I am working with some subsea cable data. I have $n$ samples that are split into three groups (sized $n_1$, $n_2$ and $n_3$). Each group goes through a different acceleration process until failure. The cables are designed to operate a nominal stress $s_0$.
In material testing it is common to use the material at an accelerated stress to induce failures more quickly than what would be seen in practice (otherwise products would take many more years (sometimes millions) to come to market). These accelerated failure times are then used to obtain the failure time under nominal stress using some physical relationships. The inverse power law is a common relationship used to infer the lifetime of a material under normal use conditions. The equation is written as:
\begin{equation} T_0 = (s_1/s_0)^n \times T_1, \end{equation}
where $s_1$ is the accelerated stress, $s_0$ is the nominal stress, $T_1$ is the failure time under accelerated conditions, and $n$ is an ageing factor that depends on the material and is known approximately for the cables $( n = 10)$. $s_1/s_0$ is called the acceleration factor.
I have seen this equation used, and proven to be accurate in practice, when $s_1/s_0$ is "small", but I am unsure if this simple relationship holds for large acceleration factors. The reason I ask is because the cable groups provide very different lifetime distributions. They are the same cables, but they undertake different acceleration procedures. If the cables were aged until failure at normal conditions, we would expect the failure times to come from the same distribution.
In short, group one is accelerated at: $3 s_0$ for two years, and then $4s_0$, $5s_0$, ... for 5 minutes each until failure. The highest level of acceleration is $18s_0$.
Group two is accelerated for one year at $3s_0$, and then 5 minutes at $4s_0$, $5s_0$, ... until failure. The highest level of acceleration is $22 s_0$.
The third group is accelerated for $3s_0$ for 5 minutes and then $4s_0$, $5s_0$, ... until failure. The highest level of acceleration is $26s_0$.
I do not believe that any of the distributions accurately describe the failure times of the cables, however, I believe that group 1 is the most accurate estimate, since these cables were accelerated for 2 years at a "reasonable" acceleration factor. By reasonable I mean, I believe if the cables were subjected to $3s_0$ until failure, we could then say
\begin{equation} T_0 = 3^{10} \times T_1. \end{equation}
However, if we just acceleration the cables until failure at $25 s_0$, I do not believe that
\begin{equation} T_0 = 25^{10} \times T_1. \end{equation}
Some of the cables in group 3 were subjected to $25s_0$ for 300 seconds. I do not believe that 300 seconds at this level corresponds to $25^{10} \times 300$ seconds (=907,224,426 years, about a billion years) at the nominal stress. I think this is why group 3 gives such poor estimates (compared to what experts believe the lifetime distributions should look like based on similar products).
Group 3 predictions are much higher than group 2 and group 2 predictions are much higher than group 1.
In some experiments, too much acceleration can introduce new failure modes that would never be seen under nominal conditions. This causes products to fail quicker than in practice. I do not have this problem, no new failure modes are being observed under high accelerations, but I do not believe that the simple power law holds for high accelerations (although it is widely accepted for small accelerations). This is interesting because some safety products are expected to last tens of millions of years and even acceleration at factors of 1000 will not provide results for thousands of years, there must be some sophisticated physics that can describe what I am seeing.
Ideally, we could use the cable at nominal stress for 10 million years and forward through time to see if we are right. This would give an accurate failure time distribution. High accelerations are used to obtain results in short periods of time but provide inaccurate predictions (about as useful as no results at all). A nice trade-off is what we aim to achieve.
So, my question: is it obvious that the inverse power law should not hold when the acceleration factor becomes sufficiently large? Is there some adjustment I should be using to describe failure times under nominal stress when the acceleration factor becomes large. Have any papers in the literature showed that this equation is not valid for large acceleration factors? Perhaps another equation is more commonly used that is more sophisticated than a simple power law?
One example of an adjustment could be
\begin{equation} T_0 = (s_1/s_0)^{n(s_1/s_0)} \times T_1, \end{equation}
where $n(s_1/s_0)$ depends on the acceleration factor. Perhaps this value should decrease as the acceleration factor increases. I have not seen any papers that have used this but maybe other adjustments could be used.
Can anyone point me in the right direction?
