Download Arrhenius Exponential

Arrhenius Relationship
This chapter includes the following subchapters:

Arrhenius Relationship Introduction

Acceleration Factor

Arrhenius-Exponential

Arrhenius-Weibull

Arrhenius-Lognormal

Arrhenius Confidence Bounds
See Also:
Contents
Introduction
Arrhenius Relationship Introduction
The Arrhenius life-stress model (or relationship) is probably the most common life-stress
relationship utilized in accelerated life testing. It has been widely used when the stimulus or
acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius
reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.
The Arrhenius reaction rate equation is given by:
where,


R is the speed of reaction,
A is an unknown nonthermal constant,
is the activation energy (eV),

K is the Boltzman’s constant (8.617385 ´

eV
),
And

T is the absolute temperature (Kelvin).
The activation energy is the energy that a molecule must have to participate in the reaction. In
other words, the activation energy is a measure of the effect that temperature has on the
reaction.
The Arrhenius life-stress model is formulated by assuming that life is proportional to the
inverse reaction rate of the process, thus the Arrhenius life-stress relationship is given by:
(1)
where:


L represents a quantifiable life measure, such as mean life, characteristic life, median life,
or B(x) life, etc.
V represents the stress level (formulated for temperature and temperature values in
absolute units i.e. degrees Kelvin or degrees Rankine}


C is one of the model parameters to be determined, (C > 0).
B is another model parameter to be determined.
Fig. 1: Graphical look at the Arrhenius life-stress relationship (linear scale) for different
life characteristics, assuming a Weibull distribution.
Since the Arrhenius is a physics-based model derived for temperature dependence, it is
strongly recommended that the model be used for temperature accelerated tests. For the
same reason, temperature values must be in absolute units (Kelvin or Rankine), even though
Eqn. (1) is unitless.
The Arrhenius relationship can be linearized and plotted on a life vs. stress plot, also called the
Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in
Eqn. (1) or,
(2)
Fig. 2: Arrhenius plot for Weibull life distribution.
In Eqn. (2) ln (C) is the intercept of the line and B is the slope of the line. Note that the inverse
of the stress, and not the stress, is the variable. In Figure 2, life is plotted versus stress and not
versus the inverse stress. This is because Eqn. (2) was plotted on a reciprocal scale. On such
a scale, the slope B appears to be negative even though it has a positive value. This is
because B is actually the slope of the reciprocal of the stress and not the slope of the stress.
The reciprocal of the stress is decreasing as stress is increasing (
increasing). The two different axes are shown in Figure 3.
is decreasing as V is
Fig. 3. An illustration of both reciprocal and non-reciprocal scales.
The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example,
in Figure 3 it is more convenient to locate the life corresponding to a stress level of 370K rather
than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life.
The shaded areas shown in Figure 3 are the imposed pdfs at each test stress level. From such
imposed pdfs one can see the range of the life at each test stress level, as well as the scatter in
life. The next figure (Figure 4) illustrates a case in which there is a significant scatter in life at
each of the test stress levels.
Fig. 4: An example of scatter in life at each test stress level.
A Look at the Parameter B
Depending on the application (and where the stress is exclusively thermal), the parameter
can be replaced by:
B
Note that in this formulation, the activation energy must be known apriori. If the activation
energy is known then there is only one model parameter remaining, C. Because in most real
life situations this is rarely the case, all subsequent formulations will assume that this activation
energy is unknown and treat B as one of the model parameters. As it can be seen in Eqn. (1),
B has the same properties as the activation energy. In other words, B is a measure of the
effect that the stress (i.e. temperature) has on the life. The larger the value of B, the higher the
dependency of the life on the specific stress (see Figure 5). Parameter B may also take
negative values. In that case, life is increasing with increasing stress (see Figure 5). An
example of this would be plasma filled bulbs, where low temperature is a higher stress on the
bulbs than high temperature.
Fig. 5: Behavior of the parameter B.
See Also:
Arrhenius Relationship
Acceleration Factor
Most practitioners use the term acceleration factor to refer to the ratio of the life (or
acceleration characteristic) between the use level and a higher test stress level or,
For the Arrhenius model this factor is,
Thus, if B is assumed to be known apriori (using an activation energy), the assumed activation
energy alone dictates this acceleration factor!
See Also:
Arrhenius Relationship
Arrhenius Exponential
The pdf for the Arrhenius relationship and the exponential distribution is given next.
The pdf of the 1-parameter exponential distribution is given by:
(3)
It can be easily shown that the mean life for the 1-parameter exponential distribution
(presented in detail in the Life Distributions chapter) is given by:
(4)
thus,
(5)
The Arrhenius-exponential model pdf can then be obtained by setting m =
Therefore,
L(V) in Eqn. (1).
Substituting for m in Eqn. (5) yields a pdf that is both a function of time and stress or,
Arrhenius Exponential Statistical Properties Summary
Mean or MTTF
The mean,
, or mean time to failure (MTTF) of the Arrhenius-exponential is given by:
(6)
Median
The median,
, of the Arrhenius-exponential is given by:
Mode
The mode,
, of the Arrhenius-exponential is given by:
Standard Deviation
The standard deviation,
, of the Arrhenius-exponential is given by:
Arrhenius-Exponential Reliability Function
The Arrhenius-exponential reliability function is given by:
This function is the complement of the Arrhenius-exponential cumulative distribution function
or,
and,
Conditional Reliability
The Arrhenius-exponential conditional reliability function is given by:
Reliable Life
For the Arrhenius-exponential, the reliable life, or the mission for a desired reliability goal,
is given by:
or,
,
Parameter Estimation
Maximum Likelihood Estimation Method
The log-likelihood function for the exponential distribution is composed of two summation
portions shown next.
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure in the

is the failure rate parameter (unknown).

