For a given calibre and chamber volumne, the way to change expansion ratio is by changing the barrel length.
For a given calibre and barrel length, the way to change expansion ratio is by changing the chamber volumne.
There is general confusion as to the meaning of the term 'Expansion ratio'. As I understand it, the first of your definitions is the strict (ballistician's) definition of the term. As you note, that introduces a firearm related variable that is quite unrelated to the cartridge design.
I have seen an unofficial term in use for the latter, the 'Effective Expansion Ratio' that is purely cartridge design related. For example how far does the bullet travel to double the volume of the combustion chamber. In a 45-70, that's a much shorter distance than in 243 Winchester.
The 'Over bore' [capacity] type metric is the ratio of case capacity to bore diameter. The easiest (and most commonly used) way to calculate this is fireformed case 'overflow water capacity' in grains weight as used in QuickLOAD, GRT etc divided by the bore size area in square inches. There is no formal / official definition of where the 'over bore capacity' boundary lies, partially because the availability of very slow burning powders keeps increasing. A previously over-bore cartridge design that gave little or no additional MV for the additional powder charge weight over lower case capacity models in P O Ackley's days when he considered IMR-4350 and 4831 as exceptionally slow-burning magnum powders is much less so today with Retumbo, Re33, 50BMG powders etc available to the handloader. Nevertheless the 7mm Remington Magnum with a case capacity to bore area ratio of 1300 was long regarded as the borderline, and is as good a boundary as any even today. (Today's 7WSM which many users define as a finicky number to load for precision use is also 1300 and can wear barrels out very quickly if loaded up to its maximum. The 300WSM using the same basic case under a bigger bore at c. 1,080 is a much better mannered beast, regularly noted for being easy to tune with wide accuracy nodes. I shoot the 300 SAUM which with c.987 is a real sweetie, very easy to tune indeed. 308 Win at the 750 mark is hugely powder and load tolerant and gives great barrel life if loads / pressures / firing rates aren't abused.)
This ratio is one of the major determinants of the optimal powder burning rate required as well as barrel life. The higher the value, the slower-burning the powder needed. Bullet weight is also a major factor too, as nearly everyone knows. More open to argument, heavier bullets also wear barrels out faster too in that more mass = greater inertia = the bullet spends longer time in the throat/leade area under maximum pressure, and ...............
Renowned ballistician Homer Powely has stated:
" High pressure is high temperature, and therefore; fast erosion. It is as simple as that ".
extreme heat for longer in the bit of the barrel that determines its life.
Another new factor which has the same effect is Swiss chemical company Nitrochemie Wimmis AG with its patented 'EI' powders as seen in a few Alliant Reloder grades and some of the Reload Swiss grades we have in Europe. 'EI' infuses the deterrent chemicals into the kernels under the surfaces which therefore work for more of the burn than traditional simple surface coatings. This extends the controlled burn period in time / bullet travel and allows the peak of the pressure curve to be longer/flatter. As with heavier bullets it means peak pressure lasts for a longer time, but also now bullet travel too. As this increases the area under the pressure curve it improves MVs - drastically in some cases such as RS60/Re17 in the 6XC and 284 Win with match loads getting boosts of a claimed 200 fps in some combinations. Extended peak pressure = extended period of peak heat energy = increased wear as people who maxed loads / pressures out with this powder soon discovered especially in BR and US F-Class with string-firing and rapid heat build-ups.
Years back somebody on this forum posted jpegs of an article written by a Finnish ordnance officer of ES and group size growth vs round count in the NRA
US Rifleman magazine using the Finnish 7.62X53R service rifle with a standard but selected barrel. 7.62X53/54R has a bore capacity ratio of c. 850, or 100 higher than 308 Win, but its nominal maximum peak chamber pressure is 390 MPa (56,565 psi) under modern CIP methods and standards. I suspect the service cartridges used in these Finnish Army tests were somewhat, possibly considerably, lower pressure. ES values, likewise groups, only started to increase slightly after the barrel was at 11,000 rounds from new, likewise MVs started slowly dropping. The tests were discontinued at over 13,000 rounds with the rifle still shooting well, but the testers worn out. I don't know what effect the qualities / hardness of Finnish chrome-moly barrel steel had on the outcomes compared to the relatively soft 'stainless' we use in match barrels, but the relatively low case capacity to bore ratio and chamber pressure values must play a large part in providing this level of barrel life. 65Grendel forum members say their AR-15 6.5 Grendel (capacity value 665 based on Lapua in my rifle's chamber); SAAMI 52,000 psi max pressure) barrels (presumably stainless) last 9-11,000 rounds - and I don't suppose most such shooters shoot strings of five rounds slowly, then let barrels cool before resuming! Hearsay I know, but I wouldn't disbelieve it. Scandi 6.5X55 shooters with their Sauer STR200s using modern chrome-moly steel barrels and mainly factory ammo which I presume sticks to the old Swedish Mauser max pressure around 50,000 psi (less?) report excellent barrel life, far better than many people I know using hot 6.5 Creedmoor handloads with heavier bullets in stainless barrels (1056 capacity ratio vs 975 for 6.5X55 and 6.5 Creedmoor).
So whichever way you play it, military scientific testing or hearsay, any of the individual metrics of high chamber pressures, high bore capacity ratio values, high rate of fire reduce barrel life all other things staying the same, but in combination will severely reduce it.