Estimating time-to-fail

How long can we expect a joint to survive? Coffin and Manson¹ suggested that the number of cycles-to-failure (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 original Coffin and Manson work related cycles to failure to the shear strain by the equation

The form of this frequently-cited 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, DT₁ and DT₂:

However, the Coffin-Manson 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 Coffin-Manson’, 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 empirically-based model known as the Arrhenius equation² predicts how time-to-fail (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 (electron-volts)

Although one of the earliest acceleration models, and the most successful, in the sense of being widely cited and used, for advanced electronics applications the model is increasingly being criticised.

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.

Table 1: Time-to-fail at lower temperatures relative to 150°C predicted by the Arrhenius equation
Temperature (°C)	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:

As the peak temperature increases, the number of cycles to failure reduces very markedly
As the cycling frequency reduces, the number of cycles to failure reduces (which is perhaps not what one might expect)
As the temperature range of the cycle increases, the number of cycles to failure will decrease rapidly.

For more information on this topic, try a Google search under “Coffin-Manson model”.

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Author: Martin Tarr

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