Materials for engineering, 3rd Edition - (Malestrom)

Determination of mechanical properties 43
Therefore,
de = dl/l o = (dl/l)(l/l o ) [2.5]
So from [2.4] and [2.5] we obtain:
de = (dl/l)(1 + e) [2.6]
Eliminating dl/l from [2.6] and [2.3] we find that, at the UTS:
dσ/de = σ u /(1 + e u ) [2.7]
where σ u is the true stress at the UTS and e u is the strain at the point of
plastic instability. Equation [2.7] thus enables us to identify these values by
means of the Considère construction, whereby a tangent to the true stress–
strain curve is drawn from a point corresponding to –1 on the strain axis
(Fig. 2.6). Additionally, the intercept of this tangent on the stress axis will
give the value of the UTS, since
σ u /UTS = A o /A = l/l o (since the volume is constant)
But from (2.4),
l/l o = 1 + e
Thus, by similar triangles, it may be seen in Fig. 2.6 that the Considère
tangent to the true stress–strain curve identifies both the point of plastic
instability, the true stress at the UTS and the value of the UTS itself.
2.3 Bend testing
Testing brittle materials in tension is difficult, due to the problems of gripping
the specimens in a tensile machine without breaking them. Furthermore, in
True stress (σ)
C
D
σ u
UTS
–I
A
I
I + e u
2.6 The Considère construction.
+I
0 B
e Conventional
u
strain