Flexoelectricity, strain gradients, and singularities in ferroelectric nanostructures

The effect of flexoelectricity on the formation and evolution of domain structures in ferroelectric materials is developed by integrating strain gradient theory into a finite element phase field model. Length scales associated with elastic strain gradients and the corresponding polarization gradient across a domain wall are integrated into the time-dependent Ginzburg–Landau theory and numerical simulated. Theoretical relations of a shear strain gradient along an electrode/dielectric interface are first solved and verified numerically using the finite element model. A singularity in the strain gradient–induced polarization is shown to occur as the elastic strain gradient length scale approaches a flexoelectric length scale. The theory and finite element modeling is then extended to quantify strain gradient electromechanics near 180° and 90° tetragonal phase domain structures. It is shown that the strain gradient length scale [Formula: see text] strongly influences changes in strain across the domain walls but has a negligible effect on the polarization. Strain gradient effects become negligible when [Formula: see text], where [Formula: see text] is the polarization domain wall length scale.