  Within the framework of Landau-Ginzburg-Devonshire (LGD) theory we studied
the role of the flexocoupling between the order parameter and elastic strain
gradients in the stability of a spatially-modulated phase (SMP) in ferroics
with commensurate and incommensurate long-range ordered phases under the
presence of squired elastic strain gradient. The squired elastic strain
gradient is required for the free energy stability to arbitrary strain
gradients. Obtained analytical expressions showed that the fundamental upper
limit for the magnitude of the static bulk flexoelectric effect strength,
established by Yudin and Tagantsev under the absence of squired elastic strain
gradient and higher order gradients terms, should be substituted by the
temperature-dependent condition on the flexoelectric coupling strength under
the presence of the gradient terms. Moreover, we established that the SMP
appears and becomes stable in commensurate ferroics if the flexocoupling
constant exceeds the critical value, defined by the reduced temperature, strain
and order parameter gradients constants, striction and expansion coefficients
in the LGD functional. We calculated the soft phonon dispersion in ferroics
with commensurate and incommensurate long-range ordered phases allowing for the
squired elastic strain gradient, as well as static and dynamic flexocoupling.
Appeared that the dispersion for the optic mode is slightly sensitive to the
flexocoupling, while the dispersion of acoustic mode strongly depends on the
coupling strength. Obtained results demonstrate that the non-trivial
differences in the dispersion of optic and acoustic modes appear under the
change of flexocoupling constant.
