What creates the shade? Trees or buildings? And what will you be using the area for - you say turf substitute - does that mean children playing on it or just that you want something that looks like grass? If you have children playing on it then I hate to suggest that you might do better with AstroTurf than with trying to find suitable planting.
I can't think of anything that meets your spec as things are - could be different if you can reduce the shade (cutting back hedges/trees) or improving the soil.

Thank you, Urbanite. The shade comes from the canopy of a huge Deodar (Indian Cedar) whose trunk is 3 m. away !! The grass grows normally once free of the canopy, and continues up the garden for 10m. It's not exactly a lawn, as there's a lot of clover daisies and dandelions, and children do play on it.
Astro Turf would be a good idea if the grass did not continue up one side of the garden, but it's difficult to see how one could join Astro Turf to the natural grass area. I suppose one could put a line of buried side-on bricks to divide the two areas, as one does for lawn edging, but I'm not sure how that would look across the width of the 'lawn'. I'll certainly think about how to make it work, and thank you very much for your comments.
Nakinaki

Could you perhaps plant something at the junction? Admittedly this would interrupt the free area for the children to play on but a border of small tough possibly evergreen shrubs might do the trick, and the kids might even enjoy having two separate areas.