Electroelastic Fields Research Articles

Electroactive differential growth and delayed instability in accelerated healing tissues

Guided by experiments contrasting electrically accelerated recovery with natural healing, this study formulates a model to investigate the importance of electroactive differential growth and morphological changes in tissue repair. It underscores the clinical potential of leveraging electroactive differential growth for improved healing outcomes. The study reveals that voltage stimulation significantly enhances the healing and growth of biological tissues, accelerating the regeneration process across various growth modalities and steering towards isotropic growth conditions that do not favor any specific growth pathways. Enhancing the electroelastic coupling parameters improves the efficacy of bioelectric devices, initiating contraction and fortification of biological tissues in alignment with the electric field. This process facilitates swift cell migration and proliferation, as well as oriented growth of tissue. In instances of strain stiffening at elevated strains, the extreme critical growth ratio aligns with the predictions of neo-Hookean models. Conversely, for tissues experiencing strain stiffening under moderate to very low strain conditions, the strain stiffening effect substantially delays the onset of electroelastic growth instability, ultimately producing a smooth, hyperelastic surface devoid of any unstable morphologies. Our investigation, grounded in nonlinear electroelastic field and perturbation theories, explores how electric fields influence differential growth and instability in biological tissues. We examine the interactions among dimensionless voltage, internal pressure, electroelastic coupling, radius ratio, and strain stiffening, revealing their effects on promoting growth and delaying instability. This framework offers insights into the mechanisms behind electroactive growth and its instabilities, contributing valuable knowledge to the tissue healing.

Phase-field fracture modelling of piezoelectric quasicrystals

In this paper, an electro-elastic phase field model is developed for brittle fracture in piezoelectric quasicrystals. The elastic field of quasicrystals is composite of a phonon field and a phason field. The phonon field, phason field and electric field are coupled with each other, and the phonon and electric loadings trigger the variation of both the phonon and phason elastic energies, which then together contribute to crack evolution. The cracks initiation and propagation in piezoelectric quasicrystals under quasi-static mechanical loading and electric loading can be well predicted. Several numerical examples are performed to evaluate the effects of electric field and anisotropic fracture toughness. Simulation results show that the electric field influences the initial force, peak force, fracture displacement and crack path. For the single-edge notched tensile test, the effects of electric field force on the initial force, peak force and fracture displacement are obvious. For the single-edge notched shear test, the electric field force has no effect on the initial force, but it affects the peak force, fracture displacement and crack path. For the porous plate tensile test, a high-intensity electric field can deflect crack substantially resulting in passing through different holes. Further simulation indicates that the anisotropic fracture toughness has a significant effect on peak force, fracture displacement, and crack path. The present model can be used to simulate various fracture problems of piezoelectric quasicrystals.

Axially Non-Uniform Properties Effected Electro-Elastic Fields in Bended Piezoelectric Semiconductor Beams

This paper studied the effects of axially non-uniform material and non-uniform doping on the electro-elastic fields in bent piezoelectric semiconductor beams. The mathematical model is established based on the phenomenological theory of piezoelectric semiconductor and Euler’s beam theory. To solve the governing equations with variable coefficients, the differential quadrature method is adopted. As two representative structures, the pure bending beam and cantilever beam are studied, respectively. From the calculated results, it can be found the varied material properties changed the effective stiffness, as a result, all the field quantities are altered. Meanwhile, the symmetry of all field quantities in a pure bending beam is broken. Considering different Gaussian doping parameters [Formula: see text] and [Formula: see text], it can be observed that the distribution rule of perturbation carrier density becomes complex. The increment of [Formula: see text] broadens the width of optimum range-producing carriers, however, the increment of [Formula: see text] describes the moving of the optimum range from left to right. Importantly, the introduction of non-uniform doping breaks the limitation of perturbation carrier density in a cantilever beam with uniform material properties, the maximum value of perturbation carrier density does not appear at the fixed end only. The obtained results could be the guidance in designing high-performance electric devices.

