Remanent Strain Research Articles

The influence of crack face electrical boundary conditions on the nonlinear behavior of ferroelectric single crystal

The nonlinear electromechanical behavior of a cracked ferroelectric single crystal subjected to pure electrical loading is investigated by a three-dimensional phase field model for different crack face electrical boundary conditions. Phase field simulations show that crack face electrical boundary conditions have significant influence on the electrical and mechanical responses of the ferroelectric single crystal to an external electric field. The coercive field for the polarization switching of a single crystal with an electrically permeable crack is about 50% larger than that for a single crystal with an electrically impermeable crack. The remanent strain and the strain variation induced by polarization switching in a single crystal with a permeable crack are larger than those with an impermeable crack. The different macroscopic nonlinear behaviors are attributed to different polarization switching processes. It is found that domain switching takes place from the surface of a single crystal with a permeable crack, while it begins from the vicinity of the crack tip when the crack is impermeable. A ferroelectric single crystal with an impermeable crack exhibits strip 90° domain switching under a negative electric field, which is consistent with experimental observation.

Nonlinear stress-strain behavior and stress-induced phase transitions in soft Pb(Zr1−xTix)O3at the morphotropic phase boundary

Temperature-dependent stress-strain behavior of various soft Pb${}_{0.98}$Ba${}_{0.01}$(Zr${}_{1\ensuremath{-}x}$Ti${}_{x}$)${}_{0.98}$Nb${}_{0.02}$O${}_{3}$, $0.40\ensuremath{\le}x\ensuremath{\le}0.60$, compositions was characterized between 25 and 400 \ifmmode^\circ\else\textdegree\fi{}C to determine the influence of crystal structure on ferroelasticity. The ferroelastic response was found to depend significantly on the crystal phase as well as the spontaneous strain, both of which varied with temperature. The remanent strain for rhombohedral materials was shown to be above the theoretical maximum allowed by the crystal spontaneous strain. This observed behavior indicates the presence of hysteretic processes in addition to ferroelasticity during mechanical compression. A phenomenological free energy analysis was used to predict the effects of stress on the stable phase in ferroelectrics and indicates the susceptibility of the rhombohedral and tetragonal structure to a stress-induced phase transition. Modeling results indicate the relative importance of such phase transitions on macroscopic stress-strain behavior, giving an indirect method to observe field-induced phase transitions in polycrystalline ferroelectrics.

Modelling of hysteretic behaviour of piezoceramic materials under external electrical and mechanical stress

The purpose of this study is to model the non-linear hysteretic behaviour of piezoceramic materials under an external (mechanical or electrical) stress. The model is constructed by bridging the characteristics of microscopic domain distribution into the macroscopic behaviour. Choosing the remanent strain and remanent polarisation as internal variables, a domain orientation distribution is used to describe the evolution of these variables and to develop a phenomenological model of ferroelasticity for electromechanical loading histories. This model is able to reproduce the (longitudinal and transversal) strain and electric displacement in function of uniaxial electrical and mechanical loading. This formulation is validated through a comparison with experimental data of the literature. The evolution of the elastic and piezoelectric constants under mechanical stress is successfully compared as well.

Ultra-low power electrically reconfigurable magnetoelectric microwave devices

We present an electrical tunable spiral inductor design and simulation results. By integrating (011) oriented single crystal ferroelectric substrate [Pb(Mg1/3Nb2/3)O3](1−x)-[PbTiO3]x (PMN-PT, x ≈ 0.32) with tunable and switchable remanent strain property, the tunability of such magnetoelectric microwave inductor can retain after releasing the electric actuation, i.e., electrically reconfigurable. A series of reconfigurable passive magnetoelectric microwave devices based on this concept are proposed and discussed. Those electrically reconfigurable magnetoelectric microwave devices have the advantage of low power consumption and fast and wide tunability, which can be potentially used for many applications.

Effect of the electric conductivity on the modeling of the poling process of ferroelectric components

Piezoceramic materials, despite being semiconductors with a high but finite ohmic resistance, are generally modeled as perfect insulators. In this work the influence of the electric conductivity on the poling process of piezoceramic materials is investigated. To solve the electromechanically coupled boundary value problem, a reduced form of the Maxwell equations is implemented inside a hybrid finite element formulation. In this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of the remanent polarization. The material model is fully ferroelectric/ferroelastic coupled, whereas the material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. The simulation of poling processes for three different components are presented. First and second, the poling of a radially poled hollow cylinder and of a stack actuator is investigated. Here, a residual electric field appears after poling, leading to significant time-dependent changes of stresses and deformation due to subsequent charge transportation processes. The third example is a bending actuator composed of two layers of hard-PZT and soft-PZT material. It is shown that considering the electric conductivity new strategies for the poling process can be developed in order to improve the bending actuation. In this way, we show the importance of considering charge transportation processes in simulations of the poling of ferroelectrics, which seems not to have been recognized so far.

