Frequency shifts in natural vibrations in pantographic metamaterials under biaxial tests

In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantographic sheet, is studied. The microstructure is characterized by two families of parallel fibers, whose deformation measures account for bending, elongation and relative rotation of the fibers. The deformation energy density of the homogenized model depends on both first and second gradients of the displacement. Modal analysis is performed in order to assess the peculiarities of the dynamic behavior of higher gradient models, and in particular the difference, with respect to classical laminae, in the dependence of the eigenfrequencies on the stiffness