is unchanged but the mass depends on the frequency and, as a result, frequency bandgaps appear.</p>
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