When fugacity and z-compressability equations instead of Gibbs free energy equations?
Elastic solids (like gelled drilling muds and elastomers) can not be treated in the same way as liquids and gases, using the z factor and fugacity (unless the loading on the solid is purely isotropic).  This is because, at equilibrium, a solid is capable of storing elastic energy (involving anisotropic loading and strains) after it is deformed.  As a result, the equilibrium state of an elastic solid cannot be described in terms of only two intensive variables (like it can for a liquid or gas).  In the case of an isotropic elastic solid, you need to specify either (a) temperature and three principal stresses or (b) temperature and 3 principal strains).  So, rather than 2 intensive variables, you need to specify 4 intensive variables.  All the thermodynamic functions, U, H, S, A, and G are functions of these 4 intensive variables.