Removing ground Ivy without chemicals

I have recently moved into a house with a large covered strawberry patch. Its not been tended for some time. Im ok removing the brambles and willow, but has anyone got any idea how to relatively easily remove the ground ivy and there is at least 30sq metres of it? MANY THANKS

 

ground ivy or ground elder?

As its an old bed and its recommended I think to replace plants every 3 - 4 years I think you should kill it all off, dig it up and start again. But not the answer you wanted. It sounds as if the bed is full of weeds.