Is having sex with a dead body classed as rape?

No, that is NECROPHILIA (http://en.wikipedia.org/wiki/Necrophilia) and Illegal in it's own right.

But the definition of rape is to have sex with somebody without their consent, if the person is dead then they can't give their consent so surely as well as being necrophilia it would also be rape

No, as it's one or the other.
How can a dead body possibly give consent.
The fact they are dead makes the difference to the crime.

But the definition of rape is to have sex with somebody without their consent, ...

A dead body isn't a "somebody". It's a thing which used to be somebody. But it isn't anymore.

But the definition of rape is to have sex with somebody without their consent, if the person is dead then they can't give their consent so surely as well as being necrophilia it would also be rape

So would it be rape if you had sex with a tomato? After all, the tomato can't give its consent.

A tomato can give consent about as well as a dead person. The fact that no dead person can give consent to sex is the very thing that separates this from rape.

Do you see the distinction?

Okay but what about a person that is 'brain dead' or unconscious, only being kept alive by a machine, their mind is practically dead so would that not be rape either?

Now you are simply being ridiculous.

I'm being logical

No, you are not being logical. Saying someone is "practically dead" is like saying someone is "sort of" pregnant. You either are or you are not. Do you think people on life support are "practically dead?"

What are you asking? Brain dead or physically dead?

If the person is LEGALLY ALIVE (and that includes brain dead people who are LEGALLY ALIVE) sex with that person is rape.

If the person is LEGALLY DEAD the criminal charge varies from State to State - in some States it's a misdemeanor, others it's a felony. Some States have very recently (within the past 4-5 years) adopted laws covering this very subject because in the past it has not been addressed by the law.

There are cases in many States which involve physical assault of an unresponsive person - those are the brain dead people who are assaulted and/or raped. The perpetrators are charged with rape - pure and simple rape.