, the life-stress relationship
is:This section presents a generalized formulation of the cumulative damage model where multiple stress types are used in the analysis and where the stresses can be any function of time.
See the following sections:
Given n
time-varying stresses , the life-stress relationship
is:
where 
and 
are model parameters.
This relationship can be further modified through the use of transformations and can be reduced to the relationships discussed previously (power, Arrhenius and exponential), if so desired. The reliability function of the unit under multiple stresses is given by:
where:
Therefore, the pdf is:
Parameter estimation can be accomplished via maximum likelihood estimation methods, and confidence intervals can be approximated using the Fisher matrix approach. Once the parameters are determined, all other characteristics of interest can be obtained utilizing the statistical properties definitions (e.g. mean life, failure rate, etc.) presented in previous chapters. The log-likelihood equation is as follows:
where:
and:
Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is
the number of suspensions in the ith group of
suspension data points.
is
the running time of the ith suspension
data group.
FI is the number of interval data groups.
is
the number of intervals in the ith group of
data intervals.
is
the beginning of the ith
interval.
is
the ending of the ith
interval.
Given n
time-varying stresses 
, the life-stress relationship
is given by:
where
and 
are model parameters.
The reliability function of the unit under multiple stresses is given by:
where:
Therefore, the pdf is:
Parameter estimation can be accomplished via maximum likelihood estimation methods, and confidence intervals can be approximated using the Fisher matrix approach. Once the parameters are determined, all other characteristics of interest can be obtained utilizing the statistical properties definitions (e.g. mean life, failure rate, etc.) presented in previous chapters. The log-likelihood equation is as follows:
where:
and:
Fe is the number of groups of exact time-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is
the number of suspensions in the ith group of
suspension data points.
is
the running time of the ith suspension
data group.
FI is the number of interval data groups.
is
the number of intervals in the ith group of
data intervals.
is
the beginning of the ith
interval.
is
the ending of the ith
interval.
Given one time-varying stress X1(t), the life-stress relationship is given by:
where
and
are model parameters.
The lognormal reliability function of the unit under a single stress is given by:
where:
and:
Therefore, the pdf is:
Parameter estimation can be accomplished via maximum likelihood estimation methods, and confidence intervals can be approximated using the Fisher matrix approach. Once the parameters are determined, all other characteristics of interest can be obtained utilizing the statistical properties definitions (e.g. mean life, failure rate, etc.) presented in previous chapters. The log-likelihood equation is as follows:
where:
and:
Fe is the number of groups of exact time-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is
the number of suspensions in the ith group of
suspension data points.
is
the running time of the ith suspension
data group.
FI is the number of interval data groups.
is
the number of intervals in the ith group of
data intervals.
is
the beginning of the ith
interval.
is
the ending of the ith
interval.
Example
A sample of 18 units of an electronic component was subjected to temperature and voltage stresses. The temperature was initially set at 100K and was then continuously increased to 200K over a period of 20 hours. The temperature was again increased at 120 hours to 300K over a 20-hour period as shown in Figure 3. The voltage was initially set at 4V and was then increased continuously to 8V over a period of 10 hours. The voltage was again increased at 110 hours to 12V over a 10-hour period as shown in Figure 4.
Fig. 3: Temperature profile
Fig. 4: Voltage profile
The failure times, as entered in ALTA PRO, are shown in the next figure.
Using an Arrhenius life-stress relationship for temperature, a Power Law life-stress relationship for voltage, and the Weibull distribution as the underlying life distribution, the estimated model parameters are shown next.
The use level reliability plot is shown in the next figure.
Fig. 5: Reliability plot at normal use conditions (100K and 4V).
See Also:
Time-Varying Stresses: The Cumulative Damage Model
Cumulative Damage Power Relationship
Cumulative Damage Arrhenius Relationship
Cumulative Damage Exponential Relationship
Time-Varying Stress Model Confidence Intervals
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