Dimensional reduction for energies with linear growth involving the bending moment

TL;DR

This paper uses Gamma-convergence to analyze the dimension reduction of 3D variational problems with linear growth, resulting in a nonlinear membrane model influenced by bending moments and involving measures and BV functions.

Contribution

It introduces a novel 3D-2D reduction framework for energies with linear growth that accounts for bending moments and Cosserat vector fields.

Findings

Derivation of a nonlinear membrane model with bending effects

Use of measure and BV space analysis for linear growth energies

Extension of dimension reduction techniques to complex loading scenarios

Abstract

A $\Gamma$-convergence analysis is used to perform a 3D-2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation.