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

80
49
THE STRESS–STRAIN dIAgRAM
70
60
Stress (ksi)
50
40
43.5 ksi
Proportional limit
30
20
∆σ
10
∆ε
0.0041 in./in.
0
0.0020 0.0060 0.0100
0.0000 0.0040 0.0080 0.0120
Strain (in./in.)
FIGURE 3.5 Proportional limit.
and normal strain. This ratio is the slope of the initial straight-line portion of the stress–
strain diagram. It is called Young’s modulus, the elastic modulus, or the modulus of
elasticity, and it is denoted by the symbol E:
E
σ
= ∆ ∆ε
(3.1)
The elastic modulus E is a measure of the material’s stiffness. In contrast to strength
measures, which predict how much load a component can withstand, a stiffness measure
such as the elastic modulus E is important because it defines how much stretching, compressing,
bending, or deflecting will occur in a component in response to the loads acting
on it.
In any experimental procedure, there is some amount of error associated with making
a measurement. To minimize the effect of this measurement error on the computed elastic
modulus value, it is better to use widely separated data points to calculate E. In the linear
portion of the stress–strain diagram, the two most widely spaced data points are the proportional
limit point and the origin. Using the proportional limit and the origin, we would
compute the elastic modulus E as
E
43.5 ksi
= = 10,610 ksi
(3.2)
0.0041 in./in.
In practice, the best value for the elastic modulus E is obtained from a least-squares fit of a
line to the data between the origin and the proportional limit. Using a least-squares analysis,
we find that the elastic modulus for this material is E = 10,750 ksi.
Work Hardening
The effect of unloading and reloading on the load–deformation plot was shown in
Figure 3.3. The effect of unloading and reloading on the stress–strain diagram is shown