Can I do a PhD in Mechanical Engineering without experimental work?
I'm an expirmentalist, and I have worked with FE modeling for a long time now. I have many publications based on tests, validated models and purely numerical "without validation". I can tell you this much, the ones that keeps me awake at night and the hardest ones to publish are those with "numerical models without any experimental tests" to back the numerical predictions up or at the least include some sort of validation. 

My MSc (structural engineering) was all numerical (but based on full scale tests done by others), so I did not have to test my own specimens (validated my models against experiments that were already published). I can tell this much, without a validation, your model is just a model! As you know FEA is just an approximation with multiple simplistic assumptions of a real phenomenon. 

However, I tend to agree that whether needing to validate your model against measured data (solely by doing your own experiment or simply against other published work) is case specific. I do not know your "thesis problem" and the fact that you have published a good number of articles (if published in high quality journals) and your advisor was fine with that "until now!" leads me to think that you may be able to get away to a "numerical" thesis. 

Perhaps you can discuss the following ideas with your advisor/committee;


If they insist on an experiment (although they are tight on funding as you have mentioned [not a very good sign in your case since you are trying to avoid that!]), try to find a somehow similar published work, use your model to validate it. I understand that this may not be a "direct validation" but at least your advisor/committee can be assured that your model is capable of simulating a real problem. In other words, your model is "not-directly" more than an FE model.
Ask if you can validate your model with some other ways i.e., analytically, hand calculations, published methods etc. (I hope you have done this by now!). You can always do a parametric study (I assume you did already), pick parameters that can relate to an actual problem (either from literature or industry) and see how your model performs. If possible/applicable, contact an industry firm (that is willing to share) and see if they have anything on hand that you can use to validate the model. 
Discuss with your advisor why did not he ask you to do this long time ago ("usually" one will do an experiment and then develop a model! [that's what I always do, keep in mind this is different that finding a problem and trying to get some funding for in which you will develop a model first to show the feasibility of your problem rather than spending money on an experiment that turns out to be "worthless"]). I understand this is too late to ask now, but if you know the "inside" of whats going on "with your committee (perhaps?)", you will know how to tackle this issue more specifically. 


Remember this, you do not have to prepare a 6 months, $ 100k experiment to validate your model. Sometimes, all what you need is a well organized set-up, few strain gauges, loading-inducing actuators and a specimen (of course this is a simplification - but you get my point).