is the exact failure time of the

time-to-failure data group.
group.
S is the number of groups of suspension data points.

is the number of suspensions in the

is the running time of the
group of suspension data points.
suspension data group.
Substituting the Arrhenius-exponential model into the log-likelihood function yields,
(7)
The solution (parameter estimates) will be found by solving for the parameters
so that
= 0 and
= 0, where,
,
See Also:
Arrhenius Relationship
Arrhenius Weibull
The pdf for the Arrhenius relationship and the Weibull distribution is given next.
The pdf for 2-parameter Weibull distribution is given by:
(8)
The scale parameter (or characteristic life) of the Weibull distribution is
Arrhenius-Weibull model pdf can then be obtained by setting
and substituting for
. The
= L(V) in Eqn. (1),
in Eqn. (8),
An illustration of the pdf for different stresses is shown in Figure 6. As expected, the pdf at
lower stress levels is more stretched to the right, with a higher scale parameter, while its shape
remains the same (the shape parameter is approximately 3 in Figure 6). This behavior is
observed when the parameter B of the Arrhenius model is positive.
Fig. 6: Behavior of the probability density function at different stresses and with the
parameters held constant.
The advantage of using the Weibull distribution as the life distribution lies in its flexibility to
assume different shapes. The Weibull distribution was presented in greater detail in the Life
Distributions chapter.
Arrhenius Weibull Statistical Properties Summary
Mean or MTTF
The mean,
given by:
(also called MTTF by some authors), of the Arrhenius-Weibull relationship is
where
is the gamma function evaluated at the value of
.
Median
The median,
for the Arrhenius-Weibull is given by:
(9)
Mode
The mode,
for the Arrhenius-Weibull is given by:
(10)
Standard Deviation
The standard deviation,
for the Arrhenius-Weibull is given by:
Arrhenius-Weibull Reliability Function
The Arrhenius-Weibull reliability function is given by:
If the parameter B is positive, then the reliability increases as stress decreases.
Fig. 7: Behavior of the reliability function at different stresses and constant parameter
values.
The behavior of the reliability function of the Weibull distribution for different values of was
illustrated in the Life Distributions chapter. In the case of the Arrhenius-Weibull model however,
the reliability is a function of stress also. A 3D plot such as in Figure 8 is now needed to
illustrate the effects of both the stress and
.
Fig. 8: Reliability function for
< 1,
= 1, and
> 1.
Conditional Reliability Function
The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:
(11)
or,
Reliable Life
For the Arrhenius-Weibull relationship, the reliable life,
and starting the mission at age zero is given by:
(12)
, of a unit for a specified reliability
This is the life for which the unit will function successfully with a reliability of
= 0.50 then
=
. If
, the median life, or the life by which half of the units will survive.
Arrhenius-Weibull Failure Rate Function
The Arrhenius-Weibull failure rate function,
(T), is given by:
Fig. 9: Failure rate function for
< 1,
= 1, and
> 1.
Parameter Estimation
Maximum Likelihood Estimation Method
The Arrhenius-Weibull log-likelihood function is composed of two summation portions,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure data points in the
time-to-failure data group.

is the Weibull shape parameter (unknown, the first of three parameters to be estimated).

B is the Arrhenius parameter (unknown, the second of three parameters to be estimated).
C is the second Arrhenius parameter (unknown, the third of three parameters to be

estimated).

is the stress level of the

is the exact failure time of the

group.
group.
S is the number of groups of suspension data points.


is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for
0 and
=0, where,
,
,
so that
= 0,
=
Example
Consider the following times-to-failure data at three different stress levels.
The data was analyzed jointly and with a complete MLE solution over the entire data set, using
ReliaSoft's ALTA. The analysis yields,
Once the parameters of the model are estimated, extrapolation and other life measures can be
directly obtained using the appropriate equations. Using the MLE method, confidence bounds
for all estimates can be obtained. Note in Figure 10 below that the more distant the
accelerated stress from the operating stress, the greater the uncertainty of the extrapolation.
The degree of uncertainty is reflected in the confidence bounds. (General theory and
calculations which underly confidence intervals are presented in Appendix A: Brief Statistical
Background. Specific calculations for confidence bounds on the Arrhenius model are
presented in Arrhenius Confidence Bounds.)
Fig. 10: Comparison of the confidence bounds for different use stress levels.
See Also:
Arrhenius Relationship
Arrhenius Lognormal
The pdf for the Arrhenius relationship and the lognormal distribution is given next.
The pdf of the lognormal distribution is given by:
(13)
Where,
and,
= mean of the natural logarithms of the times-to-failure.
= standard deviation of the natural logarithms of the times-to-failure.
The median of the lognormal distribution is given by:
(14)
The Arrhenius-lognormal model pdf can be obtained first by setting
Therefore,
or,
= L(V) in Eqn. (1).
Thus,
(15)
Substituting Eqn. (15) into Eqn. (16) yields the Arrhenius-lognormal model pdf or,
(16)
Note that in Eqn. (16), it was assumed that the standard deviation of the natural logarithms of
the times-to-failure,
is independent of stress. This assumption implies that the shape of
the distribution does not change with stress (
is the shape parameter of the lognormal
distribution).
Arrhenius-Lognormal Statistical Properties Summary
The Mean

