Exploitation of the dissipation inequality, if some balances are missing

The balance equations of continuum thermodynamics need constitutive equations in order to solve them under the constraint that the entropy production appearing in the entropy balance equation must be not negative.This dissipation inequality represents the second law of thermodynamics.There are two procedures which exploit the dissipation inequality to obtain constitutive equations which are in agreement with the second law: the Coleman-Noll and the Liu techniques.Here we use the Liu technique in the special case in which not all balance equations are taken into account when exploiting the dissipation inequality.This case is of interest because often not all balances are known, or only the energy balance is considered.It is also proved that in this abridged exploitation of the dissipation inequality, thermodynamic restrictions for the constitutive equations are obtained, so that these satisfy the second law.These restrictions represent a smaller class of materials than that obtained when all balances are taken into account.