Abstract: This work aims to derive an e ective model of the Laplacian with a metric in thin curved strips with the Neumann boundary condition. Firstly, the Neumann Laplace operator with a non-homogeneous failure will be defined as a self-adjoint operator on the Hilbert space by an associated quadratic form. Furthermore, this work shows the convergence of this operator to the one-dimensional e ective model in the spectral, strong resolvent, even in the norm-resolvent sense, all of which are illustrated with a concrete example. Finally, the rate of the convergence is derived.