Coupled Axisymmetric Thermoelectroelasticity Problem for a Round Rigidly Fixed Plate

Introduction. To describe the operation of temperature piezoceramic structures, the theory of thermoelectroelasticity is used, in which the mathematical model is formulated as a system of nonself-adjoint differential equations. The complexity of its integration in general leads to the study of problems in an unrelated formulation. This does not allow us to evaluate the effect of electroelastic fields on temperature. The literature does not present studies on these problems in a three-dimensional coupled formulation in which closed solutions would be constructed. At the same time, conducting such studies allows us to understand the interaction picture of mechanical, thermal and electric fields in a structure. To solve this problem, a new closed solution of a coupled problem for a piezoceramic round rigidly fixed plate has been constructed in this research. It provides for qualitative assessment of the cross impact of thermoelectroelastic fields in this electroelastic system.Materials and Methods. The object of the study is a piezoceramic plate. The case of unsteady temperature change on its upper front surface is considered, taking into account the convection heat exchange of the lower plane with the environment (boundary conditions of the 1st and 3rd kind). The electric field induced as a result of the thermal strain generation is fixed by connecting the electrodated surfaces to the measuring device. The thermoelectroelasticity problem includes the equations of equilibrium, electrostatics, and the unsteady hyperbolic heat equation. It is solved by the generalized method of finite biorthogonal transformation, which makes it possible to construct a closed solution of a nonself-adjoint system of equations.Results. A new closed solution of the coupled axisymmetric thermoelectroelasticity problem for a round plate made of piezoceramic material was constructed.Discussion and Conclusion. The obtained solution to the initial boundary value problem made it possible to determine the temperature, electric and elastic fields induced in a piezoceramic element under arbitrary temperature axisymmetric external action. The calculations performed provided determining the dimensions of solid electrodes, which made it possible to increase the functionality of piezoceramic transducers. Numerical analysis of the results enabled us to identify new connections between the nature of external temperature action, the deformation process, and the value of the electric field in a piezoceramic structure. This can validate a proper program of experiments under their designing and significantly reduce the volume of field studies.

Bandgap and its defect band analysis of flexoelectric effect in phononic crystal plates

The wave equations of a novel model for the electro-elastic Kirchhoff plate containing the flexoelectric effect are constructed using a variational formulation. Through an Improved Plane Wave Expansion (IPWE) approach, the proposed wave equations are then applied to examine the elastic wave bandgap and defect band in a phononic crystal plate structure without or with a defect. The results demonstrate that the flexoelectric effect on the bandgap is apparent when the structure is at the submicron scale and fades as the scale increases. Moreover, at all length scales, the magnitude of the bandgap varies dramatically with lattice length and inclusion volume ratio. This suggests that the frequency and broadness of the bandgap can be tuned by modifying the volume percentage of inclusion, plate size parameters and lattice length. The dispersion curves of the defect model and corresponding mode shape at the defect band are also studied, which can indicate the applicability of the current model incorporating flexoelectric effect in energy harvesting and wave focusing. The novel model and these findings would help the design of MEMS/NEMS devices for enhanced wave guiding when flexoelectric-based electro-elastic fields are involved.

Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

Anti-plane problem of a nano-sharp crack in one-dimensional hexagonal piezoelectric quasicrystals with the electrically semi-permeable condition

Based on the Gurtin–Murdoch surface/interface theory and the complex potential theory, the problem of an electrically semi-permeable nano-sharp crack in one-dimensional (1D) hexagonal piezoelectric quasicrystals is analyzed. By constructing conformal mapping of the sharp crack, the analytic solutions of the electroelastic field are determined. Meanwhile, the field intensity factors and the energy release rate (ERR) under the electric boundary conditions of semi-permeable are obtained. Numerical examples are given to analyze the effects of the crack size, dielectric constant, electric loading, and mechanic loadings on the dimensionless field intensity factors and the dimensionless ERR. The results show that the impact of the surface effect becomes weaker along with the growth of the crack length, but increases with the increase in the crack angle. The change of the dielectric coefficient has little influence on the dimensionless stress intensity factors (SIFs), but has a strong influence on the dimensionless electric displacement intensity factor (EDIF). The obtained results are helpful to promote the development of nano-quasicrystal composite mechanics and provide an important theoretical support for the design and application of nano-QC materials.

Model III fracture analysis of a nanoscale elliptical hole with four cracks in one-dimensional hexagonal piezoelectric quasicrystals

When the defects are at the nanoscale, the failure characteristics of the piezoelectric quasicrystals are different from the macroscale case. The mode-III fracture problem of four nano-cracks emanating from a nanoscale elliptical hole in one-dimensional hexagonal piezoelectric quasicrystals was studied based on the Gurtin-Murdoch surface elasticity model. The analytical solutions of the electro-elastic field intensity factors and the energy release rate at the crack tip were obtained. The present solutions can be reduced to the existing results. The dimensionless electro-elastic field intensity factors show obvious size dependent effect. The influences of the defects size, the multi-cracks interaction, the phonon-phason field coupling coefficient and the applied electro-elastic loads on the fracture characteristics were discussed in detail.

