A non-equilibrium thermodynamic theory due to Bree and Beevers  has been applied by Bree  to a certain class of mixtures, referred to in  as simple mixtures where it is found that global and local entropy inequalities can be obtained for the whole mixture, for each of its constituents or for any combination of its constituents, each of these entropy inequalities being derived from the single statement of the second law of thermodynamics adopted in . The present work is primarily concerned with developing the theory further in order to include the more complex class of mixtures referred to in  as heterogeneous systems. In contrast with the thermodynamics of simple mixtures, it is found that for heterogeneous systems global and local entropy inequalities can be obtained only for the whole mixture and, in connection with this, it is felt that an early controversy in the subject is resolved. It is shown how the results obtained for heterogeneous systems may be used to provide a thermodynamic foundation for two methods of representing history dependent materials. © 1990 Kluwer Academic Publishers.