scholarly journals Geometric linearization of theories for incompressible elastic materials and applications

2021 ◽  
pp. 1-32
Author(s):  
Martin Jesenko ◽  
Bernd Schmidt
Keyword(s):  
Small Displacement ◽  
Gamma Convergence ◽  
Stored Energy ◽  
Energy Functionals ◽  
Small Length Scale ◽  
Nematic Elastomers ◽  
Incompressible Materials ◽  
Geometric Linearization ◽  
Displacement Regime ◽  
Nonlinear Energy

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

scholarly journals MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION

2009 ◽  
Vol 13 (6B) ◽  
pp. 2021-2036 ◽  
Author(s):  
Mao-Sheng Chang ◽  
Shu-Cheng Lee ◽  
Chien-Chang Yen
Keyword(s):  
Gamma Convergence ◽  
Energy Functionals ◽  
Laplacian Equation

scholarly journals The Systems of Nonlinear Gradient Flows on Metric Spaces and Its Gamma-Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Mao-Sheng Chang ◽  
Bo-Cheng Lu
Keyword(s):  
Nonlinear System ◽  
Metric Spaces ◽  
Gradient Flow ◽  
Gamma Convergence ◽  
Gradient Flows ◽  
Energy Functionals ◽  
Several Variables ◽  
Flow Systems

We first establish the explicit structure of nonlinear gradient flow systems on metric spaces and then develop Gamma-convergence of the systems of nonlinear gradient flows, which is a scheme meant to ensure that if a family of energy functionals of several variables depending on a parameter Gamma-converges, then the solutions to the associated systems of gradient flows converge as well. This scheme is a nonlinear system edition of the notion initiated by Sylvia Serfaty in 2011.

MODELING OF A MEMBRANE FOR NONLINEARLY ELASTIC INCOMPRESSIBLE MATERIALS VIA GAMMA-CONVERGENCE

2006 ◽  
Vol 04 (01) ◽  
pp. 31-60 ◽  
Author(s):  
KARIM TRABELSI
Keyword(s):  
Integral Representation ◽  
Internal Energy ◽  
Elastic Membrane ◽  
Gamma Convergence ◽  
Two Dimensional ◽  
Incompressible Materials ◽  
Plate Models ◽  
Γ Convergence ◽  
Nonlinearly Elastic

In this paper, we derive nonlinearly elastic membrane plate models for hyperelastic incompressible materials using Γ-convergence arguments. We obtain an integral representation of the limit two-dimensional internal energy owing to a result of singular functionals relaxation due to Ben Belgacem [6].

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.

Subharmonicity results for the stationary solutions of isotropic energy functionals

2017 ◽  
Vol 10 (2) ◽  
pp. 169-181
Author(s):  
Aleksis Koski
Keyword(s):  
Stationary Solutions ◽  
Harmonic Mappings ◽  
Jacobian Determinant ◽  
Stored Energy ◽  
Lagrange Equations ◽  
Isotropic Energy ◽  
Energy Functions ◽  
Energy Functionals ◽  
A New Technique ◽  
Derivatives Of

AbstractWe study solutions of Euler–Lagrange equations for isotropic energy functionals, generalizing a previous result on p-harmonic mappings. We classify all stored energy functions which give rise to a first-order differential expression whose Laplacian involves no third derivatives of the stationary solution. This classification gives rise to a new technique of finding subharmonicity results for the variational equations, and we also illustrate this technique in two examples. Firstly, we prove a subharmonicity result for the Jacobian determinant in the case of weighted Dirichlet energy. Secondly, we find optimal subharmonicity results in the case of a Neohookean-type stored energy function.

scholarly journals Large elastic deformations of isotropic materials IV. further developments of the general theory

1948 ◽  
Vol 241 (835) ◽  
pp. 379-397 ◽  
Keyword(s):  
Energy Function ◽  
Equations Of Motion ◽  
Incompressible Material ◽  
Surface Forces ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Incompressible Materials ◽  
Scalar Invariants ◽  
Compressible Materials ◽  
Motion Boundary

The equations of motion, boundary conditions and stress-strain relations for a highly elastic material can be expressed in terms of the stored-energy function. This has been done in part I of this series (Rivlin 1948 a ), for both the cases of compressible and incompressible materials, following the methods given by E. & F. Cosserat for compressible materials. The stored-energy function may be defined for a particular material in terms of the invariants of strain. The form in which the equations of motion, etc., are deduced, in the previous paper, does not permit the evaluation of the forces necessary to produce a specified deformation unless the actual expression for the stored-energy function in terms of the scalar invariants of the strain is introduced. In the present paper, the equations are transformed into forms more suitable for carrying out such an explicit evaluation. As examples, the surface forces necessary to produce simple shear in a cuboid of either compressible or incompressible material and those required to produce simple torsion in a right-circular cylinder of incompressible material are derived.

A Device for Magnification Measurement

1972 ◽  
Vol 30 ◽  
pp. 622-623
Author(s):  
J.A. Eades ◽  
A. van Dun
Keyword(s):  
Electron Microscope ◽  
Small Displacement ◽  
Private Communication ◽  
Piezoelectric Crystal ◽  
Double Exposure ◽  
Standard Methods ◽  
Elegant Method ◽  
The Right ◽  
Piezo Electric

The measurement of magnification in the electron microscope is always troublesome especially when a goniometer stage is in use, since there can be wide variations from calibrated values. One elegant method (L.M.Brown, private communication) of avoiding the difficulties of standard methods would be to fit a device which displaces the specimen a small but known distance and recording the displacement by a double exposure. Such a device would obviate the need for changing the specimen and guarantee that the magnification was measured under precisely the conditions used.Such a small displacement could be produced by any suitable transducer mounted in one of the specimen translation mechanisms. In the present case a piezoelectric crystal was used. Modern synthetic piezo electric ceramics readily give reproducible displacements in the right range for quite modest voltages (for example: Joyce and Wilson, 1969).

Observation of faint contrast at planar faults in alloys

1978 ◽  
Vol 36 (1) ◽  
pp. 340-341
Author(s):  
K. Kuroda ◽  
Y. Tomokiyo ◽  
T. Kumano ◽  
T. Eguchi
Keyword(s):  
Reciprocal Lattice ◽  
Beam Theory ◽  
Al Alloys ◽  
Small Displacement ◽  
Electron Microscopic ◽  
Lattice Vector ◽  
Fringe Contrast ◽  
Planar Fault ◽  
Lattice Displacement ◽  
The Many

The contrast in electron microscopic images of planar faults in a crystal is characterized by a phase factor , where is the reciprocal lattice vector of the operating reflection, and the lattice displacement due to the fault under consideration. Within the two-beam theory a planar fault with an integer value of is invisible, but a detectable contrast is expected when the many-beam dynamical effect is not negligibly small. A weak fringe contrast is also expected when differs slightly from an integer owing to an additional small displacement of the lattice across the fault. These faint contrasts are termed as many-beam contrasts in the former case, and as ε fringe contrasts in the latter. In the present work stacking faults in Cu-Al alloys and antiphase boundaries (APB) in CuZn, FeCo and Fe-Al alloys were observed under such conditions as mentioned above, and the results were compared with the image profiles of the faults calculated in the systematic ten-beam approximation.

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