To tear or rip something (e.g. cloth, paper), creating a split or tear.
彼は古い手紙を読むとき、封を裂いて中身を取り出した。
When he read the old letter, he tore the seal and took out the contents.
不注意でコートがひっかかり、袖が裂いてしまった。
I caught my coat carelessly and ripped the sleeve.
包み紙を裂いて、プレゼントを取り出した。
He tore open the wrapping paper and took out the present.
To cut open or cleave (often with a blade); can refer to cutting open (e.g. an abdomen) in a literal or surgical sense.
外科医は患部を縦に裂いて内部を調べた。
The surgeon made a vertical incision to examine the inside of the affected area.
事故で胸が裂かれたが、命に別条はなかった。
His chest was cut open in the accident, but it wasn't life-threatening.
To forcefully separate or break apart a relationship (e.g. between lovers); to drive a wedge between people.
陰口や噂が二人の信頼を裂いてしまった。
Gossip and rumors drove a wedge between the two and destroyed their trust.
彼の裏切りが家族の絆を裂いた。
His betrayal tore apart the family's bonds.
To spare or allocate (time, money, attention, etc.). Usually written with the kanji 割く for this nuance.
忙しい中でも彼女は毎朝30分を読書に割いている。
Even though she's busy, she spares 30 minutes every morning for reading.
プロジェクトのために予算を割く必要がある。
We need to allocate budget for the project.
来週の会議のために時間を割いていただき、ありがとうございます。
Thank you for taking the time for next week's meeting.
An archaic usage meaning to have a tattoo at the corner of the eye (historical; now obsolete).
古い記録には、遊女が目の端に入れ墨を裂くと書かれている（古語）。
Old records state that courtesans had tattoos at the corners of their eyes (archaic usage).