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.

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