Exact longitudinal and torsional traveling-wave solutions to infinite, semi-infinite, and finite nonuniform functionally-graded elementary rods

Structures with unconventional designs and material configurations have gained significant attention in modern structural and acoustic fields due to their capabilities for manipulating waves. The mathematical complexity arising from their modeling often restricts the scope of analytical studies, leading to a reliance on numerical and experimental methods that compromise assessing immediate dynamic aspects. To address the analytical challenges, this work uses symmetry methods to provide exact solutions for rods with nonuniform geometries made of functionally-graded materials and modeled according to the elementary rod theory for slender structures undergoing longitudinal or torsional vibrations. Solutions originate from classification via equivalence transformations, aided by nonlocal transformations, which rewrite initial and boundary value problems for the rod’s elastodynamics equation as equivalent ones for a constant-coefficient wave equation. The equivalent problem allows for expressing exact solutions in terms of traveling waves, but restricts the extent of suitable geometric and material parameters. Corresponding inverse transformations map the wave solutions from the equivalent problem into the rod’s elastodynamics problem, making it adequate for many infinite, semi-infinite, and finite rods. Examples illustrate the obtained solution for semi-infinite and finite rods with fixed and free boundaries.