Continuum model of dispersion caused by an inherent material characteristic length

Modifications of the Helmholtz free energy and the stress associated with general constitutive equations of a simple continuum are proposed to model dispersive effects of an inherent material characteristic length. These modifications do not alter the usual restrictions on the unmodified constitutive equations imposed by the first and second laws of thermodynamics. The special case of a thermoelastic compressible Newtonian viscous fluid is considered with attention focused on uniaxial strain. Within this context, the linearized problems of wave propagation in an infinite media and free vibrations of a finite column are considered for the simple case of elastic response. It is shown that the proposed model predicts the dispersive effects observed in wave propagation through a chain of springs and masses as the wavelength decreases. Also, the nonlinear problems of steady wave propagation of a soliton in the absence of viscosity and of a shock wave in the presence of viscosity are discussed. In particular it is shown that the presence of the dispersive terms can cause the stress in a shock wave to overshoot the Hugoniot stress by as much as 50%. This phenomenon may cause an underprediction of the threshold level for failure determined by analysis of stress in shock experiments.