scholarly journals Damage-driven fracture with low-order potentials: asymptotic behavior, existence and applications

2019 ◽  
Vol 53 (4) ◽  
pp. 1305-1350 ◽  
Author(s):  
Marco Caroccia ◽  
Nicolas Van Goethem
Keyword(s):  
Asymptotic Behavior ◽  
Fracture Energy ◽  
Existence Results ◽  
Energy Functionals ◽  
Plastic Slip ◽  
Nonlinear Potentials ◽  
Γ Convergence ◽  
Physical Phenomena ◽  
Low Order

We study the Γ-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinematical constraints such as crack non-interpenetration. Existence results are also addressed.

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.

Relaxation of Ginzburg–Landau functional perturbed by continuous nonlinear lower-order term in one dimension

2014 ◽  
Vol 13 (01) ◽  
pp. 101-123 ◽  
Author(s):  
Andrija Raguž
Keyword(s):  
Asymptotic Behavior ◽  
Lower Order ◽  
Order Term ◽  
Lower Order Term ◽  
One Dimension ◽  
Young Measures ◽  
Ginzburg Landau ◽  
Carathéodory Function ◽  
Γ Convergence

We study the asymptotic behavior as ε → 0 of the Ginzburg–Landau functional [Formula: see text], where A(s, v, v′) is the nonlinear lower-order term generated by certain Carathéodory function a : (0, 1)2 × R2 → R. We obtain Γ-convergence for the rescaled functionals [Formula: see text] as ε → 0 by using the notion of Young measures on micropatterns, which was introduced in 2001 by Alberti and Müller. We prove that for ε ≈ 0 the minimal value of [Formula: see text] is close to [Formula: see text], where A∞(s) : = ½A(s, 0, -1) + ½A(s, 0, 1) and where E0 depends only on W. Further, we use this example to establish some general conclusions related to the approach of Alberti and Müller.

GINZBURG–LANDAU VORTEX LINES AND THE ELASTICA FUNCTIONAL

2009 ◽  
Vol 11 (01) ◽  
pp. 71-107
Author(s):  
ROGER MOSER
Keyword(s):  
Asymptotic Behavior ◽  
Second Order ◽  
Ginzburg Landau ◽  
Vortex Lines ◽  
Γ Convergence ◽  
Generalized Curves

We examine the asymptotic behavior of a family of second-order functionals arising in the theory of Ginzburg–Landau vortices. The results point towards Γ-convergence with the elastica functional for generalized curves as the limit.

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.

HOMOGENIZATION OF ENERGIES DEFINED ON PAIRS SET-FUNCTION

2009 ◽  
Vol 11 (03) ◽  
pp. 459-479 ◽  
Author(s):  
MARGHERITA SOLCI
Keyword(s):  
Asymptotic Behavior ◽  
Energy Density ◽  
Convergence Result ◽  
Energy Scale ◽  
Surface Energy Density ◽  
Surface Energies ◽  
Open Set ◽  
Γ Convergence ◽  
Regular Open ◽  
Set Function

In the present work, we deal with the problem of the asymptotic behavior of a sequence of non-homogeneous energies depending on a pair set-function of the form [Formula: see text] with u ∈ H1(Ω), E regular open set and the energy densities f and φ both 1-periodic in the first variable; this leads, in the Γ-limit, to a problem of homogenization. We prove a Γ-convergence result for the sequence {Fε}, showing that there is no interaction between the homogenized bulk and surface energy density; that is, even though the effect of the bulk and surface energies are at the same energy scale, oscillations in the bulk term can be neglected close to the surfaces ∂*E and S(u), where surface oscillations are dominant.

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.

scholarly journals On the Existence of Positive Solutions of a Nonlinear Differential Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Sonia Ben Othman ◽  
Habib Mâagli ◽  
Noureddine Zeddini
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Nonlinear Equation ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonnegative Function ◽  
Existence Results ◽  
Theory Approach ◽  
Existence Of Positive Solutions

We study some existence results for the nonlinear equation(1/A)(Au')'=uψ(x,u)forx∈(0,ω)with different boundary conditions, whereω∈(0,∞],Ais a continuous function on[0,ω), positive and differentiable on(0,ω),andψis a nonnegative function on(0,ω)×[0,∞)such thatt↦tψ(x,t)is continuous on[0,∞)for eachx∈(0,ω). We give asymptotic behavior for positive solutions using a potential theory approach.

scholarly journals Rigorous derivation of active plate models for thin sheets of nematic elastomers

2017 ◽  
Vol 25 (10) ◽  
pp. 1804-1830 ◽  
Author(s):  
Virginia Agostiniani ◽  
Antonio DeSimone
Keyword(s):  
Finite Elasticity ◽  
Three Dimensional ◽  
Energy Functional ◽  
Variable Degree ◽  
Energy Functionals ◽  
Nematic Order ◽  
Reduction Techniques ◽  
Nematic Elastomers ◽  
Plate Models ◽  
Γ Convergence

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension reduction techniques, based on the theory of Γ-convergence. The two-dimensional models are non-linear plate theories, in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. Three nematic textures are considered: splay-bend and twisted orientations of the nematic director, and a uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable model with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.

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