Homogenized moderately wrinkled shell theory from 3D Koiter's linear elasticity

In this paper we derive, by two$-$scale convergence, periodically wrinked shell models starting from three dimensional linear elasticity, depending of the behaviour of the small parameter $\epsilon >0$ and $p>1$, differents theories appear. We assume that the mid-surface of the shell is given by ${\displaystyle \psi (x_1,x_2)+{\epsilon }^p\theta (\frac{x_1}{\epsilon },\frac{x_2}{\epsilon })\text{vect}{a}_3(x_1,x_2)}$, where $\theta$ is $[0,1{)}^2$-periodic function and $p=2$. We also assume that the strain energy of the shell has the Koiter’s model.