Civil Engineering Reference
In-Depth Information
F
f = AB =
F
new
AC
F
new
-F
old
A
F
new
B
O
Strain
F
old
C
Figure 6.7
Factoring process for “just yielded” elements
In the event of a loading increment causing a Gauss point to go plastic for the first time,
it may be necessary to factor the matrix [
D
p
] in (6.34). A linear interpolation can be used
as indicated in Figure 6.7. Thus, instead of using [
D
p
]weuse
[
D
p
], where
f
F
new
F
new
−
F
old
f
=
(6.35)
This simple method represents a forward Euler approach to integrating the elasto-plastic
rate equations, extrapolating from the point at which the yield surface is crossed. More
complicated integrations, which are mainly relevant to tangent stiffness methods, are given
later in Section 6.9.
Although overshoot of the yield function
is an integral part of the viscoplastic algo-
rithm, a similar interpolation method to that described by (6.35) can be used if required
to compute the plastic potential derivatives in equation (6.19) using stresses corresponding
to
F
F
≈
0.
6.8 Corners on the failure and potential surfaces
For failure and potential surfaces that include “corners” as in Mohr-Coulomb (see
Figure 6.6) the derivatives required in equations (6.19) and (6.33) become indeterminate. In
the case of the Mohr-Coulomb (or Tresca) surface, this occurs when the angular invariant
θ
=±
30
◦
. The method used in the programs to overcome this difficulty is to replace the
hexagonal surface by a smooth conical surface if
|
sin
θ
|
>
0
.
49
(6.36)
30
◦
30
◦
The conical surfaces are those obtained by substituting either
θ
=
or
θ
=−
30
◦
into (6.12), depending upon the sign of
θ
as it approaches
±
(see Appendix C). It
should be noted that in the initial stress approach, both the
F
and
Q
functions must
