Complete exceptionality, constitutive laws and symmetric form for a non-linear inelastic rod

In this paper we consider quasilinear constitutive laws characterizing ‘Maxwellian Materials’ in the case of a one-dimensional inelastic rod. By using the complete exceptionality condition, we are able to determine the most general class of quasilinear constitutive relations by which a weak discontinuity wave propagating along characteristics cannot evolve into a non- linear shock. The last part of the paper is devoted to point out some properties connected with the special class of constitutive relations where the material functions do not depend on the strain. In such a case by means of a field variables transformation, we are able to reduce the fundamental system of equations to a symmetric hyperbolic system where the coefficient of the field spatial derivative is a constant matrix. This special symmetric form is very useful to study shocks.