Download HCF (HIGH-CYCLE FATIGUE) Multiaxial stress state

HCF (HIGH-CYCLE FATIGUE)
The influence of mean stress
two of these:
can be taken into account in different ways. Goodman and Gerber are
’Our’ Haigh diagram can be considered as a rationalisation of the Gerber equation, where only 2
fatigue tests (‘alternating’ and ‘pulsating’ loads ) need be performed.
Multiaxial stress state
We first deal with stress states of the type
, i.e., ‘in-phase’ stress
components. The most-used hypothesis for such cases is by Sines (1959):
In the above equations,
is the stress deviator
For a uniaxial case, we have
I.e., a Goodman equation with
Therefore, if we rewrite (3) with these values:
One can also imagine a ‘Sines’ variant’ of the Gerber equation:
or even using the Haigh diagram with the Sines’ equivalent stresses:
Example
As a simple example we look at a thin-walled cylindrical pressure vessel, loaded by a time-varying
inner pressure
The diameter is , it is assumed that Sines’ hypothesis can be used with data
and a safety factor is required against fatigue. Compute the necessary wall thickness
load cases
(a)
and
(b)
(a)
We get
,
for the two
and Sines’ hypothesis gives
(b)
In this case,
Sines now gives
With the given relations
----To compare cases (a) and (b), assume, for instance,
:
[Case (a) may be a little unrealistic, since it is unusual for a pressure vessel to be loaded by alternating
positive and negative pressure.]
General multiaxial stress states
In this section, we will deal with stress states that cannot be expressed as
Example:
For such cases, knowledge is not so established, but one can, for instance, postulate a more general
Sines’ expression:
One simple way of illustrating this is by looking at a von Mises stress space on the deviatoric plane
(i.e., looking along the space diagonal):
The
contour shown in the figure is the curve that the vector
takes through the principal stress space. When this has been tested for all possible choices of , the
minimum von Mises cylinder needed to contain all such curves is drawn (see the figure). The radius of
this cylinder is then
Example
Study the the load cycle
The definition of stress deviator gives
which, in turn, gives
Now search for the maximum of this expression!
Both these solutions lead to the same expression for
:
and the fatigue initiation criterion
becomes
With numerical values inserted, e.g.,
0
196.0
100
129.5
150
96.32
200
63.11
250
29.89
Critical-plane theories for fatigue failure
In modern fatigue design, critical-plane theories have come to more frequent use. A critical plane is a
plane on which a combination of shear stress amplitude (or shear amplitde ) and normal stress
reaches a maximum value. As an example, Findley (1959) postulated the following critical-plane
fatigue failure criterion:
In this theory,
life.
and
are material parameters, which are expected to be constants for a given fatigue
The figure below shows an example of the definition of a critical plane.
Figure 2. Definition of a critical plane. From Suresh S: Fatigue of materials