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TECHNICAL PAPERS

# Transmission of Elastic Stress Through Circular and Elliptic Cross Sections of Microstructural Elements Embedded in a Matrix Material

[+] Author and Article Information
C. M. Kennefick

Reston, VA 20191

J. Appl. Mech 72(4), 558-563 (Oct 30, 2004) (6 pages) doi:10.1115/1.1935525 History: Received May 27, 2004; Revised October 30, 2004

## Abstract

With the use of contact stress theory and complex variable methods in two dimensions, the transmission of a compressive stress through a circular cross section of a small material particle is calculated in the infinite plane of material below the circular cross section. The circular cross section of the particle is embedded in and completely bonded to an infinite plane of matrix material. It is shown that part of the stress is transmitted with a dependence of $1∕r$, where $r$ is a radial coordinate. Additionally, the stress is calculated in two dimensions for the interior of an ellipse that could model a cross section of a grain or particle. The boundary of the ellipse is loaded with the stress holding an elliptic kernel in place in an elastic matrix material after the kernel has undergone a small rotation under an applied tensile load. The resulting stresses are shown in contour plots for elliptic cross sections of varying shapes and orientations.

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## Figures

Figure 1

Diagram showing the Cartesian, cylindrical, and angular coordinates for the circular cross section

Figure 2

Diagram showing the slight rotation through an angle ε of an elliptically shaped cross section embedded in an elastic material matrix and under a remote applied tensile stress

Figure 3

Compressive stress σyy below the circular cross section as a function of distance from the center of the cross section

Figure 4

Stress σρρ in the direction perpendicular to the perimeter of the ellipse. The remote applied tensile stress is aligned with the vertical direction pointing to the top and bottom of the figure. The major axis of the ellipse makes an angle of 45deg with the applied load. The ratio of the length of the major axis of the ellipse to the length of the minor axis of the ellipse is 5 to 2.

Figure 5

Shear stress σρθ in the direction parallel to the perimeter of the ellipse. The orientation of the applied load, as well as the size and orientation of the ellipse, are the same as in Fig. 4.

Figure 6

The stress σyy is plotted for an elliptical cross section that is nearly circular. In contrast with σyy shown in Fig. 3 when a circular cross section transmits a compressive stress, shown above is an almost circular cross section that rotates slightly under a tensile load that is aligned with the vertical direction pointing to the top and bottom of the figure. The major axis of the ellipse is at an angle of 45deg from the remote applied tensile load. The ratio of the length of the major axis of the ellipse to the length of the minor axis of the ellipse is 15 to 14.

Figure 7

Stress σρρ plotted for bundles of ellipses adjacent to each other. The stress due to the rotation of a particular ellipse is plotted up to a line midway between it and the next nearest ellipse. The remote applied tensile stress is aligned with the vertical direction pointing to the top and bottom of the figure. The major axis of the ellipses in the lower bundle make and angle of 89deg with respect to the applied load. The major axes of the ellipses in the upper bundle make and angle of 60deg with respect to the applied load. The ratio of the length of the major axis of each ellipse to the length of its minor axis is 10 to 1.

Figure 8

Shading scale used for the stress contours in Figs.  4567

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