Setup and Loading Implications

The tensile test is very special in that it places a specimen of material into a state of uniaxial loading, as shown in Figure 2a, below. The actual measurements taken during the test include the force $P$ applied to the specimen as a function of time, and the change in length between two points on the undeformed geometry. In order to properly characterize the results of the test, this raw data must be interpreted and post-processed (by hand calculations or computer, for example), requires you to make some important assumptions.

Figure 2: Setup and assumptions of the tensile test: a) A cylindrical specimen with cross-sectional area $A$ is placed in uniaxial tension under a force $P$; b) Assumed state of engineering stress for a material element in the bar; c) A general state of two-dimensional stress.

ten·sor n. the mathematical idealization of a geometric or physical quantity whose analytic description, relative to a fixed frame of reference, consists of an array of numbers.

For more information on tensor quantities, visit Planet Math, or this University of California Santa Cruz site.

Stress and strain are tensor quantities, which for planar problems have three independent components (as opposed to vector quantities, which have only two). The particular loading of the tensile test allows us to focus on a single component of both - simply those associated with the orientation of the applied force or displacement. Figure 2b (above) shows the assumed state of stress on the surface of the specimen in the tensile test, which can be compared to a general state of plane stress shown in Figure 2c. When interpreting the experimental data of the tensile test, we will make two main assumptions:

1. The stress component oriented with the long axis of the specimen ( $\sigma_{yy}$ in Figure 2b) is much larger than the other components, so that they can be neglected;
2. The state of stress at any point on the cross-section is very close to the average stress in the cross-section. Since this is the only component of interest, we will simply refer to $\sigma_{yy}$ as $\sigma$ in the rest of the tutorial.

To some extent, analogous assumptions can be made about the strain tensor. If you play any one of the movies of tensile test experiments, however, you will notice that the diameter of the specimen can change appreciably toward the end of the loading. Clearly, the bar stretches along its axis and contracts in the radial direction. The effect of this lateral contraction, and the associated decrease in cross-sectional area, gives rise to a difference between true stress and engineering stress.