Phase-field simulation of piezoresponse force microscopy in consideration of different environmental conditions

Piezoresponse force microscopy (PFM) is a feasible tool which is widely used for investigating information of the domain structures of ferroelectrics. Nevertheless, one drawback of the technique may be that environmental conditions could effect the very small signal which is detected from the displacement of the tip. The present contribution addresses the simulation of PFM in consideration of environmental conditions. We employ a continuum-mechanical model based on the phase-field method which accounts for the transversely isotropic symmetry of the underlying material. The goal of this contribution is to analyze the environmental effect on the tip-sample interaction. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-field simulation of the electro-mechanical response of ferroelectrics during Piezoresponse Force Microscopy

The contribution adresses the simulation of ferroelectric matrials in the framework of the Piezoresponse Force Microscopy (PFM). Based on the PFM, ferroelectric domain structures can be analyzed in great detail by measuring the electrically induced mechanical deformations of the surface of a ferroelectric. We employ a flexible continuum-mechanical model based on the phase-field method in order to analyze the behavior of ferroelectric microstructures numerically. Since ferroelectric materials are often highly anisotropic, the phase-field formulation will account for transversely isotropic symmetry. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of domain structures in cracked ferroelectric materials

The domain structure around a crack tip plays a significant role in the fracture behavior of ferroelectrics. A continuum phase field model is used to investigate the microstructure at the crack front. The concept of the Eshelby momentum tensor and configurational forces is then generalized to account for the contributions of the polarization term. Implementation of the generalized configurational force in the Finite Element code enables us to numerically obtain the driving force at the crack tip, which corresponds to the crack-tip energy release rate. Calculations show that additional positive electric fields tend to prohibit crack growth, whereas additional negative electric fields tend to promote crack growth. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of nanoindentation experiments by means of atomic force microscopy

Nanoindentation is quite a common method for local material characterization. Values for hardness and Young's modulus can be determined directly from the recorded data. Essential for the correct determination of the material parameters is the precise measurement of the actual indentation depth of the indenter. The indenter measures the current depth by means of a Wheatstone bridge which correlates the indentation depth to a change in voltage. A possible tool for the verification of the recorded indentation depths is Atomic Force Microscopy (AFM). AFM is able to scan an area of indents for almost any surface. The deflection of the tip is measured by a laser spot reflected from the surface of the cantilever. The difference in height between the surface and the indent can directly be read off from the plotted image. However, using an AFM only allows us to measure the depth of the permanent indentation depth after unloading the indenter. Nevertheless, correlation between the remaining indentation depths measured by the explained methods allows for a first assessment of the correctness of the online recorded depth-data by the nanoindenter. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of Microstructural Components of Bitumen by Means of Atomic Force Microscopy (AFM)

Accounting for the large variation of asphalt mixes, resulting from variations of constituents and composition, and from the allowance of additives, a multiscale model for asphalt is currently developed at the Christian Doppler Laboratory for “Performance‐based optimization of flexible road pavements”. The multiscale concept allows to relate macroscopic material properties of asphalt to phenomena and material properties of finer scales of observation. Starting with the characterization of the finest scale, i.e., the bitumen‐scale, Atomic Force Microscopy (AFM) is employed. Depending on the mode of measurement (tapping versus pulsed‐force mode), the AFM provides insight into the surface topography or stiffness and adhesion properties of bitumen. The obtained results will serve as input for upscaling in the context of the multiscale model in order to obtain the homogenized material behavior of bitumen at the next‐higher scale, i.e., the mastic‐scale. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On domain switching in deformable ferroelectrics, seen as continua with microstructure

We obtain a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by a spontaneous polarization. This is accomplished by assuming the body as comprised of a continuum with vectorial microstructure: in each point of the body therefore a gross and a fine structure are superposed, the gross structure representing a non linear polarizable elastic body and the vectorial fine structure describing the spontaneous polarization.¶Among the distinctive features of ferroelectric materials, the most interesting is represented by the organization of spontaneous polarization into a domain structure, which minimizes electrostatic energy and which can be modified by the application of electric and deformation fields. This process, called "polarization reversal" or "domain switching", is associated with various hysteresis loops, the most typical being those between spontaneous polarization and electric field (dielectric hysteresis), and between strain and electric field ("butterfly" loop).¶From the balance laws of continua with vectorial microstructure and dissipation inequality we arrive at the evolution equation for the spontaneous polarization which allows for both inertial and dissipative terms and describes domain switching. We postulate a simple interaction mechanism between the spontaneous polarization and the pair electric field, deformation and arrive at, in the quasi-static case, to a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics.¶For linearized kinematics we study, in the one-dimensional case, stable relative minimizers and give a simple justification for dielectric hysteresis and butterfly loops: under the hypothesis that the domain walls are sharp interfaces, the solutions we find explain the banded twin domains morphology which is typical of many ferroelectrics.     

A finite element phase field model for relaxor ferroelectrics

A mechanically coupled phase field model is presented for the domain evolution and mesoscopic response of relaxor ferroelectrics. In the model the spontaneous polarization is treated as order parameter. The model is derived from thermodynamic analysis including the material force theory. Random field theory is adopted to take the disorder of relaxor ferroelectrics into account. Results show that the model is capable of reproducing relaxor features, such as domain miniaturization, small remnant polarization and large piezoelectric response. Dependence of these features on the random field strength is discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.     

