The SN and EN curves (and other fatigue properties) of
a material are obtained from experiment, through fully reversed rotating bending
tests. Due to the large amount of scatter that usually accompanies test results,
statistical characterization of the data should also be provided (certainty of
survival is used to estimate the worst mean log(N) according to the standard error
of the curve and a higher reliability level requires a larger certainty of
survival).
To understand these parameters, consider the SN curve as an example.
When SN testing data is presented in a log-log plot of alternating nominal stress
amplitude Sa or range SR versus cycles to failure N, the
relationship between S and N can be described by straight line segments. Normally, a
one or two segment idealization is used.
Consider the situation where SN scatter leads to variations in the
possible SN curves for the same material and same sample specimen. Due to natural
variations, the results for full reversed rotating bending tests typically lead to
variations in data points for both Stress Amplitude (S) and Life (N). Looking at the
Log scale, there are variations in Log(S) and Log(N). Specifically, looking at the
variation in life for the same Stress Amplitude applied, a set of data points may
look like this:
S
Log (S)
Log (N)
2000.0
3.3
3.9
2000.0
3.3
3.7
2000.0
3.3
3.75
2000.0
3.3
3.79
2000.0
3.3
3.87
2000.0
3.3
3.9
As with many processes, the distribution of Log(N) is assumed to be a
Normal Distribution. There is a full population of possible values of log(N) for a
particular value of log(S). The mean of this full population set is the true
population mean and is unknown. Therefore you statistically estimate the worst true
population mean of log(N) based on your input sample mean (SN curve) and Standard
Error (SE) of your sample. The SN material data input is based on the mean of the
normal distribution of the scatter in the particular sample used to generate the
data.
The experimental scatter exists in both Stress Amplitude and Life data.
The Standard Error of the scatter of log(N) is required as input (SE field for SN
curve). The sample mean is provided by the SN curve as whereas, the standard error is input via the SE
field.
If the specified SN curve is directly utilized, without any
perturbation, then the sample mean is directly used, leading to a certainty of
survival of 50%. This implies that you do not perturb the sample mean you provided.
Since a value of 50% survival certainty may not be sufficient for all applications,
SimSolid can internally perturb the SN material data
to the required certainty of survival defined by you. To accomplish this, the
following data is required:
Standard error of log(N) normal distribution SE
Certainty of Survival required for this analysis
A normal distribution or gaussian distribution is a probability density
function which implies that the total area under the curve is always equal to
1.0.
The SN curve data you defined is assumed as a normal distribution, which
is typically characterized by the following Probability Density
Function:(1)
Where:
is the data value () in the sample you defined.
is the sample mean ().
is the standard deviation of the sample (which is
unknown, as you input only Standard Error (SE).
The above distribution is the
distribution of the sample you defined, and not the full population space. Since the
true population mean is unknown, the range of the true population mean is estimated
from the sample mean and the sample SE, and then the Certainty of Survival you
defined is used to perturb the sample mean.
Standard Error is the standard
deviation of the normal distribution created by all the sample means of samples
drawn from the full population. From a single sample distribution data, the Standard
Error is typically estimated as , where is the standard deviation of the sample, and is the number of data values in the sample. The mean
of this distribution of all the sample means is actually the same as the true
population mean. The certainty of survival you provided is applied on this
distribution of all the sample means.
Generally, you convert a normal
distribution function into a standard normal distribution curve (which is a normal
distribution with mean = 0.0 and standard error = 1.0). You can then directly use
the certainty of survival values via Z-tables.
Note: The
certainty of survival is equal to the area of the curve under a probability
density function between the required sample points of interest. It is possible
to calculate the area of the normal distribution curve directly (without
transformation to standard normal curve), however, this is computationally
intensive compared to a standard lookup Z-table. Therefore, you generally first
convert the current normal distribution to a standard normal distribution and
then use Z-tables to parametrize the input survival certainty.
For
the normal distribution of all the sample means, the mean of this distribution is
the same as the true population mean , the range of which is what you want to
estimate.
Statistically, you can estimate the range of true population mean as follows:
(2)
That is,
(3)
Since the value on the left side is more conservative, use the following equation
to perturb the SN curve:
(4)
Where,
is the perturbed value
is the sample mean you defined (SN curve)
is the standard error (SE)
The value of is procured from the standard normal distribution
Z-tables based on the input value of the certainty of survival. Some typical values
of Z for the corresponding certainty of survival values are illustrated in the table
below.
Z-values (calculated)
Certainty of Survival (Input)
0.0
50.0
-0.5
69.0
-1.0
84.0
-1.5
93.0
-2.0
97.7
-3.0
99.9
Notice how the SN curve is modified to the required certainty of
survival and standard error input. This technique allows you to handle fatigue
material data scatter using statistical methods and predict data for the required
survival probability values.