The mean life of the Arrhenius-lognormal model (mean of the times-to-failure),
by:
, is given
(17)

The mean of the natural logarithms of the times-to-failure,
given by:
, in terms of
and
is
The Standard Deviation

The standard deviation of the Arrhenius-lognormal model (standard deviation of the
times-to-failure),
, is given by:
(18)

The standard deviation of the natural logarithms of the times-to-failure,
and
is given by:
, in terms of
The Mode
The mode of the Arrhenius-lognormal model is given by:
Lognormal Reliability
The reliability for a mission of time T, starting at age 0, for the Arrhenius-lognormal model is
determined by:
or,
There is no closed form solution for the lognormal reliability function. Solutions can be
obtained via the use of standard normal tables. Since the application automatically solves for
the reliability we will not discuss manual solution methods.
Reliable Life
For the Arrhenius-lognormal model, the reliable life, or the mission duration for a desired
reliability goal,
follows,
where,
and,
is estimated by first solving the reliability equation with respect to time, as
Since
= ln(T) the reliable life,
is given by:
Lognormal Failure Rate
The Arrhenius-lognormal failure rate is given by:
When Using The Lognormal Distribution in ALTA
The parameters returned for the Arrhenius-lognormal distribution are always
The returned
, C, and B.
is always the square root of the variance of the natural logarithms to failure.
Also, if the Show Mean option is checked (under the Tools menu), the returned mean value is
always the mean of the natural logarithms of the times-to-failure, given by Eqn. (15). Even
though the application denotes these values as mean and standard deviation, the user is
reminded that these are given as parameters of the distribution, and are thus the mean (a
function of stress as it can be seen in Eqn. (15)) and standard deviation of the natural
logarithms of the data. The mean life value of the times-to-failure, as well as the standard
deviation of times-to-failure (not the parameter) can be obtained through the Quick Calculation
Pad or the Function Wizard in ALTA.
Parameter Estimation
Maximum Likelihood Estimation Method
The lognormal log-likelihood function for the Arrhenius-lognormal model is composed of two
summation portions,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure data points in the

time-to-failure data group.
is the standard deviation of the natural logarithm of the times-to-failure (unknown, the
first of three parameters to be estimated).


B is the Arrhenius parameter (unknown, the second of three parameters to be estimated).
C is the second Arrhenius parameter (unknown, the third of three parameters to be
estimated).

is the stress level of the

is the exact failure time of the



group.
group.
S is the number of groups of suspension data points.
is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for
= 0, and
= 0, where
,
,
so that
= 0,
and,
See Also:
Arrhenius Relationship
Arrhenius Confidence Bounds
This subchapter is made up of the following topics:

Approximate Confidence Bounds for the Arrhenius-Exponential

Approximate Confidence Bounds for the Arrhenius Weibull

Approximate Confidence Bounds for the Arrhenius-Lognormal
See Also:
Arrhenius Relationship
Approximate Confidence Bounds for the Arrhenius Exponential
There are different methods for computing confidence bounds. ALTA utilizes confidence
bounds that are based on the asymptotic theory for maximum likelihood estimates, most
commonly referred to as the Fisher Matrix Bounds.
Confidence Bounds on the Mean Life
The Arrhenius-exponential distribution is given by Eqn. (1) by setting m = L(V) as shown in
Eqn. (6). The upper
(19)
(20)
where
is defined by:
and lower
bounds on the mean life are then estimated by:
(21)
If
= 1-
is the confidence level (i.e. 95% = 0.95), then
for the one-sided bounds. The variance of
=
for the two-sided bounds, and
is given by:
or,
(22)
The variances and covariance of B and C are estimated from the local Fisher Matrix
(evaluated at
,
) as follows,
(23)
Confidence Bounds on Reliability
The bounds on reliability for any given time, T, are estimated by:
where
and
are estimated using Eqns. (19) and (20).
Confidence Bounds on Time
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving
the reliability function with respect to time,
The corresponding confidence bounds are then estimated from,
where
and
are estimated using Eqns. (19) and (20).
See Also:
Arrhenius Confidence Bounds
Approximate Confidence Bounds for the Arrhenius Weibull
Bounds on the Parameters
From the asymptotically normal property of the maximum likelihood estimators, and since
and
,
are positive parameters, ln ( ), and ln ( ) can then be treated as normally distributed.
After performing this transformation, the bounds on the parameters can be estimated from,
Also,
and,
The variances and covariances of
(evaluated at
,
,
, B, and C are estimated from the local Fisher Matrix
, as follows,
Confidence Bounds on Reliability
The reliability function for the Arrhenius-Weibull (ML estimate) is given by:
or,
Setting,
or,
The reliability function now becomes,
The next step is to find the upper and lower bounds on
(24)
(25)
where,
,
or,
The upper and lower bounds on reliability are,
where
and
are estimated from Eqns. (24) and (25).
Confidence Bounds on Time
The bounds on time for a given reliability are estimated by first solving the reliability function
with respect to time,
or,
where
= ln
.
The upper and lower bounds on u are estimated from,
(26)
(27)
where,
or,
The upper and lower bounds on time can then found by:
where
and
are estimated using Eqns. (26) and (27).
See Also:
Arrhenius Confidence Bounds
Approximate Confidence Bounds for the Arrhenius-Lognormal
Bounds on the Parameters
The lower and upper bounds on B are estimated from,
Since the standard deviation,
ln(
, and the parameter C are positive parameters, then
) and ln(C) are treated as normally distributed. The bounds are estimated from,
and,
The variances and covariances of B, C, and
(evaluated at
,
,
, as follows,
are estimated from the local Fisher Matrix
Bounds on Reliability
The reliability of the lognormal distribution is,
Let
(t,
For t =
V; B, C,
,
=
)=
, then
, and for t =
.
,
=
. The above equation then becomes,
The bounds on z are estimated from,
where,
or,
The upper and lower bounds on reliability are,
Confidence Bounds on Time
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first
solving the reliability equation with respect to time, as follows,
where,
and,
The next step is to calculate the variance of
(V;
,
,
),
or,
The upper and lower bounds are then found by:
Solving for
and
get,
See Also:
Arrhenius Confidence Bounds
Eyring Relationship
This chapter includes the following subchapters:

Eyring Relationship Introduction

Eyring Acceleration Factor

Eyring-Exponential

Eyring-Weibull

Eyring-Lognormal

Eyring Confidence Bounds
See Also:
Contents
Introduction
Eyring Relationship Introduction
The Eyring model was formulated from quantum mechanics principles [9] and is most often
used when thermal stress (temperature) is the acceleration variable. However, the Eyring
relationship is also often used for stress variables other than temperature, such as humidity.
The relationship is given by:
(1)
where,


L represents a quantifiable life measure, such mean life, characteristic life, median life,
B(x) life, etc.,
V represents the stress level (temperature values in absolute units, i.e. degrees Kelvin or
degrees Rankine)

A is one of the model parameters to be determined,
and,

B is another model parameter to be determined.
Fig. 1: Graphical look at the Eyring relationship (linear scale), at different life
characteristics and with a Weibull life distribution.
The Eyring relationship is similar to the Arrhenius relationship. This similarity is more apparent
if Eqn. (1) is rewritten in the following way:
or,
(2)
The Arrhenius relationship is given by:
Comparing Eqn. (2) to the Arrhenius relationship, it can be seen that the only difference
between the two relationships is the
term in Eqn. (2). In general, both relationships yield
very similar results. Like the Arrhenius, the Eyring relationship is plotted on a log-reciprocal
paper.
Fig. 2: Eyring relationship plotted on Arrhenius paper.
See Also:
Eyring Relationship
Eyring Acceleration Factor
For the Eyring model the acceleration factor is given by:
See Also:
Eyring Relationship
Eyring-Exponential
The pdf for the Eyring relationship and the exponential distribution is given next.
The pdf of the 1-parameter exponential distribution is given by:
It can be easily shown that the mean life for the 1-parameter exponential distribution,
presented in detail in the Life Distributions chapter, is given by:
(3)
thus,
(4)
The Eyring-exponential model pdf can then be obtained by setting m
= L(V) in Eqn. (1),
and substituting for m in Eqn. (4),
(5)
Eyring Exponential Statistical Properties Summary
Mean or MTTF
The mean,
by:
, or mean time to failure (MTTF) for the Eyring-exponential relationship is given
Median
The median,
for the Eyring-exponential relationship is given by:
Mode
The mode,
for the Eyring-exponential relationship is
= 0.
Standard Deviation
The standard deviation,
, for the Eyring-exponential relationship is given by:
Eyring-Exponential Reliability Function
The Eyring-exponential reliability function is given by:
This function is the complement of the Eyring-exponential cumulative distribution function or,
and,
Conditional Reliability
The conditional reliability function for the Eyring-exponential relationship is given by:
Reliable Life
For the Eyring-exponential relationship, the reliable life, or the mission duration for a desired
reliability goal
or,
is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
The complete exponential log-likelihood function of the Eyring model is composed of two
summation portions,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure in the



is the stress level of the
time-to-failure data group.
group.
A is the Eyring parameter (unknown, the first of two parameters to be estimated).
B is the second Eyring parameter (unknown, the second of two parameters to be
estimated).




is the exact failure time of the
group.
S is the number of groups of suspension data points.
is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for the parameters
= 0 and
= 0 where:
and
so that
See Also:
Eyring Relationship
Eyring Weibull
The pdf for the Eyring relationship and the Weibull distribution is given next.
The pdf for 2-parameter Weibull distribution is given by:
(6)
The scale parameter (or characteristic life) of the Weibull distribution is
model pdf can then be obtained by setting
or,
Substituting for
into Eqn. (6),
= L(V) in Eqn. (1),
. The Eyring-Weibull
Eyring Weibull Statistical Properties Summary
Mean or MTTF
The mean,
where
, or mean time to failure (MTTF) for the Eyring-Weibull relationship is given by:
is the gamma function evaluated at the value of
Median
The median,
for the Eyring-Weibull relationship is given by:
(7)
Mode
The mode,
for the Eyring-Weibull relationship is given by:
(8)
Standard Deviation
The standard deviation,
for the Eyring-Weibull relationship is given by:
.
Eyring-Weibull Reliability Function
The Eyring-Weibull reliability function is given by:
Conditional Reliability Function
The Eyring-Weibull conditional reliability function at a specified stress level is given by:
or,
Reliable Life
For the Eyring-Weibull relationship, the reliable life,
starting the mission at age zero is given by:
, of a unit for a specified reliability and
(9)
Eyring-Weibull Failure Rate Function
The Eyring-Weibull failure rate function,
(T), is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
The Eyring-Weibull log-likelihood function is composed of two summation portions,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure data points in the



