Larson-Miller relation

The Larson-Miller relation, also widely known as the Larson-Miller Parameter and often abbreviated LMP, is a parametric relation used to extrapolate experimental data on creep and rupture life of engineering materials.

Background and usage

F.R. Larson and J. Miller proposed that creep rate could adequately be described by the Arrhenius type equation:

:$r\; =\; A*e^\{-Delta\; H/R*T\}$

Where r is the creep process rate, A is a constant, R is the universal gas constant, T is the absolute temperature, and $Delta\; H$ is the activation energy for the creep process. Taking the natural log of both sides:

:$ln(r)\; =\; ln(A)\; -\; Delta\; H/R*T$

With some rearrangement:

:$Delta\; H/R\; =\; T*(ln(A)\; -\; ln(r))$

Using the fact that creep rate is inversely proportional to time, the equation can be written as:

:$1/t\; =\; A\text{'}*e^\{-Delta\; H/R*T\}$

Taking the natural log:

:$ln(1/t)\; =\; ln(A\text{'})\; -\; Delta\; H/R*T$

After some rearrangement the relation finally becomes:

:$Delta\; H/R\; =\; T*(C\; +\; log(t))$

This equation is the Larson-Miller relation, and the quantity $Delta\; H/R$ is known as the Larson-Miller parameter. Using the assumption that activation energy is independent of applied stress, the equation can be used to relate the difference in rupture life to differences in temperature for a given stress. The material constant C is typically found to be in the range of 20 to 22 for metals.

The Larson-Miller model is used for experimental tests so that results at certain temperatures and stresses can predict rupture lives of time spans that would be impractical to reproduce in the laboratory.

The equation was developed during the 1950's while Miller and Larson were employed by GE performing research on turbine blade life.

References

*Hertzberg, Richard W. Deformation and Fracture Mechanics of Engineering Materials, Fourth Edition. John Wiley and Sons, Inc., Hoboken, NJ: 1996.