I don't understand the physical meaning of Weibull distribution's $k$ parameter. Here is a simplified formula of cumulative probability function of Weibull in the simplest form:
$$p(\xi \geq x) = e^{-(\frac{x}{\lambda})^k}$$
What is the physical interpretation of $k$ parameter, what kind of process would exponentiate $x$? $x$ can be interpreted as a stoppage time or a force applied to a chain/fiber, why would it be exponentiated by $k$?
For comparison: I can more or less rationalise the Gumbel distribution:
Suppose we're running $N$ queues of coin tosses of length $n$. Gumbel distribution gives the probability that the longest chain of successes, observed in any queue, is greater than x, because:
$p(\eta \geq x) = 1 - p(\eta < x) = 1 - (1-\frac{ae^{-bx}}{N})^N = e^{-ae^{-bx}}$
Here I used the fact that probability that there were no sequences of successes longer than $x$ in a Markov chain of length $n$ is distributed as $ae^{-bx}$. For example, see this application to bioinformatics.
From this derivation the interpretation of a and b parameters is pretty intuitive (please, forgive me/correct me if this derivation is not perfectly accurate, but at least it conveys some meaning).
Can you provide any rationalisation, an intuitive model that leads to a Weibull distribution, given it is associated with durability/survival?