Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

The elastic limit is the largest stress that a material can withstand without any
measurable permanent strain remaining after complete release of the stress. The procedure
required to determine the elastic limit involves cycles of loading and unloading, each time
incrementally increasing the applied stress (Figure 3.7). For instance, stress is increased to
point A and then removed, with the strain returning to the origin O. This process is repeated
for points B, C, D, and E. In each instance, the strain returns to the origin O upon unloading.
Eventually, a stress will be reached (point F) such that not all of the strain will be recovered
during unloading (point G). The elastic limit is the stress at point F.
How does the elastic limit differ from the proportional limit? Although such materials are
not common in engineered applications, a material can be elastic even though its stress–strain
relationship is nonlinear. For a nonlinear elastic material, the elastic limit could be substantially
greater than the proportional limit stress. Nevertheless, the proportional limit is generally
favored in practice since the procedure required to establish the elastic limit is tedious.
Stress
Elastic limit
F
E
D
C
B
A
51
THE STRESS–STRAIN dIAgRAM
Yielding
For many common materials (such as the low-carbon steel shown in Figure 3.4 and enlarged
in Figure 3.8), the elastic limit is indistinguishable from the proportional limit. Past the
elastic limit, relatively large deformations will occur for small or almost negligible increases
in stress. This behavior is termed yielding.
A material that behaves in the manner depicted in Figure 3.8 is said to have a yield
point. The yield point is the stress at which there is an appreciable increase in strain with no
increase in stress. Low-carbon steel, in fact, has two yield points. Upon reaching the upper
yield point, the stress drops abruptly to a sustained lower yield point. When a material yields
without an increase in stress, the material is often referred to as being perfectly plastic.
Materials having a stress–strain diagram similar to Figure 3.8 are termed elastoplastic.
Not every material has a yield point. Materials such as the aluminum alloy shown in
Figure 3.4 do not have a clearly defined yield point. While the proportional limit marks the
uppermost end of the linear portion of the stress–strain curve, it is sometimes difficult in practice
to determine the proportional limit stress, particularly for materials with a gradual transition
from a straight line to a curve. For such materials, a yield strength is defined. The yield
strength is the stress that will induce a specified permanent set (i.e., plastic deformation) in the
material, usually 0.05% or 0.2%. (Note: A permanent set of 0.2% is another way of expressing
a strain value of 0.002 in./in., or 0.002 mm/mm.) To determine the yield strength from the
stress–strain diagram, mark a point on the strain axis at the specified permanent set (Figure
3.9). Through this point, draw a line that is parallel to the initial elastic modulus line. The stress
at which the offset line intersects the stress–strain diagram is termed the yield strength.
Strain Hardening and Ultimate Strength
After yielding has taken place, most materials can withstand additional stress before
fracturing. The stress–strain curve rises continuously toward a peak stress value, which is
termed the ultimate strength. The ultimate strength may also be called the tensile strength
or the ultimate tensile strength (UTS). The rise in the curve is called strain hardening. The
strain-hardening regions and the ultimate strength points for a low-carbon steel and an
aluminum alloy are indicated on the stress–strain diagrams in Figure 3.4.
Strain
FIGURE 3.7 Elastic limit.
Stress (ksi)
80
70
60
50
40
30
20
10
O
G
Upper yield point
Low-carbon
steel
Lower
yield point
0
Strain
FIGURE 3.8 Yield point for
low-carbon steel.
Necking
In the yield and strain-hardening regions, the cross-sectional area of the specimen
decreases uniformly and permanently. Once the specimen reaches the ultimate strength,
however, the change in the specimen cross-sectional area is no longer uniform
throughout the gage length. The cross-sectional area begins to decrease in a localized