11. Vibration of Auxetic Solids

Vibration study is of great practical importance, as vibration of continuous systems with constraints implies cyclic stresses and the inevitable fatigue damage. This chapter on vibration forms the first part of elastodynamics of auxetic solids, with special emphasis on plates (both circular and rectangular) as well as shells (both cylindrical and spherical). For circular plates with free and simply supported edges, the frequency parameter changes more rapidly in the auxetic region than in the conventional region; consequently the natural vibration frequencies of these plates can be effectively reduced by choosing auxetic materials. For rectangular plates, the effect of negative Poisson’s ratio is evaluated for plates with all four sides and two sides being simply supported, as well as examples of rectangular plates with three sides being simply supported. In the case of cylindrical shells with simply supported edges, the results of frequency study imply that, when expressed in terms of flexural rigidity, the frequency is independent from the cylindrical shell radius at extreme auxetic behavior for isotropic case. In the case of spherical isotropic shells, the natural frequency diminishes as the Poisson’s ratio of the shell material approaches −1 at constant flexural rigidity and at constant shear modulus. Finally, advanced topics on vibration damping, vibration transmissibility and acoustics of auxetic solids and structures are briefly reviewed.