Flexoelectric effect in ceramic lead zirconate titanate

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Flexoelectric Effect in Solids (2013, methodology): “24 Zubko, Catalan, Tagantsev and was recently used by Ma and Cross to systematically quantify the flexoelectric response in a number of perovskite ceramics [47–51].”
Piezoelectric thin-film superlattices without using piezoelectric materials (2010, methodology): “posed as h ijkl 2 P i, j u k,l + h ijkl 2 P i, j u k,l ⎛ ⎝ ⎜ ⎞ ⎠ ⎟dx V ∫ . (17) We employ integration by parts to yield h ijkl 2 P i, j u k,l − h ijkl 2 P i u k, jl ⎛ ⎝ ⎜ ⎞ ⎠ ⎟dx V ∫ +Boundary terms. (18) Thus an energy density of the following form can be recovered h ijkl 2 P i, j u k,l −P i u () k”
Fundamentals of flexoelectricity in solids (2013, background): “Later, the flexoelectric response was characterized in a number of ferroelectric ceramics [30–34] and single crystals [22, 35–40], using direct (polarization response of a strain gradient in finite samples) or indirect (from phonon spectra or data on thermodynamically linked effects) methods.”