Maybe, maybe not...it'll all depend on the resulting design matrix of the "happenstance" data as to whether it will turn out to be well balanced or not and then how smooth the data are as a function of those variables.
A specific issue for example is that the "one at a time" technique may not have caused data to be generated in appropriate places to be able to estimate coefficients of interaction terms. Whether that is or is not significant will depend on whether there are real effects or not.
As is almost always the case, the time to have talked to a statistician was before you collected all the data; he/she could have advised you on experimental design to optimize the data collection for the purpose including, perhaps, some preliminary results to test for such things as interaction to aid in that process.
You can, of course, simply throw the data at a general regression model and see what happens; maybe you'll get lucky.
The first thing to do, however, will be to use the Matlab graphics features and visualize the data you do have and see what it looks like to get some idea of what sort of models might be suitable.
As always, there are surely other studies of the problem; what kind of models have they used for the purpose? One can imagine if you're testing to anything near failure the results are not terribly linear...
On the problem of regression in general again
Draper and Smith, Applied Regression Analysis
on response surface modeling and something about experiment design specifically for the purpose,
RH Myers, Response Surface Methodology
is an excellent introduduction/medium level exposition.
For things to think of before "just collecting data", I recommend
Box, Hunter and Hunter, Statistics for Experimenters
I particularly commend to your attention the section "Hazards of Fitting Happenstance Data" in Chapter 14. Your data aren't actually entirely "happenstance" owing to the one-at-a-time collection but the potential problems can be similar because there may be reasons from a physical basis that some of the variables aren't truly independent owing to how they effect each other in the design of the flange and its resultant behavior.
