Mechanical Boundary Conditions Research Articles

Polarization states and dielectric responses of elastically clamped ferroelectric nanocrystals

Polarization states and physical properties of ferroelectrics depend on the mechanical boundary conditions due to electrostrictive coupling between electric polarization and lattice strains. Here, we describe theoretically both equilibrium thermodynamic states and electric permittivities of ferroelectric nanocrystals subjected to the elastic three-dimensional (3D) clamping by a surrounding dielectric material. The problem is solved by the minimization of a special thermodynamic potential that describes the case of an ellipsoidal ferroelectric inclusion embedded into a linear elastic matrix. Numerical calculations are performed for BaTiO3, PbTiO3, and Pb(Zr0.5Ti0.5)O3 nanoparticles surrounded by silica glass. It is shown that, in the case of BaTiO3 and PbTiO3, elastic 3D clamping may change the order of a ferroelectric phase transition from first to second. Furthermore, the mechanical inclusion–matrix interaction shifts the temperatures of structural transitions between different ferroelectric states and even eliminates some ferroelectric phases existing in stress-free BaTiO3 and Pb(Zr0.5Ti0.5)O3 crystals. Another important effect of elastic clamping is the lowering of the symmetry of ferroelectric states in ellipsoidal inclusions, where orthorhombic and monoclinic phases may form instead of the tetragonal and rhombohedral bulk counterparts. Finally, our thermodynamic calculations show that the dielectric responses of studied perovskite ferroelectrics are sensitive to matrix-induced clamping as well. For instance, dielectric peaks occurring at structural transitions between different ferroelectric phases in BaTiO3 appear to be much higher in spherical inclusions than in the freestanding crystal. Predicted clamping-induced enhancement of certain dielectric responses at room temperature indicates that composite materials comprising nanocrystals of perovskite ferroelectrics are promising for device applications requiring the use of high-permittivity dielectrics.

Asymmetric mechanical and thermal stresses in 2D-FGPPMs hollow cylinder

ABSTRACTThe general solution of steady-state asymmetric mechanical and thermal stresses of a hollow thick cylinder made of fluid-saturated functionally graded porous piezoelectric materials based on two-dimensional equations of thermoelasticity is considered. The general form of thermal and mechanical boundary conditions is considered on the inside and outside surfaces. A direct method is used to solve the heat conduction equation and the nonhomogeneous system of partial differential Navier equations, using the complex Fourier series and the power law functions method. The material properties are assumed to depend on the radial and circumferential variables and are expressed as power law functions along the radial and circumferential direction.

Reflection and transmission of plane wave in multilayered nonlocal magneto-electro-elastic plates immersed in liquid

Taking the nonlocal effect into account, we investigate the reflection and transmission of the plane wave propagating in three-dimensional multilayered magneto-electro-elastic (MEE) plates which are immersed in liquid. Based on the basic equations with nonlocal effect, the first-order state-space system is established to describe the relations of magnitudes of the extended stress and displacement. The general solutions of the extended stress and displacement are expressed in terms of the eigenvalues and eigenvectors which are derived from the first-order state-space system. Then the stiffness matrix method is employed to find the relation between the displacement and traction on the upper and lower interfaces of the layer. After defining the mechanical, electric and magnetic boundary conditions, we derive the reflection and transmission coefficients of the plane wave propagating in the MEE plates by deriving the global stiffness matrix. Finally, numerical examples are provided to show the effect of the nonlocal parameter, stacking sequences, frequency and incident angle on the reflection and transmission coefficients.

A practical solution to the mechanical perturbations induced by non-isothermal injection into a permeable medium

Non-isothermal fluid injection into a geological formation causes alterations of the pressure and temperature fields, which affect the mechanical stability of the reservoir. This coupled thermo-hydro-mechanical behavior has become a matter of special interest because of public concern about induced seismicity. The response is complex and its evaluation often requires numerical modeling. Nevertheless, analytical solutions are useful in improving our understanding of interactions, identifying the controlling parameters, testing codes and in providing a rapid assessment of the system response to an alteration. We present analytical expressions for hydraulic and thermal driven displacements and stresses for unidirectional and radial geometries. To obtain them, an easy-to-use solution to the transient advection-conduction heat transfer problem is developed, whereas the fluid flow is assumed at steady-state. The validity is verified by comparison with numerical simulations and yields fairly accurate results. The solution is then used to illustrate some features of the poroelastic and thermoelastic response and, in particular, the sensitivity of stresses to the outer mechanical boundary conditions.

