Thermo-mechanical fatigue

The damage accumulation model is a constitutive model of TMF. It adds together the damage from the three failure mechanisms of fatigue, creep, and oxidation.

${\displaystyle {\frac {1}{N_{f}}}={\frac {1}{N_{f}^{fatigue}}}+{\frac {1}{N_{f}^{oxidation}}}+{\frac {1}{N_{f}^{creep}}}}$

where ${\displaystyle N_{f}}$ is the fatigue life of the material, that is, the number of loading cycles until failure. The fatigue life for each failure mechanism is calculated individually and combined to find the total fatigue life of the specimen.[2][3]

The life from fatigue is calculated for isothermal loading conditions. It is dominated by the strain applied to the specimen.

${\displaystyle {\frac {\Delta \epsilon _{m}}{2}}=C(2N_{f}^{fatigue})^{d}}$

where ${\displaystyle C}$ and ${\displaystyle d}$ are material constants found through isothermal testing. Note that this term does not account for temperature effects. The effects of temperature are treated in the oxidation and creep terms..

The life from oxidation is affected by temperature and cycle time.

${\displaystyle {\frac {1}{N_{f}^{oxidation}}}=\left({\frac {h_{cr}\delta _{0}}{B\Phi ^{oxidation}K_{p}^{eff}}}\right)^{\frac {-1}{\beta }}{\frac {2(\Delta {\dot {\epsilon _{m}}})^{{\frac {2}{\beta }}+1}}{\epsilon ^{1-\alpha /\beta }}}}$

where ${\displaystyle K_{p}^{eff}={\frac {1}{t_{c}}}\int _{0}^{t}D_{0}exp\left({\frac {-Q}{RT(t)}}\right)dt}$

and ${\displaystyle \Phi ^{oxidation}={\frac {1}{t_{c}}}\int _{0}^{t}exp\left[-{\frac {1}{2}}\left({\frac {({\dot {\epsilon _{th}}}/{\dot {\epsilon _{m}}})+1}{{\dot {\zeta }}^{oxidation}}}\right)^{2}\right]dt}$

Parameters are found by comparing fatigue tests done in air and in an environment with no oxygen (vacuum or argon). Under these testing conditions, it has been found that the effects of oxidation can reduce the fatigue life of a specimen by a whole order of magnitude. Higher temperatures greatly increase the amount of damage from environmental factors.[4]

${\displaystyle D^{creep}=\Phi ^{creep}\int _{0}^{t}Ae^{(-\Delta H/RT(t))}\left({\frac {\alpha _{1}{\bar {\sigma }}+\alpha _{2}\sigma _{H}}{K}}\right)^{m}dt}$

where ${\displaystyle \Phi ^{creep}={\frac {1}{t_{c}}}\int _{0}^{t}exp\left[-{\frac {1}{2}}\left({\frac {({\dot {\epsilon _{th}}}/{\dot {\epsilon _{m}}})-1}{{\dot {\zeta }}^{creep}}}\right)^{2}\right]dt}$

The damage accumulation model is one of the most in-depth and accurate models for TMF. It accounts for the effects of each failure mechanism.

The damage accumulation model is also one of the most complex models for TMF. There are several material parameters that must be found through extensive testing.[5]

Strain-rate partitioning is a phenomenological model of thermo-mechanical fatigue. It is based on observed phenomenon instead of the failure mechanisms. This model deals only with inelastic strain and ignores elastic strain completely. It accounts for different types of deformation and breaks strain into four possible scenarios:[6]

• PP – plastic in tension and compression
• CP – creep in tension and plastic in compression
• PC – plastic in tension and creep in compression
• CC – creep in tension and compression

The damage and life for each partition is calculated and combined in the model

${\displaystyle {\frac {1}{N_{f}}}={\frac {F_{pp}}{N'_{pp}}}+{\frac {F_{cc}}{N'_{cc}}}+{\frac {F_{pc}}{N'_{pc}}}+{\frac {F_{cp}}{N'_{cp}}}}$

where ${\displaystyle F_{pp}={\frac {\Delta \epsilon _{pp}}{\Delta \epsilon _{inelastic}}},F_{cc}={\frac {\Delta \epsilon _{cc}}{\Delta \epsilon _{inelastic}}},F_{pc}={\frac {\Delta \epsilon _{pc}}{\Delta \epsilon _{inelastic}}},F_{cp}={\frac {\Delta \epsilon _{cp}}{\Delta \epsilon _{inelastic}}}}$

and ${\displaystyle N'_{pp}}$ etc., are found from variations of the equation ${\displaystyle \Delta \epsilon _{inelastic}=A_{pp}(N'_{pp})^{C_{pp}}}$

where A and C are material constants for individual loading.

Strain-Rate Partitioning is a much simpler model than the damage accumulation model. Because it breaks down the loading into specific scenarios, it can account for different phases in loading.

The model is based on inelastic strain. This means that it does not work well with scenarios of low inelastic strain, such as brittle materials or loading with very low strain. This model can be an oversimplification. Because it fails to account for oxidation damage, it may overpredict specimen life in certain loading conditions.