An analysis of flexoelectric coupling associated electroelastic fields in functionally graded semiconductor nanobeams

Strain gradient with strong size dependency and structural association (geometry or microstructure) can efficiently tune the performances of semiconductors by the flexoelectric coupling effect. In this work, we studied a novel asymmetric beam-like semiconductor made by functionally graded (FG) flexoelectric materials. When being applied with pure bending loads at two ends, it can generate a relatively large inhomogeneous strain field to achieve obvious semiconducting behaviors. Unlike the analysis for piezoelectric semiconductor materials, we considered the effects of flexoelectricity and strain gradient elasticity in constitutive equations for flexoelectric semiconductor materials. Then, the complicated mutual coupling governing equations and associated boundary conditions are rederived strictly. By the Fourier series expansion and spatial integration methods, we obtained the solutions of the set of partial differential equations with non-constant coefficients. Results show that the semiconducting electromechanical coupling performances of the beam-like FG flexoelectric semiconductor depend heavily on the ratio and structural distributions of its constituent. Moreover, it is found that the inner carrier distributions and electromechanical characteristics can be significantly tuned by the strain gradient elasticity, the flexoelectricity, and the structural size. We believe this work provides a useful guideline for the practical design and manufacture of novel electromechanical semiconductor devices.

Arbitrarily loaded circular crack in a piezoelectric solid of the symmetry class 6

The paper focuses on the problem of evaluating analytically the electric and elastic fields in a piezoelectric material of the symmetry class 6 containing a penny-shaped crack loaded by both normal and tangential tractions and by electric charges. A solution of this kind has been developed for symmetry class 6 mm by Karapetian et al. (Arch. Appl. Mech. 70: 201–229, 2000). To the best of our knowledge, it has never been derived for a material of class 6. The research is motivated by needs in evaluation of damage accumulated in bone tissue, which possesses such a symmetry. The problem is solved by the method of potential functions developed by Fabrikant (Applications of Potential Theory in Mechanics: Selection of New Results. Kluwer, Dordrecht,1989). Closed form solutions are obtained for the full electric and elastic fields. The approach is first verified by comparison with the results of Karapetian et al. (Arch. Appl. Mech. 70: 201–229, 2000) for the limiting case of piezoceramic of class 6 mm (PZT-6B) and then applied to calculation of the electroelastic fields associated with a penny-shaped crack in cortical bone (material of class 6).

Electro-elastic field of a piezoelectric quasicrystal medium containing two cylindrical inclusions

Considering the piezoelectric effect, the electro-elastic field of an infinite one-dimensional quasicrystal medium with two circular cylindrical inclusions is derived under antiplane shear and inplane electric loading. The boundary value problem of the composite material with circular cylindrical inclusions is analytically solved by the use of the conformal mapping technique and analytical continuation theory. The stresses in the phonon and phason fields and the electric displacements are obtained explicitly in the form of a power series both for the matrix and the inclusions. Some typical examples are analyzed to show the effect of the geometric parameters, material properties, and electro-mechanical loading on the electro-elastic fields in the matrix, inclusions, and interfaces. The limiting cases of circular cavities and rigid circular inclusions have also been investigated and discussed.

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.

The electroelastic fields inside and outside a piezoelectric parabolic inclusion with uniform eigenstrains and eigenelectric fields

AbstractUsing the Stroh octet formalism, we obtain the electroelastic fields in an infinite homogeneous piezoelectric medium containing a parabolic Eshelby inclusion undergoing uniform eigenstrains and eigenelectric fields. We prove that the internal electroelastic field defining stresses, total strains, electric displacements, total electric fields and rigid‐body rotation inside the parabolic inclusion is uniform. The internal uniform electroelastic field is expressed explicitly in terms of the reduced generalized compliances and the imposed eigenstrains and eigenelectric fields. The decaying non‐uniform electroelastic field in the exterior of the parabolic inclusion is also obtained.