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

ABSTRACTIn this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.

Three-dimensional thermo-elastic analysis of multi-directional functionally graded rectangular plates on elastic foundation

A thermo-elastic analysis of a multi-directional functionally graded thick rectangular plate with different boundary conditions is carried out in this study. Some material properties are assumed to be temperature dependent and graded in all three spatial directions following a power law function. The differential quadrature method (DQM) is employed to obtain the temperature distribution throughout the 3D-FGM plate. The governing partial differential equations based on the three-dimensional thermo-elasticity theory are obtained, and the DQM is employed to discretize the resulting equations. Comparisons are made with the available results of 1D-FGM and 2D-FGM plates in the literature and an Abaqus model. Numerical results are obtained for different gradient indices, plate dimensions, and various thermal and mechanical boundary conditions.

Three-Dimensional Phase-Field Simulation of Li Diffusion and Stress Evolution in a Polycrystalline LiCoO2 Cathode

The evolutions of stress and strain caused by Li-extraction and -insertion in a LiCoO2 (LCO) is one of the most important phenomena which affect performance of a Li-ion battery (LIB) with the LCO cathode. However, because experimental measurement of the stress and strain evolutions in the LCO is very difficult, numerical simulation models that enable us to investigate mechanical behaviors and electrochemical reactions in LIBs have been actively studied. Recently, the phase-field method, which has been originally developed for simulating microstructure evolutions during solidification and phase transformation in various materials, is used for investigating electrochemical reactions in various electrode materials used in LIBs. In this study, three-dimensional (3D) phase-field simulations of the Li diffusion and the stress evolution in a polycrystalline LCO cathode were performed. In order to perform the 3D simulations efficiently, the stress evolution in the LCO was calculated on the basis of the phase-field microelasticity theory that employes the fast Fourier transform algorithm to solve the mechanical equilibrium equation. As an example of the simulation results, the calculated distributions of Li concentration and the von Mises stress in a polycrystalline LCO cathode are shown in Fig. 1. Through the 3D simulations, the effects of microstructural factor of the LCO cathode, such as crystal orientation and grain size, and external mechanical boundary conditions on the Li diffusion and the stress evolution in the cathode were investigated. Figure 1

New Rapid Evaluation for Long-Term Behavior in Deep Geological Repository by Geotechnical Centrifuge—Part 2: Numerical Simulation of Model Tests in Isothermal Condition

In high-level radioactive waste disposal repositories, there are long-term complex thermal, hydraulic, and mechanical (T–H–M) phenomena that involve the generation of heat from the waste, the infiltration of ground water, and swelling of the bentonite buffer. The ability to model such coupled phenomena is of particular importance to the repository design and assessments of its safety. We have developed a T–H–M-coupled analysis program that evaluates the long-term behavior around the repository (called “near-field”). We have also conducted centrifugal model tests that model the long-term T–H–M-coupled behavior in the near-field. In this study, we conduct H–M-coupled numerical simulations of the centrifugal near-field model tests. We compare numerical results with each other and with results obtained from the centrifugal model tests. From the comparison, we deduce that: (1) in the numerical simulation, water infiltration in the rock mass was in agreement with the experimental observation. (2) The constant-stress boundary condition in the centrifugal model tests may cause a larger expansion of the rock mass than in the in situ condition, but the mechanical boundary condition did not affect the buffer behavior in the deposition hole. (3) The numerical simulation broadly reproduced the measured bentonite pressure and the overpack displacement, but did not reproduce the decreasing trend of the bentonite pressure after 100 equivalent years. This indicates the effect of the time-dependent characteristics of the surrounding rock mass. Further investigations are needed to determine the effect of initial heterogeneity in the deposition hole and the time-dependent behavior of the surrounding rock mass.

