A theory of flexoelectricity with surface effect for elastic dielectrics

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Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints (2012, background): “Recently, a comprehensive model has been developed for dielectric nanomaterials by Shen & Hu (2010) with consideration of surface effects, flexoelectricity and electrostatic forces.”
Effects of surface and flexoelectricity on a piezoelectric nanobeam (2014, background): “The theories of electric field gradient and flexoelectricity with surface effect for nano-dielectrics [37, 46] provided the fundamental physical basis for investigating the flexoelectric effect.”
Electric Field Gradient Theory with Surface Effect for Nano-Dielectrics (2009, background): “Shen and Hu (2010), and Hu and Shen (2010) developed a theory for nanosized dielectrics with the flexoelectric effects as well as the surface effects.”