time-to-failure data group.
is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
A is the Eyring parameter (unknown, the second of three parameters to be estimated).
B is the second Eyring parameter (unknown, the third of three parameters to be
estimated).

is the stress level of the

is the exact failure time of the



group.
group.
S is the number of groups of suspension data points.
is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for the parameters
that
= 0,
= 0 and
= 0 where,
, A and B so
Example
Consider the following times-to-failure data at three different stress levels.
The data set was analyzed jointly and with a complete MLE solution over the entire data set
using ReliaSoft's ALTA yielding,
= 4.29186497,
= -11.08784624,
= 1454.08635742.
Once the parameters of the model are defined, other life measures can be directly obtained
using the appropriate equations. For example, the MTTF can be obtained for the use stress
level of 323K using,
or,
See Also:
Eyring Relationship
Eyring Lognormal
The pdf for the Eyring relationship and the lognormal distribution is given next.
The pdf of the lognormal distribution is given by:
(10)
where,

= ln (T),
T = times-to-failure,

and,


= mean of the natural logarithms of the times-to-failure,
= standard deviation of the natural logarithms of the times-to-failure.
The Eyring-lognormal model pdf can be obtained first by setting
(1).Therefore,
= L(V) in Eqn.
or,
Thus,
(11)
Substituting Eqn. (11) into Eqn. (10) yields the Eyring-lognormal model pdf or,
Eyring-Lognormal Statistical Properties Summary
The Mean

The mean life of the Eyring-lognormal model (mean of the times-to-failure),
(12)
, is given by:

The mean of the natural logarithms of the times-to-failure,
given by:
, in terms of
and
is
The Median

The median of the lognormal distribution is given by:
(13)
The Standard Deviation

The standard deviation of the Eyring-lognormal model (standard deviation of the
times-to-failure),
, is given by:
(14)

The standard deviation of the natural logarithms of the times-to-failure,
and
is given by:
The Mode
, in terms of

The mode of the Eyring-lognormal model is given by:
Eyring-Lognormal Reliability Function
The reliability for a mission of time T, starting at age 0, for the Eyring-lognormal model is
determined by:
or,
There is no closed form solution for the lognormal reliability function. Solutions can be
obtained via the use of standard normal tables. Since the application automatically solves for
the reliability we will not discuss manual solution methods.
Reliable Life
For the Eyring-lognormal model, the reliable life, or the mission duration for a desired reliability
goal,
where,
is estimated by first solving the reliability equation with respect to time, as follows,
and,
Since
= ln (T) the reliable life,
, is given by:
Eyring-Lognormal Failure Rate
The Eyring-lognormal failure rate is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
The complete Eyring-lognormal log-likelihood function is composed of two summation
portions,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure data points in the

time-to-failure data group.
is the standard deviation of the natural logarithm of the times-to-failure (unknown, the
first of three parameters to be estimated).


A is the Eyring parameter (unknown, the second of three parameters to be estimated).
C is the second Eyring parameter (unknown, the third of three parameters to be
estimated).

is the stress level of the

is the exact failure time of the



group.
group.
S is the number of groups of suspension data points.
is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for
= 0 and
= 0,
,
,
so that
= 0,
and,
See Also:
Eyring Relationship
Eyring Confidence Bounds
This subchapter is divided into the following topics:

