Electroelastic Problem Research Articles

Solving electrostatic and electroelastic problems with the node's residual descent method

Piezoelectric materials are extensively used in engineering for the fabrication of sensors, transducers, and actuators. Due to the coupling characteristics, anisotropy, and arbitrariness of polarization directions, the tasks of mesh generation, numerical integration, and global equation formulation involved in numerical computations are complex and nontrivial. To solve electrostatic and electroelastic problems, an easily implementable node's residual descent method (NRDM) is established in this study. The capability and accuracy of NRDM are validated through the solution of three-dimensional electrostatic, linear elastic, and electroelastic problems, achieving relative errors of 0.17%–1.60% for stress and 0.02%–0.15% for other variables. Fundamental and higher-order variables exhibit second-order accuracy, with convergence rates ranging from 1.98 to 2.27 using a first-order generalized finite difference scheme. This comprehensively validates the double derivation technique. The electroelastic coupling problems can be addressed by explicitly calculating electric displacement and stress based on the stress–charge form of piezoelectric constitutive equations during the iteration process. Additionally, the local basis coordinate technique is seamlessly integrated, enabling the solution of anisotropic piezoelectric problems with arbitrary polarization directions through explicit tensor operations.

Finite Element Modeling of a Flat Cell of Highly Porous Piezocomposite with Inclined Edges Taking into Account Nonuniform Polarization

Introduction. Highly porous composites — metal foams — are widely used due to their mechanical properties. The literature presents various methods for their mathematical modeling, including those based on periodic Gibson-Ashby cells. Piezoactive composites have a number of properties, such as high sensor sensitivity and a large bandwidth. This is the reason for the interest in their modeling. However, when constructing such models from piezoceramic materials, a certain difficulty, associated with the selection of the distribution of preliminary polarization, arises. It should be noted that this issue, specifically for highly porous piezoceramics, has not been sufficiently studied in the literature. Therefore, the objective of this work was to establish the effect of the polarization model on the characteristics of the piezoactive composite.Materials and Methods. The design material is PZT-4 piezoceramics, whose polarization depends significantly on the conditions of its guidance (model geometry, electrode arrangement). The study was divided into two steps: in the first, the residual polarization was calculated based on the theory known in the literature, the implementation of which was performed in the ACELAN package; in the second, a number of problems for a composite cell were solved, and the dependence of its properties on the polarization model was found. The finite element method implemented in the ACELAN package was used as a method for solving the corresponding boundary value problems of electroelasticity for piecewise inhomogeneous bodies.Results. The problem of determining nonuniform polarization for two types of flat cell designs of highly porous piezoceramics was solved. Some features of the obtained polarization distribution were noted, in particular, its nonuniformity and the presence of counter polarization in some edges. The problems of determining natural frequencies and vibration modes “intra cell” and their dependence on the polarization model (homogeneous and nonhomogeneous) were solved. It was noted that some frequencies differed by 10%, while the vibration modes qualitatively coincided. The dependence of the stress-strain state and output characteristics on polarization, whose difference in some values reached 15%, was analyzed.Discussion and Conclusion. The process of polarization of highly porous piezoceramics has a number of features that must be taken into account to obtain reliable information about its mechanical and electrical behavior. Auxetic properties, the difference in the mechanical and electrical response of the cell in question are directly related to these features. Thus, the polarization model has a significant impact on the characteristics of the piezoactive composite, which determines the importance of its correct selection. The results obtained should be taken into account when modeling representative volumes of highly porous piezoelectric composites to determine their effective properties, on the basis of which models of piezoelectric devices are constructed, and their output characteristics are calculated.

Instability of single polymer chain in an electroelastic problem

Under electric loading produced by compliant electrodes, a dielectric elastomer is prone to material instabilities which, in a microstructural level, may connect to single polymer chain instability. To reveal if such connection exists, we aim to use an electroelastic energy model for single polymer chain to study the chain instability. We approximate curvature shape of the chain under the electric loading by using trigonometry series in a spatial coordinate. The Rayleigh–Ritz method is then applied to solve the energy equation formed by the trigonometry series. We demonstrate in the numerical examples that the instability of the chain may occur at the value of electric field with corresponding configuration of the chain close to its full length.

Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells

In the present paper, a coupled 3D exact electro-elastic shell model for Functionally Graded (FG) and composite piezoelectric structures is proposed. Primary variables of the electro-elastic model are the electric potential and the three displacement components. The model allows to evaluate the piezoelectric effect in terms of frequencies and vibration modes. Both closed and open circuit configurations are analyzed and compared. The 3D equilibrium equations and the 3D divergence electric displacement equation for spherical shells give the set of partial differential equations for the electro-elastic problem. The proposed model for spherical shells automatically degenerates into simpler models for plates and cylindrical shells via properly considerations on radii of curvature along in-plane directions. The orthogonal mixed curvilinear coordinates α, β and z are employed. The partial differential governing equations have constant coefficients considering fictitious layers and they are solved using the Navier harmonic form and the exponential matrix method. These features lead to an exact solution for simply-supported boundary conditions. Free vibration analyses are conducted and circular frequencies for the first three thickness vibration modes are computed. After a global assessment phase to verify the correctness of the developed model, new benchmarks are proposed: different thickness ratios and material configurations are investigated. The present work can be intended as a reference general solution for those scientists interested in the study of piezoelectric structures via 2D analytical and numerical formulations.

Numerical Simulation of the Effective Piezo-Actuator Properties Based on Solution of Coupled Boundary-Value Electro-Elasticity Problems

Numerical Simulation of the Effective Piezo-Actuator Properties Based on Solution of Coupled Boundary-Value Electro-Elasticity Problems

Computer homogenization of porous piezoceramics of different ferrohardness with random porous structure and inhomogeneous polarization field

The article is concerned with the homogenization problems, in which the effective moduli of porous piezoceramic composites are determined taking into account the inhomogeneity of the polarization field. The homogenization problems are solved by the finite element method in the framework of the theory of effective moduli and the Hill energy principle using the ANSYS package. To this end, in static problems of electroelasticity, the displacements and electric potential, which are linear in spatial variables, are specified on the boundary of a representative volume to provide constant stress and electric induction fields for a homogeneous reference medium. After solving a set of boundary value problems under different boundary conditions and determining the volume-averaged stress components and the electric induction vector, a complete set of effective moduli for the piezoelectric composite is calculated. A representative volume of the piezocomposite is created in the form of a regular finite element mesh consisting of cubic elements. Pores in the representative volume are assumed to be filled with a piezoelectric material with extremely small moduli. Finite elements with pore properties are selected according to a simple random algorithm. The inhomogeneous polarization field is found by solving an electrostatic problem, in which the polarization process in the representative volume is modeled based on a simplified linear formulation. The local coordinate systems for individual finite elements of the composite matrix are specified by the directions of the polarization vectors. In the following, when solving the problems of electroelasticity, these local coordinate systems associated with the elements of the piezoelectric matrix allow recalculating the material properties according to the formulas of transformation of the tensor components as the coordinate systems rotate. In addition, consideration is given to different models describing the change in the moduli of the material from an unpolarized state to a polarized one as a function of the polarization vector. Computational experiments were carried out for three types of piezoceramics: soft ferroelectric piezoceramics PZT-5H, piezoceramics PZT-4 of medium ferrohardness, and piezoceramics PZT-8 with higher degree of ferrohardness. The dependences of the effective moduli on porosity are compared for different laws of polarization inhomogeneity and different kinds of piezoceramic material of the composite matrix.

Неосесимметричная нестационарная задача термоэлектроупругости для длинного пьезокерамического цилиндра

A new closed solution to the uncoupled non-axisymmetric problem of thermoelectroelasticity was constructed for a long hollow piezo-ceramic cylinder for the case of non-stationary temperature alteration on its inner surface, taking into account the convective heat transfer between the outer surface and the environment. Cylindrical surfaces were electroded and connected to a measurement device with large input resistance (idle mode), while the internal surface was grounded. The Fourier—Kirchhoff non-stationary heat conduction equation was considered without taking into account the influence of alteration in the body dimensions and the electric field on the temperature field. The closed solution to the heat conduction problem was constructed by the finite integral transformations (FIT) method. The quasi-static coupled electroelasticity problem at a certain temperature field was solved without taking into account the cylinder inertial properties by the FIT method. The calculated relations obtained made it possible to determine the temperature field, the stress-strain state, as well as the electric field in the long piezo-ceramic cylinder under non-stationary non-axisymmetric action in the form of a function of the temperature alteration. Numerical analysis of the results made it possible to determine the cylinder wall thickness and the region of the temperature effect alteration, where deformation could most efficiently transform into the electrical pulse.

