Materials for engineering, 3rd Edition - (Malestrom)

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Materials for engineering
presence of the neck, does not reflect the change in strength of the metal
itself, which continues to work harden to fracture. If the true stress, based on
the actual cross-section (A) of the gauge length, is used, the stress–strain
curve increases continuously to fracture, as indicated in Fig. 2.5.
The presence of a neck in the gauge length at large strains introduces local
triaxial stresses that make it difficult to determine the true longitudinal tensile
stress in this part of the curve. Correction factors have to be applied to
eliminate this effect.
Figure 2.5 illustrates the continued work hardening of the material until
fracture, but, unlike the engineering stress–strain curve, the strain at the
point of plastic instability and the UTS are not apparent. Their values may be
readily obtained from Fig. 2.5 by the following approach:
At the UTS, the load (L) passes through a maximum, i.e. dL = 0, and since
L = σA, we may write
dL = σ dA + A dσ = 0
i.e. –dA/A = dσ/σ [2.1]
The volume of the gauge length (V = Al) is constant throughout the test
(dV = 0) and
dl/l = –dA/A [2.2]
Thus, from equations [2.1] and [2.2] we may write:
dσ/σ = dl/l [2.3]
But the strain, e, is defined as
e = (l – l o )/ l o = l/ l o – 1 [2.4]
True stress
Strain
2.5 True stress–strain curve.