Group velocity formulas of anisotropic solids and their applications

This paper presents various closed-form analytic formulas that relate elastic constants of elastically anisotropic solids with orthorhombic or higher symmetry to a group velocity of quasilongitudinal (QL) or quasitransverse (QT) mode propagating in an arbitrary direction of the symmetry planes of media. Simple equations relating the direction of a group velocity to that of the corresponding wave normal are also described for both QL and QT modes. Some useful applications of these relations are discussed: first, determination of group velocity surfaces and cuspidal features; second, determination of mixed-index elastic constants, given the pure-index elastic constants obtained in the symmetry and other directions: third, determination of all elastic constants of a cubic medium with longitudinal group velocities measured at least in three different directions, Examples are provided with transversely isotropic zinc and cubic silicon crystals and an orthotropic Poly Ether Ether Ketone (PEEK) composite plate. It is demonstrated that given the numerous group velocity data, one can efficiently determine the elastic constants by first converting them into phase velocity data and then applying a least squares optimization method to the phase velocity data