Electric Displacement Intensity Factors Research Articles

Three-dimensional thermal fracture analysis of piezoelectric material by extended finite element methods

The effect of poling direction and thermal fracture analysis of penny-shaped crack (PSC) in piezoelectric material (PM) exposed to a uniform temperature and steady-state heat flow is presented using the XFEM approach. The intrinsic coupling, thermo-electro-mechanical under thermal shocks are studied and thermal stress intensity factors (TSIFs), and electrical displacement intensity factors (EDIF) are evaluated using the strain energy-based, thermal interaction integral. The discontinuous crack surfaces are treated as adiabatic and electrically impermeable. The discontinuity in both the temperature gradient and electrical field displacement over the crack surface is enriched using the Heaviside enrichment function. The fracture parameters, TSIFs, and EDIF of penny-shaped crack are a function of applied temperature fields and the thermo-electro-mechanical properties of piezoelectric material are studied. Further, the effect of poling direction, and in-plane and out-plane offset of crack position, and the inclined crack on the TSIFs and EDIF are presented. The comparison of both analytical and numerical results of the intended formulation showed excellent accuracy which has been verified on benchmark problems. The results are quite useful for the design of the sensors, transducers, and actuators in heat transfer applications.

A new framework based on XFEM for cracked semipermeable piezoelectric material

In the present work, a novel load-controlled iteration scheme within the framework of extended finite element method (XFEM) is proposed to model the semipermeable crack in a piezoelectric material. To capture the stress and electric displacement singularity at the crack tip, the electromechanical 6-fold enrichment functions are used. The domain form of electromechanical interaction integral is employed to determine the generalized intensity factor. The accuracy of a proposed scheme based on XFEM is validated against the reference solutions derived from the iterative capacitor analogy (ICA) method. The effect of several crack configurations, polarization angle, far-field mechanical and electrical traction on electric displacement intensity factor (EDIF) are analyzed for cracked semipermeable piezoelectric material.

Non-Fourier thermal shock resistance and transient thermal fracture of magneto-electro-elastic composite with a penny-shaped crack

Magneto-electro-elastic composite with a penny-shaped crack is analyzed. The exact solution of temperature field based on non-Fourier heat conduction is derived by the method of standard separation of variables. The thermal stress, electrical displacement and magnetic induction in the transversely isotropic andaxis-symmetric magneto-electro-elasticcylinder are evaluated by Hankel transform technique. The thermal stress, electrical displacement and magnetic induction intensity factors at the tip of penny-shaped crack are obtained by Abel type integral equation. The influence of the thermal relaxation time, crack length, characteristic length of materials and magneto-electro-thermo-elastic coupling on intensity factors are analyzed. Thermal shock resistance of MEE cylinder is evaluated by associating the stress based failure criterion and fracture mechanics based failure criterion.

Extended isogeometric analysis for fracture in functionally graded magneto-electro-elastic material

In this work, we present an extended isogeometric analysis (XIGA) for cracked functionally graded magneto-electro-elastic (FGMEE) material. The material properties inside the FGMEE domain vary exponentially. An electrically and magnetically impermeable crack in the FGMEE domain is modeled by adding appropriate enrichment functions into the isogeometric analysis (IGA) approximation. Heaviside function is utilized to model the displacement, electric, and magnetic potential along the crack face. In contrast, conventional four-field crack tip enrichment functions are used to mimic the crack tip singularity characteristics. The generalized intensity factors (IFs), i.e., the stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF), are assessed using magneto-electro-mechanical interaction integral. A series of numerical studies over the cracked homogeneous and FGMEE material are presented. The accuracy of XIGA is examined by comparing the obtained results with the reference solutions. Various parameters such as volume fraction, crack length, loading combination parameter (both electrical and magnetic), material non-homogeneity parameter and crack orientation are taken to examine their effects on the generalized IFs.

Anti-plane fracture problem of three nano-cracks emanating from a magnetoelectrically permeable regular triangle nano-hole in magnetoelectroelastic materials

Based on the Gurtin–Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the anti-plane fracture problem of three nano-cracks emanating from a magnetoelectrically permeable triangle nano-hole in magnetoelectroelastic materials with surface effect is studied. The exact solutions of the stress intensity factor, the electric displacement intensity factor, the magnetic induction intensity factor, and the energy release rate are obtained under the boundary conditions of magnetoelectrically permeable and impermeable. The numerical examples show the influence of surface effect on the stress intensity factor, the electric displacement intensity factor, the magnetic induction intensity factor, and the energy release rate under two different boundary conditions. It can be seen that the surface effect leads to the coupling of stress, electric and magnetic field, and with the increase of cavity size, the influence of surface effect begins to decrease until it tends to classical elasticity theory.

