Riemann–Hilbert approach-based analytical solutions for strip saturated two unequal collinear cracks in piezoelectric media

2021 ◽  
Vol 13 (4) ◽  
pp. 177-195
Author(s):  
Sandeep Singh ◽  
Kuldeep Sharma
Keyword(s):  
Analytical Solutions ◽  
Complex Potential ◽  
Crack Opening ◽  
Electric Displacement ◽  
Potential Functions ◽  
Collinear Cracks ◽  
Crack Face ◽  
Piezoelectric Media ◽  
Fracture Parameters ◽  
Poling Direction

The objective of the work is to derive analytical solutions based on the Riemann–Hilbert (R–H) approach for semipermeable strip saturated two unequal collinear cracks in arbitrary polarized piezoelectric media. We particularly consider the influence of far field electromechanical loadings, poling direction and different crack-face boundary conditions. The problem is mathematically formulated into a set of non-homogeneous R–H problems in terms of complex potential functions (related to field components) using complex variable and extended Stroh formalism approach. After solving these equations, explicit solutions are obtained for the involved unknown complex potential functions and hence, the stress and electric displacement components at any point within the domain. Furthermore, after employing standard limiting conditions, explicit expressions for some conventional fracture parameters such as saturated zone lengths (in terms of nonlinear equations), local stress intensity factors and crack opening displacement are obtained. Numerical studies are presented for the PZT-4H material to analyze the effects of prescribed electromechanical loadings, inter-cracks distance, crack-face conditions and poling direction on the defined fracture parameters.

A Study of Crack-Face Boundary Conditions for Piezoelectric Strip Cut Along Two Equal Collinear Cracks

2016 ◽  
Vol 8 (4) ◽  
pp. 573-587 ◽  
Author(s):  
R. R. Bhargava ◽  
Pooja Raj Verma
Keyword(s):  
Boundary Conditions ◽  
Crack Opening ◽  
Electric Displacement ◽  
Potential Drop ◽  
Collinear Cracks ◽  
Crack Face ◽  
Plane Shear ◽  
Fracture Parameters ◽  
Analytic Expressions ◽  
Piezoelectric Strip

AbstractA problem of two equal, semi-permeable, collinear cracks, situated normal to the edges of an infinitely long piezoelectric strip is considered. Piezoelectric strip being prescribed out-of-plane shear stress and in-plane electric-displacement. The Fourier series and integral equation methods are adopted to obtain analytical solution of the problem. Closed-form analytic expressions are derived for various fracture parameters viz. crack-sliding displacement, crack opening potential drop, field intensity factors and energy release rate. An numerical case study is considered for poled PZT–5H, BaTiO3 and PZT–6B piezoelectric ceramics to study the effect of applied electro-mechanical loadings, crack-face boundary conditions as well as inter-crack distance on fracture parameters. The obtained results are presented graphically, discussed and concluded.

Accurate fracture analysis of electrically permeable/impermeable cracks in one-dimensional hexagonal piezoelectric quasicrystal junction

2019 ◽  
Vol 24 (12) ◽  
pp. 4032-4050 ◽  
Author(s):  
Zhenting Yang ◽  
Xiong Yu ◽  
Wang Xu ◽  
Chenghui Xu ◽  
Zhenhuan Zhou ◽  
...  
Keyword(s):  
Electric Displacement ◽  
Fracture Analysis ◽  
The Finite Element Method ◽  
Crack Face ◽  
Impermeable Crack ◽  
One Dimensional ◽  
Fracture Parameters ◽  
Small Set ◽  
Singular Domain ◽  
Good Agreement

An accurate fracture analysis of a multi-material junction of one-dimensional hexagonal quasicrystals with piezoelectric effect is performed by using Hamiltonian mechanics incorporated in the finite element method. Two idealized electrical assumptions, including electrically permeable and impermeable crack-face conditions, are considered. In the Hamiltonian system, the analytical solutions to the multi-material piezoelectric quasicrystal around the crack tip (singular domain) are obtained and expressed in terms of symplectic eigensolutions. Therefore, the large number of nodal unknowns in the singular domain is reduced into a small set of undetermined coefficients of the symplectic series. The unknowns in the non-singular domain remain unchanged. Explicit expressions of phonon stresses, phason stresses, and electric displacement in the singular domain and newly defined fracture parameters are achieved simultaneously. Comparisons are presented to verify the proposed approach and very good agreement is reported. The key influencing parameters of the crack are discussed in detail. The effects of electrical assumptions and positions of the crack on the fracture parameters are discussed in detail.

