Relationships Between the Nonlinear Elastic Constants of Monocrystalline and Polycrystalline Solids of Cubic Symmetry

The relationships between the third-order elastic constants of monocrystalline and polycrystalline (quasi-isotropic, randomly oriented) solids of cubic symmetry are derived according to elastic strain energy balances assuming negligibly small elastic strain energy contribution from mismatching of the grain boundaries. The relationships between the three third-order isotropic Lamé coefficients (γ1, γ2, γ3, Brugger convention) and the six third-order single-crystal elastic constants are given by: γ1=135(C111+18C112−30C144−12C166+16C123+16C456) γ2=135(C111+4C112+19C144+2C166−5C123−12C456)γ3=135(C111−3C112−9C144+9C166+2C123+9C456). Good agreement was found between the experimentally evaluated γ1, γ2, γ3 values and those derived from the above equations using the six third-order single-crystal elastic constants for bcc columbium at room temperature.