Civil Engineering Reference
In-Depth Information
Figure 3.12
A physical model for elastic and plastic behaviour.
In Fig. 3.13 material is loaded from O
1
and is elastic until yielding occurs at Y
1
,
where the yield stress is
σ
x
1
. It is then strained further and unloaded to O
2
where
there are irrecoverable plastic strains
p
x
1
. When the material is reloaded from O
2
it is
elastic until yielding occurs at Y
2
, where the yield stress is
δε
σ
x
2
. If the material is then
strained further and unloaded to O
3
, on reloading it will have a new yield stress
σ
x
3
and so on. Thus the principal consequences of straining from Y
1
to Y
2
(or from Y
2
to Y
3
) are to cause irrecoverable plastic strains and to raise the yield point from
σ
x
1
σ
x
3
). This increase of the yield point due to plastic straining is
called hardening and the relationship between the increase in the yield stress
σ
x
2
(or from
σ
x
2
to
to
δσ
x
and
p
x
is known as a hardening law. In Fig. 3.13 there is a broken
line to the left of the first yield point, which suggests that there could be even lower
yield points for previous loadings; this simply demonstrates that the origin of strains
O
1
was arbitrarily chosen.
Yielding and plastic straining may cause hardening (i.e. an increase in the yield
stress), as shown in Fig. 3.14(a), or softening (i.e. a decrease in the yield stress), as
shown in Fig. 3.14(b). In the latter case the state has reached, and passed, a peak in
the stress-strain curve, and this is a feature commonly found in the behaviour of soils.
In each case the total strains are the sum of the elastic and plastic components and the
plastic strains are related to the change of the yield stress by a hardening law.
the plastic straining
δε