Approximate Confidence Bounds for the Eyring Exponential

Approximate Confidence Bounds for the Eyring Weibull

Approximate Confidence Bounds for the Eyring Lognormal
See Also:
Eyring Relationship
Approximate Confidence Bounds for the Eyring Exponential
Confidence Bounds on Mean Life
The mean life for the Eyring model is given by Eqn. (1) by setting m
and lower
= L(V). The upper
bounds on the mean life (ML estimate of the mean life) are estimated by:
(15)
(16)
where
If
is defined by:
is the confidence level, then
one-sided bounds. The variance of
=
for the two-sided bounds, and
is given by:
=1-
for the
or,
The variances and covariance of A and B are estimated from the local Fisher Matrix
(evaluated at
,
) as follows,
Confidence Bounds on Reliability
The bounds on reliability at a given time, T, are estimated by:
where
and
are estimated using Eqns. (15) and (16).
Confidence Bounds on Time
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving
the reliability function with respect to time,
The corresponding confidence bounds are estimated from,
where
and
are estimated using Eqns. (15) and (16).
See Also:
Eyring Confidence Bounds
Approximate Confidence Bounds for the Eyring Weibull
Bounds on the Parameters
From the asymptotically normal property of the maximum likelihood estimators, and since
is
a positive parameter, ln ( ) can then be treated as normally distributed. After performing this
transformation, the bounds on the parameters are estimated from,
Also,
and,
The variances and covariances of
(evaluated at
,
,
, A, and B are estimated from the Fisher Matrix
) as follows,
Confidence Bounds on Reliability
The reliability function for the Eyring-Weibull (ML estimate) is given by:
or,
Setting,
or,
The reliability function now becomes,
The next step is to find the upper and lower bounds on
(17)
(18)
where,
,
or,
The upper and lower bounds on reliability are,
where
and
are estimated using Eqns (17) and (18).
Confidence Bounds on Time
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving
the reliability function with respect to time,
or,
where
(19)
(20)
where,
= ln
. The upper and lower bounds on
are then estimated from,
or,
The upper and lower bounds on time are then found by:
where
and
are estimated using Eqns. (19) and (20).
See Also:
Eyring Confidence Bounds
Approximate Confidence Bounds for the Eyring Lognormal
Bounds on the Parameters
The lower and upper bounds on A and B are estimated from,
(upper bound)
(lower bound)
and
(upper bound)
(lower bound)
Since the standard deviation,
is a positive parameter, then ln (
) is treated as
normally distributed, and the bounds are estimated from,
(upper bound)
(lower bound)
The variances and covariances of A,
(evaluated at
,
,
B, and
are estimated from the local Fisher Matrix
) as follows,
where
Bounds on Reliability
The reliability of the lognormal distribution is given by:
Let
(t,
V; A, B,
)=
, then
.
For t =
,
=
becomes,
The bounds on z are estimated from,
where,
, and for t =
,
=
.The above equation then
or,
The upper and lower bounds on reliability are,
(upper bound)
(lower bound)
Confidence Bounds on Time
The bounds around time for a given lognormal percentile (unreliability) are estimated by first
solving the reliability equation with respect to time as follows,
where,
and,
The next step is to calculate the variance of
(V;
or
The upper and lower bounds are then found by:
,
,
),
Solving for
and
get,
(upper bound)
(lower bound)
See Also:
Eyring Confidence Bounds
Inverse Power Law (IPL) Relationship
This chapter includes the following subchapters:

Inverse Power Law Relationship Introduction

Acceleration Factor

IPL-Exponential

IPL-Weibull

IPL-Lognormal

IPL and Coffin Mason Relationship

IPL Confidence Bounds
See Also:
Contents
Introduction
Inverse Power Law Relationship Introduction
The inverse power law (IPL) model (or relationship) is commonly used for non-thermal
accelerated stresses and is given by:
(1)
where,



L represents a quantifiable life measure, such as mean life, characteristic life, median life,
B(x) life, etc.,
V represents the stress level,
K is one of the model parameters to be determined, (K > 0),
and,

n is another model parameter to be determined.
Fig. 1: The inverse power law relationship on linear scales at different life
characteristics and with a Weibull life distribution.
The inverse power law appears as a straight line when plotted on a log-log paper. The
equation of the line is given by:
(2)
Plotting methods are widely used in estimating the parameters of the inverse power law
relationship since obtaining K and n is as simple as finding the slope and the intercept on Eqn.
(2).
Fig. 2: Graphical look at the IPL relationship (log-log scale)
A Look at the Parameter n
The parameter n in the inverse power relationship is a measure of the effect of the stress on
the life. As the absolute value of n increases, the greater the effect of the stress. Negative
values of n indicate an increasing life with increasing stress. An absolute value of n
approaching zero indicates small effect of the stress on the life, with no effect (constant life
with stress) when n = 0.
Fig. 3: Life vs. stress for different values of
See Also:
Inverse Power Law Relationship Introduction
IPL Acceleration Factor
For the IPL model the acceleration factor is given by:
where,



is the life at use stress level,
is the life at the accelerated stress level,
is the use stress level,
n.
and

is the accelerated stress level.
See Also:
Inverse Power Law Relationship
IPL Exponential
The pdf for the Inverse Power Law relationship and the exponential distribution is given next.
The IPL-exponential model can be derived by setting m
following IPL-exponential pdf,
= L(V) in Eqn. (31), yielding the
Note that this is a 2-parameter model. The failure rate (the parameter of the exponential
distribution) of the model is simply
=
, and is only a function of stress.
Fig. 4: IPL-Exponential Failure Rate function at different stress levels.
IPL-Exponential Statistical Properties Summary
Mean or MTTF
The mean,
, or mean time to failure (MTTF) for the IPL-exponential relationship is given by:
Note that the MTTF is a function of stress only and is simply equal to the IPL relationship
(which is the original assumption), when using the exponential distribution.
Median
The median,
for the IPL-exponential relationship is given by:
Mode
The mode,
for the IPL-exponential relationship is given by:
Standard Deviation
The standard deviation,
, for the IPL-exponential relationship is given by:
IPL-Exponential Reliability Function
The IPL-exponential reliability function is given by:
This function is the complement of the IPL-exponential cumulative distribution function,
or,
Conditional Reliability
The conditional reliability function for the IPL-exponential relationship is given by:
Reliable Life
For the IPL-exponential relationship, the reliable life or the mission duration for a desired
reliability goal,
is given by:
or,
Parameter Estimation
Maximum Likelihood Parameter Estimation
Substituting the inverse power law model into the exponential log-likelihood equation yields,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure in the


is the stress level of the
time-to-failure data group.
group.
K is the IPL parameter (unknown, the first of two parameters to be estimated).





