 Plasticity, Collapse and Fatigue
If loading causes regions in a structure to become plastic (i.e.
exceed the yield stress) an analysis which includes a material with
post-yield stiffness is required to evaluate the plastic stresses.
A simple example is illustrated below. A linear analysis in which
only the Young’s Modulus (E) is included will substantially
over estimate such plastic stresses and a separate value of
post-yield stiffness (often noted as H) is required. Setting a
reliable value for H is often problematic but a value many orders
less than E (and sometimes zero – see previous articles) is
often used.
Stresses acting over cross sections can be perceived as a reaction
to the applied loads on that section. While a structure is still
mainly elastic, increases in load will cause the stresses to
increase in proportion. Eventually, small regions of plasticity
will initialise at stress concentrations (or notches). Because of
the reduced stiffness of these plastic regions, as the applied load
is increased, stresses increase much less compared with the
surrounding non-yielded material. Furthermore, the elastic, stiffer
surrounding regions control the deformation of the smaller and less
stiff plastic regions. In other words, initial plastic regions are
displacement or strain controlled, by the surrounding regions.
Although not literally true, imagining an initial yielded region as
a portion of relatively softer material may help to visualise how
it is affected by increases in applied load.
As well as giving a much improved set of stress results immediately
post-yield, plasticity analysis can provide an insight into how a
multiply-connected (multiple load-path) structure fails
(collapses), as the load is increased. Previous ‘Knowledge
Base’ articles have discussed aspects of such methods to
determine structural collapse.
Clearly a load that results in structural collapse can be applied
only once. However, lower loads, that still exceed the yield
stress, can be repeatedly applied before failure occurs (termed
cyclic, fatigue or reversed loading). A plastic material model is
also required in the analysis of such cases. (In pressure vessel
terminology this is termed shakedown and ratchetting.) In these
cases, for a given value of load, the stress at any point within
the plastic regions during unloading is different to that during
loading. This implies a hysteresis loop of cyclic stress, as
illustrated above (which shows the load and stress as being fully
reversed, i.e. having a zero mean).
At each point in the structure where the yield stress has been
exceeded, hysteresis loops of various sizes will be traversed as
the load is repeatedly applied. Prior to the advent of plasticity
in FEA, there was no generally applicable procedure for
establishing whether a given cyclic load acting on a given
structure will cause a continual growth of the plastic regions and
the hysteresis loops, resulting in eventual collapse (ratchetting),
or whether the regions will eventually stop growing and all the
hysteresis loops become stable (shakedown).
Even though a cyclic load can shakedown the structure to a stable
stress state, this does not imply that the structure can sustain
this cyclic load for an infinite number of cycles. Repeated loading
of this sort will cause the material to sustain fatigue damage, so
that some regions experiencing the same repeated value of stress,
while sustainable in the short term, will eventually start to
rupture.
Fatigue damage is not manifested as a change in the material
stiffnesses used in the FE material model and is therefore
considered as a separate analysis or post-processing step, after
the main solution. Note that fatigue damage is caused by repeated
cyclic stressing, whether the yield stress is exceeded or not, and
is a distinct phenomenon from ratchetting or plastic collapse. It
is relevant to all cyclic load scenarios.
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