# Cumulative Damage Exponential Relationship

This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the exponential relationship. Given a time-varying stress $x(t)\,\!$ and assuming the exponential relationship, the life-stress relationship is given by:

{\begin{aligned}L(x(t))=C{{e}^{bx(t)}}\end{aligned}}\,\! In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the exponential relationship:

Therefore, instead of displaying $C\,\!$ and $b\,\!$ as the calculated parameters, the following reparameterization is used:

{\begin{aligned}{{\alpha }_{0}}=\ &\ln(C)\\{{\alpha }_{1}}=\ &b\end{aligned}}\,\! ## Cumulative Damage Exponential - Exponential

Given a time-varying stress $x(t)\,\!$ and assuming the exponential life-stress relationship, the mean life is given by:

${\frac {1}{m(t,x)}}=s(t,x)={\frac {{e}^{-bx(t)}}{C}}\,\!$ The reliability function of the unit under a single stress is given by:

{\begin{aligned}R(t,x(t))={{e}^{-I(t,x)}}\end{aligned}}\,\! where:

$I(t,x)={\underset {0}{\mathop {{\overset {t}{\mathop {\int _{}^{}} }}\,} }}\,{\frac {{e}^{-bx(u)}}{C}}du\,\!$ Therefore, the pdf is:

{\begin{aligned}f(t,x)=s(t,x){{e}^{-I(t,x)}}\end{aligned}}\,\! 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:

{\begin{aligned}&\ln(L)=&\Lambda ={\overset {Fe}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,{{N}_{i}}\ln[s({{T}_{i}},{{x}_{i}})]-{\overset {Fe}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,{{N}_{i}}\left(I({{T}_{i}},{{x}_{i}})\right)-{\overset {S}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,N_{i}^{\prime }\left(I(T_{i}^{\prime },x_{i}^{\prime })\right)+{\overset {FI}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,N_{i}^{\prime \prime }\ln[R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]\end{aligned}}\,\! where:

{\begin{aligned}&R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })=&{{e}^{-I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })}}\\&R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })=&{{e}^{-I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })}}\end{aligned}}\,\! and:

• ${{F}_{e}}\,\!$ is the number of groups of exact time-to-failure data points.
• ${{N}_{i}}\,\!$ is the number of times-to-failure in the ${{i}^{th}}\,\!$ time-to-failure data group.
• ${{T}_{i}}\,\!$ is the exact failure time of the ${{i}^{th}}\,\!$ group.
• $S\,\!$ is the number of groups of suspension data points.
• $N_{i}^{\prime }\,\!$ is the number of suspensions in the ${{i}^{th}}\,\!$ group of suspension data points.
• $T_{i}^{\prime }\,\!$ is the running time of the ${{i}^{th}}\,\!$ suspension data group.
• $FI\,\!$ is the number of interval data groups.
• $N_{i}^{\prime \prime }\,\!$ is the number of intervals in the ${{i}^{th}}\,\!$ group of data intervals.
• $T_{Li}^{\prime \prime }\,\!$ is the beginning of the ${{i}^{th}}\,\!$ interval.
• $T_{Ri}^{\prime \prime }\,\!$ is the ending of the ${{i}^{th}}\,\!$ interval.

## Cumulative Damage Exponential - Weibull

Given a time-varying stress $x(t)\,\!$ and assuming the exponential life-stress relationship, the characteristic life is given by:

${\frac {1}{\eta (t,x)}}=s(t,x)={\frac {{e}^{-b\cdot x(t)}}{C}}\,\!$ The reliability function of the unit under a single stress is given by:

$R(t,x(t))={{e}^{-{{\left(I(t,x)\right)}^{\beta }}}}\,\!$ where:

$I(t,x)={\underset {0}{\mathop {{\overset {t}{\mathop {\int _{}^{}} }}\,} }}\,{\frac {{e}^{-bx(u)}}{C}}du\,\!$ Therefore, the pdf is:

$f(t,x)=\beta s(t,x){{\left(I(t,x)\right)}^{\beta -1}}{{e}^{-{{\left(I(t,x)\right)}^{\beta }}}}\,\!$ 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:

{\begin{aligned}&\ln(L)=&\Lambda ={\overset {Fe}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,{{N}_{i}}\ln[\beta s({{T}_{i}},{{x}_{i}}){{\left(I({{T}_{i}},{{x}_{i}})\right)}^{\beta -1}}]-{\overset {Fe}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,{{N}_{i}}{{\left(I({{T}_{i}},{{x}_{i}})\right)}^{\beta }}-{\overset {S}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,N_{i}^{\prime }{{\left(I(T_{i}^{\prime },x_{i}^{\prime })\right)}^{\beta }}+{\overset {FI}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,N_{i}^{\prime \prime }\ln[R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]\end{aligned}}\,\! where:

