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

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.