Criteria for Failure and Fracture

Abstract

Structural members may have geometric features (including holes, notches, and corners) that result in localized regions of high stress called stress concentrations. The ratio C of the maximum stress resulting from a stress concentration to a defined nominal stress is called the stress concentration factor. Graphs of stress concentration factors are presented for an axially loaded bar, a stepped circular bar subjected to torsion, and a rectangular beam subjected to bending. Failure in brittle materials occurs when a component breaks or fractures, whereas in ductile materials failure is usually defined to occur with the onset of yielding. For a brittle material, the maximum normal stress and Mohr’s failure criteria and for a ductile material the Tresca and von Mises yield criteria are discussed. Fatigue is subdivided into low-cycle, high-cycle, and fatigue crack growth. In low-cycle fatigue, stress levels exceed the yield stress, and the number of cycles to failure is relatively low (<103). High-cycle fatigue can occur when stress levels are lower than the yield stress, and failure may require 103 to 106 cycles. An endurance curve is a graph of the stress amplitude or fatigue strength as a function of the number of cycles to failure at zero mean stress. For some materials, there is a stress amplitude, the fatigue limit, below which fatigue life is essentially infinite. Keeping stress levels below the fatigue limit is known as safe life design. When a structural component contains a preexisting crack, fracture mechanics can be used to determine the state of stress and predict when failure will occur. The stress intensity factor K is a measure of the magnitude of the stress in the neighborhood of a crack tip. In many cases stress intensity factors can be expressed in the form \( K=\sigma \sqrt{\pi a}\;Q\left(a/W\right), \) where σ is the stress, a is the crack length, and Q(a/W) is a function called the configuration factor. The value of the stress intensity factor at which fast crack growth begins is called the fracture toughness. Under cyclic or repeated loading, cracks can grow at stress intensity factor levels that are lower than the fracture toughness. In this situation, the resulting fatigue crack growth or slow growth is governed by the Paris law da/dN = A(ΔK)n, and the number of loading cycles required for failure is determined by calculating the number of cycles necessary for the crack length to reach the critical value for fast growth.