Civil Engineering Reference
In-Depth Information
Figure 3.8
Behaviour of ideal linear elastic material.
Most texts on the strength of materials give the basic relationships between the various
elastic parameters and, for isotropic materials, these are
E
G
=
(3.30)
2(1
+
ν
)
E
K
=
(3.31)
3(1
−
2
ν
)
3.9 Perfect plasticity
When the loading has passed the yield point in Fig. 3.1 simultaneous elastic and plastic
strains occur and the stiffness decreases. During an increment of plastic deformation
the work done is dissipated and so plastic strains are not recovered on unloading.
(Bending a paper clip so it remains permanently out of shape is an example of plastic
deformation.)
At the ultimate state there are no further changes of stress (because the stress-strain
curve is horizontal) and so all the strains at failure are irrecoverable. The plastic strains
at failure in Fig. 3.1 are indeterminate; they can go on more or less for ever and so we
can talk about plastic flow. Although it is impossible to determine the magnitudes of
the plastic strains at failure, it is possible to say something about the relative rates of
different strains such as shear and volumetric strains.
Figure 3.9(a) illustrates an element of material loaded to failure with different com-
binations of some arbitrary stresses,
σ
y
. The combinations of stress that cause
failure and plastic flow are illustrated in Fig. 3.9(b) and are represented by a fail-
ure envelope. At any point on the envelope the vector of the failure stress is
σ
x
and
σ
f
and
Fig. 3.9(c) shows the corresponding plastic strains. Since the stresses remain constant
the strains accumulate with time and so the origin is arbitrary. The direction of the
vector of an increment of the plastic straining is given by
p
y
. The relationship
between the failure envelope and the direction of the vector of plastic strain is called a
flow rule.
p
x
/
δε
δε
