ON A NONLOCAL FUNCTIONAL ARISING IN THE STUDY OF THIN-FILM BLISTERING

2010 ◽  
Vol 08 (02) ◽  
pp. 109-123
Author(s):  
N. ANSINI ◽  
V. VALENTE
Keyword(s):  
Thin Film ◽  
Asymptotic Analysis ◽  
Circular Plate ◽  
One Dimensional ◽  
Von Karman ◽  
Nonlocal Functional ◽  
Γ Convergence ◽  
Von Kármán

The energy of a Von Kármán circular plate is described by a nonlocal nonconvex one-dimensional functional depending on the thickness ε. Here we perform the asymptotic analysis via Γ-convergence as the parameter ε goes to zero.

Timoshenko's beam equation as limit of a nonlinear one-dimensional von Kármán system

2000 ◽  
Vol 130 (4) ◽  
pp. 855-875 ◽  
Author(s):  
G. Perla Menzala ◽  
E. Zuazua
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Weak Limit ◽  
Beam Equation ◽  
Limit System ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Von Karman ◽  
Von Kármán System ◽  
Von Kármán

We consider a dynamical one-dimensional nonlinear von Kármán model depending on one parameter ε > 0 and study its weak limit as ε → 0. We analyse various boundary conditions and prove that the nature of the limit system is very sensitive to them. We prove that, depending on the type of boundary condition we consider, the nonlinearity of Timoshenko's model may vanish.

One-dimensional von Kármán models for elastic ribbons

Meccanica ◽  
2017 ◽  
Vol 53 (3) ◽  
pp. 659-670 ◽  
Author(s):  
Lorenzo Freddi ◽  
Peter Hornung ◽  
Maria Giovanna Mora ◽  
Roberto Paroni
Keyword(s):  
One Dimensional ◽  
Von Karman ◽  
Von Kármán

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 Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons

2022 ◽  
Vol 29 (2) ◽  
Author(s):  
Manuel Friedrich ◽  
Lennart Machill
Keyword(s):  
Metric Spaces ◽  
Three Dimensional ◽  
Gradient Flows ◽  
Dimensional Model ◽  
Discrete Approximations ◽  
Three Dimensional Model ◽  
One Dimensional ◽  
Von Karman ◽  
Von Kármán ◽  
Appl Math

AbstractWe consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin’s-Voigt’s rheology derived from a three-dimensional model at a finite-strain setting in Friedrich and Kružík (Arch Ration Mech Anal 238: 489–540, 2020). As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004) and complement the $$\Gamma $$ Γ -convergence analysis of elastic von Kármán ribbons in Freddi et al. (Meccanica 53:659–670, 2018). Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.

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