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.

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.

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