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Numerical Modeling of Electroelastic Fields in the Surface Piezoelectric Luminescent Optical Fiber Sensor to Diagnose Deformation of Composite Plates

We developed a three-dimensional numerical model of a piezoelectric luminescent optical fiber sensor fixed on a composite’s plate. The computational region of the sensor is the optical fiber with two concentric (with 6 sectors) shells of electroluminescent and piezoelectric materials, two control electrodes on interface surfaces, such as optical fiber-electroluminophore and piezoelectric-cover. The external sensor’s cover is made in the form of a semi-elliptic cylindrical polymer shell, which rectangular base is fixed on the surface of the fiberglass plate. In the piezoelectric shell sectors, the polarization directions of the PVDF transversal-isotropic polymer piezoelectric are different and non-planar for any three sectors. Deformation of the plate causes deformation of the sensor fixed on its surface, as well as the occurrence of informative piezoelectric fields in it, thus the occurrence of informative glows of electroluminescent elements. As a result, we find the requested information about the combined deformed state of the composite plate along the length of the sensor based on the digital processing of the integral intensities of the polychrome light signals at the output of the optical fiber. In simple cases of electric and mechanical loads, we present new numerical results of simulating the distribution of non-uniform electroelastic fields in the sensor multiphase volume, the sensor’s external cover and inside fragment of the composite plate. Loading of the sensor-covering-plate system is performed by controlling electric voltage on the sensor’s electrodes and the plate’s mechanical deformation by stretching along the transverse and longitudinal axes, as well as by twisting around these axes and bending in transverse and longitudinal planes. Numerical values of the control and informative transfer coefficients of the piezoelectric luminescent optical fiber sensor are determined, which makes it possible to perform a reliable and high-precision diagnostics of complex deformations of the composite plates and design sensors of this type.

Surface effects of electroelastic tip fields of multiple cracks emanating from a circular hole

The problem of piezoelectric solids containing a nanoscale multi-cracks cracked hole under antiplane mechanical load and inplane electric load is investigated theoretically. The surface electroelastic constants of the defects (crack and hole) are introduced based on the Gurtin-Murdoch surface model. An exact solution of the stress fields and electric displacement fields around the defects are obtained by constructing a conformal mapping function. Closed-form solution of the stress intensity factor and the electric displacement intensity factor at crack tips are presented. Numerical examples are provided to reveal the relationship between the electroelastic fields intensity factors and the defect size, the crack-hole interaction, and the applied mechanical-electrical loads.

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.

Analytical solutions of electroelastic fields in piezoelectric thin-film multilayer: applications to piezoelectric sensors and actuators

Analytical expressions are derived for the stresses and electric fields induced in a piezoelectric multilayer deposited on a substrate with lattice misfit and thermal expansion coefficient mismatch. The piezoelectric multilayer can be subjected to an externally applied force, moment and electric potential. The derived formulations can model any number of layers using recursive relations that minimize the computation time. A proper rotation matrix is utilized to generalize the derived expressions to accommodate various orientations allowing each layer to have hexagonal crystal symmetry. The influence of lattice misfit and thermal expansion coefficient mismatch in the presence of externally applied force, moment and electric potential on the state of the electroelastic fields in each layer is evaluated for various applications. Comparison with finite element analysis results shows excellent agreement. The analytical expressions developed here can be useful in designing electromechanical sensors, actuators and optoelectronic devices made from piezoelectric multilayers.

Fracture analysis of cracks terminating at the interface of elastic-piezoelectric bimaterials

In this paper, the fracture behaviors of a piezoelectric-elastic bimaterial with cracks terminating at the interface are investigated by a symplectic approach. In the Hamiltonian system, the Hamiltonian forms of governing equations is derived by the Hamiltonian variational principle and a total unknown vector consisted of generalized displacements and stresses. The interface fracture problem is reduced into a symplectic eigenproblem which can be directly solved by the method of separation of variables. Thus, the total unknown vector is expanded in terms of symplectic eigenfunctions. The unknown coefficients of the symplectic series can be determined from the continuity conditions at the interface and outer boundary conditions. Consequently, exact solutions for the singular electro-elastic fields and explicit expression of electric/elastic intensity factors are obtained simultaneously. Results indicate that the electro-elastic singularities and intensity factors only depend on the first few terms of symplectic eigenfunctions with non-zero eigenvalues. Numerical examples are presented to show the effects of key influencing factors on the singularity orders and intensity factors of such interface cracks. Some new results are given also.