The material is said to yield at the point where it stops behaving as an elastic material, and this point typically occurs at a small fraction of a percent of strain. After this point, if the load is removed, the specimen retains a permanent deformation called the plastic deformation. At some load the specimen breaks, and the test is over. Ductile metals neck at some point prior to failure. Until that point, the cross-sectional area of the specimen is uniform and very close to the original area before loading. The data are commonly plotted in two forms:
Figure 1: Engineering (left) and true (right) stress strain responses (not to scale).
1) engineering stress versus engineering strain, and 2) true stress versus true strain. Example plots, which aren’t to scale, are shown in Figure 1. The engineering stress, s, is the load, P, divided by the original crosssectional area of the specimen, A0, and the engineering strain, e, is the current length of the specimen, l, divided by the original length, l0,
In contrast, the true stress, σ, is the load divided by the current area, and the true strain, ε, is the log of the ratio of the original cross-sectional area to the current area at the neck,
Once the specimen starts to neck, the strain is no longer uniform in the specimen. The true stress-true strain curve describes the response at the neck, where the stress is the highest. To calculate the true strain at the neck, an equivalent length ratio is calculated by assuming that the total volume of the specimen is constant during the test,
Note that there is a peak in the engineering stress-strain plot, which is often referred to as the ultimate strength of the material. The true stressstrain plot, however, shows a steady increase in the stress after yielding to failure. A common approximation for the plastic portion of the true stressstrain curve is a straight line. In this simplied model, the yield or ow stress is a linear function of the plastic strain, ε^-p,
The plastic hardening modulus, h, is typically orders of magnitude smaller than the elastic modulus, and if it is zero, the material is said to be perfectly plastic. An important restriction on the hardening modulus is that it must be greater than, or equal to, zero. If the hardening modulus is less than zero, then the material is softening and unstable. Real materials do exhibit softening, but an additional variable, the damage D, or temperature is necessary to model the softening eect correctly. Damage mechanics, which is an important area of continuum mechanics, is beyond the scope of this discussion.
The plastic strain is a history variable that evolves with the stress during the deformation of the material. An important distinction between the plastic strain and the material strain is that while a material may be stretched and compressed to produce both positive and negative material strains, the plastic strain is always greater than, or equal to, zero and never decreases, regardless of the deformation of the material.