Variational and asymptotic analysis of a static electro-elastic problem with Coulomb’s law

The electro-elastic problem in a three-dimensional thin domain Ωɛ is considered. In modeling the contact the Coulomb free boundary friction condition is used. First, we show the variational formulation of the problem and give the existence and the uniqueness of the solution. Then, we study the asymptotic analysis when then the small parameter ɛ tends to zero. A limit problem, the uniqueness result of the limit displacement and electric potential are obtained.

РЕЗОНАНСНЫЕ ХАРАКТЕРИСТИКИ СОРБЦИОННОГО ОПТОВОЛОКОННОГО ПЬЕЗОЭЛЕКТРОЛЮМИНЕСЦЕНТНОГО ДАТЧИКА

A mathematical electro-mechanical model of a representative fragment of a gas-analytical optical fiber piezoelectroluminescent sensor with a special external absorption layer for monitoring extended areas has been developed. Informative light signal is generated as a result of mechanoluminescent effect caused by contact interaction of piezoelectric and electroluminescent cylindrical layers (coatings) of fiber in case of forced electromechanical vibrations of sensor under action of applied harmonic control electric voltage. The algorithm for finding the sorption spectrum (changes in the density of the absorption layer along the length of the sensor) is presented through the solution of the Fredholm integral equation based on the results of measuring the informative light spectrum of the intensity of the light at the output from the fiber of the sensor. Numerical solution of coupled stationary boundary value problem of electroelasticity for representative fragment of sensor is implemented. Natural frequencies and shapes of sensor oscillations were found taking into account geometric shape, relative location and anisotropy of electroelastic properties of its structural elements, including characteristics of absorption layer. Regularities of influence of diagnosed value of absorption layer density on informative values of resonance frequencies of different forms of sensor oscillations were revealed.

Multi-Component Electroacoustic Waves (Mceaw) In Piezo Crystalline Textures: Applied Opportunities

In this paper is shown, that in each sagittal plane of the piezoelectric texture, both the one-component elastic shear wave and the two-component elastic plane deformation wave can be accompanied by either oscillations of the transverse component of the electric field or oscillations of the plane electric field. Four such packets of multicomponent electroelastic waves are formed.Necessary and sufficient conditions are obtained that allow the formulation of a two-dimensional problem of electroelasticity in a vibrated sagittal plane of a piezocrystal, where separate excitation and propagation of a multicomponent quasistatic electroelastic wave is possible.

МАТЕМАТИЧНЕ ТА КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ НАПРУЖЕНОГО СТАНУ ПРИ ПЕРЕМІЩЕННІ ЖОРСТКОГО КРУГОВОГО ДИСКУ У П’ЄЗОЕЛЕКТРИЧНОМУ ПРОСТОРІ

Using the mathematical model that takes into account the coupling of force and electric fields in piezoelectric bodies, the stress distribution that occurs when a rigid thin circle disk moves along the axis of symmetry of an electroelastic transversely isotropic material is studied. The exact solution of the space problem of electroelasticity is obtained on the basis of the solution representing of the static equations of electroelasticity for a transversely isotropic body through harmonic functions and using a harmonic potential of a special type. As special cases, from the found analytical solution follow the expressions for evaluating the stress state in an elastic transversely isotropic material for the displacement of a circle hard disk. Computer simulation have been carried out, the influence of the coupling of the force and electric fields on the distribution of stresses under a rigid circle disk. 233 KEY WORDS: MATHEMATICAL MODEL, COMPUTER SIMULATION, PIEZOELECTRIC SPACE, HARD CIRCLE DISK, ELECTRICAL AND STRESS STATE, STRESS DISTRIBUTION, FIELD COUPLING EFFECT.

Displacements of the surface of a piezoelectric half-space with functionally-graded coating and circle electrode on the surface

A mathematical model describing the deformation of a material caused by the reverse piezoelectric effect is proposed. The model is based on an axisymmetric problem of electroelasticity for a half-space with a functionally-graded coating. An electric potential difference is applied across a circle area on the surface (an electrode whose thickness and elastic properties are neglected) and an infinity boundary of the half-space. Arbitrary independent variation of all electroelastic properties in depth of the coating is considered. The coating is assumed to be perfectly bonded to the substrate. The problem is solved using the bilateral asymptotic method in an approximated analytical form effective for any value of the relative thickness of the coating. Approximated analytical expressions containing finite quadratures are obtained for the distribution of radial and normal displacements and electric potential on the surface under the electrode and outside of it. Numerical calculations are provided for the distribution of radial and normal displacements and electric potential for two typical examples of functionally-graded coatings in a wide range of relative thickness values of the coating. Convergence of the results to those for a homogeneous non-coated half-space is obtained for small and large values of the relative coating thickness. Special attention is paid to the comparison of the results with those for a non-coated half-space. The redistribution of the electromechanical characteristics caused by the presence of the coating is most noticeably observed near the boundary of the electrode especially for thin and intermediate thickness coatings (in comparison with the size of the electrode).

