Three-dimensional thermo-electro-elastic field in one-dimensional hexagonal piezoelectric quasi-crystal weakened by an elliptical crack

In this paper, an analytical solution is developed for the problem of an infinite 1D hexagonal piezoelectric quasi-crystal medium weakened by an elliptical crack and subjected to mixed loads on the crack surfaces. The mixed loads comprise the phonon pressure, phason pressure, electric displacement, and temperature increment, and the crack surfaces can be electrically permeable or impermeable. Based on a general solution, combined with the generalized potential theory, the steady-state 3D thermo-electro-elastic field variables in the quasi-crystal are obtained in terms of elliptic integral functions and elementary functions. Several important physical quantities on the cracked plane, such as the generalized crack surface displacements, normal stresses, and stress intensity factors, are derived in closed forms. An illustrative numerical calculation verifies the presented analytical solution and shows the distribution of the 3D thermo-electro-elastic field. It is indicated that the influence of the phason field on the result is pronounced, especially for the electric field variables, and the electric permeability of crack surfaces has a significant effect on the electric displacement intensity factor at the crack tip.

Surface effect on two nanocracks emanating from an electrically semi-permeable regular 2n-polygon nanohole in one-dimensional hexagonal piezoelectric quasicrystals under anti-plane shear

In this paper, the conformal mapping from a regular 2[Formula: see text]-polygon hole with two collinear asymmetric cracks into a circle is constructed. Based on the Gurtin–Murdoch surface/ interface model and complex potential theory, two collinear asymmetric nanocracks emanating from an electrically semi-permeable regular 2[Formula: see text]-polygon nanohole embedded in an infinite one-dimensional hexagonal piezoelectric quasicrystals with surface effect are investigated. The size-dependent stress intensity factors of phonon field and phason field, electric displacement intensity factor at the nanocrack tip are derived for electrically semi-permeable boundary condition. Numerical examples are illustrated to show that the size of the hole, mechanical load, electric load, cracks relative size change with stress intensity factor of phonon field and electric displacement intensity factor. Also analyzed the change of the electric displacement intensity factor with different electric permeability at the nanocrack tip and the dimensionless intensity factor with [Formula: see text].

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.

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.

Effects of Maxwell Stress on Interfacial Crack Between Two Dissimilar Piezoelectric Solids

In this study, the effects of the Maxwell stress on the interfacial crack between two dissimilar piezoelectric solids are investigated. With the Stroh form and Muskhelishvili theory, the explicit expressions of generalized stresses are presented and the closed forms of the stress and electric displacement intensity factors are derived. Results show that the generalized stress field has singularities and oscillatory properties near the crack tip and the Maxwell stress has influences on the fracture characteristics. For the piezoelectric composites with the Maxwell stress, the normalized stress intensity factor KI* can be changed by both the remote stress and electric load. Such phenomenon cannot be found for the piezoelectric system without the Maxwell stress. Furthermore, the electric displacement intensity factor is more sensitive to the electric load than that to the remote stress.

Transient Dynamic Analysis of Cracked Multifield Solids with Consideration of Crack-Face Contact and Semi-Permeable Electric/Magnetic Boundary Conditions

Boundary element method (BEM) formulations for transient dynamic crack analysis intwo-dimensional (2D) multifield materials are reviwed in this paper. Both homogeneous and lin-ear piezoelectric as well as magnetoelectroelastic material models are considered. Special attentionis paid to properly modeling the non-linear crack-face contact and semi-permeable electric/magneticboundary conditions. Implementation of the corresponding time-domain BEM(TDBEM) is discussedin detail. The proposed TDBEM uses a Galerkin-method for the spatial discretization, whilst thecollocation method is considered for the temporal discretization. Iterative solution algorithms aredeveloped to compute the non-linear crack-face boundary conditions. Crack-tip elements that ac-count for the square-root local behavior of the crack opening displacements (CODs) at the crack-tipsare implemented. In this way, stress intensity factors (SIF), electric displacement intensity factor(EDIF) and magnetic induction intensity factor (MIIF) may be accurately evaluated from the nu-merically computed CODs at the closest nodes to the crack-tips. Numerical examples involving sta-tionary cracks in piezoelectric and magnetoelectroelastic solids under different combined (mechani-cal/electric/magnetic) impact loadings are investigated, in order to illustrate the effectiveness of theproposed approach and characterize the influence of the semi-permeable crack-face boundary condi-tions on the dynamic field intensity factors.

The Crack Tip Fields for Anti-Plane Interfacial Crack between Functionally Graded and Homogeneous Piezoelectric Materials

The problem of an anti-plane crack situated in the interface of functionally graded piezoelectric materials (FGPMs) and homogeneous piezoelectric materials (HPMs) is considered under the impermeable assumption of crack surfaces. The mechanical and electrical properties of the FGPMs are assumed to be exponential functions of y perpendicular to the crack. The higher order crack tip stress and electric displacement fields for FGPMs and HPMs are obtained by the eigen-expansion method. The stress intensity factor and electric displacement intensity factor are obtained explicitly.

Interaction Integrals for Fracture Analysis of Functionally Graded Piezoelectric Materials

This paper presents domain form of interaction integrals based on three independent formulations for computation of stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials. Conservation integrals of J–type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of an example.