Civil Engineering Reference
In-Depth Information
Figure 3.15
Examples of simple yield curves.
Yielding under combined stresses may be represented by a set of yield curves which
are similar to the failure envelope, as illustrated in Fig. 3.15. This shows a yield curve
for the first yield, two yield curves for subsequent yielding and a failure envelope. For
states inside the first yield curve the behaviour is elastic. The state cannot reach the
region outside the failure envelope. If the plastic strains are perfect then the vectors
of plastic strain are normal to the yield curves. Thus, for the loading path A
→
Bin
Fig. 3.16 which crosses successive yield surfaces the vectors of plastic strain are normal
to the yield surface.
Since each yield curve in Fig. 3.16 is associated with a particular plastic strain we can
use the plastic strain as a third axis to develop a yield surface, as shown in Fig. 3.17.
For any state on the yield surface there are plastic strains that are normal to the appro-
priate yield curve and are given by the movement of the stress point across the surface.
For any state inside the surface, during first loading or due to unloading, the behaviour
is elastic. Thus, for the loading and unloading O
→
A
→
B
→
C in Fig. 3.17 the
→
→
→
behaviour is elastic for the paths O
A and B
C. For the path A
B there are
simultaneous elastic and plastic strains.
It is now possible to assemble the flow rule, the hardening law and the elastic stress-
strain equations into an explicit constitutive equation for the complete range of loading
Figure 3.16
Plastic straining for loading on a yield surface.
