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# Consistent Formulations of the Interaction Integral Method for Fracture of Functionally Graded Materials

[+] Author and Article Information
Jeong-Ho Kim1

Department of Civil and Environmental Engineering, Newmark Laboratory, The University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801

Glaucio H. Paulino2

Department of Civil and Environmental Engineering, Newmark Laboratory, The University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801paulino@uiuc.edu

The FEM code I-FRANC2D was formerly called FGM-FRANC2D (19)

Here, the so-called $M$ integral should not be confused with the $M$ integral of Knowles and Sternberg (1), Budiansky and Rice (2), and Chang and Chien (3). Also, see the book by Kanninen and Popelar (4) for a review of conservation integrals in fracture mechanics.

The FEM code I-FRANC2D was formerly called FGM-FRANC2D 19

The FEM code I-FRANC2D was formerly called FGM-FRANC2D (19)

1

Present address: Department of Civil and Environmental Engineering, The University of Connecticut, 261 Glenbrook Road U-2037, Storrs, CT 06269.

2

To whom correspondence should be addressed.

J. Appl. Mech 72(3), 351-364 (Jul 27, 2004) (14 pages) doi:10.1115/1.1876395 History: Received February 24, 2003; Revised July 27, 2004

## Abstract

The interaction integral method provides a unified framework for evaluating fracture parameters (e.g., stress intensity factors and $T$ stress) in functionally graded materials. The method is based on a conservation integral involving auxiliary fields. In fracture of nonhomogeneous materials, the use of auxiliary fields developed for homogeneous materials results in violation of one of the basic relations of mechanics, i.e., equilibrium, compatibility or constitutive, which naturally leads to three independent formulations: “nonequilibrium,” “incompatibility,” and “constant-constitutive-tensor.” Each formulation leads to a consistent form of the interaction integral in the sense that extra terms are added to compensate for the difference in response between homogeneous and nonhomogeneous materials. The extra terms play a key role in ensuring path independence of the interaction integral. This paper presents a critical comparison of the three consistent formulations and addresses their advantages and drawbacks. Such comparison is made both from a theoretical point of view and also by means of numerical examples. The numerical implementation is based on finite elements which account for the spatial gradation of material properties at the element level (graded elements).

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## Figures

Figure 1

Motivation for development of alternative consistent formulations. Notice that C(x)≠Ctip for x≠0. The area A denotes a representative region around the crack tip.

Figure 2

Cartesian (x1,x2) and polar (r,θ) coordinates originating from the crack tip in a nonhomogeneous material subjected to traction (t) and displacement boundary conditions

Figure 3

A point force applied at the crack tip in the direction parallel to the crack surface

Figure 4

Conversion of the contour integral into an EDI where Γ=Γ0+Γ+−Γs+Γ−,mj=nj on Γ0 and mj=−nj on Γs

Figure 5

Crack geometry in a nonhomogeneous material, which is graded along the x1 direction

Figure 6

Example 1: FGM plate with an inclined crack with geometric angle θ¯: (a) geometry and boundary conditions (BCs) under fixed-grip loading; (b) typical finite element mesh; (c) contours for EDI computation of M integral; (d) mesh detail using 12 sectors (S12) and four rings (R4) around the crack tips (θ¯=18° counter-clockwise)

Figure 7

Example 1: comparison of J=(KI2+KII2)∕Etip for the right crack tip of an inclined crack with θ¯=18° using the M integral. The nonequilibrium formulation is used both considering and neglecting the nonequilibrium term (see Eq. 22). The incompatibility formulation is used both considering and neglecting the incompatible term (see Eq. 23)

Figure 8

Example 2: strip with an edge crack in hyperbolic-tangent materials: (a) geometry and BCs; (b) complete finite element mesh with 12 sectors (S12) and four rings (R4) around the crack tip; (c) reference configuration (d=0.0); (d) translation of material gradation to the left (d=+0.5); (e) translation of material gradation to the right (d=−0.5)

Figure 9

Example 2: variation of material properties: E11, E22, and G12 for the orthotropic case, and E for the isotropic case

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