Page 132 - Mechanical Behavior of Materials

Section 4.3 Engineering Stress–Strain Properties 133

Figure 4.13 Fractures from tension tests on 9 mm diameter specimens of hot-rolled
AISI 1020 steel (left) and gray cast iron (right). (Photos by R. A. Simonds.)

U x P x ε
u = = d = σ dε (4.10)
A i L i 0 A i L i 0

Hence, u is the work done per unit volume of material to reach a strain ε , and it is equal to the

area under the stress–strain curve up to ε . The work done is equal to the energy absorbed by the
material.
The area under the entire engineering stress–strain curve up to fracture is called the tensile
toughness, u f . This is a measure of the ability of the material to absorb energy without fracture.
Where there is considerable plastic strain beyond yielding, as for many engineering metals, some of
the energy is stored in the microstructure of the material, but most of it is dissipated as heat.
If the stress–strain curve is relatively flat beyond yielding, then u f may be approximated as the
area of a rectangle. The height is equal to the average of the yield and ultimate, and the width is
equal to the fracture strain:

σ o + σ u
u f ≈ ε f (4.11)
2
For materials that behave in a brittle manner, the gradually curving stress–strain response may be
similar to a parabolic curve with vertex at the origin, in which case u f ≈ 2σ f ε f /3.
Brittle materials have low tensile toughness, despite perhaps high strength, due to low ductility.
In low-strength ductile materials, the converse occurs, and the tensile toughness is also low. To have
a high tensile toughness, both the strength and the ductility must be reasonably high, so that a high
tensile toughness indicates a “well rounded” material.