Relaxation and Γ-Convergence of the Energy Functionals of a Two-Phase Elastic Medium

2005 ◽  
Vol 128 (5) ◽  
pp. 3177-3194
Author(s):  
A. V. Dem’yanov
Keyword(s):  
Elastic Medium ◽  
Two Phase ◽  
Energy Functionals ◽  
Γ Convergence

A justification of the Timoshenko beam model through Γ-convergence

2017 ◽  
Vol 15 (02) ◽  
pp. 261-277 ◽  
Author(s):  
Lior Falach ◽  
Roberto Paroni ◽  
Paolo Podio-Guidugli
Keyword(s):  
Linear Elasticity ◽  
Timoshenko Beam ◽  
Three Dimensional ◽  
Convergence Theory ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Energy Functionals ◽  
Γ Convergence ◽  
Suitable Sequence

We validate the Timoshenko beam model as an approximation of the linear-elasticity model of a three-dimensional beam-like body. Our validation is achieved within the framework of [Formula: see text]-convergence theory, in two steps: firstly, we construct a suitable sequence of energy functionals; secondly, we show that this sequence [Formula: see text]-converges to a functional representing the energy of a Timoshenko beam.

On micro–macro-models for two-phase flow with dilute polymeric solutions — modeling and analysis

2016 ◽  
Vol 26 (05) ◽  
pp. 823-866 ◽  
Author(s):  
G. Grün ◽  
S. Metzger
Keyword(s):  
Two Phase Flow ◽  
Type Equation ◽  
Momentum Equation ◽  
Diffuse Interface ◽  
Phase Flow ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Energy Functionals ◽  
Spring Force

By methods from nonequilibrium thermodynamics, we derive a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymers. The polymers are modeled by dumbbells subjected to general elastic spring-force potentials including in particular Hookean and finitely extensible, nonlinear elastic (FENE) potentials. Their density and orientation are described by a Fokker–Planck-type equation which is coupled to a Cahn–Hilliard and a momentum equation for phase-field and gross velocity/pressure. Henry-type energy functionals are used to describe different solubility properties of the polymers in the different phases or at the liquid–liquid interface. Taking advantage of the underlying energetic/entropic structure of the system, we prove existence of a weak solution globally in time for the case of FENE-potentials. As a by-product in the case of Hookean spring potentials, we derive a macroscopic diffuse-interface model for two-phase flow of Oldroyd-B-type liquids.

scholarly journals Plate theory as the variational limit of the complementary energy functionals of inhomogeneous anisotropic linearly elastic bodies

2017 ◽  
Vol 23 (8) ◽  
pp. 1119-1139
Author(s):  
François Murat ◽  
Roberto Paroni
Keyword(s):  
Dual Problem ◽  
Plate Theory ◽  
Energy Functionals ◽  
Elastic Bodies ◽  
Hyper Elastic ◽  
Cylindrical Region ◽  
Anisotropic Bodies ◽  
Γ Convergence ◽  
Variational Limit

We consider a sequence of linear hyper-elastic, inhomogeneous and fully anisotropic bodies in a reference configuration occupying a cylindrical region of height [Formula: see text]. We study, by means of Γ-convergence, the asymptotic behavior as [Formula: see text] goes to zero of the sequence of complementary energies. The limit functional is identified as a dual problem for a two-dimensional plate. Our approach gives a direct characterization of the convergence of the equilibrating stress fields.

scholarly journals Incompressible limit for a magnetostrictive energy functional

2013 ◽  
Vol 61 (4) ◽  
pp. 1025-1030
Author(s):  
B. Gambin ◽  
W. Bielski
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Elastic Behaviour ◽  
Incompressible Limit ◽  
Elastic Deformations ◽  
Energy Functionals ◽  
Second Gradient ◽  
Non Local ◽  
Γ Convergence ◽  
Non Locality

Abstract The modern materials undergoing large elastic deformations and exhibiting strong magnetostrictive effect are modelled here by free energy functionals for nonlinear and non-local magnetoelastic behaviour. The aim of this work is to prove a new theorem which claims that a sequence of free energy functionals of slightly compressible magnetostrictive materials with a non-local elastic behaviour, converges to an energy functional of a nearly incompressible magnetostrictive material. This convergence is referred to as a Γ -convergence. The non-locality is limited to non-local elastic behaviour which is modelled by a term containing the second gradient of deformation in the energy functional.

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