Page 143 - Mechanical Behavior of Materials

144 Chapter 4 Mechanical Testing: Tension Test and Other Basic Tests

Since the area A decreases as a tension test proceeds, true stresses increasingly rise above the
corresponding engineering stresses. Also, there is no drop in stress beyond an ultimate point, which
is expected, as this behavior in the engineering stress–strain curve is due to the rapid decrease in
cross-sectional area during necking. These trends are evident in Fig. 4.18.
For true strain, let the length change be measured in small increments, L 1 , L 2 , L 3 , etc.,
and let the new gage length, L 1 , L 2 , L 3 , etc., be used to compute the strain for each increment. The
total strain is thus

L 1 L 2 L 3 L j
˜ ε = + + + ··· = (4.13)
L 1 L 2 L 3 L j
where L is the sum of these L j .Ifthe L j are assumed to be infinitesimal—that is, if L is
measured in very small steps—the preceding summation is equivalent to an integral that defines true
strain:
L
dL L
˜ ε = = ln (4.14)
L L i
L i
Here, L = L i + L is the final length. Note that ε = L/L i is the engineering strain, leading to
the following relationship between ε and ˜ε:

L i + L L
˜ ε = ln = ln 1 + = ln (1 + ε) (4.15)
L i L i

4.5.2 Constant Volume Assumption

For materials that behave in a ductile manner, once the strains have increased substantially beyond
the yield region, most of the strain that has accumulated is inelastic strain. Since neither plastic
strain nor creep strain contributes to volume change, the volume change in a tension test is limited
to the small amount associated with elastic strain. Thus, it is reasonable to approximate the volume
as constant:
A i L i = AL (4.16)
This gives

A i L L i + L
= = = 1 + ε (4.17)
A L i L i
Substitution into Eqs. 4.12(b) and 4.14 then gives two additional equations relating true and
engineering stress and strain:

˜ σ = σ (1 + ε) (4.18)

A i
˜ ε = ln (4.19)
A
For members with round cross sections of original diameter d i and final diameter d, the last equation
maybeusedinthe form