You can also think of it in simplified math therms. Simplify the square anvil into a round one then look at the equations for torsional stiffness of a round shaft. This site, picked by the internet search gods, provides the equations
Torsional stress is the shear stress that results from the action of a twist on a transverse cross-section. It is defined as the shear stress brought on by twisting a shaft.
byjusexamprep.com
Since we are just comparing two anvil sizes we can assume things like material properties are the same. We can also simplify our square shaft into a round shaft that has the same diameter as the flats of the square.
One way to look at this is as a pure stiffness question. From this link we get the following:
This article will walk product designers and engineers through the methods used to analyze and improve torsional rigidity in design. Learn more!
www.fictiv.com
The deflection under load equation is:
theta= TL/JG, theta is the amount a beam twists for a given applied torque
where:
T=toque applied
L=length of beam
J=Polar moment of inertia ->this is the part of the equation that cares about the cross section of the part
G=Shear modulus -> this is the part of the equation that cares if we use aluminum vs steel.
Since we are only doing a comparison between two things we can assume T, L and G are the same in both cases (L isn't likely the same when we are talking about a 3/8 vs 1/2 anvil but we will assume since the difference is relatively small compared to the J part.
J is the part that cares about the shape of the anvil. Using the round shaft assumption J=πD^4/32. So increasing diameter (D) really increases stiffness. A 1/2" anvil is 3.16x as stiff as a 3/8" shaft. BTW, this is also why that hole down the center of a ratchet doesn't affect the stiffness much. Assume that's a 1/8" hole in a 3/8" shaft... (3/8)^4=0.01978, (1/8)^4=0.00024. The effective diameter of the 3/8" shaft with a 1/8 whole is (0.01978-0.00024)^.25= 0.3738" or a 2.99/8" diameter. The shaft with the whole is 98.8% as stiff as one without.
But, our maximum torque is a function of stress/strain in the part, not just stiffness. Since diameter also impacts torsional stress, the increase in diameter hurts us. Net result is instead of D^4 we have D^3.
τmax=16T/πD^3
τ is stress in the part (this is our material limit).
T is applied torque
D is again diameter of the assumed round shaft.
Since all we care about is maximum T we can shift things around like this:
T= τmax*πD^3/16
Since we are only comparing the D^3 part is all that matters. The rest stay the same. So .5^3 / (3/8)^3 = 2.37x stronger.
Incidentally, if we want to add that quick release hole then the effective diameter looks like this (Do^4-Di^4)/Do.
So for a 3/8 drive with a 1/8 hole, we have (.375^4-.125^4)/.375 = 0.05208 vs .375^3 = 0.0527. That change in effective diameter is a peek load of 96.5% of the 3/8 solid shaft. So that hole down the center costs you 3.5% of what the ratchet could have handled.