n is the second IPL parameter (unknown, the second of two parameters to be estimated).
is the exact failure time of the
group.
S is the number of groups of suspension data points.
is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for the parameters
= 0 and
,
so that
= 0, where,
See Also:
Inverse Power Law Relationship
IPL Weibull
The pdf for the Inverse Power Law relationship and the Weibull distribution is given next.
The IPL-Weibull model can be derived by setting
pdf,
= L(V), yielding the following IPL-Weibull
This is a three-parameter model. Therefore it is more flexible but it also requires more
laborious techniques for parameter estimation. The IPL-Weibull model yields the
IPL-exponential model for
= 1.
IPL-Weibull Statistical Properties Summary
Mean or MTTF
The mean,
where
, (also called MTTF), of the IPL-Weibull relationship is given by:
is the gamma function evaluated at the value of
Median
The median,
of the IPL-Weibull relationship is given by:
(3)
Mode
The mode,
of the IPL-Weibull relationship is given by:
(4)
Standard Deviation
The standard deviation,
of the IPL-Weibull relationship is given by:
IPL-Weibull Reliability Function
.
The IPL-Weibull reliability function is given by:
Conditional Reliability Function
The IPL-Weibull conditional reliability function at a specified stress level is given by:
or,
Reliable Life
For the IPL-Weibull relationship, the reliable life,
starting the mission at age zero is given by:
, of a unit for a specified reliability and
(5)
IPL-Weibull Failure Rate Function
The IPL-Weibull failure rate function,
(T), is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
Substituting the inverse power law model into the Weibull log-likelihood function yields,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure data points in the



is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
K is the IPL parameter (unknown, the second of three parameters to be estimated).
n is the second IPL parameter (unknown, the third of three parameters to be estimated).

is the stress level of the

is the exact failure time of the

time-to-failure data group.
group.
group.
S is the number of groups of suspension data points.


is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for
0 and
= 0, where,
, K, n so that
= 0,
=
Example 1
Consider the following times-to-failure data at two different stress levels.
The data set was analyzed jointly and with a complete MLE solution over the entire data set
using ReliaSoft's ALTA. The analysis yields,
= 2.61647,
= 0.00102241,
= 1.32729123.
See Also:
Inverse Power Law Relationship
IPL Lognormal
The pdf for the Inverse Power Law relationship and the lognormal distribution is given next.
The pdf of the lognormal distribution is given by:
(6)
where,
= ln(T),
T = times-to-failure,
and,
= mean of the natural logarithms of the times-to-failure,
= standard deviation of the natural logarithms of the times-to-failure.
The median of the lognormal distribution is given by:
(7)
The IPL-lognormal model pdf can be obtained first by setting
= L(V) in Eqn. (30). Therefore,
or
Thus,
(8)
Substituting Eqn. (8) into Eqn. (6) yields the IPL- lognormal model pdf or,
IPL-Lognormal Statistical Properties Summary
The Mean

The mean life of the IPL-lognormal model (mean of the times-to-failure),
, is given by:
(9)

The mean of the natural logarithms of the times-to-failure,
given by:
The Standard Deviation
, in terms of
and
is

The standard deviation of the IPL-lognormal model (standard deviation of the
times-to-failure),
, is given by:
(10)

The standard deviation of the natural logarithms of the times-to-failure,
and
is given by:
, in terms of
The Mode

The mode of the IPL-lognormal is given by:
IPL-Lognormal Reliability
The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is
determined by:
or,
Reliable Life
The reliable life, or the mission duration for a desired reliability goal,
solving the reliability equation with respect to time, as follows,
is estimated by first
where,
and,
Since
= ln(T) the reliable life,
, is given by:
Lognormal Failure Rate
The lognormal failure rate is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
The complete IPL-lognormal log-likelihood function is composed of two summation portions,
where:

is the number of groups of exact times-to-failure data points.

is the number of times-to-failure data points in the
time-to-failure data group.

is the standard deviation of the natural logarithm of the times-to-failure (unknown, the
first of three parameters to be estimated).

K is the IPL parameter (unknown, the second of three parameters to be estimated).
n is the second IPL parameter (unknown, the third of three parameters to be estimated).


is the stress level of the

is the exact failure time of the



group.
group.
S is the number of groups of suspension data points.
is the number of suspensions in the
is the running time of the
group of suspension data points.
suspension data group.
The solution (parameter estimates) will be found by solving for
= 0 and
= 0, where (for FI = 0),
,
,
so that
= 0,
and,
See Also:
Inverse Power Law Relationship
IPL and Coffin Manson Relationship
In accelerated life testing analysis, thermal cycling is commonly treated as a low-cycle fatigue
problem, using the inverse power law relationship. Coffin and Manson suggested that the
number of cycles-to-failure of a metal subjected to thermal cycling is given by [28],
(11)
where,



N is the number of cycles to failure,
C is a constant, characteristic of the metal,
is another constant, also characteristic of the metal,
and

