Fundamental solutions of two 3D rectangular semi-permeable cracks in transversely isotropic piezoelectric media based on the non-local theory

2020 ◽  
Vol 16 (6) ◽  
pp. 1497-1520
Author(s):  
Haitao Liu ◽  
Liang Wang
Keyword(s):  
Electric Displacement ◽  
Transversely Isotropic ◽  
Elastic Behavior ◽  
Classical Solutions ◽  
Local Theory ◽  
Dual Integral Equations ◽  
Piezoelectric Media ◽  
Content Type ◽  
Present Solution ◽  
Non Local

PurposeThe paper aims to present the non-local theory solution of two three-dimensional (3D) rectangular semi-permeable cracks in transversely isotropic piezoelectric media under a normal stress loading.Design/methodology/approachThe fracture problem is solved by using the non-local theory, the generalized Almansi's theorem and the Schmidt method. By Fourier transform, this problem is formulated as three pairs of dual integral equations, in which the elastic and electric displacements jump across the crack surfaces. Finally, the non-local stress and the non-local electric displacement fields near the crack edges in piezoelectric media are derived.FindingsDifferent from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack edges in piezoelectric media.Originality/valueAccording to the literature survey, the electro-elastic behavior of two 3D rectangular cracks in piezoelectric media under the semi-permeable boundary conditions has not been reported by means of the non-local theory so far.

Dynamic analysis of two collinear permeable Mode-I cracks in piezoelectric materials based on non-local piezoelectricity theory

2019 ◽  
Vol 15 (6) ◽  
pp. 1274-1293
Author(s):  
Haitao Liu ◽  
Shuai Zhu
Keyword(s):  
Piezoelectric Materials ◽  
Electric Displacement ◽  
Local Stress ◽  
Classical Solutions ◽  
Mode I ◽  
Dual Integral Equations ◽  
Content Type ◽  
Present Solution ◽  
Crack Tips ◽  
Non Local

Purpose Based on the non-local piezoelectricity theory, this paper is concerned with two collinear permeable Mode-I cracks in piezoelectric materials subjected to the harmonic stress wave. The paper aims to discuss this issue. Design/methodology/approach According to the Fourier transformation, the problem is formulated into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. Findings Finally, the dynamic non-local stress and the dynamic non-local electric displacement fields near the crack tips are obtained. Numerical results are provided to illustrate the effects of the distance between the two collinear cracks, the lattice parameter and the circular frequency of the incident waves on the entire dynamic fields near the crack tips, which play an important role in designing new structures in engineering. Originality/value Different from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack tips in piezoelectric materials. It is found that the maximum stress and maximum electric displacement can be used as a fracture criterion.

Basic solution of two collinear mode-I cracks in the orthotropic medium by use of the non-local theory

2017 ◽  
Vol 13 (1) ◽  
pp. 100-115 ◽  
Author(s):  
Haitao Liu
Keyword(s):  
Stress Singularity ◽  
Local Theory ◽  
Mode I ◽  
Basic Solution ◽  
Orthotropic Medium ◽  
Dual Integral Equations ◽  
Content Type ◽  
Present Solution ◽  
Crack Tips ◽  
Non Local

Purpose The purpose of this paper is to present the basic solution of two collinear mode-I cracks in the orthotropic medium by the use of the non-local theory. Design/methodology/approach Meanwhile, the generalized Almansi’s theorem and the Schmidt method are used. By the Fourier transform, it is converted to a pair of dual integral equations. Findings Numerical examples are provided to show the effects of the crack length, the distance between the two collinear cracks and the lattice parameter on the stress field near the crack tips in the orthotropic medium. Originality/value The present solution exhibits no stress singularity at the crack tips in the orthotropic medium.

The Non-Local Theory Solution of a Crack in the Functionally Graded Piezoelectric Materials Subjected to the Harmonic Anti-Plane Shear Stress Waves

2007 ◽  
Vol 353-358 ◽  
pp. 258-262
Author(s):  
Zhen Gong Zhou ◽  
Lin Zhi Wu
Keyword(s):  
Shear Stress ◽  
Piezoelectric Materials ◽  
Electric Displacement ◽  
Stress Waves ◽  
Functionally Graded ◽  
Local Theory ◽  
Plane Shear ◽  
Functionally Graded Piezoelectric Materials ◽  
Non Local ◽  
Functionally Graded Piezoelectric

In this paper, the non-local theory of elasticity was applied to obtain the dynamic behavior of a Griffith crack in functionally graded piezoelectric materials under the harmonic anti-plane shear stress waves. The problem can be solved with the help of a pair of dual integral equations. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips, thus allows us to use the maximum stress as a fracture criterion.

Nonlinear Behavior and Critical State of a Penny-Shaped Dielectric Crack in a Piezoelectric Solid

2006 ◽  
Vol 74 (5) ◽  
pp. 852-860 ◽  
Author(s):  
Chun-Ron Chiang ◽  
George J. Weng
Keyword(s):  
Critical State ◽  
Nonlinear Response ◽  
Electric Displacement ◽  
Nonlinear Behavior ◽  
Transversely Isotropic ◽  
Universal Relation ◽  
Dual Integral Equations ◽  
Impermeable Crack ◽  
Ceramic Property ◽  
Dielectric Crack

