Uniaxial Fatigue Analysis, using S-N (stress-life) and E-N (strain-life) approaches for predicting the life (number
of loading cycles) of a structure under cyclical loading may be performed by using HyperLife.
Strain-life analysis is based on the fact that many critical locations such as notch
roots have stress concentration, which will have obvious plastic deformation during
the cyclic loading before fatigue failure. Thus, the
elastic-plastic strain results are essential for performing strain-life analysis.
Multiaxial Fatigue Analysis, using S-N (stress-life), E-N (strain-life), and Dang Van Criterion (Factor
of Safety) approaches for predicting the life (number of loading cycles) of a structure under cyclical
loading may be performed by using HyperLife.
Seam Weld Fatigue analysis is available to facilitate Fatigue analysis for seam welded structures. It allows you to
simulate the Fatigue failure at the seam weld joints to assess the corresponding fatigue failure characteristics like
Damage and Life.
Uniaxial Fatigue Analysis, using S-N (stress-life) and E-N (strain-life) approaches for predicting the life (number
of loading cycles) of a structure under cyclical loading may be performed by using HyperLife.
Strain-life analysis is based on the fact that many critical locations such as notch
roots have stress concentration, which will have obvious plastic deformation during
the cyclic loading before fatigue failure. Thus, the
elastic-plastic strain results are essential for performing strain-life analysis.
Strain-life analysis is based on the fact that many critical
locations such as notch roots have stress concentration,
which will have obvious plastic deformation during the
cyclic loading before fatigue failure. Thus, the
elastic-plastic strain results are essential for performing
strain-life analysis.
Neuber Correction
Neuber correction is the most popular practice to correct elastic analysis results into elastic-plastic results.
In order to derive the local stress from the nominal stress that is easier to obtain, the
concentration factors are introduced such as the local
stress concentration factor , and the local strain
concentration factor .(1)
(2)
Where, is the local stress,
is the local strain, is the nominal stress,
and is the nominal strain.
If nominal stress and local stress are both
elastic, the local stress concentration factor is
equal to the local strain concentration factor.
However, if the plastic strain is present, the
relationship between and no long holds.
Thereafter, focusing on this situation, Neuber
introduced a theoretically elastic stress
concentration factor defined
as:(3)
Substitute Equation 1 and Equation 2 into Equation 3, the theoretical stress
concentration factor can be rewritten
as:(4)
Through linear static FEA, the local stress instead of nominal stress is provided, which implies
the effect of the geometry in Equation 4 is removed, thus you can set as 1 and rewrite Equation 4 as:(5)
Where, , is locally elastic
stress and locally elastic strain obtained from
elastic analysis, , the stress and strain at the
presence of plastic strain. Both
and can be calculated from Equation 5 together with the equations
for the cyclic stress-strain curve and hysteresis
loop.
Monotonic Stress-Strain Behavior
Relative to the current configuration, the true stress and strain relationship can be defined
as:(6)
(7)
Where, is the current
cross-section area, is the current objects
length, is the initial objects
length, and and are the true stress and
strain, respectively, Figure 1 shows the monotonic stress-strain
curve in true stress-strain space. In the whole
process, the stress continues increasing to a
large value until the object fails at C.
The curve in Figure 1 is comprised of two typical
segments, namely the elastic segment OA and
plastic segment AC. The segment OA keeps the
linear relationship between stress and elastic
strain following Hooke Law:(8)
Where, is elastic modulus and is elastic strain. The
formula can also be rewritten as:(9)
by expressing elastic strain in terms of stress. For most of materials, the relationship between
the plastic strain and the stress can be
represented by a simple power law of the
form:(10)
Where,
is plastic strain, is strength
coefficient, and is work hardening
coefficient. Similarly, the plastic strain can be
expressed in terms of stress as:(11)
The total strain induced by loading the object up to point B or D is the sum of plastic strain
and elastic strain:(12)
Cyclic Stress-Strain Curve
Material exhibits different behavior under
cyclic load compared with that of monotonic load.
Generally, there are four kinds of response.