Influence of spatial static and dynamic inhomogeneities on the distribution of electroelastic fields and electronic processes in piezoceramic composites

The electroelasticity problems for piezoceramic bodies with inhomogeneities were examined. The short review of model situations considered in the previous works of the authors, a piezoceramic body with an internal elliptical crack, a piezoceramic body containing an external elliptical crack, a piezoceramic body with a spherical cavity, was carried out. In all these cases, the coupled electroelastic field in the neighborhood of inhomogeneity for the particular case of static loading was investigated. The formulas to calculate the stress intensity factors were analyzed. The dependences of stresses and electric displacement on the shape of the cavity for the cases of electrical loading and the mechanical loading only were examined. The possibility to apply the obtained results to solving of practical problems, not only static but also dynamic nature, was considered.

Soft dielectric elastomer tubes in an electric field

Soft dielectric elastomers with high relative permittivity, very low modulus and high electric breakdown strength have emerged as promising materials for various applications as sensors, actuators and in energy harvesting and soft robotics. We study the intricate deformation behaviour of a soft dielectric elastomer tube of finite length and closed ends, that carries a dead load, is internally pressurised by an injected fluid and has a high electric potential applied across its walls. As for soft tubes in the absence of a potential, this electro-hyperelastic problem involving very large deformations, exhibits a multitude of possibilities, including homogeneous deformation, inhomogeneous bifurcation, snap-through instabilities and post bifurcation behaviour in the form of propagation of axisymmetric bulges. We develop a coupled Finite Element procedure for the situation where the domains of the mechanical and electrostatic problems coincide. The procedure can handle volume flow rate controlled electro-elastic problems. We use it to study the many aspects of the deformation behaviour and limitations placed by competing failure mechanisms on practical utilisation of the large areal strains and axial actuations that can be produced. If electrical breakdown can be avoided by ingenious design of loading sequences, the large volume increase in the tube due to the triggering of instabilities can be harnessed and has far-reaching technological implications.

Numerical analysis of the effective properties of inhomogeneously polarized porous piezoelectric ceramics with ni-doped pore walls taking into account the influence of volume fractions of metal and pores

The paper considers a porous piezoelectric composite with metal layers deposited on the interface between the piezoelectric and vacuum phases. Such metal layers can be added technologically to improve the mechanical and electromechanical properties of the composite. To find the effective moduli, we designed a simple representative cubic volume of a unit cell consisting of a piezoelectric matrix with a compound spherical pore in its center. In turn, the compound pore includes the pore itself and a hollow metal sphere on its surface. The composite's three phases were all modeled as piezoelectric materials. The conducting interface layer was modeled as a piezoelectric material with very high dielectric constants, small piezoelectric moduli, and elastic properties of the employed metal, while the vacuum pore was modeled as a piezoelectric material with marginal moduli. The mathematical formulation of boundary value homogenization problems with full contact conditions on the interface boundaries, based on the Hill energy criterion, was described. By solving nine boundary value problems of electroelasticity with different boundary conditions for displacements and electric potential by the finite element method, a complete set of effective moduli of the piezoelectric composite is determined. The importance of considering the inhomogeneous polarization due to the presence of pores and metallic inclusions was discussed. An approximate method was proposed for determining the inhomogeneous polarization field in a piezoceramic matrix, based on a preliminary solution of the electrostatic problem for dielectrics and locating the element coordinate systems rotated along the polarization vector. The paper describes the results of computational experiments for a piezocomposite consisting of PZT-5H piezoceramic matrix, pores, and nickel layers on the pore surfaces. Comparisons between the effective properties of this composite with different volume fractions of metal and the conventional porous piezocomposite were provided, depending on the porosity and taking into account the inhomogeneous polarization. Significant differences were observed for some piezomoduli and dielectric constants, which are promising for various practical applications of the considered composites in piezoactuators using the phenomenon of the transverse piezoelectric effect.

