Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-07 GTBL042-Callister-v2 August 9, 2007 13:52

208 • Chapter 7 / Mechanical Properties

Strain

Engineering

Corrected

True

M

M

Stress

Figure 7.16 A comparison of typical

tensile engineering stress–strain and

true stress–strain behaviors. Necking

begins at pointMon the engineering

curve, which corresponds toM′on the

true curve. The “corrected” true

stress–strain curve takes into account

the complex stress state within the neck

region.

true strain Furthermore, it is occasionally more convenient to represent strain astrue strain

T, defined by

T=ln

li

l 0

Definition of true (7.16)

strain

If no volume change occurs during deformation—that is, if

Aili=A 0 l 0 (7.17)

true and engineering stress and strain are related according to

σT=σ(1+) (7.18a)

Conversion of

engineering stress to

true stress

T=ln(1+) (7.18b)

Conversion of

engineering strain to

true strain

Equations 7.18a and 7.18b are valid only to the onset of necking; beyond this point

true stress and strain should be computed from actual load, cross-sectional area, and

gauge length measurements.

A schematic comparison of engineering and true stress–strain behaviors is made

in Figure 7.16. It is worth noting that the true stress necessary to sustain increasing

strain continues to rise past the tensile pointM′.

Coincident with the formation of a neck is the introduction of a complex stress

state within the neck region (i.e., the existence of other stress components in addition

to the axial stress). As a consequence, the correct stress (axial) within the neck is

slightly lower than the stress computed from the applied load and neck cross-sectional

area. This leads to the “corrected” curve in Figure 7.16.

For some metals and alloys the region of the true stress–strain curve from the on-

set of plastic deformation to the point at which necking begins may be approximated

by

σT=KTn (7.19)

True stress-true

strain relationship in

plastic region of

deformation (to

point of necking)

In this expression,Kandnare constants; these values will vary from alloy to alloy, and

will also depend on the condition of the material (i.e., whether it has been plastically

deformed, heat treated, etc.). The parameternis often termed thestrain-hardening

exponentand has a value less than unity. Values ofnandKfor several alloys are

contained in Table 7.4.