Free-vibration acoustic resonance of nonlinear hyperelastic materials

The free-vibration acoustic resonance (FVAR) of nonlinear hyperelastic materials was investigated within the framework of nonlinear elasticity and the calculus of variations. Variational analysis revealed that displacement due to FVAR satisfies the following conditions: time reversal symmetry, time periodicity, periodic vanishing of displacement velocity, and surface traction-free conditions. Numerical analysis of two-dimensional St. Venant-Kirchhoff hyperelastic materials revealed the existence and structure of the color symmetry embedded therein. The nonlinear FVAR modes showed non-monotonic amplitude dependence of resonance frequency. The symmetries of three-dimensional nonlinear FVAR modes for triclinic, monoclinic, and orthorhombic elastic media are shown on the basis of the magnetic point groups. This classification shows excellent agreement with that obtained by the overtone analysis of molecular vibration.