Effective thermodynamic elastic constants under finite deformation

Nonlinear elasticity is required to account for mechanical and acoustic properties of bodies under finite elastic deformation. R. N. Thurston and K. Brugger gave equations [Phys. Rev. 133, A1604 (1964)] relating experimental acoustic velocities in a strained elastic body to the thermodynamic third-order elastic constants cIJK given in data books. However, no correct and complete set of expressions has been given for the general case, to obtain effective elastic constants from the cIJK to solve mechanical or physical problems. By considering two deformations applied to an elastic body, effective thermodynamic second-order constants cIJ under finite strain are defined thermodynamically and correctly related to the Lagrangian and Eulerian third-order elastic stiffness constants, cIJK.