Phenomenological theory of surface effects in ferroelectrics and its relationship with transverse Ising model

The relationship between the transverse field Ising model and the Landau phenomenological theory for ferroelectrics is analyzed, and the Landau free energy expression for ferroelectrics having surfaces is derived. It is pointed out that the traditional expression in which the surface integral has only a term of the square polarization is valid only for special cases, in general a term of the polarization to the four should be included as well. By use of the newly derived free energy expression, the thickness-dependence of the spontaneous polarization and Curie temperature of ferroelectric films is calculated; thereby some experimental results incompatible with the traditional phenomenological theory are successfully explained.     

Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin-Huxley neurons

An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction-diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin-Huxley neurons is measured by imposing an additive term V_E on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 × 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive.     

A combined molecular dynamics-phase-field modelling approach to fracture

In order to better understand and ease the determination of material and model parameters required for the macroscopic modelling of brittle fracture, a bottom-up comparative study between molecular dynamics (MD) simulations and the continuum phase-field modelling (PFM) is carried out. In particular, based on the MD simulations of fracture of a highly brittle material, a number of PFM parameters such as the width of the transition zone between the damaged and the undamaged material, the crack resistance and the elasticity modulus are estimated. This study opens the door for an efficient way for multi-scale modelling of fracture. To illustrate this approach, a comparative two-dimensional numerical initial-boundary-value problem (IBVP) for the highly brittle aragonite (CaCO₃) is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.     

Modeling of Electromechanical Contact on a Microscopic Lengthscale

This paper treats the simulation of the contact mode of an atomic force microscope (AFM). The contact forces between the tip and the sample surface is calculated via the van derWaals force in a Finite Element approach, the integration of the contact force a FEM environment is discussed and numerical examples are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear dynamics of tapping mode atomic force microscopy in the bistable phase

Nonlinear dynamics of amplitude modulation atomic force microscopy (AFM) is studied employing a reduced-order model based on a differential quadrature method (DQM). The AFM microcantilever is assumed to be operating in the dynamic contact or tapping mode while the microcantilever tip being initially located in the bistable region. We have found that the DQM is capable of precise prediction of the static bifurcation diagram and natural frequencies of the microcantilever. We have used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region.     

A constitutive and damage model for high cycle fatigue of tetragonal ferroelectrics

Reliability and life time of smart materials are crucial features for the development and design of actuator and sensor devices. Being widely used and exhibiting brittle failure characteristics, ceramic ferroelectrics are of particular interest in this field. Due to manifold interactions of the complex nonlinear constitutive behavior on the one hand and the damage evolution in terms of microcrack growth on the other, modeling and simulation are inevitable to investigate influence parameters on strength, reliability and life time. A condensed approach is used for the simulations considering just one characteristic point in the material, nonetheless accounting for polycrystalline grain interactions. On this basis, a model to predict the life time in terms of high cycle fatigue under electromechanical loading conditions is introduced. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of hydraulic fracture of porous materials using the phase-field modeling approach

In this contribution, the numerical simulation of hydraulic fracture of fluid-saturated porous materials is carried out on a continuum-mechanical scale using the theory of porous media (TPM), extended by a phase-field modeling (PFM) approach. Following this, behaviors such as crack nucleation and propagation, solid matrix deformation and interstitial-fluid flow change from Darcy to Stokes-like flow in the cracked region can be realized. Moreover, permanent changes of the local physics due to occurrence of the crack, such as of the volume fractions and the permeability, are taken into consideration. The mathematical modeling of this problem yields a strongly coupled system of differential algebraic equations (DAE). Thus, special descretization schemes for a stable and efficient solution are needed. To reveal the ability of the proposed model to simulate the important features of hydraulic cracking, a two-dimensional example using the finite element method is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Layered elastic solid method for the generation of unstructured dynamic mesh

The layered elastic solid method (LESM), a modified elastic solid method (ESM) was put forward in the present study. In LESM, the computational zone is divided into several layers and material properties of these layers, which stay a certain value in ESM, are changed with mesh deformation. The deformation capability of mesh in LESM is better than ESM and the quality of the mesh generated by LESM is superior to that generated by ESM when they undergo the same deformation. In ESM, the main influence factors on mesh quality and deformation capability are Poisson’s ratio and single-step rotation angle. In LESM, mesh quality and deformation capability reach a largest value with an increase in Young’s modulus. Meanwhile, the mesh can achieve a larger deformation capability when single-step rotation angle is 0.25°. Finally, numerical simulation on a two-dimensional aerofoil using LESM was carried out. It is found that the results of LESM show a better agreement with experiment results.     

Variational treatment of electrostatic interaction force in atomic force microscopy

In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips.      

A strip yield model solution for an internally cracked piezoelectric strip

An analysis of the crack closure and fatigue crack growth rate have been carried out for an infinitely long poled piezoelectric ceramic strip weakened by a straight hair line internal crack. The ceramic under consideration is assumed to be mechanically more brittle. The crack faces are perpendicular to the poled direction of the strip. The crack faces open in Mode-I deformation on account of in-plane tension applied to the edges of the strip together with either an in-plane electric displacement prescribed on edges of the strip or a uniform constant electric field prescribed on its edges. As a result, a yield zone is formed ahead of each tip of the crack. The yield zones developed are then arrested by applying a normal, cohesive, linearly varying yield point-stress to their rims. For each case, the Fourier transform method is used to find a solution. The resulting integral equations are solved numerically. Expressions are derived for the crack opening displacement and the crack growth rate. The variations in these quantities are plotted in relation to the affecting parameters, viz., the strip thickness, the yield zone length, the electric displacement, and material constants. A case study is presented graphically for PZT-4, PZT-5H, and BaTiO₃ ceramics. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 647–664, September–October, 2008.