Compatibility of weak rigidity with some types of elastic schemes

The behavior of the hypoelastic-Synge, hypoelastic-Maugin, and hypoelastic-Carter and Quintana almost-thermodynamic material schemes, under weak rigidity hypotheses, is studied. In every case, the absence of principal transverse shock waves (or the vanishing of the corresponding speeds) is obtained. The same result follows for the longitudinal shock waves when the Lame coefficient μ does not vanish. A definition of an elastic almost-thermodynamic material scheme based on the Fermi--Walker transport is proposed and compared with the above elastic schemes. The speeds of the principal shock waves associated are attained and its compatibility with the Ferrando--Olivert incompressibility condition is proved. In the presence of weak rigidity the elastic schemes here defined lead (assuming μnot =0) to the Born-rigidity condition