How long can we expect a joint to survive? Coffin and Manson^{1} suggested that the number of cyclestofailure (N_{f}) of a metal subjected to thermal cycling is given by:
where
C = a constant, characteristic of the metal
g = another constant, also characteristic of the metal, but typically 2
DT = the range of the thermal cycle
The form of this frequentlycited equation makes it clear that the time to failure will depend critically on characteristics of the material, and that fatigue will result in much earlier failure when the joint experiences wider temperature excursions. The most useful derivative of this equation is probably the relationship between the number of cycles to failure with two different thermal ranges, DT_{1} and DT_{2}:
However, the CoffinManson equation has been criticised as a means of estimating the thermal fatigue life of solders because it was developed for temperatures below 0.5T_{m}, where T_{m} is the melting temperature in Kelvin. As explained at this link, solders generally operate at high homologous temperatures. A number of alternative models, generally referred to by the phrase ‘modified CoffinManson’, have been used with more or less success to model crack growth in solder due to repeated temperature cycling. One such power cycling model takes the form
where
f = the cycling frequency
a = the cycling frequency exponent (typical value 0.33)
DT = the range of the thermal cycle
b = the temperature range exponent (typical value 1.9–2.0)
The final term, G_{Tmax}, is an ‘Arrhenius’ term evaluated at the maximum temperature reached in each cycle. The empiricallybased model known as the Arrhenius equation^{2} predicts how timetofail (t_{f}) varies with temperature and takes the form:
where
A = a numerical constant characteristic of the system
T = the temperature of the failure process in Kelvin
k = Boltzmann’s constant (8.617×10^{–5} eV/K)
E = the ‘activation energy’ in eV (electronvolts)
The Arrhenius activation energy (E) is the critical parameter in the model. Its value depends on the failure mechanism and the materials involved, and typically ranges from 0.3 up to 1.5 (and sometimes higher). As the value of E increases, the acceleration factor between two temperatures increases exponentially, as can be seen from Table 1. For G_{Tmax}, the value of E is about 1.25.
Temperature (°C) 
Temperature (°C) 

0.4 
0.7 
1.0 
1.5 

0 
413 
37,815 
3,463,487 
6,445,703,012 
50 
29.8 
380 
4,844 
337,108 
100 
4.3 
13.1 
39.4 
247 
150 
1 
1 
1 
1 
Don’t worry about the maths! The overall implications are that:
For more information on this topic, try a Google search under “CoffinManson model”.