Strain Gradient Finite Element Formulation of Flexoelectricity in Ferroelectric Material Based on Phase-Field Method

Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics,regardless of point group symmetry.However,the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect.In this study,we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect.A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions.The microstructure evolution of domains is described by an independent order parameter,namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory.The model is developed based on a thermodynamic framework,in which a set of microforces is introduced to construct the constitutive relation and evolution equation.For the flexoelectric part of electric enthalpy,the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space.The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution.The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids.A comparison is made between the polarization distribution in dielectrics and ferroelectrics.The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube,and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect.

Evolution of domain structure in 0.96(K_(0.48)Na_(0.52))(Nb_(0.96)Sb_(0.04))O_(3)0.04(Bi_(0.50)Na_(0.50))ZrO_(3) ceramics with poling and temperature

Domain structure often has significant influences on both piezoelectric properties and piezoelectric temperature stability of a ferroelectric ceramic.In-depth studies on the characters of domain structure should be helpful for the better understanding of piezoelectric performance.In this work,the evolution of domain structure in large-d_(33)0.96(K_(0.48)Na_(0.52))(Nb_(0.96)Sb_(0.04))O_(3)-0.04(Bi_(0.50)Na_(0.50))ZrO_(3) ceramics with poling and temperature was systematically investigated via comparing the various domain patterns that are obtained by acid-etching.It was found that domain structure changes greatly upon poling and varies largely with temperature.Complex domain patterns consisting of long narrow parallel stripes or herringbone structure separated by 180°domain boundaries are observed in the unpoled ceramics at room temperature.Domain patterns become less complicated upon poling,due to the collective polarization reversals of parallel-stripe domain clusters and banded fine-stripe domain segments.Parallel stripes and herringbone bands become much wider upon poling,as some narrow stripes and herringbone bands coalesce into broad ones,respectively.Hierarchical domain structure is commonly seen in the domain patterns acid-etched at room temperature,but is less frequently recognized at elevated temperatures.Schematic models of domain configurations were proposed to explain the domain structure and its evolution with poling.

HIGH PIEZOELECTRIC PERFORMANCE AND RELEVANT PHYSICAL MECHANISM OF CuO-MODIFIED Ba(Ti_(0.96)Sn_(0.04))O_(3)CERAMICS

Ba(Ti_(0.96)Sn_(0.04))O_(3)and CuO-modified Ba(Ti_(0.96)Sn_(0.04))O_(3)ceramics were prepared by the solid state reaction technique.Their piezoelectric properties were investigated and compared with those of the recently obtained high-d_(33)BaTiO_(3)ceramic.It has been found that simply substituting Ti4t with Sn4t worsens severely the piezoelectric properties whereas a combined usage of CuO additive greatly improves the overall piezoelectric performance.CuO-modified BaeTi_(0.96)Sn_(0.04)TO_(3)ceramic shows excellent piezoelectric properties of d_(33)=390 pC=N;kp=0.49 and k33=0.67 at room temperature.Furthermore,it possesses weak temperature dependences of electromechanical coe±cients between
20 and 55 and a good thermal aging stability down to a low experimental temperature limit of
50℃and up to 90.Detailed analysis suggests that its high piezoelectric performance should be largely ascribed to the ideal microstructure of high relative density and small grains and the corresponding domain configurations.