Marc Kamlah

Finite element analysis of piezoceramic components taking into account ferroelectric hysteresis behavior

Abstract A simplifying macroscopic constitutive law for ferroelectric and ferroelastic hysteresis effects of piezoceramics is presented. After summarizing the uniaxial formulation motivated elsewhere (Kamlah, M., Tsakmakis, C., 1999. Int. J. Solids Struct. 36, 669–695; Kamlah, M., Bohle, U., Munz, D., Tsakmakis, Ch., 1997. Smart Structures and Materials 1997: Mathematics and Control in Smart Structures, Proceedings of SPIE, vol. 3039, 144–155), it is generalized to a three-dimensional tensorial formulation. The model has been implemented in the public domain finite element code PSU of Stuttgart University. The finite element analysis is carried out in a two-step scheme: First the purely dielectric boundary value problem is solved for the history of the electric potential. Second, prescribing this electric potential, the electro-mechanical stress analysis for the mechanical boundary conditions yields the electro-mechanical fields as, for instance, the mechanical stress field. In order to verify the capabilities of our tool, a multilayer-like actuator geometry is analyzed. It is shown that the remanent polarization remaining after poling gives rise to a non-vanishing distribution of the electric potential even it is reduced to zero at the electrodes. Concerning the residual stresses present after poling, a tensile stress field perpendicular to the direction of the electrodes can be found in the passive region of the actuator where so-called poling cracks are known to occur. It is concluded that our finite element tool is suitable for studying the influence of geometry and material parameters on the stresses in critical regions of piezoceramic devices.

Constraint-induced crack initiation at electrode edges in piezoelectric ceramics

A strain incompatibility arises between the electrically active and inactive parts of a partially electroded piezoelectric material. This leads to cracking perpendicular to the electrode edge. Symmetric partial electrodes of different widths were applied to lead zirconate titanate (PZT) plates to study the development of these cracks. Different cracking patterns appeared upon application of an electric field, depending on the degree of coverage and on the thickness of the plates. The apparent coercive fields are correlated to the degree of clamping and the amount of cracking in the specimens. Non-linear finite element modelling is used to analyse the material response.

Rate dependence of soft PZT ceramics under electric field loading

Polarization and longitudinal strain of the commercial soft PZT piezoceramic PIC151 were measured as a function of amplitude and frequency of an AC electric field. The range of frequencies considered was selected in the quasi-static range from 0.01Hz to 1.0 Hz. The electric field was selected as triangular loading. Besides the standard hysteresis loops for polarization (P) and strain (S) versus electric field (E), strain versus polarization curves (S-P curves) were plotted in separate diagrams. It was shown that both polarization and strain were frequency dependent. The coercive field increased with the loading frequency. Furthermore, a significant hysteresis was observed for S-P curves at a loading frequency below 1Hz. At a frequency of 1Hz, however, the S-P plots were nearly close to an idealized parabolic curve without hysteresis. A tentative explanation shall be given for these observations in terms of the rate effects of the domain switching process.

Size-dependent polarization distribution in ferroelectric nanostructures: Phase field simulations

From phase field simulations, we investigate the size-dependent polarization distribution in ferroelectric nanostructures embedded in a nonferroelectric medium. The simulation results exhibit that vortex structures of polarizations and single-domain structures are formed in ferroelectric nanodots and nanowires, respectively. Furthermore, a single-vortex structure is formed in the ferroelectric nanodots if the aspect ratio of thickness to lateral size is less than a critical value, whereas the ferroelectric nanodots are in a multivortex state if the aspect ratio exceeds the critical value. When the aspect ratio approaches infinity, nanodots will become nanowires, in which polarizations are homogeneous.

Three-dimensional finite element modeling of polarization switching in a ferroelectric single domain with an impermeable notch

Ferroelectric materials often possess defects, such as notches and voids. They are also brittle and susceptible to cracking at all scales ranging from electric domains to devices. Under mechanical and electrical loadings, the intensified stresses and electric fields in the vicinity of notches or voids may cause polarization switching in ferroelectric materials. Due to the polarization switching, the linear piezoelectric fracture mechanics cannot predict the nonlinear fracture behavior of ferroelectrics under mechanical and electrical loadings. To investigate the polarization switching and the nonlinear behavior of ferroelectrics, a three-dimensional nonlinear finite element model for ferroelectric materials is developed based on a principle of virtual work. The finite element model includes three physical fields, namely the polarization field, electric field and strain field, and the coupling between them. The developed finite element model is employed to investigate the polarization distribution near an impermeable notch in a ferroelectric single domain subjected to mechanical and electrical loadings. It is found that the polarization switching takes place near the notch tip if the mechanical loadings exceed a critical value. Furthermore, the simulation results show that an electric field parallel to the initial polarization increases the critical value while an electric field antiparallel to the initial polarization decreases it. In the simulation, we do not make any prior assumptions, i.e. without any switching criterion, on the polarization switching. The polarization switching is a result of the minimization of the total energy in the simulated ferroelectrics.

