Prof. C. Boutin
ENTPE, Université de Lyon, FRANCE
LOCAL: Centro de Tecnologia, Bloco G, sala 205
DATA: 13/08/2012 (2ª. feira)
Abstract: This communication deals with the design of elastic composites behaving as generalized media with non-local effects in space and/or time at the leading order. This issue requires to relate the physics at the scale of the internal microstructure to the macroscopic behaviour. Hence, the theories of generalized continua postulated at the macroscale, e.g. , are not appropriate for this question. Conversely, the homogenization method of periodic media, combined with a systematic use of scaling is an efficient approach for our purpose. The present study is developed in this framework, based on the key assumption of micro-macro scale separation.
It as been established by high order homogenisation , that elastic composites present non-local effects in space and time. However, this latter effects appears as correctors of the classical Cauchy behaviour at the leading order. Therefore, to enhance the non local effects up to the leading order, the physics at the local scale must be modified, either by introducing a contrast in the “geometry” of the constituents (instead of O(1) aspect ratio as in ), or by increasing the contrast of elastic properties (taken O(1) in ). These options are analysed by considering reticulated materials or highly contrasted composites.
Reticulated materials that present an high contrast between shear and compression deformability are investigated through the homogenization method of discrete media. The conditions of second gradient behaviour (non local in space at the leading order) are identified according to the parameters of the cell and the dimensions of the shearing zone . Further, weakening the 3D- periodicity condition to 1D periodicity enables to obtain Cosserat description related to generalized beam behaviours. In dynamics, inner resonance may also arise and induces a non-Newtonian description. As in meta-materials, this non locality in time leads to dispersive P waves, with cut-off frequencies .
Similar results are established for elastic composites made of a soft matrix periodically reinforced by linear slender inclusions of high shear modulus. One obtains a second gradient continuum accounting for the inner bending of inclusions and the shear behaviour of the matrix  (experiments on specimens designed following these theoretical studies confirm this description). In accordance with a non local spatial effect at the leading order, such materials present an internal intrinsic length much larger than the period size. In dynamics, the strong contrast induces a phenomenon of inner resonance of the soft matrix. This results in an apparent mass depending on the frequency, i.e. a non-Newtonian description corresponding to “meta-materials”.
To conclude, both analyses show that a key ingredient to reach a generalized behaviour at the leading order is the large contrasts of mechanical properties within the microstructure. In appropriate loading conditions, it enables enriched local kinematics (global rotation and/or inner deformation i.e. non locality in space) but also inner resonance (non locality in time).
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 Chesnais, C., Boutin, C., Hans, S., On the Application of Generalized Continua Models to Structural Mechanics, “Mechanics of Generalized Continua”, Eds. Altenbach, Maugin, Erofeev - Advanced Structured Materials, Vol. 7, Springer, 2011.