Failure Analysis

When a specimen is loaded until its failure point in the tensile test, a distinct surface is formed on each of the resulting two pieces. In general, it is possible to predict the failure surface given the state of stress and the material characterization. Upon knowing whether the material is brittle or ductile, we can predict the failure surface reasonably well. Failure takes place along the plane of maximum shear stress for ductile materials. Brittle materials fail on planes of maximum tensile normal stress. Thus the failure surface can be identified by:

1. Locating the point where the stress state is most critical (i.e., the largest in magnitude and the correct sign),
2. Determining the state of stress at this point (drawing Mohr's circle),
3. Identifying the correct plane based upon the class of material.

Mohr's Circle

A very useful way of expressing and visualizing the plane stresses in a loaded structural element is method known as Mohr's Circle, developed by a German engineer, Otto Mohr (1835 - 1918).

For a biography of Mohr, click on his portrait. For helpful links, visit a site of the University of Wisconsin, or the University of Valladolid. An interactive java applet can be found on a site of Iowa State University.

For the tensile test, the state of stress is similar along the length of the specimen. For this basic analysis, we will ignore the difference in the state of stress between the region where necking occurs and the rest of the specimen. We will also assume that the stress component which is perpendicular to the specimen axis (and thus the direction of the applied load) is negligible in comparison to the component along the axis. Under these simplifying assumptions,the Mohr's circle for a specimen in pure tension is as shown in Figure 10. Plane A falls on the origin in the diagram and plane B (with coordinates $(\sigma_o,0)$) is as indicated in the figure. Once we have constructed Mohr's circle for the tensile test, all we need to do is to consider the failure criteria for brittle and ductile materials to determine the plane along which failure takes place. This process is shown in Figure 11, and we provide details for a brittle material. Brittle materials fail on planes of maximum tensile normal stress, which is identified as point $2$ in Figure 11. So a brittle specimen would fail along the plane represented by point $2$, which is the plane perpendicular to the longitudinal axis of the tensile specimen. A similar argument can be used to verify that a ductile specimen will fail on a plane with a $45^0$ inclination to the longitudinal axis of the specimen. Remember that a $2\theta$ angle between points in Mohr's circle corresponds to an inclination of $\theta$ between the corresponding planes.

In ductile materials, the failure surface obtained through the tensile test usually resembles a ``cup and cone'' fracture. This can be seen clearly in the videos of the tests on ductile specimens. Rupture occurs along a cone-shaped surface which forms an angle of approximately $45^0$ with the original surface of the specimen. If you examine Mohr's circle in Figure 10 again, you should notice that the shear stress is a maximum at angles of $45^0$ from the loading axis. Thus, the notion that ductile materials fail primarily in shear leads us to a reliable prediction of the failure surface in the tensile test.

Figure 11: Failure surface determination for brittle and ductile materials.