Gradient Elasticity Theory for a Mode III Crack in a Functionally Graded Material

1999 ◽  
Vol 308-311 ◽  
pp. 971-976 ◽  
Author(s):  
Glaucio H. Paulino ◽  
A.C. Fannjiang ◽  
Y.-S. Chan
Keyword(s):  
Elasticity Theory ◽  
Functionally Graded Material ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Mode Iii ◽  
Mode Iii Crack ◽  
Graded Material ◽  
Gradient Elasticity Theory

scholarly journals Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials—Part I: Crack Perpendicular to the Material Gradation

2003 ◽  
Vol 70 (4) ◽  
pp. 531-542 ◽  
Author(s):  
G. H. Paulino ◽  
A. C. Fannjiang ◽  
Y.-S. Chan
Keyword(s):  
Elasticity Theory ◽  
Fourier Transforms ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Surface Strain ◽  
Mode Iii ◽  
Graded Materials ◽  
Graded Material ◽  
Gradient Elasticity Theory ◽  
Cartesian Coordinate

Anisotropic strain gradient elasticity theory is applied to the solution of a mode III crack in a functionally graded material. The theory possesses two material characteristic lengths, l and l′, which describe the size scale effect resulting from the underlining microstructure, and are associated to volumetric and surface strain energy, respectively. The governing differential equation of the problem is derived assuming that the shear modulus is a function of the Cartesian coordinate y, i.e., G=Gy=G0eγy, where G0 and γ are material constants. The crack boundary value problem is solved by means of Fourier transforms and the hypersingular integrodifferential equation method. The integral equation is discretized using the collocation method and a Chebyshev polynomial expansion. Formulas for stress intensity factors, KIII, are derived, and numerical results of KIII for various combinations of l,l′, and γ are provided. Finally, conclusions are inferred and potential extensions of this work are discussed.

Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials—Part II: Crack Parallel to the Material Gradation

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
Youn-Sha Chan ◽  
Glaucio H. Paulino ◽  
Albert C. Fannjiang
Keyword(s):  
Elasticity Theory ◽  
Strain Gradient ◽  
Crack Opening ◽  
Crack Problem ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Numerical Results ◽  
Surface Strain ◽  
Mode Iii ◽  
Gradient Elasticity Theory

A Mode-III crack problem in a functionally graded material modeled by anisotropic strain-gradient elasticity theory is solved by the integral equation method. The gradient elasticity theory has two material characteristic lengths ℓ and ℓ′, which are responsible for volumetric and surface strain-gradient terms, respectively. The governing differential equation of the problem is derived assuming that the shear modulus G is a function of x, i.e., G=G(x)=G0eβx, where G0 and β are material constants. A hypersingular integrodifferential equation is derived and discretized by means of the collocation method and a Chebyshev polynomial expansion. Numerical results are given in terms of the crack opening displacements, strains, and stresses with various combinations of the parameters ℓ, ℓ′, and β. Formulas for the stress intensity factors, KIII, are derived and numerical results are provided.

Buckling Analysis of a Bi-Directional Strain-Gradient Euler–Bernoulli Nano-Beams

2020 ◽  
Vol 20 (11) ◽  
pp. 2050114
Author(s):  
Murat Çelik ◽  
Reha Artan
Keyword(s):  
Potential Energy ◽  
Gradient Elasticity ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Minimum Potential Energy ◽  
End Conditions ◽  
Graded Material ◽  
Minimum Potential ◽  
Gradient Elasticity Theory ◽  
Basic Equations

Investigated herein is the buckling of Euler–Bernoulli nano-beams made of bi-directional functionally graded material with the method of initial values in the frame of gradient elasticity. Since the transport matrix cannot be calculated analytically, the problem was examined with the help of an approximate transport matrix (matricant). This method can be easily applied with buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on gradient elasticity theory. Basic equations and boundary conditions are derived by using the principle of minimum potential energy. The diagrams and tables of the solutions for different end conditions and various values of the parameters are given and the results are discussed.

Higher-Order Terms for the Mode-III Stationary Crack-Tip Fields in a Functionally Graded Material

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
Linhui Zhang ◽  
Jeong-Ho Kim
Keyword(s):  
Functionally Graded Material ◽  
Crack Tip ◽  
Functionally Graded ◽  
Mode Iii ◽  
Stationary Crack ◽  
Crack Tip Field ◽  
Homogeneous Solutions ◽  
Variable Approach ◽  
Graded Material ◽  
Plane Displacement

This paper provides full asymptotic crack-tip field solutions for an antiplane (mode-III) stationary crack in a functionally graded material. We use the complex variable approach and an asymptotic scaling factor to provide an efficient procedure for solving standard and perturbed Laplace equations associated with antiplane fracture in a graded material. We present the out-of-plane displacement and the shear stress solutions for a crack in exponentially and linearly graded materials by considering the gradation of the shear modulus either parallel or perpendicular to the crack. We discuss the characteristics of the asymptotic solutions for a graded material in comparison with the homogeneous solutions. We address the effects of the mode-III stress intensity factor and the antiplane T-stress onto crack-tip field solutions. Finally, engineering significance of the present work is discussed.

scholarly journals Nanomechanics of a screw dislocation in a functionally graded material using the theory of gradient elasticity

2013 ◽  
Vol 21 (5-6) ◽  
pp. 187-194 ◽  
Author(s):  
Kamyar M. Davoudi ◽  
Hossein M. Davoudi ◽  
Elias C. Aifantis
Keyword(s):  
Screw Dislocation ◽  
Functionally Graded Material ◽  
Dislocation Line ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Short Article ◽  
Graded Material ◽  
The Fourier Transform ◽  
Stress Function Method ◽  
Analytical Expressions

AbstractThe modest aim of this short article is to provide some new results for a screw dislocation in a functionally graded material within the theory of gradient elasticity. These results, based on a displacement formulation and the Fourier transform technique, complete earlier findings obtained with the stress function method and extends them to the case of the second strain gradient elasticity. Rigorous and easy-to-use analytical expressions for the displacements, strains, and stresses are obtained, which are free from singularities at the dislocation line.

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