How valid is it to assume the initial crack size for fatigue crack growth analysis as equal to the size of the smallest element in the critical region of the finite element model?
I would appreciate any references and papers regarding this .
I haven't heard anything as you explained in your question. As far as I know, you can define the crack size based on the criteria you define (it can be based on an experiment or any hypothesis for crack growth), then refine the mesh size accordingly on the crack location. Basically, you should have an idea of how much would be the crack growth at it each iteration and adjust the mesh size according to that. In XFEM method, software does this on its own since it uses adaptive mesh control algorithm.
I haven't seen any work using that approach. As you may know, the element size around crack should be small enough to capture the stress gradient field. If you have performed a mesh sensitivity analysis, you can rely on the model.
Just keep in mind that initial crack size in your case is only an auxiliary way to start the crack propagation and it shouldn't affect the global crack growth path as well as the total FCG life. As Hamidreza have correctly stated, what's important is the crack increment size for each iteration.
Are you attempting to do a simulation just to see if the simulation works or are you trying to model a real scenario? In the first case, I have no opinion but others have offered some suggestions already.
In the second case, "it's complicated." Damage tolerance analysis of aerospace parts is based on assumed flaw sizes which are a function of the material, geometry, inspection method, and inspection process capability. I've seen inspection systems that have trouble reliably finding cracks that are 0.200 inch surface length and have heard reports of systems capable of finding 0.010 inch surface length cracks.
For more info, Google "ENSIP crack size." The third link has a lot on NDT, flaw sizes, and looks like it also has some stuff on FEA simulations. There is also probably quite a bit on the subject if you look for proceedings of ASIP conferences.