T is the range of the thermal cycle.
This model is basically the inverse power law relationship, where instead of the stress,
V, the
range V is substituted into Eqn. (30). This is an attempt to simplify the analysis of a
time-varying stress test by using a constant stress model. It is a very commonly used
methodology for thermal cycling and mechanical fatigue tests. However, by performing such a
simplification, the following assumptions and shortcomings are inevitable. First the
acceleration effects due to the stress rate of change are ignored. In other words, it is assumed
that the failures are accelerated by the stress difference and not by how rapidly this difference
occurs. Secondly, the acceleration effects due to stress relaxation and creep are ignored.
Example
In this example the use of Eqn. (11) will be illustrated. This is a very simple example that can
be repeated at any time. The reader is encouraged to perform this test.
Product: ACME Paper Clip Model 4456
Reliability Target: 99% at a 90% confidence after 30 cycles of 45°.
After consulting with our paper-clip engineers, the acceleration stress was determined to be
the angle to which the clips are bent. Two bend stresses of 90° and 180° where used. A
sample of six paper clips was tested to failure at both 90° and 180° bends with the
following data obtained.
The test was performed as shown in the next figures (a side-view of the paper clip is shown).
Using the IPL lognormal model, determine whether the reliability target was met.
Solution
By using the IPL relationship to analyze the data, we are actually using a constant stress
model to analyze a cycling process. Caution must be taken when performing the test. The rate
of change in the angle must be constant and equal for both the 90° and 180° bends and
constant and equal to the rate of change in the angle for the use life of 45° bend. Rate effects
are influencing the life of the paper clip. By keeping the rate constant and equal at all stress
levels, we can then eliminate these rate effects from our analysis. Otherwise the analysis will
not be valid.
The data are entered and analyzed using ReliaSoft's ALTA.
The parameters of the IPL-lognormal model were estimated to be,
= 0.198533,
K = 0.000012,
n = 1.856808.
Using the QCP, the 90% lower 1-sided confidence bound on reliability after 30 cycles for a
45° bend was estimated to be 99.6%, as shown below.
This meets the target reliability of 99%.
See Also:
Inverse Power Law Relationship
IPL Confidence Bounds
This subchapter is made up of the following topics:

Approximate Confidence Bounds on IPL-Exponential

Approximate Confidence Bounds on IPL-Weibull

Approximate Confidence Bounds on IPL- Lognormal
See Also:
Inverse Power Law Relationship
Approximate Confidence Bounds on IPL-Exponential
Confidence Bounds on the Mean Life
From the inverse power law relationship the mean life for the exponential distribution is given
by Eqn. (30) by setting m = L(V). The upper (
(ML estimate of the mean life) are estimated by:
) and lower (
) bounds on the mean life
(12)
(13)
where
If
is defined by:
is the confidence level, then
one-sided bounds. The variance of
or,
=
for the two-sided bounds, and
is given by:
= 1-
for the
The variances and covariance of K and n are estimated from the Fisher Matrix (evaluated at
,
) as follows,
Confidence Bounds on Reliability
The bounds on reliability at a given time, T, are estimated by:
where
and
are estimated using Eqns. (12) and (13).
Confidence Bounds on Time
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving
the reliability function with respect to time,
The corresponding confidence bounds are estimated from,
where
and
are estimated using Eqns. (12) and (13).
See Also:
IPL Confidence Bounds
Approximate Confidence Bounds on IPL-Weibull
Bounds on the Parameters
Using the same approach as previously discussed (
and
positive parameters),
and
The variances and covariances of
(evaluated at
,
,
, K, and n are estimated from the local Fisher Matrix
) as follows:
Confidence Bounds on Reliability
The reliability function (ML estimate) for the IPL-Weibull model is given by:
or,
Setting,
or,
(14)
The reliability function now becomes,
The next step is to find the upper and lower bounds on
(15)
(16)
where,
,
or,
The upper and lower bounds on reliability are,
where
and
are estimated using Eqns. (15) and (16).
Confidence Bounds on Time
The bounds on time for a given reliability (ML estimate of time) are estimated by first solving
the reliability function with respect to time,
or,
where
(17)
(18)
where,
= ln
. The upper and lower bounds on u are estimated from,
or,
The upper and lower bounds on time are then found by:
Where
and
are estimated using Eqns. (17) and (18).
See Also:
IPL Confidence Bounds
Approximate Confidence Bounds on IPL- Lognormal
Bounds on the Parameters
Since the standard deviation,
, and
are positive parameters, then ln (
are treated as normally distributed, and the bounds are estimated from,
(upper bound)
(lower bound)
And,
(upper bound)
(lower bound)
The lower and upper bounds on n, are estimated from,
(upper bound)
) and ln ( )
(lower bound)
The variances and covariances of A,
(evaluated at
,
,
B, and
are estimated from the local Fisher Matrix
), as follows,
where,
Bounds on Reliability
The reliability of the lognormal distribution is,
Let
(t,
V; K, n,
For t =
becomes,
,
=
)=
, then =
, and for t
=
,
.
=
. The above equation then
The bounds on z are estimated from,
where,
or,
The upper and lower bounds on reliability are,
(upper bound)
(lower bound)
Confidence Bounds on Time
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first
solving the reliability equation with respect to time, as follows,
where,
and
The next step is to calculate the variance of
(V;
,
,
),
or,
The upper and lower bounds are then found by:
Solving for
and
we get,
(upper bound)
(lower bound)
See Also:
IPL Confidence Bounds