Elastic Region

In the context of material behavior, a structural component is said to behave elastically if during loading/unloading the deformation is reversible. In other words, when the loads are released the specimen will return to its original, undeformed configuration. On the contrary, if the material does not return to its initial undeformed state after unloading it is said to behave in-elastically. During tensile testing, the region shaded dark green in Figure 5 corresponds to elastic behavior. Thus if the load were removed within this region, the tensile specimen would return to its undeformed length.

Young's Modulus E is defined as:

or the slope of the stress-strain graph. E is the change in stress divided by the change in strain, so the units are psi per unit strain. Since strain is a unitless quantity, the units for Young's modulus are simply psi.

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Most of the elastic region shown in Figure 5 corresponds to linear-elastic response. Linear elasticity refers to the notion that stress is linearly proportional to strain. So if you double the stress in this region, the change in length of the specimen will also double. In the elastic region, the slope of the stress-strain curve is theYoung's Modulus $E$. So we have:

$$\sigma = E \cdot \epsilon$$ (7)

It is also possible for materials to behave elastically but not linear-elastically. A small portion of the elastic region shown in Figure 5 is above the proportionality limit, where strain increases nonlinearly with increasing stress. You should recognize, however, that there are some materials for which the elastic region is mostly nonlinear.

As loads are increased and the stress in the specimen continues to rise, the material eventually reaches the elastic limit. Beyond this limit, any additional loading will result in some permanent change to the specimen geometry upon unloading. For many materials, this point is indistinguishable from the proportionality limit.

Exercise: Use the java applet for the tensile test to load and unload the bar in the elastic region. What do you think the applet is using for a value for Young's Modulus? What happens if you load just a little past the yield point, and then unload?

Quick Quiz: What's the Young's Modulus of a steel bar that has a cross-sectional area of .73 in^2, is 4 inches long, and supports a load of 3.9e+7 lbs, deforming .2 percent?

E = 9.36e-6 psi
E = 2.67e+10 psi
E = 3.74e-11 psi
E = 2.67e+10 lbs.