Inverse Power Law Relationship - Revision history

Lisa Hacker: /* A Look at the Parameter n\,\! */

2023-06-27T22:03:40Z

A Look at the Parameter n\,\!

← Older revision		Revision as of 22:03, 27 June 2023
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	[[Image:ALTA8.2.png\|center\|450px\|Graphical look at the IPL relationship (log-log scale)]]		[[Image:ALTA8.2.png\|center\|450px\|Graphical look at the IPL relationship (log-log scale)]]

	===A Look at the Parameter ~~<math>~~n~~\,\!</math>~~===		===A Look at the Parameter n===

	The parameter <math>n\,\!</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n\,\!</math> increases, the greater the effect of the stress. Negative values of <math>n\,\!</math> indicate an increasing life with increasing stress. An absolute value of <math>n\,\!</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.\,\!</math>		The parameter <math>n\,\!</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n\,\!</math> increases, the greater the effect of the stress. Negative values of <math>n\,\!</math> indicate an increasing life with increasing stress. An absolute value of <math>n\,\!</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.\,\!</math>

Sharon Honecker: /* Conditional Reliability Function */ changed R(T,t,V) to R((t|T),V)

2017-02-08T21:10:25Z

Conditional Reliability Function: changed R(T,t,V) to R((t|T),V)

← Older revision		Revision as of 21:10, 8 February 2017
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	The IPL-Weibull conditional reliability function at a specified stress level is given by:		The IPL-Weibull conditional reliability function at a specified stress level is given by:

	::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left[ K{{V}^{n}}\left( T+t \right) \right]}^{\beta }}}}}{{{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}}}\,\!</math>		::<math>R((t\|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left[ K{{V}^{n}}\left( T+t \right) \right]}^{\beta }}}}}{{{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}}}\,\!</math>

	:or:		:or:

	::<math>R(T,t,V)={{e}^{-\left[ {{\left( K{{V}^{n}}\left( T+t \right) \right)}^{\beta }}-{{\left( K{{V}^{n}}T \right)}^{\beta }} \right]}}\,\!</math>		::<math>R((t\|T),V)={{e}^{-\left[ {{\left( K{{V}^{n}}\left( T+t \right) \right)}^{\beta }}-{{\left( K{{V}^{n}}T \right)}^{\beta }} \right]}}\,\!</math>

	===Reliable Life===		===Reliable Life===

Sharon Honecker: /* Conditional Reliability */ changed R(T,t,V) to R((t|T),V)

2017-02-08T21:09:46Z

Conditional Reliability: changed R(T,t,V) to R((t|T),V)

← Older revision		Revision as of 21:09, 8 February 2017
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	The conditional reliability function for the IPL-exponential model is given by:		The conditional reliability function for the IPL-exponential model is given by:

	::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-K{{V}^{n}}t}}\,\!</math>		::<math>R((t\|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-K{{V}^{n}}t}}\,\!</math>

	===Reliable Life===		===Reliable Life===

Nicolette Young at 15:21, 17 December 2015

2015-12-17T15:21:10Z

← Older revision		Revision as of 15:21, 17 December 2015
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	Plotting methods are widely used in estimating the parameters of the inverse power law relationship since obtaining <math>K\,\!</math> and <math>n\,\!</math> is as simple as finding the slope and the intercept in the above equation.		Plotting methods are widely used in estimating the parameters of the inverse power law relationship since obtaining <math>K\,\!</math> and <math>n\,\!</math> is as simple as finding the slope and the intercept in the above equation.

	[[Image:ALTA8.2.png\|center\|~~400px~~\|Graphical look at the IPL relationship (log-log scale)]]		[[Image:ALTA8.2.png\|center\|450px\|Graphical look at the IPL relationship (log-log scale)]]

	===A Look at the Parameter <math>n\,\!</math>===		===A Look at the Parameter <math>n\,\!</math>===

Lisa Hacker at 23:37, 23 June 2015

2015-06-23T23:37:29Z

← Older revision		Revision as of 23:37, 23 June 2015
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	& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}		& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
	\end{align}\,\!</math>		\end{align}\,\!</math>

	~~[[Category: Completed Theoretical Review]]~~

Lisa Hacker at 00:51, 31 January 2013

2013-01-31T00:51:12Z

← Older revision		Revision as of 00:51, 31 January 2013
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	[[Image:ALTA8.3.png\|center\|400px\|Life vs. Stress for different values of n.]]		[[Image:ALTA8.3.png\|center\|400px\|Life vs. Stress for different values of n.]]

