Elastic constants of solids and fluids with initial pressure via a unified approach based on equations-of-state

The second and third-order Brugger elastic constants are obtained for liquids and ideal gases having an initial hydrostatic pressure p1. For liquids the second-order elastic constants are C11 = A + p1, C12 = A − p1, and the third-order constants are C111 = −(B + 5A + 3p1), C112 = −(B + A − p1), and C123 = A − B − p1, where A and B are the Beyer expansion coefficients in the liquid equation of state. For ideal gases the second-order constants are C11 = p1γ + p1, C12 = p1γ − p1, and the third-order constants are C111 = −p1(γ2 + 4γ + 3), C112 = −p1(γ2 − 1), and C123 = −p1 (γ2 − 2γ + 1), where γ is the ratio of specific heats. The inequality of C11 and C12 results in a nonzero shear constant C44 = (1/2)(C11 − C12) = p1 for both liquids and gases. For water at standard temperature and pressure the ratio of terms p1/A contributing to the second-order constants is approximately 4.3 × 10−5. For atmospheric gases the ratio of corresponding terms is approximately 0.7. Analytical expressions that include initial stresses are derived for the material 'nonlinearity parameters' associated with harmonic generation and acoustoelasticity for fluids and solids of arbitrary crystal symmetry. The expressions are used to validate the relationships for the elastic constants of fluids.