I think you can use Inkscape (vector svg files) to skew the text shapes to an arc as in your image and then export as png file into Gimp to texture, or just do the effects all in Inkscape. Select the leter shape using the Edit Paths by Nodes tool > Object > Transform > Skew tab. Skew each letter individually, but first it best to draw a guide ;)

Inkscape is free (vector) - so too is Gimp (raster).

I roll with Adobe products, but I do have Inkscape loaded on one machine and skewed some text but didn't distress any (looked at a few Filters though). Although I did notice that Inkscape has a D-map (displacement map) function. It's found via Filter > Filter Editor > Add Effects > select Displacement Map from the drop down. D-maps are awesome.

As for exporting your most awesome art work, my advice is to talk to your t-shirt guy. Some will work with .svg files and some not to happy about that. My experience is Adobe .ai files are well received.

Another option is to simply download a 30 day trail of Adobe Illustrator or Photoshop and skew the text as mentioned above (Skew). Photoshop will give a broader distress ability (including negative texturing) and you could also download machine wash filters for both. Great ones are to be found at Mister Retro and there's tons of tutes on the interwebz for those.

Good luck! Hope that helps :)