On the Existence of Solutions to the Generalized Marguerre-von Kármán Equations

2006 ◽  
Vol 11 (1) ◽  
pp. 83-100 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Liliana Gratie
Keyword(s):  
Operator Equation ◽  
Shallow Shell ◽  
Existence Of Solutions ◽  
Vertical Displacement ◽  
Existence Of A Solution ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Nonlinearly Elastic Plate ◽  
Von Kármán ◽  
Nonlinearly Elastic

Using techniques from asymptotic analysis, the second author has recently identified equations that generalize the classical Marguerre-von Kármán equations for a nonlinearly elastic shallow shell by allowing more realistic boundary conditions, which may change their type along the lateral face of the shell. We first reduce these more general equations to a single “cubic” operator equation, whose sole unknown is the vertical displacement of the shell. This equation generalizes a cubic operator equation introduced by M. S. Berger and P. Fife for analyzing the von Kármán equations for a nonlinearly elastic plate. We then establish the existence of a solution to this operator equation by means of a compactness method due to J. L. Lions.

ON THE GENERALIZED VON KÁRMÁN EQUATIONS AND THEIR APPROXIMATION

2007 ◽  
Vol 17 (04) ◽  
pp. 617-633 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE ◽  
SRINIVASAN KESAVAN
Keyword(s):  
Original Problem ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Von Karman ◽  
Strict Positivity ◽  
Von Kármán Equations ◽  
Conforming Finite Element ◽  
Nonlinearly Elastic Plate ◽  
Von Kármán ◽  
Nonlinearly Elastic

We consider here the "generalized von Kármán equations", which constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions "of von Kármán type" only on a portion of its lateral face, the remaining portion being free. As already shown elsewhere, solving these equations amounts to solving a "cubic" operator equation, which generalizes an equation introduced by Berger and Fife. Two noticeable features of this equation, which are not encountered in the "classical" von Kármán equations are the lack of strict positivity of its cubic part and the lack of an associated functional. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J. L. Lions and on Brouwer's fixed point theorem. This convergence proof provides in addition an existence proof for the original problem.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

scholarly journals Elastohydrodynamics of a pre-stretched finite elastic sheet lubricated by a thin viscous film with application to microfluidic soft actuators

2019 ◽  
Vol 862 ◽  
pp. 732-752 ◽  
Author(s):  
Evgeniy Boyko ◽  
Ran Eshel ◽  
Khaled Gommed ◽  
Amir D. Gat ◽  
Moran Bercovici
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Finite Domain ◽  
Viscous Film ◽  
Elastic Sheet ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Wide Range ◽  
Soft Actuators ◽  
Von Kármán

The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Föppl–von Kármán equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and time scale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Föppl–von Kármán equations and present the deformation behaviour as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics.

On group analysis of the von Kármán equations

1982 ◽  
Vol 6 (8) ◽  
pp. 845-853 ◽  
Author(s):  
Karen A. Ames ◽  
W.F. Ames
Keyword(s):  
Group Analysis ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán
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