scholarly journals BMO and Elasticity: Korn’s Inequality; Local Uniqueness in Tension

2021 ◽  
Author(s):  
Daniel E. Spector ◽  
Scott J. Spector
Keyword(s):  
Positive Definite ◽  
Second Variation ◽  
Local Uniqueness ◽  
Equilibrium Solutions ◽  
Principal Stresses ◽  
Korn’S Inequality ◽  
The Second Variation ◽  
Strain Space ◽  
Vector Valued ◽  
Uniqueness Of Equilibrium

AbstractIn this manuscript two BMO estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the BMO-seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the BMO-seminorm of the symmetric part of its gradient, that is, a Korn inequality in BMO. The uniqueness of equilibrium for a finite deformation whose principal stresses are everywhere nonnegative is then considered. It is shown that when the second variation of the energy, when considered as a function of the strain, is uniformly positive definite at such an equilibrium solution, then there is a BMO-neighborhood in strain space where there are no other equilibrium solutions.

Initial post-buckling analysis of a fixed–fixed strut

2019 ◽  
Vol 24 (11) ◽  
pp. 3403-3409
Author(s):  
M Jin ◽  
HY Qi
Keyword(s):  
Potential Energy ◽  
Critical Load ◽  
Stability Limit ◽  
Positive Definite ◽  
Second Variation ◽  
Buckling Mode ◽  
Lateral Deflection ◽  
Post Buckling ◽  
The Stability ◽  
The Second Variation

In this paper, the initial post-buckling of a fixed–fixed strut in compression at the first bifurcation point is analyzed. Using the Fourier series of the lateral deflection, the second variation of the potential energy is proved, analytically, to be semi-positive definite when the compression is equal to the Euler critical load. The fourth variation of the potential energy is positive when the disturbance of the lateral deflection matches the buckling mode. Based on Koiter’s initial post-buckling theory, the equilibrium of the straight state of the strut is stable at the stability limit; when the compression slightly exceeds the Euler critical load, the curved shape at initial post-buckling is stable.

The second variation and neighboring optimal feedback guidance for multiple burn orbit transfers

1995 ◽  
Author(s):  
C-HGoodson, Chuang, , Troy D ◽  
Laura Ledsinger ◽  
John Hanson
Keyword(s):  
Second Variation ◽  
Optimal Feedback ◽  
The Second Variation

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

scholarly journals Autoparallel Deviation in the Geometry of Lyra

1960 ◽  
Vol 3 (3) ◽  
pp. 263-271 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Calculus Of Variations ◽  
Finsler Manifold ◽  
Geometrical Approach ◽  
Second Variation ◽  
Geodesic Deviation ◽  
The Difference ◽  
The Second Variation

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

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