  The Ising one-dimensional (1D) chain with spin $S=1/2$ and magnetoelastic
interactions is studied with the lattice contribution included in the form of
elastic interaction and thermal vibrations simultaneously taken into account.
The magnetic energy term and the elastic (static) energy term based on the
Morse potential are calculated exactly. The vibrational energy is calculated in
the Debye approximation, in which the anharmonicity is introduced by the
Gr{\"u}neisen parameter. The total Gibbs potential, including both the magnetic
field, as well as the external force term, is constructed and from its minimum
the equation of state is derived.
  From the Gibbs energy all the thermodynamic properties are calculated in a
self-consistent manner. The comprehensive numerical calculations are performed
in a full temperature range, i.e., from zero temperature up to the vicinity of
melting. In particular, a role of magneto-elastic coupling is emphasized and
examined. The numerical results are illustrated in figures and discussed.
