Effective elastic constants in nonlinear elasticity

The Lagrangian elastic constants cIJ and cIJK which are given in data-books are not stress-strain relationships. The relationships between them and any experimental observables are far from straightforward to evaluate under conditions of finite deformation when elasticity is nonlinear. We present effective second- and third-order elastic constants, which are thermodynamic stress-strain relationships and which express real material properties. They are derived by applying two and three independent strains to an elastic body and expressed in terms of the Lagrangian cIJ and cIJK. Experimental observables are readily calculated using the effective constants. Their application is demonstrated by deriving the Murnaghan equation of state, by calculating the stress-strain relationships of a pseudomorphic epitaxial strained layer, and by deriving the observable changes in acoustic velocities under stress.