The name is very, very rare in Germany.
Its two elements are:
TRAUT: originally Germanic *thruthi (power, sreength), later interpreted as "traut" (familiar, trusted)
LIEB: from Old High
German "liob" (friendly, beloved); with Jewish names sometimes from
Levi or Löb (O umlaut), "lion".
The HDV (Historisches Deutsches Vornamenbuch, W.Seibicke) only has TRAUTLIEBE (f.) and says it is a modern coinage. His only reference for LIEBTRAUD is from the 20th century.
Andy ;—)