On the energy harvesting via doubly curved piezoelectric panels

This article presents an analytical solution for power output from a doubly curved piezoelectric energy harvester. The energy harvester is made of an elastic core layer coupled with one or two surface-bonded piezoelectric layers. Five mechanical equations of motion together with Gauss’s equation are derived on the basis of first-order shell theory and solved simultaneously for simply supported mechanical boundary conditions. The influence of structural damping is taken into account using Rayleigh damping. The electromechanical frequency response functions that relate the power output and circuit load resistance are identified from the exact solutions. Finally, the performance of the system is analyzed extensively for different parameters such as type of energy harvester including unimorph and bimorph panel in series or parallel connections of piezoelectric layers, circuit load resistance, geometrical parameters, and material properties of core and piezoelectric layers.

Identification of crystal variants in shape-memory alloys using molecular dynamics simulations

Shape-memory alloys (SMA) have interesting behaviors and important mechanical properties due to the solid–solid phase transformation. These phenomena are dominated by the evolution of microstructures. In recent years, the microstructures in SMAs have been studied extensively and modeled using molecular dynamics (MD) simulations. However, it remains difficult to identify the crystal variants in the simulation results, which consist of large numbers of atoms. In the present work, a method is developed to identify the austenite phase and the monoclinic martensite crystal variants in MD results. The transformation matrix of each lattice is calculated to determine the corresponding crystal variant. Evolution of the volume fraction of the crystal variants and the microstructure in Ni-Ti SMAs under thermal and mechanical boundary conditions are examined. The method is validated by comparing MD-simulated interface normals with theoretical solutions. In addition, the results show that, in certain cases, the interatomic potential used in the current study leads to inconsistent monoclinic lattices compared with crystallographic theory. Thus, a specific modification is applied and the applicability of the potential is discussed.

Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory

Literature researching the active vibration control of functionally graded material (FGM) plates with piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) using any of the element-free methods does not exist. To the best of the authors’ knowledge, this paper is the first to use Reddy’s HSDT with the element-free IMLS-Ritz method to investigate this problem. In this study, seven mechanical degrees of freedom (DOF) and one additional electrical DOF are considered for each node of the discretized domain. The natural frequency results of two FGM plates with top and bottom piezoelectric layers are compared with the literature in terms of various electrical and mechanical boundary conditions, volume fraction exponent (n) and dimension ratios, with obvious agreement. Furthermore, parametric studies are performed, for the first time, to study the effects of mechanical boundary conditions, n value, FGM plate thickness-to-width ratio and piezoelectric layer thickness to FGM plate thickness ratio on the natural frequency increment between open and closed circuit conditions. For the purpose of active vibration control, a constant velocity feedback approach is utilized. The effectiveness of two proposed positions, of piezoelectric sensor and actuator layers, to control the vibration of FGM plates is investigated.

Agglomeration effects on electro‐magneto‐thermo elastic behavior of nano‐composite piezoelectric cylinder

Agglomeration effect on the electro‐magneto‐thermo elastic response of a hollow piezoelectric circular cylinder reinforced with carbon nanotubes (CNTs) is the main contribution of present work. The cylinder is subjected to an internal pressure, a distributed temperature field due to steady state heat conduction with convective boundary condition, a magnetic field in axial direction and a constant electric potential difference between its inner and outer surfaces or combination of these loadings. All the equivalent mechanical, thermal and piezoelectric properties of nano‐composite cylinder are obtained using Mori–Tanaka model considering agglomeration effects. The governing equation in polarized form is shown to reduce to a second‐order ordinary differential equation for the radial displacement. Considering mechanical boundary condition, this differential equation is analytically solved. The electro‐magneto‐thermo‐mechanical stresses, the electric potential and radial displacement distributions in the structure are discussed in detail. The presented results indicate that the agglomeration and volume percent of CNTs have significant influences on the electro‐magneto‐thermo‐mechanical behavior of the nano‐composite cylinder. POLYM. COMPOS., 39:1594–1603, 2018. © 2016 Society of Plastics Engineers