	===Acceleration Factor====		===Acceleration Factor===
	For the IPL relationship the acceleration factor is given by:		For the IPL relationship the acceleration factor is given by:

Lisa Hacker: moved Inverse Power Law (IPL) Relationship to Inverse Power Law Relationship

2013-01-31T00:50:42Z

moved Inverse Power Law (IPL) Relationship to Inverse Power Law Relationship

← Older revision	Revision as of 00:50, 31 January 2013
(No difference)

Richard House at 22:21, 21 September 2012

2012-09-21T22:21:13Z

Show changes

Richard House: /* IPL and the Coffin-Manson Relationship */

2012-09-04T20:38:16Z

IPL and the Coffin-Manson Relationship

← Older revision		Revision as of 20:38, 4 September 2012
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	=IPL and the Coffin-Manson Relationship=		=IPL and the 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 [[~~Reference Appendix D~~: ~~References~~\|[28]]]:		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 Nelson [[Appendix_E:_References\|[28]]]:

	::<math>N=\frac{C}{{{\left( \Delta T \right)}^{\gamma }}}</math>		::<math>N=\frac{C}{{{\left( \Delta T \right)}^{\gamma }}}</math>

Richard House at 03:00, 30 August 2012

2012-08-30T03:00:47Z

← Older revision		Revision as of 03:00, 30 August 2012
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	=IPL-Exponential=		=IPL-Exponential=
	The IPL-exponential model can be derived by setting <math>m=L(V)\,\!</math> in the exponential ~~<math>~~pdf~~</math>~~, yielding the following IPL-exponential ~~<math>~~pdf~~</math>~~ :		The IPL-exponential model can be derived by setting <math>m=L(V)\,\!</math> in the exponential ''pdf'', yielding the following IPL-exponential ''pdf'':

	::<math>f(t,V)=K{{V}^{n}}{{e}^{-K{{V}^{n}}t}}</math>		::<math>f(t,V)=K{{V}^{n}}{{e}^{-K{{V}^{n}}t}}</math>
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	=IPL-Weibull=<!-- THIS SECTION HEADER IS LINKED TO: Inverse Power Law Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->		=IPL-Weibull=<!-- THIS SECTION HEADER IS LINKED TO: Inverse Power Law Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
	The IPL-Weibull model can be derived by setting <math>\eta =L(V)\,\!</math> in the Weibull ~~<math>~~pdf~~</math>~~, yielding the following IPL-Weibull ~~<math>~~pdf~~</math>~~:		The IPL-Weibull model can be derived by setting <math>\eta =L(V)\,\!</math> in the Weibull ''pdf'', yielding the following IPL-Weibull ''pdf'':

	::<math>f(t,V)=\beta K{{V}^{n}}{{\left( K{{V}^{n}}t \right)}^{\beta -1}}{{e}^{-{{\left( K{{V}^{n}}t \right)}^{\beta }}}}</math>		::<math>f(t,V)=\beta K{{V}^{n}}{{\left( K{{V}^{n}}t \right)}^{\beta -1}}{{e}^{-{{\left( K{{V}^{n}}t \right)}^{\beta }}}}</math>
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	=IPL-Lognormal=		=IPL-Lognormal=
	The ~~<math>~~pdf~~</math>~~ for the Inverse Power Law relationship and the lognormal distribution is given next.		The ''pdf'' for the Inverse Power Law relationship and the lognormal distribution is given next.

	The ~~<math>~~pdf~~</math>~~ of the lognormal distribution is given by:		The ''pdf'' of the lognormal distribution is given by:

	::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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	::<math>\breve{T}=e^{\overline{T}'}</math>		::<math>\breve{T}=e^{\overline{T}'}</math>

	The IPL-lognormal model ~~<math>~~pdf~~</math>~~ can be obtained first by setting <math>\breve{T}=L(V)\,\!</math> in the lognormal ~~<math>~~pdf~~</math>~~. Therefore:		The IPL-lognormal model ''pdf'' can be obtained first by setting <math>\breve{T}=L(V)\,\!</math> in the lognormal ''pdf''. Therefore:

	::<math> \breve{T}=L(V)=\frac{1}{K \cdot V^n}</math>		::<math> \breve{T}=L(V)=\frac{1}{K \cdot V^n}</math>
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	::<math>\overline{T}'=-ln(K)-n ln(V) </math>		::<math>\overline{T}'=-ln(K)-n ln(V) </math>

	So the IPL-lognormal model ~~<math>~~pdf~~</math>~~ is:		So the IPL-lognormal model ''pdf'' is:

	::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+ln(K)+n ln(V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+ln(K)+n ln(V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>