  By introducing a realistic model of nanogranular superconductors (NGS) based
on 2D arrays of Josephson nanocontacts (created by a network of twin-boundary
dislocations with strain fields acting as insulating barriers between hole-rich
domains), in this Chapter we present some novel phenomena related to
mechanical, magnetic, electric and transport properties of NGS in underdoped
single crystals. In particular, we consider chemically induced magnetoelectric
effects and flux driven temperature oscillations of thermal expansion
coefficient. We also predict a giant enhancement of the nonlinear thermal
conductivity of NGS reaching up to 500% when the intrinsically induced
chemoelectric field (created by the gradient of the chemical potential due to
segregation of hole producing oxygen vacancies) closely matches the externally
produced thermoelectric field. The estimates of the model parameters suggest
quite an optimistic possibility to experimentally realize these promising and
important for applications effects in non-stoichiometric NGS and artificially
prepared arrays of Josephson nanocontacts.