Nonlinear analysis of a crack in 2D piezoelectric semiconductors with exact electric boundary conditions

In this paper, taking the exact electric boundary conditions into account, we propose a double iteration method to analyze a crack problem in a two-dimensional piezoelectric semiconductor. The method consists of a nested loop process with internal and outside circulations. In the former, the electric field and electron density in governing equations are constantly modified with the fixed boundary conditions on crack face and the crack opening displacement; while in the latter, the boundary conditions on crack face and the crack opening displacement are modified. Such a method is verified by numerically analyzing a crack with an impermeable electric boundary condition. It is shown that the electric boundary condition on crack face largely affects the electric displacement intensity factor near a crack tip in piezoelectric semiconductors. Under exact crack boundary conditions, the variation tendency of the electric displacement intensity factor versus crack size is quite different from that under an impermeable boundary condition. Thus, exact crack boundary conditions should be adopted in analysis of crack problems in a piezoelectric semiconductor.

Antiplane Fracture Problem of Three Nanocracks Emanating from an Electrically Permeable Hexagonal Nanohole in One-Dimensional Hexagonal Piezoelectric Quasicrystals

Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals**Project supported by the National Key R&D Program of China (Grant No. 2017YFC1405605), the Innovation Youth Fund of the Ocean Telemetry Technology Innovation Center of the Ministry of Natural Resources, China (Grant No. 21k20190088), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2018MS01005), and

By constructing a new conformal mapping function, we study the surface effects on six edge nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional (1D) hexagonal piezoelectric quasicrystals under anti-plane shear. Based on the Gurtin–Murdoch surface/interface model and complex potential theory, the exact solutions of phonon field, phason field and electric field are obtained. The analytical solutions of the stress intensity factor of the phonon field, the stress intensity factor of the phason field, the electric displacement intensity factor and the energy release rate are given. The interaction effects of the nano-cracks and nano-hole on the stress intensity factor of the phonon field, the stress intensity factor of the phason field and the electric displacement intensity factor are discussed in numerical examples. It can be seen that the surface effect leads to the coupling of phonon field, phason field and electric field. With the decrease of cavity size, the influence of surface effect is more obvious.

Fracture Analysis of Griffith Interface Crack in Fine-Grained Piezoelectric Coating/Substrate under Thermal Loading

Coating often plays a role in monitoring and protecting substrates in engineering applications. Interface cracks between the coating and the substrate can lead to crack growth under the action of external loading and will cause device failure. In this paper, the behavior of a fine-grained piezoelectric coating/substrate with a Griffith interface crack under steady-state thermal loading is studied. The temperature field, displacement field, and electric field of the coupling of thermal and electromechanical problems are constructed via integral transformation and the principle of superposition. Thus, problems are transformed into a system of singular integral equations, and the expressions of thermal intensity factor, thermal stress intensity factor, and electric displacement intensity factor are obtained. We used a numerical calculation and a system of singular equations to obtain the relationship of strength factor with material parameters, coating thickness, and crack size.

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.

In-Plane Analysis of an FGP Plane Weakened by Multiple Moving Cracks

In this paper, the analytical solution of an electric and Volterra edge dislocation in a functionally graded piezoelectric (FGP) medium is obtained by means of complex Fourier transform. The system is subjected to in-plane mechanical and electrical loading. The material properties of the medium vary exponentially with coordinating parallel to the crack. In this study, the rate of the gradual change of the shear moduli and mass density is assumed to be same. At first, the Volterra edge dislocation solutions are employed to derive singular integral equations in the form of Cauchy singularity for an FGP plane containing multiple horizontal moving cracks. Then, these equations are solved numerically to obtain dislocation density functions on moving crack surfaces. Finally, the effects of the crack moving velocity, material properties, electromechanical coupling factor and cracks arrangement on the normalized mode I and mode II stress intensity factors and electric displacement intensity factor are studied.

