# Derivation of a homogenized von Kármán shell theory

@inproceedings{Hornung2012DerivationOA,
title={Derivation of a homogenized von K{\'a}rm{\'a}n shell theory},
author={Peter Hornung and Igor Vel{\vc}i{\'c}},
year={2012}
}
• Published 31 October 2012
• Mathematics, Physics
We derive the model of homogenized von K\'arm\'an shell theory, starting from three dimensional nonlinear elasticity. The original three dimensional model contains two small parameters: the oscillations of the material $\e$ and the thickness of the shell $h$. Depending on the asymptotic ratio of these two parameters, we obtain different asymptotic theories. In the case $h\ll\e$ we identify two different asymptotic theories, depending on the ratio of $h$ and $\e^2$. In the case of convex shells…
13 Citations
On the derivation of homogenized bending plate model
We derive, via simultaneous homogenization and dimension reduction, the $$\Gamma$$Γ-limit for thin elastic plates of thickness $$h$$h whose energy density oscillates on a scale $$\varepsilon Derivation of a homogenized nonlinear plate theory from 3d elasticity • Mathematics, Physics • 2012 We derive, via simultaneous homogenization and dimension reduction, the$$\Gamma $$Γ-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much Non-periodic homogenization of bending–torsion theory for inextensible rods from 3D elasticity • Mathematics • 2016 We derive, by means of$$\Gamma $$Γ-convergence, the equations of homogenized bending rod starting from 3D nonlinear elasticity equations. The main assumption is that the energy behaves like Homogenization of the nonlinear bending theory for plates • Mathematics, Physics • 2013 We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of$$\Gamma $$Γ-convergence. In contrast to what one naturally would On the general homogenization of von Karman plate equations from 3D nonlinear elasticity Starting from 3D elasticity equations we derive the model of the homogenized von K\'arm\'an plate by means of \Gamma-convergence. This generalizes the recent results, where the material Homogenization of bending theory for plates; the case of elastic laminates • Physics, Mathematics • 2014 In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the direction of Homogenization of bending theory for plates; the case of oscillations in the direction of thickness • Mathematics • 2015 In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the direction of Derivation of a Homogenized Bending–Torsion Theory for Rods with Micro-Heterogeneous Prestrain • Physics, Mathematics • 2019 In this paper we investigate rods made of nonlinearly elastic, composite–materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to General homogenization of bending theory for plates from 3D elasticity; the case of elastic laminates • Materials Science • 2014 In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the thickness direction on On effective material parameters of thin perforated shells under static loading • Mathematics • 2020 Abstract One of the defining properties of thin shell problems is that the solution can be viewed as a linear combination of local features, each with its own characteristic thickness-dependent ## References SHOWING 1-10 OF 75 REFERENCES On the derivation of homogenized bending plate model We derive, via simultaneous homogenization and dimension reduction, the$$\Gamma $$Γ-limit for thin elastic plates of thickness$$h$$h whose energy density oscillates on a scale$$\varepsilon
DERIVATION OF A HOMOGENIZED VON-KÁRMÁN PLATE THEORY FROM 3D NONLINEAR ELASTICITY
• Mathematics
• 2013
We rigorously derive a homogenized von-Karman plate theory as a Γ-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an
The membrane shell model in nonlinear elasticity: A variational asymptotic derivation
• 1996
SummaryWe consider a shell-like three-dimensional nonlinearly hyperelastic body and we let its thickness go to zero. We show, under appropriate hypotheses on the applied loads, that the deformations
Derivation of a homogenized nonlinear plate theory from 3d elasticity
• Mathematics, Physics
• 2012
We derive, via simultaneous homogenization and dimension reduction, the $$\Gamma$$Γ-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much
A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity
• Physics, Mathematics
• 2002
The energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a 0-limit of
Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity
• Mathematics, Physics
• 2008
We discuss the limiting behavior (using the notion of Γ-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the
A nonlinear model for inextensible rods as a low energy Γ-limit of three-dimensional nonlinear elasticity
• Mathematics
• 2004
Abstract Using a variational approach we rigorously deduce a nonlinear model for inextensible rods from three-dimensional nonlinear elasticity, passing to the limit as the diameter of the rod goes to
Homogenization of linear elastic shells
Homogenization techniques were used by Duvaut (1976,1978) in asymptotic analyse of 3-dimensional periodic continuum problems and periodic von Kármán plates.In this paper we homogenize
Homogenization, linearization and dimension reduction in elasticity with variational methods
The objective of this thesis is the derivation of effective theories for thin elastic bodies with periodic microstructure. The main result is the rigorous, ansatz free derivation of a homogenized
Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence
• Mathematics, Physics
• 2002
Abstract We show that the nonlinear bending theory of shells arises as a Γ -limit of three-dimensional nonlinear elasticity. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I