Today I tried putting a piece
of nylon rope
where the trimmer
line usually goes.
It worked pretty good till it
got frayed, and it still kept cutting, but not as fierce.
Maybe soaking the rope in
some sort of adhesive would extend the amount of cutting each piece can
do before it needs replacing?
Try melting the free end of the nylon string together before you put it in the trimmer. That should stop it from coming apart.
Adhesives probably won't work well because nylon doesn't adhere well. In composites manufacturing nylon films are often used as vacuum or autoclave bagging films because they don't stick to most kinds of resins.
To make adhesives stick to nylon, the nylon has to be given a flame or corona treatment. And the effects of those treatments generally don't last long.
While crimp rings are often used on steel cable, I haven't seen them used on cables made from nylon or other plastics. There are at least two reasons I can think of. First, thermoplastics have the tendency to flow under a constant load. This is called creep. I would guess that a crimp ring around a nylon cable would come loose in time. Second, a metal crimp ring would create a local increase in stiffness in the cable. Such combinations of stiff and flexible materials are often failure points.
Steel cable might be a bit too heavy. And I wonder if steel would be flexible enough. I would guess that strands might break on impact. Do a small test on a piece of plastic like the safety cover. Anything that can chew up the safety cover shouldn't be used.
It could well be that the ring might fly off the line when using the mower. Due to creep the line material will deform and the friction between the ring and the line will lessen over time.
But as to failure mode, I would expect the outer strands to break at the interface between the ring and the line. In cases like these (dissimilar materials) much depends on the characteristics of the materials in question and the details of the interface; like if the edge of the ring where it meets the line is sharp or rounded. If the edge of the ring is sharp, it is like you're cutting into the line with a knife.
Finding the points with the highest stresses in a structure given its intended loads is an imported part of engineering calcuations and analysis. Given that failures to do this properly can be costly and even lethal, there are experience-based standards for e.g. constructing houses.
In general engineering we set the nominal load that a part is designed for, and then multiply that with a safety factor of three. The part is then designed so that the maximum occurring stress in in case of the triple load does not exceed whatever is allowable for the material and environment conditions. For heavy duty equipment like cranes, offshore platforms et cetera a safety factor of ten is often used.