  The mixed spin-1/2 and spin-S Ising model on a decorated planar lattice
accounting for lattice vibrations of decorating atoms is treated by making use
of the canonical coordinate transformation, the decoration-iteration
transformation, and the harmonic approximation. It is shown that the
magnetoelastic coupling gives rise to an effective single-ion anisotropy and
three-site four-spin interaction, which are responsible for the anomalous spin
frustration of the decorating spins in virtue of a competition with the
equilibrium nearest-neighbor exchange interaction between the nodal and
decorating spins. The ground-state and finite-temperature phase diagrams are
constructed for the particular case of the mixed spin-1/2 and spin-1 Ising
model on a decorated square lattice for which thermal dependencies of the
spontaneous magnetization and specific heat are also examined in detail. It is
evidenced that a sufficiently strong magnetoelastic coupling leads to a
peculiar coexistence of the antiferromagnetic long-range order of the nodal
spins with the disorder of the decorating spins within the frustrated
antiferromagnetic phase, which may also exhibit double reentrant phase
transitions. The investigated model displays a variety of temperature
dependencies of the total specific heat, which may involve in its magnetic part
one or two logarithmic divergences apart from one or two additional round
maxima superimposed on a standard thermal dependence of the lattice part of the
specific heat.
