Temperature cycling accelerated life test analysis based on Norris-Landzberg model

In Printed Circuit Board Assemblies, solder joint failures can occur due to fatigue caused by temperature cycling. Norris-Landzberg model have been used to model fatigue failure in solder joints due to repeated temperature cycling as the device is switched on and off.

In this article, we will analyze accelerated test data using the Norris-Landzberg model for lead-free solder joint low-cycle fatigue, and solve for its model parameters by using the general log-linear (GLL) model implemented Weibull Toolbox.

Norris-Landzberg model

The number of cycles to failure in the Norris-Landzberg model is represented as:


Nf is the number of cycles to failure

C is a coefficient

f is the cycling frequency (cycles/day)

ΔT is the temperature range during a cycle (K)

Tmax is the maximum temperature during each cycle (K)

Ea is the activation energy (eV)

K is the Boltzmann's constant (8.617 x 10-5 eV/K)

m and n are the model parameters

It models 3 simultaneous stresses impacting on solder joint (low-cycle fatigue) life, Nf :

Cycling frequency, f with Inverse Power Law (IPL) Life-Stress Relationship,

Temperature range, ∆T with Inverse Power Law (IPL) Life-Stress Relationship and,

Cycle maximum temperature, Tmax with Arrhenius Life-Stress Relationship.

Acceleration Factor (AF), the ratio of Life at use-stress (Nf,use) to Life at accelerated-stress (Nf,acc), can be expressed as:

General Log-Linear model

In the general log-linear (GLL) model, the characteristic life L with 3 stresses, is given by:

The life characteristic, L, can represent any percentile of the assumed underlying life distribution. It can be the scale parameter eta for Weibull distribution, or median for Lognormal distribution.

To match the Norris-Landzberg model above, logarithmic transformation to f and ΔT, and a reciprocal transformation to Tmax:


X2=ln(∆T), and

X3=1/Tmax, such that,

By comparing it with the Norris-Landzberg model equation, the parameter relationships can be derived:

C = eα0

m = -α1

n = -α2

Ea/K =α3

The following is an accelerated life test using GLL life-stress relationship with 3 stresses for a particular lead-free solder joint type. The figure shows the results of thermal cycling test with different stress level settings. For a 3-stresses model, the minimum number of experiments is 4 (because there are 4 unknowns to be solved). Each test uses 10 units. The units are inspected every 100 cycles.

Figure 1, Thermal cycling test results of 4 experiments, each with unique accelerated stress settings

The underlying distribution is assumed to be lognormal. The stress model for f (cycling frequency) and ∆T (temperature range during cycling) are set to Inverse Power Law, and Tmax is set to Arrhenius model as shown on the right panel in below (figure 2).

Figure 2, The dataset is entered into Weibull Toolbox ALTA module.

The analysis results can be used to derive the parameters of Norris-Landzberg model:

C = eα0 = 56,387

m = -α1 = 0.2955

n = -α2 = 1.789

Ea/K = α3 = 1419

Hence, the acceleration factor for this lead-free solder joint temperature cycling life is:

Assuming you conducted an accelerated life with the following settings:

and further assume that you obtain the mean Life of 1500 (cycles) from your dataset.

From the above formular,

The projected mean life is 1500 x 8.79 = 13,185 cycles at use-level stress.


The article demonstrates how quantitative accelerated life test analysis can be used to calculate the Norris-Landzberg model for solder joint open circuit failure resulting from thermal cycling conditions. Similar process can be applied to other commonly used accelerated life stress test for electronic devices like Hallberg-Peck model.


K. C. Norris and A. H. Landzberg, "Reliability of controlled collapse interconnections," IBM Journal of Research and Development 13, no. 3 (1969): 266–271.