What you're failing to take into consideration is that fuses blow because of excessive heat, not excessive current.
That is not exactly correct. Fuses are designed using 2 factors, current and AIC (ampere interruting capacity). Excessive current passing thru the fuse creates heat which if maintained will eventually burn out/melt the fuse...how long depends on type/class of fuse.
The whole concept of fuses and circuit breakers is based on current and time, the idea is to prevent the circuit from overheating.
As for the t=o thing I am not sure I understood where Terry was going.
Since we are leaving out XC and XL I assume we are speaking strictly of a DC ckt.
1. If its a purely resistive load then current will be (more or less) instantaneous...I= V/R. There is no time constants, transients etc etc. Technically this applies to AC as well.
Fast acting (Normal) fuses are used primarily for resistive load OR to protect sensitive devices from the heat caused by excessive current, some may be current limiting. These provide overload protection at 135% and 200% of rated value. To use these with motors they need to be rated 200-300% over load current but now they only provide short ckt protection and not overload protection.
Time delay or Slow blow fuses are designed to allow temporary overloads as high as 500% for 10-12 seconds. These fuses can be selected closer to a ckts actual operating current which provides overload and short ckt protection.
All fuses have an inverse time characteristic...ie as the current increases the opening time decreases.
In the US NEC an UL governs classification and actions for fuses.