If there's evidence of damage, Vodafone won't be able to carry out a warranty repair for you, so you'll need to go direct to Sony. If the problem is clearly nothing to do with the cracks, they may look kindly on you.
Before you do this, there are some things you could try. The first is a soak (24 hour) charge. If that doesn't work, try starting the phone with it plugged in. If this works, but not permanently, then I'd suspect a battery problem. I think Sony still guarantee them for two years so, if yours is less old than that, you may be in luck. Remember that allowing lithium-based batteries to go flat will considerably shorten the life (if you've been doing that).
The other thing is to locate the reset button, which is under the SIM card flap and may be red, yellow or orange. It's very small and you'll need to press it with something like the point of a pencil. I don't know whether it needs a quick press or a long one, so try both.
