We explore topological properties of one-dimensional (1D) bilayered elastic rods (i.e., phononic crystals) using the phason as a virtual dimension and subsequently analyze topological interface modes. Following adoption of the phason, we use a two-dimensional (2D) topological invariant—the Chern number—and show that its differences for various crystals predict the number of gapless interface states in $(1+1)\mathrm{D}$ systems. The latter constitutes a manifestation of the bulk-boundary correspondence principle. We develop equations of periodic systems that depend explicitly on the phason, hence facilitate understanding and bring new insight into wave propagation and dynamics of topological states in these media. For the design and analysis of interface modes in systems with two virtual parameters, we introduce the phason-phason space. We find that the interface states mark strips in this space, separated by linear functions with slopes related to the Chern numbers. Consequently, this space can be used to directly predict the Chern numbers of both adjoined phononic crystals as well as the number of gapless interface modes for different phason changes.

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• Band gap
• Elastic waves
• Topological effects in acoustic systems
• 1-dimensional systems
• Mechanical metamaterials
• Phononic crystals
• Topological materials
• Bloch-Floquet theorem