Nanoscale Insight Into Lead‐Free BNT‐BT‐xKNN

AbstractPiezoresponse force microscopy (PFM) is used to afford insight into the nanoscale electromechanical behavior of lead‐free piezoceramics. Materials based on Bi1/2Na1/2TiO3 exhibit high strains mediated by a field‐induced phase transition. Using the band excitation technique the initial domain morphology, the poling behavior, the switching behavior, and the time‐dependent phase stability in the pseudo‐ternary system (1–x)(0.94Bi1/2Na1/2TiO3‐0.06BaTiO3)‐xK0.5Na0.5NbO3 (0 <= x <= 18 mol%) are revealed. In the base material (x = 0 mol%), macroscopic domains and ferroelectric switching can be induced from the initial relaxor state with sufficiently high electric field, yielding large macroscopic remanent strain and polarization. The addition of KNN increases the threshold field required to induce long range order and decreases the stability thereof. For x = 3 mol% the field‐induced domains relax completely, which is also reflected in zero macroscopic remanence. Eventually, no long range order can be induced for x >= 3 mol%. This PFM study provides a novel perspective on the interplay between macroscopic and nanoscopic material properties in bulk lead‐free piezoceramics.

Simulation of the shape memory effect in a NiTi nano model system

The shape memory behavior of a NiTi nanoparticle is analyzed by molecular dynamics simulations. After a detailed description of the equilibrium structures of the used model potential, the multi variant martensitic ground state, which depends on the geometry of the particle, is discussed. Tensile load is applied, changing the variant configuration to a single domain state with a remanent strain after unloading. Heating the particle leads to a shape memory effect without a phase transition to the austenite, but by variant reorientation and twin boundary formation at a certain temperature. These processes are described by stress–strain and strain–temperature curves, together with a visualization of the microstructure of the nanoparticle. Results are presented for five different Ni concentrations in the vicinity of 50%, showing for example, that small deviations from this ideal composition can influence the critical temperature for shape recovery significantly.

Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls

In this paper a new material model is proposed for the description of stress-softening observed in cyclic tension tests of collagenous soft tissues such as arterial walls, for applied loads beyond the physiological level. The modeling framework makes use of terms known from continuum damage mechanics and the concept of internal variables introducing a scalar-valued variable for the representation of fiber damage. A principle is given for the construction of damage models able to reflect remanent strains as a result of microscopic damage in the reinforcing collagen fiber families. Particular internal variables are defined able to capture the nature of arterial tissues that no damage occurs in the physiological loading domain. By application of this principle, specific models are derived and fitted to experimental data. Finally, their applicability in numerical simulations is shown by some representative examples where the damage distribution in arterial cross-sections is analyzed.

Configurational forces in a multiscale approach to piezoelectric materials

AbstractDue to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, a two‐scale classical homogenization of an electro‐mechanically coupled material using a FE2‐approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. In the homogenization different representative volume elements (RVEs), which capture the micro‐structure of the inhomogeneous material, are used to represent the macroscopic material response. Two different schemes are considered. In the first case, domain wall movement is not allowed, but in the second case the movement of the domain walls is taken into account using thermodynamic considerations. Later this technique is used to determine the macroscopic and microscopic configurational forces on defects [2]. These defect situations include the driving force on a crack tip. The effect of the applied electric field on configurational forces at the crack tip is investigated. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.

Compositional Dependence of R‐curve Behavior in Soft Pb(Zr1−xTix)O3 Ceramics

The compositional dependent fracture behavior of soft PZT (Pb0.99Ba0.01(Zr1−xTix)0.98Nb0.02O3) ceramics in the vicinity of the morphotropic phase boundary was characterized using compact‐tension specimens. The compositions with 0.40 ≤ x ≤ 0.55 displayed an increasing fracture resistance with crack extension (R‐curve behavior). It was observed that the rhombohedral composition and the compositions near the morphotropic phase boundary showed the largest toughness enhancement. R‐curve behavior was found to be influenced by the ferroelastic coercive stress, saturated remanent strain, and elastic modulus, which were experimentally measured for each composition. X‐ray diffraction measurements were performed and compared to the fracture results to investigate the impact of phase and lattice distortion on ferroelastic toughening behavior.