Scaling Effects in Perovskite Ferroelectrics: Fundamental Limits and Process‐Structure‐Property Relations

Ferroelectric materials are well‐suited for a variety of applications because they can offer a combination of high performance and scaled integration. Examples of note include piezoelectrics to transform between electrical and mechanical energies, capacitors used to store charge, electro‐optic devices, and nonvolatile memory storage. Accordingly, they are widely used as sensors, actuators, energy storage, and memory components, ultrasonic devices, and in consumer electronics products. Because these functional properties arise from a noncentrosymmetric crystal structure with spontaneous strain and a permanent electric dipole, the properties depend upon physical and electrical boundary conditions, and consequently, physical dimension. The change in properties with decreasing physical dimension is commonly referred to as a size effect. In thin films, size effects are widely observed, whereas in bulk ceramics, changes in properties from the values of large‐grained specimens is most notable in samples with grain sizes below several micrometers. It is important to note that ferroelectricity typically persists to length scales of about 10 nm, but below this point is often absent. Despite the stability of ferroelectricity for dimensions greater than ~10 nm, the dielectric and piezoelectric coefficients of scaled ferroelectrics are suppressed relative to their bulk counterparts, in some cases by changes up to 80%. The loss of extrinsic contributions (domain and phase boundary motion) to the electromechanical response accounts for much of this suppression. In this article, the current understanding of the underlying mechanisms for this behavior in perovskite ferroelectrics is reviewed. We focus on the intrinsic limits of ferroelectric response, the roles of electrical and mechanical boundary conditions, grain size and thickness effects, and extraneous effects related to processing. In many cases, multiple mechanisms combine to produce the observed scaling effects.

A phase-field approach to fracture coupled with diffusion

A phase-field model is proposed to describe cracking and its coupling with diffusion. While our specific interest is on diffusion of chloride ions in hardened cement paste, the proposed model has more general applicability to any porous material potentially developing cracks. The model is formulated in Gurtin’s framework and first applied to a two-dimensional plate subject to mechanical and diffusion boundary conditions, where an analytical solution can be found. Several more complex examples solved with the finite element method demonstrate the ability of the model to capture the effect of cracking on the local and on the macroscopically measured diffusivities. Experimental results from the literature are used to calibrate the relationship between local diffusivity and phase-field parameter. The calibrated relationship is applied to a unit cell problem, which serves as the first step towards the coupled diffusion–mechanical analysis at the mesoscale of concrete.

Equivalent mechanical boundary conditions for single layer graphene sheets

To describe and analyse mechanical structures due to external loads, equation of motion must be supplemented with appropriate boundary conditions. When nanomechanical structures (especially layered systems) are considered, satisfying the boundaries is more challenging. It may be essential to develop a specific methodology for any system in such scale and configuration. This work introduces a general approach for boundary conditions of monolayer graphene sheets as the most important part of future sensors and resonators in nanoelectromechanical systems. Comparing with the experiments, it has been demonstrated that the graphene sheets on the adhesive surfaces should be assumed as a hinged edge condition and not as a clamped one. As a result, inaccuracy of the commonly used rigid base as a clamped condition is proven due to comparison with a reported experiment. It is suggested that a flexible substrate can be replaced to have better accordance. A flexible silicon base with equilibrium distance equal to s = 2.5 Å and depth of potential well equal to ε = 0.02 eV is obtained as the best substrate for a square graphene sheet under doubly clamped edge conditions.

Time-Dependent Electro–Magneto–Thermoelastic Stresses of a Poro-Piezo-Functionally Graded Material Hollow Sphere

In this paper, analytical solution for time-dependent electro–magneto–thermoelastic stresses of a hollow sphere made of a fluid-saturated functionally graded porous piezoelectric material (FGPPM) is presented. All material properties, except Poisson's ratio, vary through the radial direction of the FGPPM spherical structure according to a simple power-law. The general form of thermal, mechanical, and electric potential boundary conditions is considered on the internal and external surfaces of the sphere, and the sphere is under constant electrical and magnetic fields. Stress–strain and strain–displacement relations are used to obtain stress–displacement equations, and then by putting stress–displacement equations in the equilibrium equation, Navier equation is acquired. The homogenous differential heat conduction equation is solved. The nonhomogenous differential Navier equation is solved for two cases. At first, creep strains are ignored and the initial electro–magneto–thermoelastic stresses are obtained. Then considering creep strains singly, the creep stress rates are obtained. Finally, time-dependent creep stress distributions at any time ti are attained.

