Homogenized moderately wrinkled shell theory from 3D Koiter's linear elasticity
Pedro Hernández-Llanos, Rajesh Mahadevan, Ravi Prakash
TL;DR
This work derives a rigorous homogenization framework for periodically wrinkled shells within 3D linear elasticity by applying two-scale convergence in the moderately wrinkled regime ($p=2$). The macro-scale behavior is captured by a Koiter-type shell equation, augmented with micro-scale corrections that arise from the periodic wrinkles, while the micro-scale is characterized through well-posed local (cell) problems that determine effective coupling through $M^y_{αβ}$–type terms. The result is a decoupled system: a macroscopic variational problem for the displacement $u^0$ coupled to a set of microscopic cell problems that supply the effective properties, enabling computation of homogenized shell behavior under wrinkle-induced heterogeneity. Overall, the paper provides a rigorous derivation of the homogenized shell model with both membrane and bending (Koiter) responses influenced by surface microstructures, offering a pathway to compute effective properties for design and analysis of corrugated shells.
Abstract
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 $\varepsilon>0$ and $p>1$, differents theories appear. We assume that the mid-surface of the shell is given by $\displaystyle ψ(x_1,x_2)+\varepsilon^pθ\left(\frac{x_1}{\varepsilon},\frac{x_2}{\varepsilon}\right)\vect{a}_{3}(x_1,x_2)$, where $θ$ is $[0,1)^2$-periodic function and $p=2$. We also assume that the strain energy of the shell has the Koiter's model.