what calculations (possibly stress analysis) are needed when analysing
the forces used in mashing potatoes for all the existing models

Hmmm...all my models use calculations for mashing sweet potatoes and
bananas. I don't think it's a match- however i have experienced
success with minimal stress when i went ahead and mashed my potatoes
without the analysis. A bold move I know but- the potatoes tasted fine
and i was none worse or the wear.

Balcrob, 
Dancingbear is, of course, absolutely right, but for a serious answer,
it is necessary to know what kind of potatoes you want to mash (D-B
touched on this) and also how long you boil them.  Some varieties of
potatoes (and some of them have nice girl's names like Gerlinde [maybe
just nice names, like Gerlinde]) and some are noted for remaining firm
when boiled, ideal for salads, while other get soft and mushy (we're
not talking about Gerlinde) if they are boiled too long.
But with these, of course, less force is required to mash them.
There seem to be more variables than you originally considered.
And if you mash them too thoroughly, your guests may suspect that you
avoided the whole question and used powdered potatoes, but this can be
countered by including bits of peeling (an old Army recipe, another
one adds eggshell to powdered scrambled eggs).
That said, to my wife's chagrin, our son prefers Maggi's (?) powdered
mash potatoes.  (She uses a mixer, anyway.)