scholarly journals The Föppl-von Kármán equations for plates with incompatible strains

2010 ◽  
Vol 467 (2126) ◽  
pp. 402-426 ◽  
Author(s):  
Marta Lewicka ◽  
L. Mahadevan ◽  
Mohammad Reza Pakzad
Keyword(s):  
Three Dimensional ◽  
Active Materials ◽  
Von Karman ◽  
Solvent Absorption ◽  
Von Kármán Equations ◽  
Residual Strains ◽  
Rigorous Bounds ◽  
Low Dimensional ◽  
Von Kármán ◽  
Recent Formulation

We provide a derivation of the Föppl-von Kármán equations for the shape of and stresses in an elastic plate with incompatible or residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three-dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials with complex non-Euclidean geometries.

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.

Description of transient states of von Kármán vortex streets by low‐dimensional differential equations

1990 ◽  
Vol 2 (4) ◽  
pp. 479-481 ◽  
Author(s):  
F. Ohle ◽  
P. Lehmann ◽  
E. Roesch ◽  
H. Eckelmann ◽  
A. Hübler
Keyword(s):  
Differential Equations ◽  
Transient States ◽  
Von Karman ◽  
Vortex Streets ◽  
Karman Vortex ◽  
Low Dimensional ◽  
Von Kármán

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

Instabilities of the von Kármán Boundary Layer

2015 ◽  
Vol 67 (3) ◽  
Author(s):  
R. J. Lingwood ◽  
P. Henrik Alfredsson
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Rotating Disk ◽  
Direct Numerical Simulations ◽  
Layer Model ◽  
Complex Flow ◽  
Von Karman ◽  
Dimensional Boundary ◽  
Flow Configurations ◽  
Von Kármán

Research on the von Kármán boundary layer extends back almost 100 years but remains a topic of active study, which continues to reveal new results; it is only now that fully nonlinear direct numerical simulations (DNS) have been conducted of the flow to compare with theoretical and experimental results. The von Kármán boundary layer, or rotating-disk boundary layer, provides, in some senses, a simple three-dimensional boundary-layer model with which to compare other more complex flow configurations but we will show that in fact the rotating-disk boundary layer itself exhibits a wealth of complex instability behaviors that are not yet fully understood.

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