Civil Engineering Reference
In-Depth Information
Stress
F>0
F=0
F<0
F= f(stress, material properties)
Strain
Figure 6.3
Elastic-perfectly plastic stress-strain law
During plastic straining, the material may flow in an “associated” manner, that is the
vector of plastic strain increment may be normal to the yield or failure surface. Alterna-
tively, normality may not exist and the flow may be “non-associated”. Associated flow
leads to various mathematically attractive simplifications and, when allied to the von Mises
or Tresca failure criterion, correctly predicts zero plastic volume change during yield for
undrained clays. For frictional materials, whose ultimate state is described by the Mohr-
Coulomb criterion, associated flow leads to physically unrealistic volumetric expansion or
dilation during yield. In such cases, non-associated flow rules are preferred in which plastic
straining is described by a plastic potential function
Q
. This function may be geometri-
cally similar to the failure function
F
but with the friction angle
φ
replaced by a dilation
angle
. The implementation of the plastic potential function will be described further in
Sections 6.6 and 6.7.
Before outlining some commonly used failure criteria and their representations in prin-
cipal stress space, some useful stress invariant expressions are reviewed briefly.
ψ
6.3 Stress invariants
The Cartesian stress tensor defining the stress conditions at a point within a loaded body
is given by:
σ
x
σ
y
σ
z
τ
xy
τ
yz
τ
zx
T
(6.1)
which can be shown to be equivalent to three principal stresses acting on orthogonal planes:
σ
1
σ
2
σ
3
]
T
[
(6.2)
Principal stress space is obtained by treating the principal stresses as three-dimensional
coordinates and is a useful way of representing a stress state at a point. It may be noted
that although principal stress space defines the magnitudes of the principal stresses, it gives
no indication of their orientation in physical space.