{\begin{aligned}&R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })=&{{e}^{-{{\left(I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })\right)}^{\beta }}}}\\&R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })=&{{e}^{-{{\left(I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })\right)}^{\beta }}}}\end{aligned}}\,\! and:

• ${{F}_{e}}\,\!$ is the number of groups of exact time-to-failure data points.
• ${{N}_{i}}\,\!$ is the number of times-to-failure in the ${{i}^{th}}\,\!$ time-to-failure data group.
• ${{T}_{i}}\,\!$ is the exact failure time of the ${{i}^{th}}\,\!$ group.
• $S\,\!$ is the number of groups of suspension data points.
• $N_{i}^{\prime }\,\!$ is the number of suspensions in the ${{i}^{th}}\,\!$ group of suspension data points.
• $T_{i}^{\prime }\,\!$ is the running time of the ${{i}^{th}}\,\!$ suspension data group.
• $FI\,\!$ is the number of interval data groups.
• $N_{i}^{\prime \prime }\,\!$ is the number of intervals in the ${{i}^{th}}\,\!$ group of data intervals.
• $T_{Li}^{\prime \prime }\,\!$ is the beginning of the ${{i}^{th}}\,\!$ interval.
• $T_{Ri}^{\prime \prime }\,\!$ is the ending of the ${{i}^{th}}\,\!$ interval.

## Cumulative Damage Exponential - Lognormal

Given a time-varying stress $x(t)\,\!$ and assuming the exponential life-stress relationship, the median life is:

${\frac {1}{{\breve {T}}(t,x)}}=s(t,x)={\frac {{e}^{-bx(t)}}{C}}\,\!$ The reliability function of the unit under a single stress is given by:

{\begin{aligned}R(t,x(t))=1-\Phi (z)\end{aligned}}\,\! where:

$z(t,x)={\frac {\ln I(t,x)}{\sigma _{T}^{\prime }}}\,\!$ and:

$I(t,x)={\underset {0}{\mathop {{\overset {t}{\mathop {\int {}^{}} }}\,} }}\,{\frac {{e}^{-bx(u)}}{C}}du\,\!$ Therefore, the pdf is:

$f(t,x)={\frac {s(t,x)\varphi (z(t,x))}{\sigma _{T}^{\prime }I(t,x)}}\,\!$ 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:

{\begin{aligned}&\ln(L)=&\Lambda ={\overset {Fe}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,{{N}_{i}}\ln[{\frac {s({{T}_{i}},{{x}_{i}})\varphi (z({{T}_{i}},{{x}_{i}}))}{\sigma _{T}^{\prime }I({{T}_{i}},{{x}_{i}})}}]{\overset {S}{\mathop {{\underset {i=1}{\mathop {+{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,N_{i}^{\prime }\ln \left(1-\Phi (z(T_{i}^{\prime },x_{i}^{\prime }))\right)+{\overset {FI}{\mathop {{\underset {i=1}{\mathop {{\underset {}{\overset {}{\mathop {\sum } }}}\,} }}\,} }}\,N_{i}^{\prime \prime }\ln[\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]\end{aligned}}\,\! where:

{\begin{aligned}&z_{Ri}^{\prime \prime }=&{\frac {\ln I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}}\\&z_{Li}^{\prime \prime }=&{\frac {\ln I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}}\end{aligned}}\,\! and:

• ${{F}_{e}}\,\!$ is the number of groups of exact times-to-failure data points.
• ${{N}_{i}}\,\!$ is the number of times-to-failure in the ${{i}^{th}}\,\!$ time-to-failure data group.
• ${{T}_{i}}\,\!$ is the exact failure time of the ${{i}^{th}}\,\!$ group.$S\,\!$ • $S\,\!$ is the number of groups of suspension data points.
• $N_{i}^{\prime }\,\!$ is the number of suspensions in the ${{i}^{th}}\,\!$ group of suspension data points.
• $T_{i}^{\prime }\,\!$ is the running time of the ${{i}^{th}}\,\!$ suspension data group.
• $FI\,\!$ is the number of interval data groups.
• $N_{i}^{\prime \prime }\,\!$ is the number of intervals in the ${{i}^{th}}\,\!$ group of data intervals.
• $T_{Li}^{\prime \prime }\,\!$ is the beginning of the ${{i}^{th}}\,\!$ interval.
• $T_{Ri}^{\prime \prime }\,\!$ is the ending of the ${{i}^{th}}\,\!$ interval.