The manuscript offers a methodology to solve an analytical model of a heterogeneous elastic problem for curvilinear layered structures, using the two scales asymptotic homogenization method (AHM). The local problems and the mechanical properties of the local functions were derived. The analytical modeling for the linear elastic problem considering quasi-periodic multi-layered curvilinear composites and the corresponding homogenized problem were obtained. The analytic expression of the effective stress for curvilinear composites is presented. In order to validate the presented model, comparisons with a computational modeling and experimental results for Fibonacci laminated composite and wavy laminated structure are given. The methodology is applied to composites with thickness variation where the effective coefficients were computed and a comparison between the results reported by AHM and numerical analysis given by finite element method (FEM) is presented. Finally, the aorta is studied as a curvilinear laminated shell composite and the above results were used to determinate the effective elastic tensor for healthy and unhealthy aorta using AHM and FEM.