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.

Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity

2014 ◽  
Vol 24 (10) ◽  
pp. 2085-2153 ◽  
Author(s):  
Elisa Davoli
Keyword(s):  
Three Dimensional ◽  
Scaling Factor ◽  
Thin Plates ◽  
Quasistatic Evolution ◽  
Plastic Energy ◽  
Von Karman ◽  
Finite Plasticity ◽  
Plastic Dissipation ◽  
Γ Convergence ◽  
Von Kármán

In this paper we deduce by Γ-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by ε the thickness of the plate, we study the case where the scaling factor of the elasto-plastic energy is of order ε2α-2, with α ≥ 3. These scalings of the energy lead, in the absence of plastic dissipation, to the Von Kármán and linearized Von Kármán functionals for thin plates. We show that solutions to the three-dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on α.

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.

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 Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence

2009 ◽  
Vol 139 (5) ◽  
pp. 1037-1070 ◽  
Author(s):  
Lucia Scardia
Keyword(s):  
Nonlinear Elasticity ◽  
Three Dimensional ◽  
Curved Beam ◽  
Rigorous Derivation ◽  
One Dimensional ◽  
Asymptotic Models ◽  
Linearized Elasticity ◽  
Curved Rods ◽  
Γ Convergence ◽  
The One

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity.

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.

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