Electric Intensity Factors Research Articles

In-plane stress analysis in a cracked functionally graded piezoelectric plane

The fracture behavior of a cracked functionally graded piezoelectric plane was investigated under in-plane mechanical and electrical loading. The electro-elastic properties of the plane exponentially varied along the y-axis. Employing the Fourier transform, stress analysis containing a Volterra edge dislocation in a functionally graded piezoelectric plane was carried out. These solutions were used to derive a system of Cauchy singular integral equations for analyzing the straight and curved cracks. Linear, arc, parabolic and sine wave-shaped cracks were evaluated. The integral equations were numerically solved using the Lobatto–Chebyshev integration formula for dislocation density functions on the crack face which were employed to calculate field intensity factors. The effects of the shaped crack, loading parameters, and non-homogeneity parameter on the stress and electric intensity factors were considered.

The frictional contact problem of a rigid punch sliding over thermoelectric materials

A theoretical frictional contact model of thermoelectric materials with nonlinear thermoelectric coupling loaded by a sliding punch is considered in which the punch is also assumed to be an electrical and thermal conductor. For the electric fields and the temperature fields, some analytical results are obtained by Fourier transform technique, such as the full field closed-form formulas of the electrochemical potential, the temperature, the electric current density and the energy flux. The distributions of normal electric current density and normal energy flux at subsurface are graphically given. For the elastic fields, the thermoelastic contact problem is reduced to a singular integral equation of the second kind in which thermoelectric effects are reflected in the right end of the resulting equation. The present results demonstrate that the normal electric current density, the normal energy flux and the contact pressure on the contact surface of thermoelectric materials exhibit high singularity in the vicinity of the edges of the punch. Furthermore, the effects of the friction coefficient and the thermoelectric loads on the critical stresses, the electric current density intensity factor, the energy flux intensity factor and the stress intensity factor are provided. The finding of the presented model is helpful for accessing the surface wear resistance of thermoelectric materials.

Interaction of Screw Dislocation with Edge Interfacial Crack on Fine-Grained Piezoelectric Coating/Substrate

The interaction between micro- and macrocracks in a fine-grained piezoelectric coating/substrate under remote antiplane mechanical and in-plane electrical loadings was studied. The principle of superposition and a mapping function method was used to transform the fine-grained coating/substrate structure containing the screw dislocation and the edge interfacial crack into the right semi-infinite plane piezoelectric bimaterial with screw dislocation to simplify the problem. Furthermore, the electric field, displacement field, intensity factors, and image force of these two problems were established. In addition, numerical calculations were then given graphically to study the effects of the elastic modulus of the material, the size of the crack, the thickness of the coating, and the screw dislocation angle on the edge interface crack and dislocation.

Exact solutions to magneto-electro-thermo-elastic fields for a cracked cylinder composite during thermal shock

This report derives the exact solutions for the problem of a magneto-electro-elastic cylinder with a penny-shaped and embedded crack subjected to transient thermal load. Thermal cracking is analysed in the theoretical framework of linear magneto-electro-thermo-elasticity. The heat conduction equation for a magneto-electro-thermo-elastic cylinder with a finite size is solved using the standard method of separation variables. The coupling magneto-electro-thermo-elastic fields are determined in a stationary case via the Hankel integral transform. Based on Abel’s integral equation and the dual integral equation, the mathematical formulations for the permeable crack conditions are derived. Solutions to the elastic, electric, and magnetic intensity factors are obtained. Due to the explicitness of these solutions, they are very interesting for the design and analysis of magneto-electro-thermo-elastic composites.

Analysis of Mode Ⅲ Interface Fracture for Magneto-Electro-Elastic Materials

A novel finite element discretized symplectic method is developed for analyzing interface fracture of magneto-electro-elastic (MEE) materials under anti-plane loads. The overall cracked body is meshed by conventional finite elements and divided into a finite size singular region near the crack tip (near field) and a regular region far away from the crack tip (far field). In the near field, a based-Hamiltonian model is introduced to find the analytical series expressions, and the large number nodal unknowns are condensed into a small set of the undetermined coefficients of the symplectic series by a transformation. The nodal unknowns in the far field remain unchanged. The stress, electric and magnetic intensity factors, energy release rates (ERRs) and explicit expressions of singular field variables in the near field are simultaneously obtained without any processing.

