Cauchy-like relation between elastic constants in amorphous materials

Abstract

It is well known that in addition to the longitudinal modulus, viscoelastic liquids show a shear stiffness at sufficiently high probe frequencies due to structural relaxations. For probe frequencies that are large compared to the structural relaxation frequency, the measured elastic longitudinal and shear moduli become so-called clamped properties ${(c}_{11}^{\ensuremath{\infty}}$ and ${c}_{44}^{\ensuremath{\infty}},$ respectively). During freezing or polymerization of amorphous liquids, these clamped moduli behave in a strongly nonlinear fashion as a function of temperature or polymerization time. Based on Brillouin spectroscopy data we will show that there exists a linear relation between ${c}_{11}^{\ensuremath{\infty}}$ and ${c}_{44}^{\ensuremath{\infty}}$ over a large temperature or polymerization time range. Surprisingly, the parameters of this linear relation between the elastic moduli vary only little for different materials. Implications for the nonlinear elastic behavior at the glass transition will be discussed on the basis of mode Gr\"uneisen parameters.