Inhomogeneity and material configurational forces in three dimensional ferroelectric polycrystals

Abstract Ferroelectrics are polycrystalline materials consisting of multi-grains with their individual orientations. Polarization switching under an applied electric field is predicted by a micromechanical nonlinear constitutive model for ferroelectric polycrystals proposed by Huber et al. [J. Mech. Phys. Solids 47 (1999) 1663]. A three dimensional finite element method is used to simulate the macroscopic mechanical and electrical response over the whole multi-grains body. Moreover, the microscopic behavior inside each separate grain is studied. Due to the heterogeneity highly fluctuating inhomogeneous stresses and strains are observed to be distributed in ferroelectric polycrystal body. High stresses are developed especially around the grain boundaries. This phenomenon is attributed to different lattice orientations of grains as well as grain-to-grain interactions in polycrystalline ceramic. Furthermore, a consistent thermodynamical framework is outlined to develop the theory of material configurational forces which takes into account the contribution of remanent polarization and strain due to domain switching. The conservation laws of configurational forces are concluded, i.e. the summation of three components of configurational forces on all nodes over the whole ferroelectric body is always conserved.

Electrical control of reversible and permanent magnetization reorientation for magnetoelectric memory devices

We report giant reversible and permanent magnetic anisotropy reorientation between two perpendicular easy axes in a magnetoelectric polycrystalline Ni thin film and (011) oriented [Pb(Mg1/3Nb2/3)O3](1−x)-[PbTiO3]x (PMN-PT) heterostructure. The PMN-PT is partially poled prior to Ni film deposition to provide a remanent strain bias. Following Ni deposition and full poling of the sample, two giant remanent strains of equal and opposite values are used to reversibly and permanently reorient the magnetization state of the Ni film. These experimental results are integrated into micromagnetic simulation to demonstrate the usefulness of this approach for magnetoelectric based magnetic random access memory.

Domain engineered switchable strain states in ferroelectric (011) [Pb(Mg1/3Nb2/3)O3](1−x)-[PbTiO3]x (PMN-PT, x≈0.32) single crystals

The ferroelectric properties of (011) [Pb(Mg1/3Nb2/3)O3](1−x)-[PbTiO3]x (PMN-PT, x≈0.32) single crystals with focus on piezoelectric strain response were reported. Two giant reversible and stable remanent strain states and tunable remanent strain properties are achieved by properly reversing the electric field from the depolarized direction. The unique piezoelectric strain response, especially along the [100] direction, mainly stems from the non-180° ferroelectric polarization reorientation in the rhombohedral phase crystal structure. Such giant strain hysteresis with tunable remanent strain properties may be useful for magnetoelectric based memory devices as well as a potential candidate for other applications.

Micromechanical modelling of remanent properties of morphotropic PZT

The present paper investigates the capability of micromechanical material models to predict the ferroelectric behaviour of morphotropic PZT ceramics in a rate-independent approximation based on realistic microscopic material parameters. Starting point is a three-dimensional tetragonal model, which builds on the model of Pathak and McMeeking [2008. Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading. Journal of the Mechanics and Physics of Solids 56, 663–683]. Volume fractions of the crystallographic variants represent the domain structure inside the grains. Interactions between the grains are taken into account by means of a representative volume element of the grain compound. A simplified set of realistic microscopic material parameters of the lattice in terms of Young's modulus, Poisson's ratio, dielectric constant, and spontaneous strain and polarisation is derived from experimental data and theoretical results given in the literature. The simulation of the macroscopic remanent polarisation and strain response due to two load cases shows explicitly that the tetragonal model is not capable to reproduce the behaviour of morphotropic PZT. Therefore, the model is extended by the rhombohedral phase, allowing a mixture of both phases with varying quantities inside the grains. A comparison of our results with experimental data shows a remarkably good agreement, revealing the capability of the extended model.

CuO as a sintering additive for (Bi 1/2Na 1/2)TiO 3–BaTiO 3–(K 0.5Na 0.5)NbO 3 lead-free piezoceramics

CuO as a sintering additive was utilized to explore a low-temperature sintering of 0.92(Bi 1/2Na 1/2)TiO 3–0.06BaTiO 3–0.02(K 0.5Na 0.5)NbO 3 lead-free piezoceramic which has shown a promise for actuator applications due to its large strain. The sintering temperature guaranteeing the relative density of greater than 98% is drastically decreased with CuO addition, and saturates at a temperature as low as ∼930 °C when the addition level exceeds ca. 1 mol.%. Two distinguished features induced by the addition of CuO were noted. Firstly, the initially existing two-phase mixture gradually evolves into a rhombohedral single phase with an extremely small non-cubic distortion. Secondly, a liquid phase induced by the addition of CuO causes an abnormal grain growth, which can be attributed to the grain boundary reentrant edge mechanism. Based on these two observations, it is concluded that the added CuO not only forms a liquid phase but also diffuses into the lattice. In the meantime, temperature dependent permittivity measurements both on unpoled and poled samples suggest that the phase stability of the system is greatly influenced by the addition of CuO. Polarization and strain hysteresis measurements relate the changes in the phase stability closely to the stabilization of ferroelectric order, as exemplified by a significant increase in both the remanent strain and polarization values. Electron paramagnetic resonance (EPR) spectroscopic analysis revealed that the stabilization of ferroelectric order originates from a significant amount of Cu 2+ diffusing into the lattice on B-site. There, it acts as an acceptor and forms a defect dipole in association with a charge balancing oxygen vacancy.