Electric Current Dependent Fracture in GaN Piezoelectric Semiconductor Ceramics.

In this paper, the fracture behavior of GaN piezoelectric semiconductor ceramics was investigated under combined mechanical and electric loading by using three-point bending tests and numerical analysis. The experimental results demonstrate that, in contrast to traditional insulating piezoelectric ceramics, electric current is a key factor in affecting the fracture characteristics of GaN ceramics. The stress, electric displacement, and electric current intensity factors were numerically calculated and then a set of empirical formulae was obtained. By fitting the experimental data, a fracture criterion under combined mechanical and electrical loading was obtained in the form of an ellipsoid function of intensity factors. Such a fracture criterion can be extended to predict the failure behavior of other piezoelectric semiconductors or devices with a crack, which are useful in their reliability design and applications.

Heat Conduction in Multilayered Thermoelectric Materials with Multiple Cracks

Multilayer thin-film thermoelectric materials are of technological importance. This paper describes a method to analyze the heat conduction in a multilayered thermoelectric plate containing some non-collinear cracks. The material properties in one layer may be different from those in another even though each layer may still be homogeneous. Using the Fourier integral transforms, the boundary value problem is reduced to a system of general singular integral equations. The model is sufficiently general to account for any number of layers and any number of cracks. As a numerical illustration, the electric flux intensity factor, energy flux intensity factor and thermal flux intensity factor for a three-layer plate specimen with two cracks are presented. The effects of strip width on the electric flux intensity factor and thermal flux intensity factor are studied.

Analytic solutions of thermoelectric materials containing a circular hole with a straight crack

The two-dimensional problem of thermoelectric materials under the action of uniform electric current and uniform total energy flux is studied by means of the complex variable function method and the conformal mapping technique. The model of infinite thin plate containing a circular hole with a straight crack is considered, and the boundary conditions of the inner surface are assumed to be electrically and thermally impermeable. The analytic solutions of the electric current density and total energy flux are derived. In addition, the electric current density intensity factor (EIF), the total energy flux density intensity factor (UIF) and the stress intensity factor (SIF) near the crack tip are obtained, which have a great significance for the engineering application of thermoelectric materials. It is found that, on the one hand, the EIF and UIF are in proportion to the applied far-field electric current loads and the total energy flux loads, respectively. On the other hand, all these field's intensity factors are closely related to the geometrical characteristics of the inner boundary (i.e. the circular hole radius R and crack length L). When the radius tends to zero, the degenerated results are identical to the pre-existing solutions in case of the Griffith crack. Numerical results are also made to mainly discuss the effects of radius and crack length on these nondimensionalized field's intensity factors.

Phase field analysis of crack tip parameters in ferroelectric polycrystals under large-scale switching

The determination of fracture parameters at cracks in ferroelectrics under large-scale domain switching is still a great challenge. In order to evaluate the crack-tip mechanical and electrical field intensity factors, the interaction integral (I-integral) technique is further developed to cope with the heterogeneous microstructures based on the polarization distribution and crack tip fields from phase field simulations. The enhanced I-integral exhibits several merits over previous techniques for determining the crack-tip intensity factors. First, small-scale switching assumption is unnecessary. Second, the intensity factors of different modes are decoupled. Third, it is independent of integration area size, regardless of the presence of grain boundaries and domain walls. These advantages ensure the successful application of the enhanced I-integral to study toughening effects in ferroelectric polycrystals due to large-scale domain switching. Using this approach, a tensile test of PbTiO3 ferroelectric polycrystals with an impermeable crack is simulated. Simulations show that domain switching initiates not only from the crack tip but also from the grain boundaries due to high polarization gradient and stress concentration. Large-scale switching triggered by a critical load reduces the crack-tip stress intensity factors anomalously. In comparison to single crystals, the critical load for polycrystals is lower and even vanishes due to grain orientations. The results also demonstrate that the I-integral method possesses satisfactory accuracy and good area-independence for grain boundaries and domain walls.

