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Special Issue Honoring Professor Fazil Erdogan’s Contributions to Mixed Boundary Value Problems of Inhomogeneous and Functionally Graded Materials

# A Generalized Interaction Integral Method for the Evaluation of the $T$-Stress in Orthotropic Functionally Graded Materials Under Thermal Loading

[+] Author and Article Information
Jeong-Ho Kim1

Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Road, U-2037, Storrs, CT 06269jhkim@engr.uconn.edu

Amit KC

GM2 Associates Inc., 730 Hebron Avenue, Glastonbury, CT 06033

The FEM code FGM-FRANC2D is upgraded based on I-FRANC2D (31) at the University of Illinois at Urbana-Champaign and also FRANC2D (54-55) at Cornell University.

1

Corresponding author.

J. Appl. Mech 75(5), 051112 (Jul 22, 2008) (11 pages) doi:10.1115/1.2936234 History: Received June 05, 2007; Revised December 28, 2007; Published July 22, 2008

## Abstract

The interaction integral method that is equipped with the nonequilibrium formulation is generalized to evaluate the nonsingular $T$-stress as well as mixed-mode stress intensity factors in orthotropic functionally graded materials under thermomechanical loads. This paper addresses both Mode-I and mixed-mode fracture problems and considers various types of orthotropic material gradation. The orthotropic thermomechanical material properties are graded spatially and integrated into the element stiffness matrix using the direct Gaussian formulation. The types of orthotropic material gradation considered include exponential, power-law, and hyperbolic-tangent functions, and the numerical formulation is generalized for any type of smooth material gradation. The $T$-stress and mixed-mode stress intensity factors are evaluated by means of the interaction integral method developed in conjunction with the finite element method. The accuracy of numerical results is assessed by means of thermomechanically equivalent problems.

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## Figures

Figure 1

A crack in an orthotropic FGM. Notice that C(x)≠Ctip for x≠0. The area A denotes a representative region around the crack tip.

Figure 2

A point force applied at the crack tip in the direction parallel to the crack surface in a homogeneous orthotropic body where material orthotropy directions are aligned with the global coordinates

Figure 3

Example 1: (a) An exponentially graded orthotropic strip with an edge crack under thermal loads, (b) complete finite element mesh, (c) mesh detail showing 12 sectors (S12) and 4 rings (R4) around the crack tip employed in the 2D analysis, and (d) mesh detail showing 10 sectors (S10) and 14 rings (R14) around the crack tip employed in the 3D analysis

Figure 4

Example 2: (a) An orthotropic FGM plate with an inclined crack with geometric angle θ¯ subjected to thermal loads, (b) mechanically equivalent fixed-grip loading, (c) typical finite element mesh, and (d) mesh detail using 12 sectors (S12) and 4 rings (R4) around the crack tips and four contour surrounding four domains used for interaction integrals (θ¯=30deg counterclockwise)

Figure 5

Example 3: (a) A crack in an orthotropic functionally graded TBC, (b) complete finite element mesh, and (c) mesh detail using 16 sectors (S16) and 4 rings (R4) around the crack tip

Figure 6

Example 3: Variations of the thermal conductivity coefficient (κ11(X1)) and the resulting normalized temperature field (θ(X1)∕θ0)

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