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Research Papers

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

[+] Author and Article Information
Youn-Sha Chan

Department of Computer and Mathematical Sciences, University of Houston-Downtown, One Main Street, Houston, TX 77002

Glaucio H. Paulino

Department of Civil and Environmental Engineering, University of Illinois, 2209 Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801

Albert C. Fannjiang

Department of Mathematics, University of California, Davis, CA 95616

According to the geometry of the problem (see Fig. 3), it is the upper half-plane that is considered in the formulation. The crack is sitting on the $x$-axis, which is on the boundary of the upper half-plane. Thus, the outward unit normal should be $(0,−1,0)$, and not (0, 1, 0). Based on Eq. 5, or the last equation in Eq. (5) of Ref. 4, the sign in front of $ℓ′$ in the expression for both $μyxz$ and $μyyz$ should be “−” instead of “+.”

J. Appl. Mech 75(6), 061015 (Aug 21, 2008) (11 pages) doi:10.1115/1.2912933 History: Received February 23, 2007; Revised February 27, 2007; Published August 21, 2008

## Abstract

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.

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## Figures

Figure 1

A geometric comparison of the material gradation with respect to the crack location

Figure 2

A schematic illustration of a continuously graded microstructure in FGMs

Figure 3

Geometry of the crack problem and material gradation

Figure 4

Full crack displacement profile for homogeneous material (β̃=0) under uniform crack surface shear loading σyz(x,0)=−p0 with choice of (normalized) ℓ̃=0.2 and ℓ̃′=0

Figure 5

Classical LEFM, i.e., ℓ̃=ℓ̃′→0. Crack surface displacement in an infinite nonhomogeneous plane under uniform crack surface shear loading σyz(x,0)=−p0 and shear modulus G(x)=G0eβx. Here, a=(d−c)∕2 denotes the half crack length.

Figure 6

Classical LEFM, i.e., ℓ̃=ℓ̃′→0. Crack surface displacement in an infinite nonhomogeneous plane under uniform crack surface shear loading σyz(x,0)=−p0 and shear modulus G(x)=G0eβx. Here, a=(d−c)∕2 denotes the half crack length.

Figure 7

Crack surface displacement in an infinite nonhomogeneous plane under uniform crack surface shear loading σyz(x,0)=−p0 and shear modulus G(x)=G0eβx with choice of (normalized) ℓ̃=0.10 and ℓ̃′=0.01. Here, a=(d−c)∕2 denotes the half crack length.

Figure 8

Crack surface displacement in an infinite nonhomogeneous plane under uniform crack surface shear loading σyz(x,0)=−p0 and shear modulus G(x)=G0eβx with choice of (normalized) ℓ̃=0.10 and ℓ̃′=0.01. Here, a=(d−c)∕2 denotes the half crack length.

Figure 9

Strain ϕ(x∕a) along the crack surface (c,d)=(0,2) for β̃=0.5, ℓ̃′=0, and various ℓ̃ in an infinite nonhomogeneous plane under uniform crack surface shear loading σyz(x,0)=−p0 and shear modulus G(x)=G0eβx. Here, a=(d−c)∕2 denotes the half crack length.

Figure 10

Stress σyz(x∕a,0)∕G0 along the ligament for β̃=0.5, ℓ̃′=0, and various ℓ̃. Crack surface (c,d)=(0,2) located in an infinite nonhomogeneous plane is assumed to be under uniform crack surface shear loading σyz(x,0)=−p0 and shear modulus G(x)=G0eβx. Here, a=(d−c)∕2 denotes the half crack length.

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