Thermodynamics analysis of thermoelectric materials: Influence of cracking on efficiency of thermoelectric conversion

This paper provides a theoretical model to analyze the thermoelectric conversion efficiency of a cracked thermoelectric material with finite height and width based on the nonlinearly coupled transport equations of electricity and heat. The crack is assumed to be filled with air-like medium, and the electric current density in crack is neglected since air is a very good electric insulator. The thickness of crack is very small in general, and the heat conductivity of thermoelectric material is only larger than that of air by 1–3 orders of magnitude, therefore the heat flux on the upper and lower surfaces of the crack is considered. Using Fourier transform technique and singular integral equation method, analytical expressions for the distribution of electric potential and temperature fields, and electric current density and energy flux intensity factors at crack tips are obtained to aid in determination of thermoelectric conversion efficiency. Numerical results show that higher conversion efficiency can be achieved in a cracked thermoelectric material. This paper can be served as a starting point for more sophisticated analytic and computational treatments of defects in thermoelectric materials, and the analysis thus point to a new direction in developing high thermoelectric performance of engineer hybrid composites with inclusion structures.

Iterative boundary element method for crack analysis of two-dimensional piezoelectric semiconductor

Based on the well-developed boundary element methods for piezoelectric media and conductors, we present an iterative boundary element method to solve the boundary value problems in two-dimensional piezoelectric semiconductors (PSCs). The proposed method is verified by analyzing a piezoelectric semiconductor plate under multi-field load. Two typically important boundary value problems, a hole and a crack, are studied in PSC plates by using the proposed method. The stress concentration near the edge of an elliptical hole in a finite piezoelectric semiconductor plate is studied by using the single-domain boundary element method. Also, by using the sub-domain boundary element method, we analyzed how the mechanical load, electrical load, electric current density, and initial electron density affected the stress, electric displacement and electric current intensity factors near the crack tip.

Effect of cracking on the thermoelectric conversion efficiency of thermoelectric materials

Analytical solutions for a rectangular thermoelectric plate with a crack under combined electrical and temperature loadings are obtained. The electric current density and energy flux intensity factors at the crack tip are calculated. The effective thermoelectric properties are obtained. From the results, it is found that both effective electric and heat conductivities are reduced by increasing the crack size. However, the thermoelectric conversion efficiency of the thermoelectric plate is independent of the crack size if the crack face boundary conditions are assumed to be electrically and thermally insulated. In addition, the effect of thermoelectric properties and size of an inclusion on the thermoelectric conversion efficiency is also discussed, and the condition for high efficiency thermoelectric materials is identified. This is the first theoretical paper to study the effect of cracking on the thermoelectric properties by a rigorous inference of mathematics and physics.

Penny-shaped cracks in three-dimensional piezoelectric semiconductors via Green’s functions of extended displacement discontinuity

In this article, a penny-shaped crack in the isotropic plane of three-dimensional transversely isotropic piezoelectric semiconductors is analyzed via the displacement discontinuity boundary element method. The general solutions are derived based on the generalized Almansi’s theorem and the operator theory. Green’s functions are derived using the Hankel transformation under uniformly distributed extended displacement discontinuities (including displacement discontinuities, potential discontinuity, and electron density discontinuity) on a penny-shaped crack in the isotropic plane of three-dimensional transversely isotropic piezoelectric semiconductors. Using the extended displacement discontinuity boundary element method, penny-shaped cracks in transversely isotropic plane of three-dimensional piezoelectric semiconductors are studied, and the stress, electric displacement, and electric current intensity factors under uniform mechanical-electric-current loads applied on the penny-shaped crack surface are calculated. The fracture behaviors of piezoelectric semiconductors are studied.

Extended displacement discontinuity method for analysis of cracks in 2D piezoelectric semiconductors

Piezoelectric semiconductors are materials which have the properties of both piezoelectrics and semiconductors. Many of these materials' properties are very sensitive to defects such as cracks. In this paper, we study cracks in two-dimensional n-type piezoelectric semiconductors by using the displacement discontinuity method. Based on the general solution and Fourier transform, the fundamental solutions for a line crack under uniformly distributed extended displacement discontinuities on the crack surface are derived, in which the extended displacement discontinuity denotes the displacement discontinuities along the x-axis and z-axis, electric potential discontinuity, electron density discontinuity, respectively. By using the obtained fundamental solutions, the extended displacement discontinuity boundary element method is developed to calculate the stress, electric displacement and electric current intensity factors at crack tip. Numerical calculations show how the mechanical, electrical and current loadings affect these intensity factors.

