There is nothing I am aware of that you could use that would be less expensive then either the steel or aluminum edging and still do the job. If that grass you wish to keep from moveing in is either Quack Grass or Johnston Grass that barrier needs to go much deeper then the usual 4 inches most edging is sold as.

I,ve used this in the past under fences and thinking about it again for some new areas it works and can easily be kept up with a little weed killer ever few months. Buy or collect leftover roof shingles and lay under the fence line. 2nd,s of course are cheaper, which could just be a busted open package. Lays flat, mow on edge, weed killer once in awhile for anything creeping over. Any bad comments about using this anyone else.

These asphalt based materials may keep top growth from happening but since the will not stop the roots, and instead will create an environment the grass roots find real nice and will then promote root growth, that would not be a barrier to grass growing into a planting bed.