  We prove the two-scale transformation method which allows rigorous
homogenisation of problems defined on locally periodic domains by
transformation on periodic domains. The idea to consider periodic substitute
problems was originally proposed by M. A. Peter for the homogenisation on
evolving microstructure and is applied in several works. However, only the
homogenisation of the periodic substitute problems was proven, whereas the
method itself was just postulated (i.e. the equivalence to the homogenisation
of the actual problem had to be assumed). In this work, we develop this idea
further and formulate a rigorous two-scale convergence concept for microscopic
transformation to prove this method. Moreover, we show a new two-scale
transformation rule for gradients which allows to derive new limit problems
that are now transformationally independent.