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

2020 ◽  
pp. 1045389X2096316
Author(s):  
MingHao Zhao ◽  
XinFei Li ◽  
Chunsheng Lu ◽  
QiaoYun Zhang
Keyword(s):  
Boundary Condition ◽  
Intensity Factor ◽  
Boundary Conditions ◽  
Crack Opening ◽  
Electric Displacement ◽  
Opening Displacement ◽  
Crack Face ◽  
Electric Boundary Condition ◽  
Electric Displacement Intensity Factor ◽  
Piezoelectric Semiconductors

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.

scholarly journals Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Bogdan Rogowski
Keyword(s):  
Magnetic Induction ◽  
Electric Displacement ◽  
Dual Integral Equations ◽  
Crack Face ◽  
Poling Direction ◽  
Electric And Magnetic Fields ◽  
Crack Tips ◽  
Square Root Singularity ◽  
Field Intensity Factors ◽  
Magnetic Potentials

This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic medium subjected to electro-magneto-mechanical loads. The bimaterial medium contains a crack which lies at interface and is parallel to their poling direction. Fourier transform technique is used to reduce the problem to three pairs of dual integral equations. These equations are solved exactly. The semipermeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, cracks displacement, electric and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Obtained results indicate that the stress field and electric and magnetic fields near the crack tips exhibit square-root singularity.

scholarly journals Determination of fracture parameters for interface cracks in transverse isotropic magnetoelectroelastic composites

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Jun Lei ◽  
Pengbo Sun ◽  
Tinh Quoc Bui
Keyword(s):  
Crack Opening ◽  
Crack Plane ◽  
Interfacial Cracks ◽  
Fracture Parameters ◽  
Extrapolation Formula ◽  
Poling Direction ◽  
The Matrix ◽  
Transverse Isotropic ◽  
Crack Faces ◽  
Displacement Extrapolation

AbstractTo determine fracture parameters of interfacial cracks in transverse isotropic magnetoelectroelastic composites, a displacement extrapolation formula was derived. The matrix-form formula can be applicable for both material components with arbitrary poling directions. The corresponding explicit expression of this formula was obtained for each poling direction normal to the crack plane. This displacement extrapolation formula is only related to the boundary quantities of the extended crack opening displacements across crack faces, which is convenient for numerical applications, especially for BEM. Meantime, an alternative extrapolation formula based on the path-independent J-integral and displacement ratios was presented which may be more adaptable for any domain-based numerical techniques like FEM. A numerical example was presented to show the correctness of these formulae.

Uncertainty Characterization of the CrCOD Circumferential Crack Opening Displacement Module for xLPR

2015 ◽  
Author(s):  
Richard Olson ◽  
Paul Scott
Keyword(s):  
Crack Opening Displacement ◽  
Crack Opening ◽  
Opening Displacement ◽  
Crack Face ◽  
Pipe Surface ◽  
Circumferential Crack ◽  
The Us ◽  
Experimental Values ◽  
Metal Weld

The US NRC/EPRI xLPR (eXtremely Low Probability of Rupture) probabilistic pipe fracture analysis program uses deterministic modules as the foundation for the calculation of the probability of pipe leak or rupture as a consequence of active degradation mechanisms, vibration or seismic loading. The circumferential crack opening displacement module, CrCOD, estimates crack opening displacement (COD) at the inside pipe surface, at the mid-wall thickness location, and at the outside pipe surface using a combined tension/crack face pressure/bending GE/EPRI-like solution. Each module has an uncertainty beyond the uncertainty of the xLPR data inputs. This paper documents the uncertainty for CrCOD. Using 36 pipe fracture experiments, including: base metal, similar metal weld, and dissimilar metal weld experiments; bend only and pressure and bend loading; static and dynamic load histories; cracks that range from short to long, the uncertainty of the CrCOD methodology is characterized. Module uncertainty is presented in terms mean fit and standard deviation between prediction and experimental values.

Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

2018 ◽  
Vol 24 (6) ◽  
pp. 1821-1848 ◽  
Author(s):  
Yuan Li ◽  
CuiYing Fan ◽  
Qing-Hua Qin ◽  
MingHao Zhao
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Analytical Solutions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Elliptical Crack ◽  
Two Dimensional ◽  
Crack Face ◽  
Penny Shaped Crack ◽  
Crack Problems

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

A Simple Approach for Including Weld Residual Stresses in the Calculation of Pipe Circumferential Through-Wall Crack Opening Displacements

2016 ◽  
Author(s):  
Richard Olson
Keyword(s):  
Residual Stresses ◽  
Axial Force ◽  
Analytic Solution ◽  
Crack Opening ◽  
Bending Moment ◽  
Generalized Solution ◽  
Finite Element Analyses ◽  
Opening Displacement ◽  
Crack Face ◽  
Basic Equations

Current methodologies for predicting the crack opening displacement (COD) of circumferentially through-wall cracked pipe do not include the effect of weld residual stresses (WRS). Even the most advanced COD prediction methodology only includes the effect of applied axial force, bending moment, and crack face pressure. For some years, it has been known that weld residual stresses do alter the COD, but there has been no convenient way to include them in a COD prediction without doing case-specific finite element analyses. This paper documents a generalized solution for including WRS effects on COD. The model uses a closed-form analytic solution to approximate the crack face rotations that the WRS would induce which, subsequently, can be added to the typical axial force-bending-crack face pressure COD solution. The methodology is described and the basic equations for the solution are presented. Following this, application to cases to evaluate the efficacy of the approach are presented which show a mixture of results ranging from amazingly good to “of questionable value” with respect to the FEA results.

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