Temperature and electric potential fields of an interface crack in a layered thermoelectric or metal/thermoelectric material

The interface crack problem in a layered thermoelectric or metal/thermoelectric material subjected to thermoelectric loadings is studied based on the nonlinear governing equations and complex variable method. The electric impermeable and heat semi-permeable crack boundary conditions are adopted. Explicit and closed-form solutions of temperature and electric potential on the interface crack surfaces, and crack tip thermoelectric fields are obtained. The influence of crack thickness to length ratio and thermal conductivity of air inside the crack on electric current density and heat flux intensity factors at crack tip is discussed. It is found that electric current density and heat flux intensity factors at the interface crack tip between metal interconnector and thermoelectric material are much bigger than those in homogenous or layered thermoelectric material. This explains why the failure (or microcrack initiation) of a thermoelectric generation/cooler element always appears at the interface between the metal electrode and thermoelectric material.

Singularity analysis of planar cracks in three-dimensional piezoelectric semiconductors via extended displacement discontinuity boundary integral equation method

The displacement discontinuity boundary integral equation method is extended to analyze the singularity of near-border fields of the planar crack of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic piezoelectric semiconductor. The hyper-singular boundary integral equations are derived in terms of the displacement, electric potential and carrier density discontinuities across the crack faces, in which body integrals for the carrier density are introduced. Based on the finite-part integrals, singularity exponents and asymptotic expressions of the crack border fields are obtained. The stress, electric displacement and electric current intensity factors are given in terms of the displacement, electric potential and carrier density discontinuities. Finite element results for penny-shaped and line cracks based on the piezoelectric-conductor iterative method are used to verify the derivations of the intensity factors.

Explicit solutions of an elliptic hole or a crack problem in thermoelectric materials

The generalized two-dimensional problem of an elliptic hole or a crack in a thermoelectric material subjected to uniform electric current density and energy flux at infinity is investigated based on the complex variable method and the conformal mapping technique. Firstly, the exact solutions of electric potential, temperature and stress fields are presented with the assumption that the surfaces of the elliptic hole are electrically and thermally insulated. It is shown that both the concentration factors of electric current density and stress at hole rim increase as the value of major to minor axis ratio of the elliptic hole increases. Then, explicit solutions are also obtained in closed-form when the elliptic hole degenerates into a crack. It is found that all fields exhibit an inverse square-root singularity at the crack tip, and electric current density and stress intensity factors are defined according to the traditional way. The results show that the mode-I and mode-II stress intensity factors are dependent on the applied electric current density and energy flux, respectively. Furthermore, the mode-I stress intensity factor induced by Joule heating effect has a non-linear relationship with the remote electric loads.

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

SummaryThe dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.

Analysis of cracks in 3D piezoelectric media with various electrical boundary conditions

A weakly singular, symmetric Galerkin boundary element method capable of solving problems of isolated cracks in three-dimensional, linear anisotropic piezoelectric, infinite media with various types of crack-face boundary conditions including impermeable, permeable, semi-permeable, and the energetically consistent boundary condition introduced by Landis (Int J Solids Struct 41:6291–6315, 2004) is established. The key governing boundary integral equation used in the formulation possesses several crucial features including its desirable symmetric weak-form, weakly singular nature, and ability to treat general material anisotropy, arbitrary crack configurations and any type of boundary condition on the crack surface. The positive consequence of utilizing the singularity-reduced integral equations in the modeling, is that all involved singular integrals can be interpreted in the sense of Riemann and their validity requires only continuous crack-face data allowing $$\hbox {C}^{0}$$ -interpolation functions to be employed everywhere in the numerical discretization. Special crack-tip elements with appropriate square-root functions are adopted in a local region along the crack front to accurately approximate the relative crack-face displacement and electric potential. With use of these crack-tip elements, the stress and electric intensity factors can be extracted directly in terms of crack-front nodal data. A system of nonlinear algebraic equations resulting from semi-permeable and energetically consistent boundary conditions is solved by standard Newton–Raphson iterative scheme. Various numerical examples of both planar and non-planar cracks under different types of electrical boundary conditions are considered and the proposed technique is found promising and computationally robust. In addition, it was determined that using crack-tip elements along the crack front significantly enhances the computational performance and that the stress and electric intensity factors can be obtained accurately using relatively coarse meshes.

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.