Page 148 - Mechanical Behavior of Materials

Section 4.5 True Stress–Strain Interpretation of Tension Test 149

in Fig. 4.18. This is seen to represent the corrected data very well over all strains that are beyond
the anomalous upper/lower yield region of this material. Where the stress–strain curve exhibits
a smooth, gradual yielding behavior, Eq. 4.25 may provide a good fit for all strains. For some
materials, Eq. 4.25 does not fit very well, and some cases of this type can be handled by two power
laws that fit different portions of the stress–strain curve.
If cross-sectional area (diameter) measurements are not available for a tension test, then
Eq. 4.25 may still be employed, but the fit is restricted to data that are not beyond the engineering
ultimate stress point. The Bridgman correction then becomes unnecessary, so that Eq. 4.23 applies,
and ˜σ B in Eq. 4.25 is replaced by the uncorrected ˜σ. Also, since the ultimate point is not exceeded,
˜ σ and ˜ε may be calculated simply from Eqs. 4.18 and 4.15.

4.5.6 True Stress–Strain Properties

Additional materials properties obtained from tension tests may be defined on the basis of true stress
and strain. The true fracture strength, ˜σ f , is obtained simply from the load at fracture and the final
area, or from the engineering stress at fracture:

P f A i
˜ σ f = = σ f (4.26)
A f A f
Since the Bridgman correction is generally needed, the value obtained should be converted to ˜σ fB
by using Eq. 4.21, but note the limitation of Eq. 4.22 to steels.
The true fracture strain, ˜ε f , may be obtained from the final area or from the percent reduction
in area:
A i 100
˜ ε f = ln , ˜ ε f = ln (a, b) (4.27)
A f 100 − %RA
The second equation follows readily from the first. Note that ˜ε f cannot be computed if only the
engineering strain at fracture (percent elongation) is available.
The true toughness, ˜u f , is the area under the true stress–strain curve up to fracture. Assume that
the material is quite ductile, so that the elastic strains are small, compared with the plastic strains,
over most of the stress–strain curve. Hence, let the plastic and total strains be taken as equivalent,
n
˜ ε p ≈˜ε, and consequently replace Eq. 4.24 with ˜σ B = H ˜ε . On this basis, the true toughness is
n+1
˜ ε f ˜ ε f n f ˜ σ fB ˜ε f
H ˜ε
˜ u f = ˜ σ B d˜ε = H ˜ ε d˜ε = = (4.28)
0 0 n + 1 n + 1
The various engineering and true stress–strain properties from tension tests are summarized
by the categorized listing of Table 4.5. Note that the engineering fracture strain ε f and the percent
elongation are only different ways of stating the same quantity. Also, the %RA and ˜ε f can each be
calculated from the other by using Eq. 4.27(b). The strength coefficient H determines the magnitude
of the true stress in the large strain region of the stress–strain curve, so it is included as a measure
of strength. The strain hardening exponent n is a measure of the rate of strain hardening for the true