Electro-thermo-mechanical nonlinear free vibration of nanobeam resting on the winkler-pasternak foundations based on nonlocal elasticity using differential transform method

This study presents electro-thermo-mechanical nonlinear vibration of nanobeam resting on the Winkler-Pasternak foundations. The nanobeam is considered as an Euler–Bernoulli beam together with von Karman assumptions are employed to model the geometrical nonlinearity where embedded on an elastic foundation which is simulated by spring constant of the Winkler-model and the shear constant of the Pasternak-model. The corresponding governing equations are obtained using Hamilton’s principle considering nonlinear strains. In order to obtain the nonlinear frequency for simply supported mechanical boundary condition at two ends of the nanobeam, differential transform method (DTM) is used in conjunction with a program. By applying this technique, the nanobeam’s natural frequencies can be easily obtained and a rapidly convergent sequence is obtained during the solution. A detailed parametric study is conducted to investigate the influences of nonlocal parameter, temperature change, spring constants and imposed electric potential on the nonlinear free vibration characteristics of nanobeam. To verify the results some comparisons are presented between differential transform method results and open literature to show the accuracy of this new approach. It has been discovered that the DTM does not necessitate small perturbation and is also suitably precise to both linear and nonlinear problems in physics and engineering.

Numerical prediction and analysis of electromagnetic vibration and noise of claw pole alternator

With the improvement of internal combustion engine noise, the noise of claw pole alternator which is universally used on modern automobiles is becoming more distinct. Extensive experimental results show that the noise of claw pole alternator is annoying at low to middle speed and mainly originates from the electromagnetic noise. In this paper, a complete methodology for electromagnetic vibration and noise prediction of claw pole alternator is presented. It is a three-step multiphysics simulation. The first step is to calculate the magnetic force exerted on the inner surface of the stator using 3D FEM. The second step is the structural analysis, which is to investigate the natural resonant frequencies and modal shapes of the alternator system taking into account the real mechanical boundary conditions. The third is the electromagnetic vibration and noise calculation by using modal superposition method and boundary-element method, respectively. Simulation results agree well with those measured by experiments. Furthermore, the mechanisms of electromagnetic vibration and noise are analyzed by using the electromagnetic theory. The method can be used in the initial design stage of claw pole alternator and is also suitable for other types of motor.

Energy harvesting via shallow cylindrical and spherical piezoelectric panels using higher order shear deformation theory

In this article an analytical solution is presented for power output from a piezoelectric shallow shell energy harvester using higher order shear deformation theory (HSDT). The energy harvester is made of an elastic substrate layer coupled with one or two surface bonded piezoelectric layers. Mechanical equations of motion with Gauss’s equation are derived on the basis of HSDT and solved simultaneously for simply-supported mechanical boundary conditions. The electromechanical frequency response functions that relate the power output and circuit load resistance are identified from the exact solutions. Using Rayleigh damping the influence of structural damping is taken into account. Also performance of the system is analyzed extensively for different parameters such as type of energy harvester including unimorph and bimorph panel in series or parallel connections of piezoelectric layers, circuit load resistance, geometrical parameters and material properties of core and piezoelectric layers.

Modelling and diagnostic of an ultrasonic piezoelectric actuator

ABSTRACTModeling of piezoelectric motors is a difficult task because their characteristics are affected by various factors such as materials properties, electrical and mechanical boundary conditions. This work presents the modeling of piezoelectric motor via bond graph method and used for the diagnostic. This method is an innovative way to analyse the effects of different design variables on the objective function but can be also considered as an optimization stage of the study. The validation and the development of bond graph models are based on physical insight to aid in structural damage detection and use the technique of optimal sensors placement.