If we consider a tooth as beam in bending, then we can say the section modulus of the tooth is proportional to the square of the tooth's base width. The increase in strength due to the section modulus is partly counteracted by the fact that the distance to the point of force application (and hence bending moment) scales with tooth size. Therefore, bending stress calculated by the Lewis method scales with the (^-1) power of module.
In a spline, unlike a gear (with only one or two teeth loaded at a time), the smaller the teeth become, the more teeth there are to share the load; the two effects appear to cancel each other out
That being said, since spline teeth are short and stubby (with typical 30° pressure angle), my understanding is that they don't typically break in bending, making the above bending analysis moot.
I'm looking at
When Splines Need Stress Control by Dudley:
https://dokumen.tips/download/link/when ... udley.html
(you can download this file for reference)
Machinery's Handbook has a section on spline stresses, but it's just Dudley's article cut-and-pasted.
Dudley suggests studying four types of stresses (copied verbatim from article):
1. Shear stresses in spline shaft
2. Shear stresses in spline teeth
3. Compressive stresses in spline teeth
4. Bursting stresses in internal spline parts
My thoughts on module with respect to each of the four stresses:
1. The bigger the teeth, the smaller the root diameter of the shaft, and the less capable the shaft is of transmitting torque.
2. Shearing is assumed to occur along the pitchline of the teeth. The cylindrical area of metal to be sheared is not a function of the number of teeth; the number of teeth simply determines among how many segments the cylindrical area is split.
3. Bigger teeth with more flank surface area can only happen with a reduction in the number of teeth, so this appears to be a wash, unless a greater percentage of the flank happens to be in contact, as I mentioned above.
4. If the OD of the female part is already small, then bigger teeth leave a thinner rim more susceptible to cracking.
I have a helper who spends much of his time doing Ansys simulations, but I'm not sure how we would simulate the contact between the imperfect male and female parts, and have reliable results. Right now he's studying something that's more straightforward, and that I know will be a productive use of his time.
Whenever published guidelines exist for analyzing something standard, I prefer to use those, while focusing the FEA type analysis on the bespoke/weird areas of the design.