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.

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.

scholarly journals Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory

Vibration ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 51-69
Author(s):  
Reza Moheimani ◽  
Hamid Dalir
Keyword(s):  
Power Index ◽  
Linear Equations ◽  
Local Effect ◽  
Functionally Graded ◽  
Local Theory ◽  
Classical Elasticity ◽  
Graded Materials ◽  
Nano Scale ◽  
Non Local ◽  
External Loads

Functionally graded materials (FGMs) have wide applications in different branches of engineering such as aerospace, mechanics, and biomechanics. Investigation of the mechanical behaviors of structures made of these materials has been performed widely using classical elasticity theories in micro/nano scale. In this research, static, dynamic and vibrational behaviors of functional micro and nanobeams were investigated using non-local theory. Governing linear equations of the problem were driven using non-local theory and solved using an analytical method for different boundary conditions. Effects of the axial load, the non-local parameter and the power index on the natural frequency of different boundary condition are assessed. Then, the obtained results were compared with those obtained from classical theory. These results showed that a non-local effect could greatly affect the behaviors of these beams, especially at nano scale.

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.

Antiplane Crack Problem in Functionally Graded Piezoelectric Materials

2002 ◽  
Vol 69 (4) ◽  
pp. 481-488 ◽  
Author(s):  
Chunyu Li ◽  
G. J. Weng
Keyword(s):  
Integral Equations ◽  
Piezoelectric Material ◽  
Electric Fields ◽  
Crack Problem ◽  
Functionally Graded ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Functionally Graded Piezoelectric Material ◽  
Functionally Graded Piezoelectric Materials ◽  
Functionally Graded Piezoelectric

In this paper the problem of a finite crack in a strip of functionally graded piezoelectric material (FGPM) is studied. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permitivity of the FGPM vary continuously along the thickness of the strip, and that the strip is under an antiplane mechanical loading and in-plane electric loading. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. The near-tip singular stress and electric fields are obtained from the asymptotic expansion of the stresses and electric fields around the crack tip. It is found that the singular stresses and electric displacements at the tip of the crack in the functionally graded piezoelectric material carry the same forms as those in a homogeneous piezoelectric material but that the magnitudes of the intensity factors are dependent upon the gradient of the FGPM properties. The investigation on the influences of the FGPM graded properties shows that an increase in the gradient of the material properties can reduce the magnitude of the stress intensity factor.

Behaviour of three parallel non-symmetric mode III cracks in a functionally graded material plane

2008 ◽  
Vol 222 (12) ◽  
pp. 2311-2330 ◽  
Author(s):  
P-W Zhang ◽  
Z-G Zhou ◽  
L-Z Wu
Keyword(s):  
Integral Equations ◽  
Functionally Graded Material ◽  
Jacobi Polynomials ◽  
Functionally Graded ◽  
Shielding Effect ◽  
Mode Iii ◽  
Dual Integral Equations ◽  
Plane Shear ◽  
Graded Material ◽  
Material Plane

In this article, the behaviour of three parallel non-symmetric finite-length cracks in an infinite functionally graded material plane subjected to anti-plane shear stress loading was studied by the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The results show that the stress intensity factors depend on the crack lengths, spacing of cracks, and the material parameters. It was also revealed that the crack shielding effect is present in functionally graded materials.

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