Response properties of lattice metamaterials under periodically distributed boundary loads

We discuss response properties of lattice metamaterials to sinusoidal distributed static loads applied on a material edge. Analytical displacement solutions are obtained using a Fourier domain transfer matrix for both essential and natural boundary conditions, which are valid for a state of plane strain in orthorhombic lattices, as well as for planar metasurfaces. These solutions give sinusoidal displacement profiles in the material interior of the same wavenumber (spatial frequency) as the boundary load. Their amplitudes decay in the material interior in one of two possible ways, exponential or oscillatory exponential, depending on the lattice design and on the spatial frequency of the load. There is a tendency for the oscillatory exponential behavior to occur at lower wavenumbers representing smoother boundary loads. As explained on a specific example, comparing the metamaterial's response amplitudes with those of a homogenous material also allows studying an effective, wavenumber-dependent shear modulus of the lattice, which can take both positive and negative values.