Stress concentration at a notch tip in unidirectional metal matrix composites
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I.
INTRODUCTION
W H E N notched composites are pulled in tension, high stress concentration is exerted on the fibers at the notch tip. For two-dimensional (2-D) composites, the stress concentration can be calculated by the shear lag analysis method t~-5] first put forth by Hedgepeth. tl] This method has been demonstrated to give a good description of the stress concentration factor. [6'7] For metal matrix composites, Zweben tz] and Goree and Gross t3] calculated the stress concentration for a nonstrain-hardenable matrix and Reedy t4] for a strainhardenable matrix. Their analyses were based on the assumption that the width of the notch 2A is zero, as schematically shown in Figure l(a). However, the artificial notch has a finite width. For the rectangular notch shown in Figure l(b), Dharani et al. tS~calculated the stress concentrations for a nonstrain-hardenable matrix, but the calculation results on the influence of the width of the notch on the stress concentrations have not been presented in detail. The aim of the present work is to present an approximate method of calculating the stress concentrations in the fibers at the tip of a rectangular notch for a strainhardenable matrix and to predict the influence of the width of the notch on the stress concentrations by modifying the shear lag analysis. Another aim is to try to describe the load-COD curves and notched strength of boron/ aluminum and alumina/aluminum composites.
II. C A L C U L A T I O N M E T H O D O F T H E STRESS CONCENTRATION FACTOR OF THE FIBERS AT T H E N O T C H T I P In this section, the calculation method is presented for double-edge notch. This method can be applied also to center notch.
SHOJIRO OCHIAI, Associate Professor, KOZO OSAMURA, Professor, and KENJI TOKINORI, Graduate Student, are with the Department of Metallurgy, Kyoto University, Sakyo-ku, Koyoto 606, Japan. MITSUHISA NAKATANI, Deputy Manager, and KOJI YAMATSUTA, Research Associate, are with Sumitomo Chemicals Co. Ltd., Chuuo-ku, Tokyo 103, Japan. Manuscript submitted July 5, 1990. METALLURGICAL TRANSACTIONS A
A. Modeling Figure 2(a) shows a schematic representation of the transverse cross section of a composite, in which the fiber array is assumed to be a square. This three-dimensional (3-D) composite was converted to a 2-D one, as shown in Figure 2(b), whose longitudinal section is shown in Figure 2(c). In this conversion, the width of the fiber in the 2-D model, df, was given by
df = d~ ( vrVl /4) '/2
[11
where d~ is the fiber diameter and Vf is the fiber volume fraction. The width of the matrix region in the 2-D model, din, was given by
dm= ds VmlV:
[2]
where V,, is the volume fraction of the matrix. The present 2-D model with the width W consists of N total number of fibers, among which n fibers are cut by the introduction of the double-edge notch with a total length 2C, as shown in Figure 2(e). The artificial notch was modeled as rectangular. The half-notch width was taken to be A, as shown in Figure 3. The n cut fibers (named as "1" in Figure 3) are separated from
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