Cumulative Damage General Log-Linear Relationship

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:

Cumulative Damage General Log-Linear-Exponential

Given n time-varying stresses MATH, the life-stress relationship is:

MATH

where $\alpha _{0}$ and $\alpha _{j}$ 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:

MATH

where:

MATH

Therefore, the pdf is:

MATH

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:

MATH

where:

MATH

and:  

Cumulative Damage General Log-Linear-Weibull

Given n time-varying stresses MATH, the life-stress relationship is given by:

MATH

where $\alpha _{0}$ and $\alpha _{j}$ are model parameters.

The reliability function of the unit under multiple stresses is given by:

MATH

where:

MATH

Therefore, the pdf is:

MATH

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:

MATH

where:

MATH

and:  

Cumulative Damage General Log-Linear-Lognormal

Given one time-varying stress X1(t), the life-stress relationship is given by:

chapter12__325a.gif

where $\alpha _{0}$ and $\alpha _{j}$ are model parameters.

The lognormal reliability function of the unit under a single stress is given by:

chapter12__329.gif

where:

chapter12_333.gif

and:

chapter12__334.gif

Therefore, the pdf is:

chapter12__335.gif

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:

chapter12__337.gif

where:

chapter12__338.gif

and:

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.

ch12-3.gif

Fig. 3: Temperature profile

ch12-4.gif

Fig. 4: Voltage profile

The failure times, as entered in ALTA PRO, are shown in the next figure.

cumdamagemulfactdata.gif

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.

chapter12_new.gif

The use level reliability plot is shown in the next figure.

ch12-5.gif

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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