MacBook Pro won't Power up?

Yesterday morning I went to wake up my MacBook Pro (Non- Retina, 13inch mid 2012 model) from sleep to find that it had switched itself off overnight.

When I press the power button there is no response at all, apart from the sleep indicator light blinking once. The power indicator lights still work and show full battery, as does the MagSafe light which appears green when it is plugged in.

I have tried resetting the SMC although this doesn't seem to work. Interestingly it does seem to acknowledge the keyboard sequence as the sleep indicator light doesn't light up when the keys & power button is pressed whilst resetting the SMC. Force restarting doesn't seem to help either.

This actually the second time it's done this, the first being a couple of weeks ago. Then after being off for about 36 hours I tried pressing the power button repeatedly and from there it went into sleep mode. It then went between sleep mode and powers off when I pressed the power button again, before eventually restarting.

Has anyone had any experience of this before and know what is going on?