Page 64 - Shigley's Mechanical Engineering Design

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Materials 39
The material now has a higher yield point, is less ductile as a result of a reduction in
strain capacity, and is said to be strain-hardened. If the process is continued, increasing
p , the material can become brittle and exhibit sudden fracture.
It is possible to construct a similar diagram, as in Fig. 2–6b, where the abscissa is
the area deformation and the ordinate is the applied load. The reduction in area corre-
sponding to the load P f , at fracture, is defined as

A 0 − A f A f
R = = 1 − (2–12)
A 0 A 0
where A 0 is the original area. The quantity R in Eq. (2–12) is usually expressed in per-
cent and tabulated in lists of mechanical properties as a measure of ductility. See
Appendix Table A–20, for example. Ductility is an important property because it mea-
sures the ability of a material to absorb overloads and to be cold-worked. Thus such
operations as bending, drawing, heading, and stretch forming are metal-processing
operations that require ductile materials.
Figure 2–6b can also be used to define the quantity of cold work. The cold-work
factor W is defined as
A 0 − A A 0 − A i
W = i ≈ (2–13)
A 0 A 0
where A corresponds to the area after the load P i has been released. The approxima-

i
tion in Eq. (2–13) results because of the difficulty of measuring the small diametral
changes in the elastic region. If the amount of cold work is known, then Eq. (2–13) can
be solved for the area A . The result is

i

A = A 0 (1 − W) (2–14)
i
Cold working a material produces a new set of values for the strengths, as can
3
be seen from stress-strain diagrams. Datsko describes the plastic region of the true
stress–true strain diagram by the equation
σ = σ 0 ε m (2–15)
where σ = true stress
σ 0 = a strength coefficient, or strain-strengthening coefficient
ε = true plastic strain
m = strain-strengthening exponent
4
It can be shown that
(2–16)
m = ε u
provided that the load-deformation curve exhibits a stationary point (a place of zero
slope).

3 Joseph Datsko, “Solid Materials,” Chap. 32 in Joseph E. Shigley, Charles R. Mischke, and Thomas
H. Brown, Jr. (eds.), Standard Handbook of Machine Design, 3rd ed., McGraw-Hill, New York, 2004. See also
Joseph Datsko, “New Look at Material Strength,” Machine Design, vol. 58, no. 3, Feb. 6, 1986, pp. 81–85.
4 See Sec. 5–2, J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, 6th ed., McGraw-Hill,
New York, 2001.