ПРОГНОЗИРОВАНИЕ ФУНКЦИОНАЛЬНЫХ ПАРАМЕТРОВ ТЕНЗОЧУВСТВИТЕЛЬНОГО НАНОСТРУКТУРИРОВАННОГО ПОЛИМЕР-КЕРАМИЧЕСКОГО ПОКРЫТИЯ НА ОСНОВЕ АНОДНЫХ ОКСИДОВ МЕТАЛЛОВ

Strength and strain sensitivity of a thin polymer-ceramic coating in the form of a nanoporous anodic aluminum oxide impregnated with a piezoelectric polymer are studied. The coating is considered as a unidirectionally reinforced composite containing cylindrical polymer fibers oriented perpendicular to the coating surface. A three-phase micromechanical model of the specified material is proposed and the stress-strain state of the coating under the influence of uniformly distributed pressure is analyzed. As a result of solving the related problem of electroelasticity, calculated estimates of the specific piezoelectric sensitivity of the coating used as a pressure sensor and the maximum allowable pressure were obtained according to the strength criteria of the ceramic matrix and the plastic flow of the polymer filler. The article shows dependences of the indicated parameters on the volumetric content of the polymer for the coating, adhesively bonded to a non-deformable foundation and freely (without friction) lying on the foundation. At low volume content of polymer, the strength loss of coating is caused by local failure of matrix. At high filler content the transition of polymer into plastic state precedes to the beginning of matrix failure. After increasing the filler content above 80 % the value of maximal pressure according to yield criteria for polymer filler scarcely changes.

Modeling the Deformation of a Plate Using Piezoelectric Elements Located on its Surface

One of the directions of using piezoelectric elements in modern technology is related to their use for controlling the shape of a structure under the influence of operational loads. The issue of a change in the shape (geometry) of a structure can be caused, for example, by the need for minimization of displacements in certain regions or, vice versa, for maximization of displacements to ensure the stability of a structural shape under operational loads. Due to the presence of the inverse piezoelectric effect in piezoelectric materials, this problem can be solved by applying a predetermined electric voltage to the electroded surfaces of piezoelectric elements. Elastic structures with elastic piezoelectric elements attached to their surfaces become electroelastic. It is necessary to first assess the abilities of piezoelectric elements to have an effect on deformations caused by external actions of different types in order to use different strategies of control over their mechanical behavior. This means that piezoelectric elements must provide a controlled form change depending not only on the characteristics of the elements themselves (the size, physico-mechanical properties of the material, and position) but also on the parameters of the structure (its geometry, size, boundary conditions, and physico-mechanical characteristics) and acting loads. The influences of different factors on the deformation of an electroelastic structure under the action of an electric voltage applied to the piezoelectric elements in this work was established numerically based on mathematical modeling by solving the static problem of electroelasticity. The numerical implementation was carried out by the finite-element method in the ANSYS software package. The possibilities for using piezoelectric elements for changing the shape of a structure for different numbers of elements and different options of their layout on the surface are demonstrated based on the example of a cantilevered fixed plate.

Plane electroelastic problem for a cracked piezoelectric half-space subject to remote electric field action

Plane electroelastic problem is considered for a longitudinal crack in a transversely-isotropic piezoelectric half-space subject to remote electric field perpendicular to the flat surface of the half-space. The flat surface is fully covered by thin electrode and free of mechanical load. The Lekhnitskii’ generalized complex potentials are used for the solution representation. For a case of an internal crack, boundary conditions on the flat surface are satisfied analytically by use of the Cauchy integration technique. The least-square method is applied for satisfying boundary conditions on the crack surface. An edge crack is considered. Numerical investigations are carried out for a internal or an edge crack in dependency on crack orientation and location. The results showed that one of the most dangerous crack is parallel to the flat surface. A crack perpendicular to the flat surface does not cause any significant electroelastic field in a piezoelectric half-space. Location of a crack relative the flat surface influences on electromechanical coupling near its tips.

Asymptotic method in two-dimensional problems of electroelasticity

Asymptotic method in two-dimensional problems of electroelasticity

Development of a structurally similar element of a helicopter blade with an active twist system

A mathematical model is formulated for calculating the stress-strain state (SSS) of piezoelectric layered polymer composite materials (PCM) equipped with a control piezoactuator. The model implements the principle of thermo-piezoelectric analogy, which allows to move from the coupled boundary value problem of electroelasticity to the boundary value problem of thermoelasticity. The mathematical model was tested with a numerical calculation of the mechanical behavior of a sample structurally similar to a rotor blade, equipped with control piezo-actuators and made of equal strength and unidirectional fiberglass.