Free energies for materials with memory in terms of state functionals

The aimof thiswork is to determinewhat free energy functionals are expressible as quadratic forms of the state functional I^t which is discussed in earlier papers. The single integral form is shown to include the functional ΨF proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a newsystematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to uncover new free energy functionals. However, while, in principle, the same method should apply to free energieswhich can be given by quadratic forms in terms of It , it emerges that this requirement is very restrictive; indeed, only the minimum free energy can be expressed in such a manner.