Dynamic intensity factors for multiple permeable cracks in piezoelectric materials

AbstractThis article presents the dynamic intensity factors for multiple permeable Mode‐I cracks in piezoelectric materials plane by means of the generalized Almansi's theorem. The Fourier transform is used and the present problem is formulated into two pairs of dual integral equations. The dual integral equations are solved by the Schmidt method. The dynamic stress intensity factors (DSIFs) and the dynamic electric displacement intensity factor (DEDIF) at the crack tips are given. It is shown that the effects of the crack length, the distance between two parallel cracks and the circular frequency of incident wave near the crack tips on the DSIFs and DEDIF.

Multiple mixed-mode cracks in a magneto-electro-elastic half-plane under in-plane loading

The mixed-mode problem of cracked magneto-electro-elastic (MEE) half-plane subjected to in-plane magneto-electro-mechanical loads is considered. The crack is modelled as distributed dislocations with unknown density and the closed form solution of the generalized dislocation is obtained by using the Fourier transform. Furthermore, stress, electric displacement and magnetic induction fields in the intact MEE half-plane, under magneto-electro-mechanical point forces are determined. These solutions are used to construct singular integral equations in the MEE half-plane with multiple cracks. The integral equations are solved numerically to obtain dislocation density functions on a crack surface. These functions are used to determine field intensity factors for cracks. The results show that the stress, electric displacement and magnetic induction intensity factors at the crack tips depend on crack geometry, patterns and the loading conditions. Moreover, the effect of the boundary of the MEE half-plane as well as interaction between cracks upon field intensity factors is investigated.

Analysis of cracked functionally graded piezoelectric material using XIGA

In this paper, an extended isogeometric analysis (XIGA) is presented to model the crack in functionally graded piezoelectric materials (FGPMs). The material properties of functionally graded piezoelectric material (FGPM) vary in an exponential manner along the length of the domain. To model a crack in the FGPM, the isogeometric analysis (IGA) approximation is enriched with the Heaviside and crack tip enrichment functions. Conventional four-fold enrichment functions of isotropic materials are employed to capture the crack tip singularity. To determine the fracture parameters, the domain form of electromechanical interaction integral is used. The accuracy of the proposed XIGA approach is highlighted by validating the several cracked homogeneous and functionally graded piezoelectric material against the analytical/reference solution available in the literature. Several example problems of cracked FGPM domain are solved by XIGA. The influence of material non-homogeneity parameter, loading combination parameter and crack length are examined on the stress intensity factors (SIFs) and electrical displacement intensity factor (EDIF).

Stack of non-uniformly loaded shear cracks in magnetoelectroelastic materials

Detailed expressions are derived in closed-form for the field components at a general point in the deformation of magnetoelectroelastic media by a stack of parallel, antiplane shear cracks to which non-uniform mechanical, electric, and magnetic loads are applied, by exploiting a generalized technique of continuous distributions of dislocations. Their angular variations around a crack tip are deduced; from which comprehensive, novel representations for the appropriate stress, electric displacement, and magnetic induction intensity factors follow. Representative numerical results are illustrated graphically. The phenomenon of the so-called crack shielding effect is exhibited. Limiting special cases are considered as validation of this analysis.

Anti-plane fracture problem of four nano-cracks emanating from a regular 4<i>n</i>-polygon nano-hole in magnetoelectroelastic materials

According to the conformal mapping from the exterior region of the regular <i>n</i>-polygon hole to the exterior region of a unit circle and from the exterior region of four cracks emanating from a circle to the interior region of a unit circle, a new conformal mapping is constructed to map the exterior region of four cracks emanating from a regular 4<i>n</i>-polygon hole to the interior of a unit circle. Then, based on the Gurtin-Murdoch surface/interface model and complex method, the anti-plane fracture of four nano-cracks emanating from a regular 4<i>n</i>-polygon nano-hole in magnetoelectroelastic material is studied. The exact solutions of stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor, and energy release rate are obtained under the boundary condition of magnetoelectrically impermeable with considering the surface effect. Without considering the effect of the surface effect, the exact solution of four cracks emanating from a regular 4<i>n</i>-polygon hole in a magnetoelectroelastic material can be obtained. The numerical results show the influences of surface effect and the size of defect on the stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor and energy release rate under the magnetoelectrically impermeable boundary condition. It can be seen that the stress intensity factor, electric displacement intensity factor, and magnetic induction intensity factor are significantly size-dependent when considering the surface effects of the nanoscale defects. And when the size of defect increases to a certain extent, the influence of surface effect begins to decrease and finally tends to follow the classical elasticity theory. When the distance between the center and the vertex of the regular 4<i>n</i>-polygon nano-hole is constant, the dimensionless field intensity factor decreases gradually with the increase of the number of edges, and approaches to the conclusion of a circular hole with four cracks. With the increase of the relative size of the crack, the dimensionless field intensity factor increases gradually. The dimensionless energy release rate of the nanoscale cracked hole has a significant size effect. The increase of mechanical load will increase the normalized energy release rate. The normalized energy release rate first decreases and then increases with electrical load increasing. The normalized energy release rate decreases with magnetic load increasing.

