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TECHNICAL PAPERS

# Null-Field Approach for the Multi-inclusion Problem Under Antiplane Shears

[+] Author and Article Information
Jeng-Tzong Chen

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwanjtchen@mail.ntou.edu.tw

An-Chien Wu

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan

J. Appl. Mech 74(3), 469-487 (May 22, 2006) (19 pages) doi:10.1115/1.2338056 History: Received November 15, 2005; Revised May 22, 2006

## Abstract

In this paper, we derive the null-field integral equation for an infinite medium containing circular holes and/or inclusions with arbitrary radii and positions under the remote antiplane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. By moving the null-field point to the boundary, singular and hypersingular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. Besides, the boundary-layer effect is eliminated owing to the introduction of degenerate kernels. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The method is basically a numerical method, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The exact solution for a single inclusion is derived using the present formulation and matches well with the Honein ’s solution by using the complex-variable formulation (Honein, E., Honein, T., and Hermann, G., 1992, Appl. Math., 50, pp. 479–499). Several problems of two holes, two inclusions, one cavity surrounded by two inclusions and three inclusions are revisited to demonstrate the validity of our method. The convergence test and boundary-layer effect are also addressed. The proposed formulation can be generalized to multiple circular inclusions and cavities in a straightforward way without any difficulty.

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## Figures

Figure 4

Vector decomposition for the potential gradient in the hypersingular equation

Figure 1

(a) Infinite antiplane problem with arbitrary circular inclusions under the remote shear, (b) infinite medium with circular holes under the remote shear, (c) interior Laplace problems for each inclusion, (d) infinite medium under the remote shear, and (e) exterior Laplace problems for the matrix

Figure 2

Graph of the degenerate kernel for the fundamental solution, s=(10,π∕3)

Figure 3

(a) Sketch of the null-field integral equation for a null-field point in conjunction with the adaptive observer system (x∉D,x→Bk) and (b) sketch of the boundary integral equation for a domain point in conjunction with the adaptive observer system (x∊D,x→Bk)

Figure 5

(a) Two equal-sized holes (r1=r2) with centers on the x axis, (b) stress concentration of the problem containing two equal-sized holes, (c) convergence test of the problem containing two equal-sized holes (d=1.0), and (d) tangential stress in the matrix near the boundary (d=1.0)

Figure 6

(a) Two identical inclusions with centers on the x axis and (b) average shear stress of inclusion versus fiber spacing

Figure 9

(a) Two circular inclusions embedded in a matrix under the remote antiplane shear in two directions, (b) stress distributions along the x axis when d=0.1, (c) stress distributions along the x axis when d=0.4, (d) stress distributions along the x axis when d=1.0, (e) normal stress distributions along the contour (1.001,θ), (f) tangential stress distributions along the contour (1.001,θ), (g) variations of stresses at the point (1.001,0deg), and (h) stress distributions along the x axis when the two inclusions touch each other

Figure 10

(a) One hole surrounded by two circular inclusions, (b) tangential stress distribution along the hole boundary with β=0deg, (c) tangential stress distribution along the hole boundary with β=90deg, (d) stress concentration as a function of the spacing d∕r1 with β=0deg, and (e) stress concentration as a function of the spacing d∕r1 with β=90deg

Figure 11

(a) Three identical inclusions forming an equilateral triangle, and (b) tangential stress distribution around the inclusion located at the origin

Figure 8

(a) Two circular inclusions with centers on the y axis, (b) stresses around the circular boundary of radius r1, (c) convergence test of the two-inclusions problem, (d) radial stress in the matrix near the boundary, and (e) tangential stress in the matrix near the boundary

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