Page 61 - Shigley's Mechanical Engineering Design

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36 Mechanical Engineering Design
In addition to providing strength values for a material, the stress-strain diagram
provides insight into the energy-absorbing characteristics of a material. This is because
the stress-strain diagram involves both loads and deflections, which are directly related
to energy. The capacity of a material to absorb energy within its elastic range is called
resilience. The modulus of resilience u R of a material is defined as the energy absorbed per
unit volume without permanent deformation, and is equal to the area under the stress-
strain curve up to the elastic limit. The elastic limit is often approximated by the yield
point, since it is more readily determined, giving

y
∼ (2–8)
u R = σd
0
where y is the strain at the yield point. If the stress-strain is linear to the yield point,
then the area under the curve is simply a triangular area; thus
1 1 S 2 y
∼ (2–9)
u R = S y y = (S y )(S y /E) =
2 2 2E
This relationship indicates that for two materials with the same yield strength, the
less stiff material (lower E), will have a greater resilience, that is, an ability to absorb
more energy without yielding.
The capacity of a material to absorb energy without fracture is called toughness.
The modulus of toughness u T of a material is defined as the energy absorbed per unit
volume without fracture, which is equal to the total area under the stress-strain curve up
to the fracture point, or
f

u T = σd (2–10)
0
where f is the strain at the fracture point. This integration is often performed graphi-
cally from the stress-strain data, or a rough approximation can be obtained by using the
average of the yield and ultimate strengths and the strain at fracture to calculate an area;
that is,

S y + S ut
∼ (2–11)
u T = f
2
3
3
The units of toughness and resilience are energy per unit volume (lbf in/in or J/m ),
which are numerically equivalent to psi or Pa. These definitions of toughness and
resilience assume the low strain rates that are suitable for obtaining the stress-strain
diagram. For higher strain rates, see Sec. 2–5 for impact properties.

2–2 The Statistical Significance of Material Properties
There is some subtlety in the ideas presented in the previous section that should be pon-
dered carefully before continuing. Figure 2–2 depicts the result of a single tension test
(one specimen, now fractured). It is common for engineers to consider these important
stress values (at points pl, el, y, u, and f ) as properties and to denote them as strengths
with a special notation, uppercase S, in lieu of lowercase sigma σ, with subscripts
added: S pl for proportional limit, S y for yield strength, S u for ultimate tensile strength
(S ut or S uc , if tensile or compressive sense is important).
If there were 1000 nominally identical specimens, the values of strength obtained
would be distributed between some minimum and maximum values. It follows that the