Effect of K0.5Na0.5NbO3 on Properties at and off the Morphotropic Phase Boundary in Bi0.5Na0.5TiO3–Bi0.5K0.5TiO3 Ceramics

Lead-free (1-y)(Bi1/2Na1/2TiO3–xBi1/2K1/2TiO3)–yK0.5Na0.5NbO3 [(1-y)(BNT–xBKT)–yKNN] (0.1≤x≤0.6 and 0≤y≤0.05) ceramics were prepared with compositions both at as well as off the morphotropic phase boundary (MPB) of x=0.2 and y=0. It was found that KNN addition destabilizes a ferroelectric order in BNT–BKT at zero electric-field, resulting in low remanent polarization, remanent strain and effective small signal d33eff [23 pC/N for (x,y)=(0.2,0.05), 60 pC/N for (0.2, 0.02)], while it does not affect the maximum polarisation and strain at high fields and thus enhances large signal piezoelectric coefficient, d33* [200 pm/V for (0.2,0.05), 217 pm/V for (0.2,0.02)]. This effect of KNN was found to be independent from the MPB and thus equally high d33* values were obtained for compositions at and far off the MPB. Thus, adding KNN to BNT–BKT removes the restriction to remain at exactly MPB position and provides more flexibility while offering equal or even improved strain.

Effect of K0.5Na0.5NbO3on Properties at and off the Morphotropic Phase Boundary in Bi0.5Na0.5TiO3–Bi0.5K0.5TiO3Ceramics

Lead-free (1-y)(Bi1/2Na1/2TiO3–xBi1/2K1/2TiO3)–yK0.5Na0.5NbO3 [(1-y)(BNT–xBKT)–yKNN] (0.1≤x≤0.6 and 0≤y≤0.05) ceramics were prepared with compositions both at as well as off the morphotropic phase boundary (MPB) of x=0.2 and y=0. It was found that KNN addition destabilizes a ferroelectric order in BNT–BKT at zero electric-field, resulting in low remanent polarization, remanent strain and effective small signal d33eff [23 pC/N for (x,y)=(0.2,0.05), 60 pC/N for (0.2, 0.02)], while it does not affect the maximum polarisation and strain at high fields and thus enhances large signal piezoelectric coefficient, d33* [200 pm/V for (0.2,0.05), 217 pm/V for (0.2,0.02)]. This effect of KNN was found to be independent from the MPB and thus equally high d33* values were obtained for compositions at and far off the MPB. Thus, adding KNN to BNT–BKT removes the restriction to remain at exactly MPB position and provides more flexibility while offering equal or even improved strain.

Electric-poling-induced magnetic anisotropy and electric-field-induced magnetization reorientation in magnetoelectric Ni/(011) [Pb(Mg1/3Nb2/3)O3](1-x)-[PbTiO3]x heterostructure

This study reports the influence of poling a PMN-PT single crystal laminated structure on the magnetic properties of a 35 nm polycrystalline Ni thin film. During the poling process, a large anisotropic remanent strain is developed in the PMN-PT that is transferred to the ferromagnetic film creating a large predefined magnetic anisotropy. Test results show that operating the PMN-PT substrate in the linear regime following poling produces sufficient anisotropic strain to reversibly reorient the magnetization toward an easy axis oriented 90° to the magnetic easy axis induced during poling. The influence of poling prestress on the magnetic anisotropy field, coercive field and magnetic remanence is discussed.

A rate-dependent incremental variational formulation of ferroelectricity

This paper presents a variational-based modeling and computational implementation of the non-linear, rate-dependent response of piezoceramics under electro-mechanical loading. The point of departure is a general internal variable formulation that describes the hysteretic electro-mechanical response of the material as a standard dissipative solid. Consistent with this type of dissipative continua, we develop a variational formulation of the coupled electro-mechanical boundary-value-problem based on incremental potentials for the stresses and the electric displacement. We specify the variational formulation to a model that describes time-dependent, electric polarizations accompanied by remanent strains. It is governed by a dual dissipation function formulated in terms of the internal driving forces. The model reproduces experimentally observed dielectric and butterfly hystereses, which are characteristic for ferroelectric materials. It accounts for the rate-dependency of the hystereses and the macroscopically non-uniform distribution of the polarization in the solid. An important aspect of our treatment is the numerical implementation of the coupled problem. The monolithic discretization of the two-field problem appears, as a consequence of the proposed variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a spectrum of benchmark problems.