The effects of axisymmetric radial, circumferential and longitudinal eigenstrains on the traveling wave solution in a neo-Hookean cylindrical rod. (1st June 2021) - Motaghian, Seyedemad; Rahimian, Mohammad
Highlights: The final nonlinear equation is derived by solving a Bernoulli differential equation. All traveling waves formed in the rod are investigated. Cylindrical eigenstrains can increase or decrease the wave velocity. In particular conditions, the governing equation is satisfied by any function. Abstract: This study deals with the impact of cylindrical eigenstrains on the traveling wave solutions of a neo-Hookean cylindrical rod. For this purpose, we consider an isotropic, incompressible neo-Hookean rod with the symmetrical distribution of radial, circumferential and longitudinal eigenstrains. To establish the momentum balance equations, we construct a Riemannian manifold as the reference configuration, and then place it in Euclidean space. Assuming that the rod has an axisymmetric region with uniform eigenstrains, we extend and simplify the governing equations to come up with the final nonlinear differential equation. After thorough analysis of this equation, we introduce a traveling wave which is not observed in an eigenstrain-free rod and also explain how the cylindrical eigenstrains affect the velocities. In addition, we propose an important solution, stating that with special quantities of the eigenstrains, any arbitrary function can be a traveling wave in the rod (provided that it is physically acceptable). To substantiate this claim, we find those eigenstrain parameters by which the equilibrium equation is satisfied automatically. Proving that these waves are of equal velocities, we can say that this solution is similar to d'Alembert's solution in linear approaches.