Linear Models of a Stiffened Plate via $\Gamma$-convergence

TL;DR

This paper uses $$-convergence to analyze the asymptotic behavior of scaled 3D stiffened plates, deriving 23 limit models that reveal how the stiffener influences the plate's elastic response as size diminishes.

Contribution

It introduces a $$-convergence framework to derive multiple limit models for scaled stiffened plates, highlighting the effects of different scaling regimes on the asymptotic behavior.

Findings

Derivation of 23 limit problems depending on scaling regimes

Identification of the interplay between plate and stiffener effects

Asymptotic models for various relative stiffness configurations

Abstract

We consider a family of three-dimensional stiffened plates whose dimensions are scaled through different powers of a small parameter $\varepsilon$. The plate and the stiffener are assumed to be linearly elastic, isotropic, and homogeneous. By means of $\Gamma$-convergence, we study the asymptotic behavior of the three-dimensional problems as the parameter $\varepsilon$ tends to zero. For different relative values of the powers of the parameter $\varepsilon$, we show how the interplay between the plate and the stiffener affects the limit energy. We derive twenty-three limit problems.