Phase-field modeling of stress generation in electrode particles of lithium ion batteries

Many cathode materials in lithium ion batteries show capacity fade due to particle crackings even at low applied charge and discharge current (C-rate). The promising candidate material LixMn2O4 exhibits such effects on the 4 V-plateau when the state of charge 0<x<1, hence the crystalline host remains cubic spinel. Single particle failure in this range of values indicates the existence of very large stresses, which may occur due to phase segregation. In this letter, we employ a phase-field model coupled to mechanics to demonstrate the relationship between phase segregation and high values of the mechanical stresses thus explaining numerous experimental results.

High-field dielectric and piezoelectric performance of soft lead zirconate titanate piezoceramics under combined electromechanical loading

Piezoelectric actuators normally have complicated structures and work under severe loading conditions, e.g., high driving electric field and significant compressive preload. This study is focused on the experimental investigation of the electromechanical properties of a commercial soft lead zirconate titanate material under loading conditions simulating the in-service environment of high-strain actuators. The polarization and strain responses were first measured under a constant-stress preload. A significant enhancement of the dielectric and piezoelectric performance is observed within a small prestress range. At much higher preload levels, the predominant mechanical depolarization effect makes the material exhibits hardly any piezoeffect. In the other two series of tests, the specimen was subjected to cyclic mechanical load with different mean stresses and amplitudes. When the stress is applied in-phase with electrical loading, the polarization and strain outputs are found to monotonically decrease with ...

Determination of room-temperature creep of soft lead zirconate titanate piezoceramics under static electric fields

This study focuses on the experimental investigation of the time-dependent effects of a commercial soft lead zirconate titanate material at room temperature. Samples in initially unpoled states were subjected to a cyclic stepwise electric field which was kept constant at different levels for 300 s. Due to ferroelectric domain switching, significant nonlinearity and hysteresis were observed in the overall polarization and strain response. In particular, the material exhibited creep behavior as the applied electric field was held constant over extended periods of time. This creep was caused by microscopic domain switching processes induced gradually during the holding time. The creep was of primary or transient type in nature and depended strongly on the magnitude of the load applied. Most pronounced creep was observed when holding the field close to the coercive field. Logarithmic representation of the polarization or strain versus time curves indicated that the creep behavior could be quantified approxima...

Phase field simulations of ferroelectric nanoparticles with different long-range-electrostatic and -elastic interactions

Two-dimensional phase field simulations of ferroelectric nanoparticles with different long-range (LR)-electrostatic and -elastic interactions and different domain wall energy densities are conducted based on the time-dependent Ginzburg–Landau equation. The phase field simulations exhibit vortex patterns of polarizations, which have purely toroidal moments of polarizations and macroscopically negligible averaged polarizations, in nanoparticles without or with weak LR-elastic interactions when LR-electrostatic interactions are fully taken into account. However, a single-domain structure without any toroidal moment of polarizations is formed in small nanoparticles if LR-electrostatic interactions are completely ignored or LR-elastic interactions are fully taken into account. The polarization gradient energy or domain wall energy density plays also an important role in the formation of polarization structure. The vortex structure transits from a multivortex structure to a single-vortex structure as the domain...

Domain structures of ferroelectric nanotubes controlled by surface charge compensation

Domain structures in ferroelectric nanotubes (FNTs) under different electrical boundary conditions are predicted through a phase field model. Simulation results show that domain structures are highly dependent on the compensation of polarization-induced surface charges. In order to reduce the depolarization energy, polarizations in FNTs form a vortex structure under an open-circuit boundary condition. When surface charges are compensated on the inner and outer surfaces, a multidomain structure is formed in FNTs as a result of competition between the long-range electrostatic and elastic interactions. However, a single-domain structure is energically favorable in FNTs if the upper and lower surfaces are short-circuited.