scholarly journals Triangular buckling patterns of twisted inextensible strips

2010 ◽  
Vol 467 (2125) ◽  
pp. 285-303 ◽  
Author(s):  
A. P. Korte ◽  
E. L. Starostin ◽  
G. H. M. van der Heijden
Keyword(s):  
Boundary Value Problem ◽  
Critical Load ◽  
Critical Loads ◽  
Equilibrium Shape ◽  
Boundary Value ◽  
Mode Number ◽  
Force Extension ◽  
Higher Modes ◽  
Post Buckling ◽  
Good Agreement

When twisting a strip of paper or acetate under high longitudinal tension, one observes, at some critical load, a buckling of the strip into a regular triangular pattern. Very similar triangular facets have recently been found in solutions to a new set of geometrically exact equations describing the equilibrium shape of thin inextensible elastic strips. Here, we formulate a modified boundary-value problem for these equations and construct post-buckling solutions in good agreement with the observed pattern in twisted strips. We also study the force–extension and moment–twist behaviour of these strips by varying the mode number n of triangular facets and find critical loads with jumps to higher modes.

scholarly journals Benchmark for the Coupled Magneto-Mechanical Boundary Value Problem in Magneto-Active Elastomers

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2380
Author(s):  
Philipp Metsch ◽  
Raphael Schiedung ◽  
Ingo Steinbach ◽  
Markus Kästner
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
Diffuse Interface ◽  
Two Dimensional ◽  
Benchmark Problem ◽  
Material Models ◽  
Lagrangian Finite Element ◽  
Good Agreement

Within this contribution, a novel benchmark problem for the coupled magneto-mechanical boundary value problem in magneto-active elastomers is presented. Being derived from an experimental analysis of magnetically induced interactions in these materials, the problem under investigation allows us to validate different modeling strategies by means of a simple setup with only a few influencing factors. Here, results of a sharp-interface Lagrangian finite element framework and a diffuse-interface Eulerian approach based on the application of a spectral solver on a fixed grid are compared for the simplified two-dimensional as well as the general three-dimensional case. After influences of different boundary conditions and the sample size are analyzed, the results of both strategies are examined: for the material models under consideration, a good agreement of them is found, while all discrepancies can be ascribed to well-known effects described in the literature. Thus, the benchmark problem can be seen as a basis for future comparisons with both other modeling strategies and more elaborate material models.

scholarly journals The effect of influence of conservative and tangential axial forces on transverse vibrations of tapered vertical columns

2017 ◽  
Vol 4 (20) ◽  
pp. 333-342
Author(s):  
Jerzy Jaroszewicz ◽  
Leszek Radziszewski ◽  
Łukasz Dragun
Keyword(s):  
Boundary Value Problem ◽  
Natural Frequency ◽  
Critical Loads ◽  
Cauchy Function ◽  
Axial Forces ◽  
Transverse Vibrations ◽  
Cantilever Length ◽  
Tangential Forces ◽  
Elastically Supported ◽  
Good Agreement

The Cauchy function and characteristic series were applied to solve the boundary value problem of free transverse vibrations of vertically mounted, elastically supported tapered cantilever columns. The columns can be subjected to universal axial point loads which considerate – conservative and follower /tangential/ forces, and to distributed loads along the cantilever length. The general form of characteristic equation was obtained taking into account the shape of tapered cantilever for attached and elastically secured. Bernstein-Kieropian double and higher estimators of natural frequency and critical loads were calculated based on the first few coefficients of the characteristic series. Good agreement was obtained between the calculated natural frequency and the exact values available in the literature.

Remarks on the equilibrium shape of a Tokamak plasma

1993 ◽  
Vol 123 (6) ◽  
pp. 1059-1070
Author(s):  
Yang Jianfu
Keyword(s):  
Free Boundary ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Equilibrium Shape ◽  
Boundary Value ◽  
Tokamak Plasma ◽  
Free Boundary Value Problem

SynopsisThe equilibrium of a Tokamak plasma not confined inside a conducting shell is governed by a free boundary value problem. The existence of solutions of the free boundary value problem is discussed.

scholarly journals Boundary Value Problem of an Infinite Array of Loaded Apertures

2015 ◽  
Vol 3 ◽  
pp. 11-15
Author(s):  
Luis Alejandro Iturri-Hinojosa ◽  
Alexander E. Martynyuk ◽  
Mohamed Badaoui
Keyword(s):  
Mathematical Model ◽  
Reflection Coefficient ◽  
Plane Wave ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Incident Plane Wave ◽  
Incident Plane ◽  
Infinite Array ◽  
The Mathematical Model ◽  
Good Agreement

A mathematical model of the scattering by a periodically arranged apertures in conducting plates is presented. The boundary value problem of an infinite array of loaded apertures is formulated for an arbitrary incident plane wave. The reflection coefficient for some array geometries is obtained and the calculated values are in good agreement with the measurements in a previously published researches. All the rectangular apertures in the array are assumed to be identical and infinitesimally thin. The mathematical model is based on Floquet’s theorem that specifies the requirement of periodicity by the electromagnetic fields.

Dynamic Snap-Through of Imperfect Viscoelastic Shallow Arches

1968 ◽  
Vol 35 (2) ◽  
pp. 289-296 ◽  
Author(s):  
N. C. Huang ◽  
W. Nachbar
Keyword(s):  
Lower Bound ◽  
Critical Load ◽  
Critical Loads ◽  
Jump Discontinuity ◽  
Closed Form Expression ◽  
Antisymmetric Mode ◽  
Form Expression ◽  
Shallow Arches ◽  
Voigt Model ◽  
Good Agreement

Dynamic snap-through or dynamic buckling of imperfect viscoelastic shallow arches with hinged ends is considered under step loads of infinite duration. Attention is principally devoted to the influence both of small imperfections and of small amounts of damping, acting together, on the critical loads. For the problem considered, the Voigt model is used for viscoelasticity, the deflection is represented by the first two harmonic modes, and imperfections have the shape of the second (antisymmetric) mode. Results obtained by numerical integration of the differential equations show that the critical load exhibits a jump discontinuity in the limit for vanishing imperfection. Critical loads for slightly imperfect and elastic (inviscid) arches are slightly higher than those from the saddle-point formula of Hoff and Bruce [1, p. 276], confirming that the formula gives a lower bound on the critical load. However, critical loads for arches with slight imperfection and slight viscosity are considerably higher than for the slightly imperfect elastic arches. Another closed-form expression is shown to be in good agreement with these results. For finite amounts of viscosity, the critical loads tend rapidly to the values obtained for infinite viscosity, which are the same as the critical loads for quasi-static buckling. Apart from the jump discontinuity at zero, the critical load for any viscosity decreases continuously and monotonically with imperfection.

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