This work is devoted to the study of the wave propagation in infinite two-dimensional structures made up of the periodic repetition of frames. Such materials are highly anisotropic and, because of lack of bracing, can present a large contrast between the shear and compression deformabilities. Moreover, when the thickness to length ratio of the frame elements is small, these elements can resonate in bending at low frequencies, when compressional waves propagate in the structure. The frame size being small compared to the wavelength of the compressional waves, the homogenization method of periodic discrete media is extended to situations with local resonance and it is applied to identify the macroscopic behavior at the leading order. In particular, the local resonance in bending leads to an effective mass different from the real mass and to the generalization of the Newtonian mechanics at the macroscopic scale. Consequently, compressional waves become dispersive and frequency bandgaps occur. The physical origin of these phenomena at the microscopic scale is also presented. Finally, a method is proposed for the design of such materials.

This paper deals with the dynamic behaviour of periodic reticulated beams and materials. Through the homogenization method of periodic discrete media the macro-behaviour is derived at the leading order. With a systematic use of scaling, the analysis is performed on the archetypical case of symmetric unbraced framed cells. Such cells can present a high contrast between shear and compression deformability, conversely to "massive" media. This opens the possibility of enriched local kinematics involving phenomena of global rotation, inner deformation or inner resonance, according to studied configuration and frequency range.

Through the analysis of the wave propagation in infinite twodimensional periodic frame materials, this paper illustrates the complexity of their dynamic behavior. Assuming the frame size is small compared to the wavelength, the homogenization method of periodic discrete media coupled with normalization is used to identify the macroscopic behavior at the leading order.The method is applied on a frame material with the vertical elements stiffer than the horizontal elements. Such a material is highly anisotropic and presents a large contrast between the rigidities of the possible mechanisms. Thus the waves associated with different kinematics appear in different frequency ranges.Moreover, the stiffer elements can deform in bending at the macroscopic scale.The equivalent continuum is a second grade medium at the leading order and shear waves can be dispersive. A criterion is proposed to easily determine when this bending effect has to be taken into account.