The Time-Dependent Von Kármán Shell Equation as a Limit of Three-Dimensional Nonlinear Elasticity

2020 ◽  
Author(s):  
Yizhao Qin ◽  
Peng-Fei Yao
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional ◽  
Time Dependent ◽  
Von Karman ◽  
Shell Equation ◽  
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.

On the general homogenization of von Kármán plate equations from three-dimensional nonlinear elasticity

2016 ◽  
Vol 15 (01) ◽  
pp. 1-49 ◽  
Author(s):  
Igor Velčić
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional ◽  
Von Karman ◽  
Elasticity Equations ◽  
Plate Equations ◽  
Von Kármán Plate ◽  
Von Kármán ◽  
Three Dimensional Elasticity

Starting from three-dimensional elasticity equations we derive the model of the homogenized von Kármán plate by means of [Formula: see text]-convergence. This generalizes the recent results, where the material oscillations were assumed to be periodic.

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.

scholarly journals von Kármán–Howarth equation for three-dimensional two-fluid plasmas

2016 ◽  
Vol 93 (6) ◽  
Author(s):  
N. Andrés ◽  
P. D. Mininni ◽  
P. Dmitruk ◽  
D. O. Gómez
Keyword(s):  
Three Dimensional ◽  
Von Karman ◽  
Von Kármán ◽  
Two Fluid

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 Numerical and experimental study of the time-dependent states and the slow dynamics in a von Kármán swirling flow

2009 ◽  
Vol 103 (2-3) ◽  
pp. 163-177 ◽  
Author(s):  
E. Crespo Del Arco ◽  
J. J. Sánchez-Álvarez ◽  
E. Serre ◽  
A. De La Torre ◽  
J. Burguete
Keyword(s):  
Experimental Study ◽  
Swirling Flow ◽  
Time Dependent ◽  
Slow Dynamics ◽  
Von Karman ◽  
Von Kármán
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