  The influence of an external magnetic field on the static shear strain and
the effective shear modulus of a magnetoactive elastomer (MAE) is studied
theoretically in the framework of a recently introduced approach to the
single-particle magnetostriction mechanism [V. M. Kalita et al, Phys. Rev. E
93, 062503 (2016)]. The planar problem of magnetostriction in an MAE with soft
magnetic inclusions in the form of a thin disk (platelet) having the magnetic
anisotropy in the plane of this disk is solved analytically. An external
magnetic field acts with torques on magnetic filler particles, creates
mechanical stresses in the vicinity of inclusions, induces shear strain and
increases the effective shear modulus of these composite materials. It is shown
that the largest effect of the magnetic field on the effective shear modulus
should be expected in MAEs with soft elastomer matrices, where the shear
modulus of the matrix is less than the magnetic anisotropy constant of
inclusions. It is derived that the effective shear modulus is non-linearly
dependent on the external magnetic field and approaches the saturation value in
magnetic fields exceeding the field of particle anisotropy. It is shown that
model calculations of the effective shear modulus correspond to a
phenomenological definition of effective elastic moduli and magnetoelastic
coupling constants. Obtained theoretical results compare well with known
experimental data. Determination of effective elastic coefficients in MAEs and
their dependence on magnetic field is discussed. The concentration dependence
of the effective shear modulus at higher filler concentrations has been
estimated using the method of Pad\'e approximants, which correctly predicts
that both the absolute and relative changes of the magnetic-field dependent
effective shear modulus will significantly increase with the growing
concentration of filler particles.
