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Hysteresis in Fatigue
If the yield stress is exceeded at notches in a structure,
hysteresis loops of various sizes will be traversed at points
around the notches. Providing the load is not too large, shakedown
is achieved, in which the loops stabilise at all points, after a
small number of cycles. But sustained repetition of this load of
typically, between 102 and 107 cycles, will have to consider the
separate and longer term phenomenon of fatigue damage.
Aside from Young’s Modulus and Poisson’s Ratio, the
material properties required for a shakedown analysis are the yield
stress and the post-yield stiffness. Fatigue behaviour is described
by further material constants, but these are not required in the
main FEA solution phase.
In fatigue, loads are cyclic and fatigue damage at a point is
strongly related to the range of stress encountered during a given
load cycle, rather than the peak stress due to the extreme of the
cycle. Each stress cycle will damage the material by a small
amount, eventually causing cracks to form. Generally, no account is
made of the shape of each individual peak to peak load (and
corresponding stress) and, for a sequence of different load ranges,
no account is made of the order in which the loads are imposed.
Figure 1 shows common terminology used to describe load or stress
cycles.
It will be obvious that although the range of stress might be the
most important determinant of fatigue damage, the mean stress value
must have some effect. This can be conveniently considered
separately however; the main effort in a typical fatigue analysis
is expended in isolating the discrete stress ranges within a
complex load cycle, with each cycle’s mean stress accounted
for separately.
The fatigue life of ferrous and some aluminium alloys can be
predicted based on the following assumptions. For other material
including cast irons and other aluminium alloys, at least some of
these assumptions are not valid, so fatigue lifing may not always
be viable for such materials:
Life is defined by material properties, that is, quantities that
are assumed constant for a given material. These are obtained from
cyclic loading or straining of specimens of the said material.
There is a greater amount of scatter of fatigue life data from
material specimens than of say, stiffness. Such scatter may be
accounted for by taking the mean minus two standard deviation life
or other statistical approaches.
For a fixed, repeatedly applied load cycle, the hysteresis
stress-strain curve at any point will not change, until failure
begins. This is termed cyclic stability. In reality, hardening or
softening may occur initially, so a specimen hysteresis loop
changes shape, but this stabilises after a few cycles and is one
aspect of shakedown behaviour, already described.
The memory effect underpins the range counting techniques used in
fatigue analysis. This implies that any complex sequence of load
cycles will give rise to several complete (closed) hysteresis loops
corresponding to each peak to peak excursion. Thus, the damage due
to each loop is related only to the size of that loop,
independently of the other loops or their relative position. This
is explained in more detail below.
The hysteresis loops described in previous articles used straight
line portions, indicative of the typical bilinear stiffness
representations used in FEA. Real hysteresis loops are curved, as
shown in Figure 3. This figure shows the hysteresis loop B-C-B
enclosed within the loop O-A-D-O.
Figure 2 shows a load cycle in which several stress ranges could
possibly be considered in a damage calculation, such as O-A, C-D
and D-E. But the actual ranges chosen for the damage calculation
will be those that give rise to closed hysteresis loops. At point B
in Figure 2, the load reversal causes a new hysteresis loop to be
started in Figure 3. At some point between C and D in Figure 2, the
B-C-B loop in Figure 3 will be closed. The stress strain curve does
not follow the continuation of the CB curve, shown dotted, but
instead resumes the continuation of the A-B loop, until the load at
D is reached. This is the memory effect. Hence the ranges to
consider in the ensuing fatigue calculation are A-D and B-C.
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