2,350 research outputs found
On interpretations and constructions of classical dynamical r-matrices
In this note we complement recent results on the exchange -matrices
appearing in the chiral WZNW model by providing a direct, purely
finite-dimensional description of the relationship between the monodromy
dependent 2-form that enters the chiral WZNW symplectic form and the exchange
-matrix that governs the corresponding Poisson brackets. We also develop the
special case in which the exchange -matrix becomes the `canonical' solution
of the classical dynamical Yang-Baxter equation on an arbitrary self-dual Lie
algebra.Comment: 8 pages, LaTeX, based on a talk given by L.F. at the QTS2 symposium,
18-21 July 2001, Krakow, Poland. References are updated, and a typo is
removed in v2; a misprint in equation (A.13) is corrected in v
Disappearance of stretch-induced wrinkles of thin sheets: a study of orthotropic films
A recent paper (Healey et al., J. Nonlin. Sci., 2013, 23:777-805.) predicted
the disappearance of the stretch-induced wrinkled pattern of thin, clamped,
elastic sheets by numerical simulation of the F\"oppl-von K\'arm\'an equations
extended to the finite in-plane strain regime. It has also been revealed that
for some aspect ratios of the rectangular domain wrinkles do not occur at all
regardless of the applied extension. To verify these predictions we carried out
experiments on thin 20 micrometer thick adhesive covered), previously
prestressed elastomer sheets with different aspect ratios under displacement
controlled pull tests. On one hand the the adjustment of the material
properties during prestressing is highly advantageous as in targeted strain
regime the film becomes substantially linearly elastic (which is far not the
case without prestress). On the other hand a significant, non-ignorable
orthotropy develops during this first extension. To enable quantitative
comparisons we abandoned the assumption about material isotropy inherent in the
original model and derived the governing equations for an orthotropic medium.
In this way we found good agreement between numerical simulations and
experimental data.
Analysis of the negativity of the second Piola-Kirchhoff stress tensor
revealed that the critical stretch for a bifurcation point at which the
wrinkles disappear must be finite for any aspect ratio. On the contrary there
is no such a bound for the aspect ratio as a bifurcation parameter. Physically
this manifests as complicated wrinkled patterns with more than one highly
wrinkled zones on the surface in case of elongated rectangles. These
arrangements have been found both numerically and experimentally. These
findings also support the new, finite strain model, since the F\"oppl-von
K\'arm\'an equations based on infinitesimal strains do not exhibit such a
behavior.Comment: 16 pages, 5 figure
The Mullins effect in the wrinkling behavior of highly stretched thin films
Recent work demonstrates that finite-deformation nonlinear elasticity is
essential in the accurate modeling of wrinkling in highly stretched thin films.
Geometrically exact models predict an isola-center bifurcation, indicating that
for a bounded interval of aspect ratios only, stable wrinkles appear and then
disappear as the macroscopic strain is increased. This phenomenon has been
verified in experiments. In addition, recent experiments revealed the following
striking phenomenon: For certain aspect ratios for which no wrinkling occurred
upon the first loading, wrinkles appeared during the first unloading and again
during all subsequent cyclic loading. Our goal here is to present a simple
pseudo-elastic model, capturing the stress softening and residual strain
observed in the experiments, that accurately predicts wrinkling behavior on the
first loading that differs from that under subsequent cyclic loading. In
particular for specific aspect ratios, the model correctly predicts the
scenario of no wrinkling during first loading with wrinkling occurring during
unloading and for all subsequent cyclic loading.Comment: 15 pages, 9 figure
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