rut
Pare (a homophone of pear, not just a rhyming word) means to trim or cut away, and tear might work as well.
Amputate. Truncate.
butt
The word that means cut into two parts is bisect. The root of the word comes from "bi" meaning two and "secare" meaning to cut.
"Bisect" is a word that means cut into two halves which starts with bi.
A trench in the earth made by, or as by, a plow., Any trench, channel, or groove, as in wood or metal; a wrinkle on the face; as, the furrows of age., To cut a furrow in; to make furrows in; to plow; as, to furrow the ground or sea., To mark with channels or with wrinkles.
There are no three letter words that rhyme with art.
Bisect.
slice
slash
valley
The word that has a shell and rhymes with "cut" is "nut." A nut is a hard-shelled fruit or seed that typically contains an edible kernel. Nuts are a good source of healthy fats, protein, and various vitamins and minerals.