On shock wave propagation in stressed isotropic nonlinearly elastic media

Abstract

The relationships in weak shock waves propagating over an arbitrary elastic medium in lightly deformed (stressed) state are analyzed in nonlinear approximation. Hugoniot curves that correspond to waves propagating over an arbitrary stressed state are investigated using relations at discontinuities in quasilongitudinal and quasitransverse shock waves. Generally, when the initial deformation ahead of the shock wave disrupts the isotropy in planes parallel to the wave front, the Hugoniot curve represents for both the quasitransverse and quasilongitudinal waves a certain curve which is investigated below. Points that correspond to shock waves accompanied by entropy rise are indicated on segments of that curve. In the case of quasitransverse waves this made necessary and taking into account in computations fourth order terms with respect to discontinuity amplitude. The shock wave velocity behavior and its relation to small perturbation velocities is investigated. Segments of the Hugoniot curve are indicated at whose points the conditions of /shock wave/ evolution are satisfied. It is shown that in the case of quasitransverse waves the conditions of evolution and those of entropy increase correspond to two different sets on the Hugoniot curve. Only shock waves that correspond to the intersection of these sets can actually exist. Shock waves in nonlinearly elastic media were investigated in /1–5/. (The numerous publications in which purely longitudinal shock waves, i.e. Waves in which only the normal component becomes discontinuous and those in which discontinuities were analyzed in linear approximation, are not mentioned here.) Weak shock waves propagating in an Isotropie elastic medium in unstressed state were the subject of detailed analysis in /1/. Some of the results obtained in /2–5/ are relevant to the present investigation. The dependence of shock wave velocity on the discontinuity of the derivative of the displacement normal vector component along the normal to the wave, uniquely related to density, was investigated, and the change of the entropy sign for quasilongitudinal waves was clarified. When the initial deformation is small relative to the discontinuity amplitude, the third approximation does not indicate a change of entropy in quasilongitudinal waves. The equation of momentum conservation and the defining equations of the medium showed the existence of particular types of shock waves: pure longitudinal and pure transverse. Certain corollaries were obtained in the form of inequalities. The Hugoniot curve was not considered as a whole in /2–5/, and shock wave evolution was not investigated.