Abstract
In this paper we describe a three-scales homogenization process which we use to determine a macroscopic model for pantographic metamaterials. The smallest scale refers to the length at which the considered deformable mechanical system can be modeled as a Cauchy’s continuum. Of course, at this scale, its geometry is rather complex. The meso-scale refers to a length at which the system can be modeled as a Hencky-type discrete system constituted by masses interconnected by extensional and rotational springs. At macro-scale the model to be used is a generalized plate whose deformation energy depends on geodesic curvature. While the direct identification from the smallest scale to the macro-scale seems rather diffcult, the identification from smallest scale to meso-scale can be successfully obtained. The geometrical properties, along with Young and Poisson coeffcients of the used isotropic material, at the smallest scale determine the extensional and rotational stiffnesses to be used at the meso-scale. On the other hand, the Piola-type identification process allows us to determine the stiffnesses of the macroscopic generalized plate model, via a simple asymptotic expansion. We have observed that this process is valid in both cases when the smallest scale is of the order of microns and when the smallest scale is of the order of tenth of millimeters. Some experimental and numerical results supporting this statement are exhibited.
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Turco, E. (2019). How the Properties of Pantographic Elementary Lattices Determine the Properties of Pantographic Metamaterials. In: Abali, B., Altenbach, H., dell'Isola, F., Eremeyev, V., Öchsner, A. (eds) New Achievements in Continuum Mechanics and Thermodynamics. Advanced Structured Materials, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-030-13307-8_33
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