On Rayleigh waves in elastic lattices

A mathematical framework is presented to guide the search for Rayleigh waves in lattice materials based on periodic structure theory and the Bloch theorem. Architected materials with a periodic microstructure are distinguished from crystals in continuum anisotropic elasticity by the presence of at least one length scale and a band structure with partial and complete gaps for Bloch wave propagation. Non-affine bending deformations at or below the characteristic cell size are included by considering the unit cell as a framework of Timoshenko beams. We show that a quadratic eigenvalue problem, with a Hermitian palindrome structure , emerges from the force equilibrium and displacement compatibility relations for a propagating Bloch wave along any chosen orientation of the free edge/surface. Waves propagating along the free edge and penetrating to a finite depth into the medium are a partial set of eigensolutions of the nonlinear eigenproblem, or its linearized symplectic form. These partial eigenwaves are used as the basis vectors to expand any arbitrary boundary displacements and force vectors, which then constitute a complex asymmetric semi-infinite dynamic stiffness matrix. Surface and Rayleigh waves exist in its null space. Traction-free boundary conditions are used to show that the secular equation for Rayleigh waves is a real polynomial equation, consistent with Stroh’s formulation for a length-scale independent anisotropic continuum crystal elasticity. Significant differences arising from the periodic structure are highlighted. Computational issues in the numerical solution of the structured eigenvalue problem for surface waves in lattices are addressed. Our formulation is applicable to any arbitrary lattice with complex unit cells and material architectures. Surface waves in a planar square lattice are found to emerge from the gaps for bulk waves in the band structure.