Mary, not a non-chemical method, but one which will kill the vine without damaging the rhubarb: Better done in the fall but may still work now-with lopping shears, small saw, etc, cut vine stems near ground. Immediately apply high-concentration glyphosate, ie 40% solution, on cut surfaces. This requires a tiny amount of chemical and offers near-complete control of what plants get that herbicide. The reason it works best in fall/winter is that at those times of year, sap in plants is tending to get pulled down into the roots, so this aids the applicator in directing that chemical to where it will kill entire plant. we use this methodology all the time to control invasive woodland shrubs like buckthorn and non-native honeysuckles. It really works well.

+oM

Wisconsitom, thank you for your response. I will write that down for future use. Have the vines torn/cut down and now need to chop away at the stumps! Thanks again!

Just remember-if you elect to use this method-to apply the chemical to the cut end immediately after cutting. Otherwise, the plant will form a sort of callus layer and the herbicide will not be absorbed so well. It really takes a tiny amount of product to do this. there are also non-glyphosate materials which will do as or better of a job at this. Garlon is one well-known chemical and it has numerous generic equivalents. Brush B Gone is one such item.

+oM

I have been successful in getting rid of a Trumpet Vine by cutting it, drilling holes, vertically, into the freshly cut stump, and filling the holes with herbicide. All this in an area where I am busy propagating many plants. The poison, not being sprayed, only affected the Trumpet Vine, to my knowledge, as it died and all else seemed normal.