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

physical review. b./physical review. b

The influence of elastic strain gradient on the upper limit of  flexocoupling strength, spatially-modulated phases and soft phonon dispersion  in ferroics