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.

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.

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 A hierarchy of multilayered plate models

2020 ◽  
Author(s):  
Miguel de Benito Delgado ◽  
Bernd Schmidt
Keyword(s):  
Nonlinear Elasticity ◽  
Curvature Tensor ◽  
Elastic Constants ◽  
Three Dimensional ◽  
Spontaneous Curvature ◽  
Multilayered Plate ◽  
Von Karman ◽  
Plate Models ◽  
Γ Convergence ◽  
Von Kármán

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of Γ-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the Γ-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von Kármán, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.

scholarly journals Asymptotic variational analysis of incompressible elastic strings

2020 ◽  
pp. 1-28
Author(s):  
Dominik Engl ◽  
Carolin Kreisbeck
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Differential Constraint ◽  
Energy Functionals ◽  
Elastic Strings ◽  
Non Linear ◽  
Applied Forces ◽  
Linear Differential ◽  
Γ Convergence ◽  
External Forces

Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

scholarly journals Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets

2019 ◽  
Vol 25 ◽  
pp. 24 ◽  
Author(s):  
Virginia Agostiniani ◽  
Alessandro Lucantonio ◽  
Danka Lučić
Keyword(s):  
Curvature Tensor ◽  
Plate Theory ◽  
Finite Elasticity ◽  
Thin Sheet ◽  
Three Dimensional ◽  
Elastic Plates ◽  
Three Dimensional Model ◽  
Spontaneous Strain ◽  
Piecewise Constant ◽  
Γ Convergence

We rigorously derive a Kirchhoff plate theory, via Γ-convergence, from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables x′. At the same time, the spontaneous strain is h-close to the identity, where h is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into ℝ3, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from an x′-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.

Count and Symmetry of Global and Local Minimizers of the Cahn-Hilliard Energy Over Cylindrical Domains

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Arnaldo Simal do Nascimento ◽  
João Biesdorf ◽  
Janete Crema
Keyword(s):  
Three Dimensional ◽  
Unique Continuation ◽  
Energy Functional ◽  
Circular Cylinders ◽  
Cylindrical Domain ◽  
Local Minimizers ◽  
Global Minimizers ◽  
Cylindrical Domains ◽  
Γ Convergence ◽  
Global And Local

AbstractWe address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion.The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

A JUSTIFICATION OF THE REISSNER–MINDLIN PLATE THEORY THROUGH VARIATIONAL CONVERGENCE

2007 ◽  
Vol 05 (02) ◽  
pp. 165-182 ◽  
Author(s):  
ROBERTO PARONI ◽  
PAOLO PODIO-GUIDUGLI ◽  
GIUSEPPE TOMASSETTI
Keyword(s):  
Plate Theory ◽  
Scaling Law ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Energy Functional ◽  
Mindlin Plate ◽  
Variational Convergence ◽  
Mindlin Plate Theory ◽  
Γ Convergence ◽  
Three Dimensional Elasticity

We provide a justification of the Reissner–Mindlin plate theory, using linear three-dimensional elasticity as framework and Γ-convergence as technical tool. Essential to our developments is the selection of a transversely isotropic material class whose stored energy depends on (first and) second gradients of the displacement field. Our choices of a candidate Γ-limit and a scaling law of the basic energy functional in terms of a thinness parameter are guided by mechanical and formal arguments that our variational convergence theorem is meant to validate mathematically.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals Internal constraints and arrested relaxation in main-chain nematic elastomers

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Takuya Ohzono ◽  
Kaoru Katoh ◽  
Hiroyuki Minamikawa ◽  
Mohand O. Saed ◽  
Eugene M. Terentjev
Keyword(s):  
Mechanical Properties ◽  
Main Chain ◽  
Polymer Networks ◽  
Nematic Order ◽  
Nematic Elastomers ◽  
Nematic Director ◽  
Highly Nonlinear ◽  
Nonlinear Mechanical ◽  
Liquid Crystal Elastomers ◽  
Memory Applications

AbstractNematic liquid crystal elastomers (N-LCE) exhibit intriguing mechanical properties, such as reversible actuation and soft elasticity, which manifests as a wide plateau of low nearly-constant stress upon stretching. N-LCE also have a characteristically slow stress relaxation, which sometimes prevents their shape recovery. To understand how the inherent nematic order retards and arrests the equilibration, here we examine hysteretic stress-strain characteristics in a series of specifically designed main-chain N-LCE, investigating both macroscopic mechanical properties and the microscopic nematic director distribution under applied strains. The hysteretic features are attributed to the dynamics of thermodynamically unfavoured hairpins, the sharp folds on anisotropic polymer strands, the creation and transition of which are restricted by the nematic order. These findings provide a new avenue for tuning the hysteretic nature of N-LCE at both macro- and microscopic levels via different designs of polymer networks, toward materials with highly nonlinear mechanical properties and shape-memory applications.

scholarly journals From nonlinear to linearized elasticity via Γ-convergence: The case of multiwell energies satisfying weak coercivity conditions

2014 ◽  
Vol 25 (01) ◽  
pp. 1-38 ◽  
Author(s):  
V. Agostiniani ◽  
T. Blass ◽  
K. Koumatos
Keyword(s):  
Coercivity Conditions ◽  
Coercivity Condition ◽  
Nematic Elastomers ◽  
Linearized Elasticity ◽  
Γ Convergence

Linearized elasticity models are derived, via Γ-convergence, from suitably rescaled nonlinear energies when the corresponding energy densities have a multiwell structure and satisfy a weak coercivity condition, in the sense that the typical quadratic bound from below is replaced by a weaker p bound, 1 < p < 2, away from the wells. This study is motivated by, and our results are applied to, energies arising in the modeling of nematic elastomers.

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