This is a really tough one. All the etiquette books fight over this one all the time. Basically I'm compiled for you the differences they say.
Emily Post-No 100% no. Thats like saying Please send a gift but I really don't want you to actually attend
Ann Landers/Dear Abby-No its rude
Miss Manners-Terribly rude
Many other books (Including Wedding Planning the Easy Way and The Everything Wedding Book) say that the difference between invitations and announcements is that invitations are sent prior and announcements should be sent after the wedding (as in you should probably mail them a day or 2 before the wedding so that it will arrive on the day of or the day after). and that it is perfectly ok but it should be worded in such a way to explain:
This is the annoucement of the wedding of Sarah and Andrew
who were married in a private ceremony with the immediate family on September 19, 2006.
The couple is residing at 709 Springfield Rd.
Most books and manners experts (well that felt it was appropriate to annouce) tend to agree that you need to specify Why they weren't invited (it was a small, private ceremony--even if you did invite 40-50 people). And to include a new residence address.
I know that personally I'd like to know if my sisters friend got married--even though there is no reason I should be invited. But as I send out christmas cards, new names and new addresses are very helpful with that process.
Though to appease the many experts that say it is rude to solicit gifts like that I would place at the bottom
"In lieu of gifts, please make a donation in the couples name to St Jude's Children's Hospital or the Dr. Phil Foundation"
Pick 2 well known charities that mean something to you (I live in a military town so they pick USO; a couple of my friends suggested the SPCA in NYC) That way people don't look at the annoucement as a plea for gifts and money and you help out a charity at the same time.
So I know this is long and I play devil's advocate for both sides but the short of it is--yes many people consider it rude (especially if you are southern) but since so many couples elope or have destination weddings but come back for a reception, I dont see why it is not perfectly acceptable to inform people of the change in status. Though just be careful if you chose not to have the "In lieu of gifts" line, that it might seem greedy. But yes If I were having a small wedding, I would send out annoucements to everyone. But really only if you have a small wedding.
I hope this helps! Good luck. and I'm sure everything will be beautiful.