Stable state
Cyclically hardening
Cyclically softening
Softening or hardening depending on strain
range
Which response will occur depends on its nature
and initial condition of heat treatment. Figure 2 illustrates the effect of cyclic
hardening and cyclic softening where the first two
hysteresis loops of two different materials are plotted. In
both cases, the strain is constrained to change in fixed
range, while the stress is allowed to change arbitrarily. If
the stress range increases relative to the former cycle
under fixed strain range, as shown in the upper portion of
Figure 2, it is called cyclic
hardening; otherwise, it is called
cyclic softening, as shown in the
lower portion of Figure 2. Cyclic response of material can
also be described by specifying the stress range and leaving
strain unconstrained. If the strain range increases relative
to the former cycle under fixed stress range, it is called
cyclic softening; otherwise, it is
called cyclic hardening. In fact, the cyclic
behavior of material will reach a steady-state after a short
time which generally occupies less than 10 percent of the
material total life. Through specifying different strain
ranges, a series of hysteresis loops at steady-state can be
obtained. By placing these hysteresis loops in one
coordinate system, as shown in Figure 3, the line connecting all the
vertices of these hysteresis loops determine cyclic
stress-strain curve which can be expressed in the similar
form with monotonic stress-strain curve as:
(13)
Where,
Cyclic strength coefficient
Strain cyclic hardening exponent
Hysteresis Loop Shape
Bauschinger observed that after the initial load had caused plastic strain, load reversal caused
materials to exhibit anisotropic behavior. Based on
experiment evidence, Massing put forward the hypothesis that
a stress-strain hysteresis loop is geometrically similar to
the cyclic stress strain curve, but with twice the
magnitude. This implies that when the quantity () is two times of (), the stress-strain
cycle will lie on the hysteresis loop. This can be expressed
with formulas:(14)
(15)
Expressing in terms of Δσ,
in terms of Δε, and
substituting it into Equation 13, the hysteresis loop formula
can be calculated as:(16)
Almost a century ago, Basquin observed the linear relationship between stress and fatigue life in
log scale when the stress is limited. He put
forward the following fatigue formula controlled
by stress:(17)
Where, is the stress
amplitude, is the fatigue
strength coefficient, and is the fatigue
strength exponent. Later in the 1950s, Coffin and Manson
independently proposed that plastic strain may also be
related with fatigue life by a simple power
law:(18)
Where, is the plastic strain
amplitude, is the fatigue
ductility coefficient, and is the fatigue
ductility exponent. Morrow combined the work of
Basquin, Coffin and Manson to consider both
elastic strain and plastic strain contribution to
the fatigue life. He found out that the total
strain has more direct correlation with fatigue
life. By applying Hooke Law, Basquin rule can be
rewritten as:(19)
Where, is elastic strain
amplitude. Total strain amplitude, which is the
sum of the elastic strain and plastic stain,
therefore, can be described by applying Basquin
formula and Coffin-Manson formula:(20)
Where, is the total strain
amplitude, the other variable is the same with
above.
Mean Stress Correction
The fatigue experiments carried out in the
laboratory are always fully reversed, whereas in
practice, the mean stress is inevitable, thus the
fatigue law established by the fully reversed
experiments must be corrected before applied to
engineering problems.
Morrow:
Morrow is the first to consider
the effect of mean stress through introducing the
mean stress in fatigue strength
coefficient by:(21)
Thus, the entire fatigue
life formula becomes:(22)
Morrow's equation is
consistent with the observation that mean stress
effects are significant at low value of plastic
strain and of little effect at high plastic
strain.
MORROW2:
Improves the MORROW method by ignoring the effect
of negative mean stress.
Smith, Watson and Topper:
Smith, Watson
and Topper proposed a different method to account
for the effect of mean stress by considering the
maximum stress during one cycle (for convenience,
this method is called SWT in the
following). In this case, the damage parameter is
modified as the product of the maximum stress and
strain amplitude in one cycle.(23)
The SWT method will
predict that no damage will occur when the maximum
stress is zero or negative, which is not
consistent with the reality.
When comparing
the two methods, the SWT method predicted
conservative life for loads predominantly tensile,
whereas, the Morrow approach provides more
realistic results when the load is predominantly
compressive.
Damage Accumulation Model
In the E-N approach, use the same damage accumulation model as the S-N approach, which is Palmgren-Miner's linear damage summation rule.