By means of the Hankel transform and dual-integral equations, the nonlinear response of a penny-shaped dielectric crack with a permittivity κ0 in a transversely isotropic piezoelectric ceramic is solved under the applied tensile stress σzA and electric displacement DzA. The solution is given through the universal relation, Dc∕σzA=KD∕KI=MD∕Mσ, regardless of the electric boundary conditions of the crack, where Dc is the effective electric displacement of the crack medium, and KD and KI are the electric displacement and the stress intensity factors, respectively. The proportional constant MD∕Mσ has been derived and found to have the characteristics: (i) for an impermeable crack it is equal to DzA∕σzA; (ii) for a permeable one it is only a function of the ceramic property; and (iii) for a dielectric crack with a finite κ0 it depends on the ceramic property, the κ0 itself, and the applied σzA and DzA. The latter dependence makes the response of the dielectric crack nonlinear. This nonlinear response is found to be further controlled by a critical state (σc,DzA), through which all the Dc versus σzA curves must pass, regardless of the value of κ0. When σzA<σc, the response of an impermeable crack serves as an upper bound, whereas that of the permeable one serves as the lower bound, and when σzA>σc the situation is exactly reversed. The response of a dielectric crack with any κ0 always lies within these bounds. Under a negative DzA, our solutions further reveal the existence of a critical κ*, given by κ*=−RDzA, and a critical D*, given by D*=−κ0∕R (R depends only on the ceramic property), such that when κ0>κ* or when ∣DzA∣<∣D*∣, the effective Dc will still remain positive in spite of the negative DzA.

Analysis of the Dynamic Behavior of a Moving Crack in FGMs Using Non-Local Theory

2011 ◽  
Vol 58-60 ◽  
pp. 186-191
Author(s):  
Xian Shun Bi ◽  
Cai Song Luo ◽  
De Kui Wang
Keyword(s):  
Integral Equations ◽  
Crack Surface ◽  
Numerical Study ◽  
Crack Tip ◽  
Functionally Graded ◽  
Local Theory ◽  
Dynamic Crack ◽  
Dual Integral Equations ◽  
Non Local ◽  
Crack Tip Stress

A theoretical and numerical study has been conducted to investigate the dynamic crack propagation in functionally graded materials (FGMs) by making use of non-local theory. The variation of the shear modulus and mass density of the FGMs are modeled by a exponential increase along the direction perpendicular to the crack surface. The Poisson’s ratio is assumed to be constant. The mixed boundary value problem is reduced to a pair dual integral equations through Fourier. In solving the dual integral equations, the crack surface displacement is expanded in a series using Jacobi’s polynomials and Schmidt’s method is used. Contrary to the classical elasticity solution, the crack-tip stress fields does not retains the inverse square root singularity. The analysis revals that the peak values of crack-tip stress increase with the the crack velocity as characteristic length is decreased.

Crack-Tip Stress Fields in FGMs under Anti-Plane Shear Impact Loading Using the Non-Local Theory

2007 ◽  
Vol 348-349 ◽  
pp. 821-824
Author(s):  
Xian Shun Bi ◽  
Xue Feng Cai ◽  
Jian Xun Zhang
Keyword(s):  
Integral Equations ◽  
Impact Loading ◽  
Infinite Plate ◽  
Stress Singularity ◽  
Crack Tip ◽  
Functionally Graded ◽  
Local Theory ◽  
Dual Integral Equations ◽  
Plane Shear ◽  
Non Local

A crack in an infinite plate of functionally graded materials (FGMs) under anti-plane shear impact loading is analyzed by making use of non-local theory. The shear modulus and mass density of FGMs are assumed to be of exponential form and the Poisson’s ratio is assumed to be constant. The mixed boundary value problem is reduced to a pair dual integral equations through the use of Laplace and Fourier integral transform method. In solving the dual integral equations, the crack surface displacement is expanded in a series using Jacobi’s polynomials and Schmidt’s method is used. The numerical results show that no stress singularity is present at the crack tip. The stress near the crack tip tends to increase with time at first and then decreases in amplitude and the peak values of stress decreases with increasing the graded parameters.

Interaction between a Rigid Cylinder with a Piezoelectric Half-Space with Partial Adhesion

2008 ◽  
Vol 33-37 ◽  
pp. 333-338 ◽  
Author(s):  
Zuo Rong Chen ◽  
Shou Wen Yu
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Integral Transform ◽  
Mixed Boundary ◽  
Piezoelectric Materials ◽  
Contact Region ◽  
Electric Displacement ◽  
Transversely Isotropic ◽  
Dual Integral Equations ◽  
Slip Region

An axisymmetric problem of interaction of a rigid rotating flat ended punch with a transversely isotropic linear piezoelectric half-space is considered. The contact zone consists of an inner circular adhesion region surrounded by an outer annular slip region with Coulomb friction. Beyond the contact region, the surface of the piezoelectric half-space is free from load. With the aid of the Hankel integral transform, this mixed boundary value problem is formulated as a system of dual integral equations. By solving the dual integral equations, analytical expressions for the tangential stress and displacement, and normal electric displacement on the surface of the piezoelectric half-space are obtained. An explicit relationship between the radius of the adhesion region, the angle of the rotation of the punch, material parameters, and the applied loads is presented. The obtained results are useful for characterization of piezoelectric materials by micro-indentation and micro-friction techniques.

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

2017 ◽  
Vol 34 (5) ◽  
pp. 1754-1770 ◽  
Author(s):  
Zaher Rahimi ◽  
Wojciech Sumelka ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Calculus ◽  
Free Vibration ◽  
Governing Equation ◽  
Stress Gradient ◽  
Local Theory ◽  
Small Scale ◽  
Content Type ◽  
Non Linear ◽  
Non Local ◽  
Fractional Parameter

Purpose Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems. Design/methodology/approach In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition. Findings The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio. Originality/value A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

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