Periodic interfacial cracks in dissimilar piezoelectric materials under the influence of Maxwell stress

This paper presents an analytical method to investigate the dissimilar piezoelectric materials containing periodic cracks in the presence of uniform electromechanical loads with consideration of the Maxwell stress. Using Stroh’s formalism and conformal mapping, electric and stress fields are obtained for the semi-permeable cracks with a constant dielectric parameter. An approximate method is developed to obtain the analytical solution to the crack opening displacement, and which is validated by the pre-existing solution and with numerical integral. Using the method developed in this paper, the results of the electric displacement field in cracks, the stress, and electric displacement intensity factors are obtained. The numerical results show uniform electric displacement field inside in cracks, and it is independent of the distribution or the size of cracks when under static uniform electromechanical loads. Additionally, it is found that the field intensity is roughly the same as that in the case of a single interfacial crack when the gap of periodic crack exceeds four times the crack length.

Anti-plane crack problem of a functionally graded piezoelectric materials strip with arbitrarily distributed properties

This paper treats the anti-plane crack problem of a functionally graded piezoelectric material (FGPM) strip with arbitrarily distributed properties. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric constant of the FGPM vary continuously along the thickness of the medium, and the strip is under anti-plane mechanical and in-plane electric impact loadings. By using the Fourier transform and defining unknown discontinuous functions across the crack surfaces, the anti-plane crack problem of FGPM is reduced to a group of singular integral equations, which are solved numerically. The stress and electric displacement intensity factors are presented, and the influences of the nonhomogeneous and geometric parameters on stress and electric displacement intensity factors are also included.

Non-uniformly loaded row of moving shear cracks in magnetoelectroelastic media

Derivations and discussions are obtained in detail of the closed-form representations of the components of the deformation fields at a general point created by a moving row of smart magnetoelectroelastic shear cracks that are subjected to non-uniform mechanical, electric, and magnetic loadings. The creative analysis exploits a generalization of the basic dislocation layer method. Near a crack tip, their angular variations and the corresponding stress, electric displacement and magnetic induction intensity factors are deduced. Graphical displays of some illustrative numerical data are presented. As a particular case, the results for an analogous stationary row of non-constantly loaded magnetoelectroelastic cracks are derived. The correctness and validity of the results of this novel analysis are examined by the verification of their agreement with those previously presented for various limiting particular cases.

Interaction of multiple cracks in a nonhomogeneous piezoelectric rectangular plane under an electromechanical loading

PurposeIn accord with the literature reviews, there is not a promising examination regarding the several straight and curved cracks interaction with arbitrary arrangement in the rectangular FGP plane. The purpose of this paper is to consider the effect of crack length, position of the point load, material non-homogeneity constant and also the arrangement of cracks on the resulting field intensity factors.Design/methodology/approachFirst of all, in order to obtain a set of Cauchy singular integral equations, both the dislocation method and the finite Fourier cosine transform technique are applied. Using the corresponding solution to these equations, the dislocation densities on the crack surfaces are then obtained. Considering the results, both the stress intensity factors (SIFs) and electric displacement intensity factors (EDIFs) for a vertical crack and the interaction between two straight and curved cracks, which have an arbitrary configuration, are determined.FindingsThe numerical examples are represented in order to illustrate the interesting mechanical and electrical coupling phenomena induced by multi-crack interactions. At the end, the effects of the material non-homogeneity constant, the crack length and the cracks arrangements on the SIFs and EDIFs are investigated.Originality/valueThe solutions are obtained in series expansion forms which may be considered as Green’s functions in an FGP rectangular plane possessing multiple cracks. The technique of Green’s function provides the ability to analyze multiple cracks having any smooth configuration.