Littlewood–Richardson rule: Difference between revisions

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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 (deformation)|creep]] and rupture life of engineering materials.
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==Background and usage==
 
F.R. Larson and J. Miller proposed that creep rate could adequately be described by the Arrhenius type equation:
 
:<math>r = A \cdot e^{-\Delta H/(R \cdot T)}</math>
 
Where r is the creep process rate, A is a constant, R is the [[universal gas constant]], T is the [[absolute temperature]], and <math>\Delta H</math> is the [[activation energy]] for the creep process. Taking the natural log of both sides:
 
:<math>ln(r) = ln(A) - \Delta H/(R \cdot T)</math>
 
With some rearrangement:
 
:<math>\Delta H/R = T \cdot (ln(A) - ln(r))</math>
 
Using the fact that creep rate is inversely proportional to time, the equation can be written as:
 
:<math>\Delta l/\Delta t = A' \cdot e^{-\Delta H/ ( R \cdot T )}</math>
 
Taking the natural log:
 
:<math>\ln(\Delta l/\Delta t) = ln(A') - \Delta H/(R \cdot T)</math>
 
After some rearrangement the relation finally becomes:
 
:<math>\Delta H/R = T \cdot (B + ln(\Delta t))</math>, where B =  <math>\ln(A'/\Delta l)</math>
 
This equation is of the same form as the Larson–Miller relation.
 
:<math>LMP = T \cdot (C + log(t))</math>
 
where the quantity LMP 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 when time is expressed in hours and temperature in degrees Rankine.
 
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 1950s while Miller and Larson were employed by GE performing research on turbine blade life.
 
==See also==
* [[Creep (deformation)]]
 
==References==
 
*Hertzberg, Richard W.  Deformation and Fracture Mechanics of Engineering Materials, Fourth Edition. John Wiley and Sons, Inc., Hoboken, NJ: 1996.
*Larson, Frank R. and Miller, James: A Time-Temperature Relationship for Rupture and Creep Stresses. Trans. ASME, vol. 74, pp.&nbsp;765−775.
* Fuchs, G. E. High Temperature Alloys, ''Kirk-Othmer Encyclopedia of Chemical Technology''
* Smith & Hashemi, ''Foundations of Material Science and Engineering''
* Dieter, G.E. ''Mechanical Metallurgy, Third Edition'', McGraw-Hill Inc., 1986, p 461-465, ISBN 0-07-016893-8.
 
{{DEFAULTSORT:Larson-Miller relation}}
[[Category:Materials science]]

Latest revision as of 22:47, 20 November 2014

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