Electric Intensity Vector Research Articles

Influence of flexoelectricity on interface crack problems under a dynamic load

In the present paper, the influence of flexoelectricity on behavior of the interface crack between two dissimilar dielectric materials under a dynamic mechanical load is investigated. The induced electric field affects the distribution and evolution of mechanical fields in dielectric materials. Large strain gradients induce the electric polarization in the direct flexoelectricity. Due to the large strain gradients at the crack tip vicinity it is needed to consider the strain gradient theory model. Governing equations in this theory contain higher-order derivatives than in the conventional continuum mechanics approach. The mixed finite element method (FEM) is developed here for a general boundary value problem, where the standard C0 continuous finite elements are applied for independent approximations of displacements and strains. The constraints between them are satisfied by collocation at appropriate internal points in elements. Interface cracks are observed in layered structures frequently due to a poor adhesion of layers. The incorporation of flexoelectricity and micro-inertial effects is needed into the failure analysis of interface cracks in nano-sized structures under dynamic loading. In numerical examples, we discuss the influence of flexoelectricity coefficients as well as the ratio of elastic coefficients and the geometrical size to microstructural (micro-stiffness and micro-inertia) length scale parameters of the bilayer composite on the crack opening displacement, stresses ahead the crack tip and induced electric intensity vector.

Analysis of a curved Timoshenko nano-beam with flexoelectricity

A Timoshenko beam model is applied for the analysis of the flexoelectric effect for a curved nano-beam. This theory transforms the beam to a 1-d problem along the beam axis. The electric intensity vector for the open-circuit condition can be expressed by mechanical quantities, namely strains and strain gradients. The variational principle is applied to derive the system of ordinary differential equations (ODEs) for the beam deflection, cross-section rotation, and in-plane displacement. A numerical solution based on the Taylor series expansion with Cartesian coordinate along the beam is proposed for the system of ODEs. A convergence analysis with respect to the number of terms in the Taylor series expansion is performed. Numerical results for the beam deflection, rotation, in-plane displacement, and induced electric intensity vector are presented for various flexoelectric coefficients and the beam curvature.

A Numerical Method of Determining Permittivity from the Modulus of the Electric Intensity Vector of an Electromagnetic Field

The system of equations of electrodynamics is considered for a nonmagnetic nonconducting medium. For this system, the problem is under study of determining the permittivity e from the given modulus of the electric intensity vector of the electromagnetic field that is the result of the interference of the two fields generated by some point sources of an extraneous current. It is assumed that permittivity is different from a given positive constant e0 only inside a compact domain Ω0 ⊂ ℝ3, whereas the modulus of the electric field intensity vector is given for all frequencies starting from a fixed frequency ω0 on the boundary S of some domain Ω that includes Ω0. It is shown that this information allows us to reduce the original problem to the well-known inverse kinematic problem of determining the refraction index inside Ω by the travel time of the electromagnetic wave between the points of the boundary of this domain. The algorithm for numerical solution of the inverse problem is constructed, and the test calculations on the simulated data are presented.

A numerical method of determining permittivity from the modulus of the electric intensity vector of an electromagnetic field

A numerical method of determining permittivity from the modulus of the electric intensity vector of an electromagnetic field

The MLPG for crack analyses in composites with flexoelectricity effects

The meshless Petrov-Galerkin (MLPG) method is developed to analyse 2-D crack problems where the electric field and displacement gradients exhibit a size effect. The size-effect phenomenon in micro/nano electronic structures is described by the strain- and electric field-gradients. Both the electric intensity vector and strain gradients are considered in the constitutive equations of the material and the governing equations are derived with the corresponding boundary conditions using the variational principle. The coupled governing partial differential equations (PDE) for stresses and electric displacement field are satisfied in a local weak-form on small fictitious subdomains. All field quantities are approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, we obtain a system of algebraic equations for the nodal unknowns.

The nonlocal and gradient theories for a large deformation of piezoelectric nanoplates

The von Karman large deformations are considered in the Mindlin plate theory described by the nonlocal and gradient elasticity for piezoelectric nanoplates. It is shown that electric intensity vector can be expressed by mechanical quantities. The governing equations for bending moments, normal and shear stresses are derived from the variational principle. The finite element method is developed for considered governing equations. Differences of both theories are presented.

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

ABSTRACTA meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.

Bending of a porous piezoelectric cylinder under a thermal load

A meshless method based on the local Petrov–Galerkin approach is proposed to analyze bending of a porous piezoelectric cylinder under thermal loading. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors for solid and fluid phases. The influence of thermal expansion coefficients in solid and fluid phases on the plate deflection and on the induced electric potential is investigated via the local integral equation method developed in this paper. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares (MLS) scheme. The heat conduction equation is considered as uncoupled with respect to the mechanical and electrical fields.