A mixed variational principle for the Föppl–von Kármán equations

Brunetti, Matteo; Favata, Antonino; Paolone, Achille; Vidoli, Stefano

doi:10.1016/j.apm.2019.10.041

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…Various localization phenomena occurring in nonlinear slender structures have been analysed recently based on 1d models, including the necking of hyper-elastic cylinders or bars [4], the bulging in axisymmetric balloons [5], or the folding of tape-springs [6,7]. The 1d models that have been proposed for these different phenomena are all mathematically similar.…”

Section: Introductionmentioning

confidence: 99%

A one-dimensional model for elasto-capillary necking

Lestringant

Audoly

2020

Proc. R. Soc. A.

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We derive a nonlinear one-dimensional (1d) strain gradient model predicting the necking of soft elastic cylinders driven by surface tension, starting from three-dimensional (3d) finite-strain elasticity. It is asymptotically correct: the microscopic displacement is identified by an energy method. The 1d model can predict the bifurcations occurring in the solutions of the 3d elasticity problem when the surface tension is increased, leading to a localization phenomenon akin to phase separation. Comparisons with finite-element simulations reveal that the 1d model resolves the interface separating two phases accurately, including well into the localized regime, and that it has a vastly larger domain of validity than 1d models proposed so far.

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Section: Introductionmentioning

confidence: 99%

A one-dimensional model for elasto-capillary necking

Lestringant

Audoly

2020

Proc. R. Soc. A.

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“…Other benefits of this method include its generalisability for solving higher-order and nonlinear problems. For example, in this paper, we have described how the ANN framework proposed can be utilised for solving the Navier-Stokes equation [13,15,28] and the Von-Kármán equation [8].…”

Section: Discussionmentioning

confidence: 99%

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

2021

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Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

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“…For more details on the derivation and meaning of the FvK equations, we refer the interested reader to [39]. In addition, we note that general boundary conditions for the problem (3.1)–(3.2) are treated in several papers (see [40]); in the following, we will consider only the special case corresponding to our problem.…”

Section: Dimensional Reduction From the Föppl–von Kármán Shell Modelmentioning

confidence: 99%

“…In [40,41], we have shown that equations (3.1)–(3.2) can be deduced by enforcing the following mixed variational problem: minw∈scriptWmaxφ∈scriptSscriptFfalse(φ,wfalse), where scriptW and scriptS are two suitable subsets of H 2 ( Ω ) and the functional scriptFfalse(φ,wfalse) is given by the splitting scriptFfalse(φ,wfalse)=Ebfalse(normal∇normal∇wfalse)−Emfalse(φfalse)+12∫Ωbold-italicNfalse(φfalse)⋅(normal∇w⊗normal∇w−normal∇w0⊗normal∇w0). In particular, for the case under consideration the Airy function φ is required to vanish on ∂ Ω , so that scriptS=H02false(normalΩfalse)=falsefalse{f∈H2false(normalΩfalse)…”

Section: Dimensional Reduction From the Föppl–von Kármán Shell Modelmentioning

confidence: 99%

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Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability

2020

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We deduce a one-dimensional model of elastic planar rods starting from the Föppl–von Kármán model of thin shells. Such model is enhanced by additional kinematical descriptors that keep explicit track of the compatibility condition requested in the two-dimensional parent continuum, that in the standard rods models are identically satisfied after the dimensional reduction. An inextensible model is also proposed, starting from the nonlinear Koiter model of inextensible shells. These enhanced models describe the nonlinear planar bending of rods and allow to account for some phenomena of preeminent importance even in one-dimensional bodies, such as formation of singularities and localization (d-cones), otherwise inaccessible by the classical one-dimensional models. Moreover, the effects of the compatibility translate into the possibility to obtain multiple stable equilibrium configurations.

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A mixed variational principle for the Föppl–von Kármán equations

Cited by 6 publications

References 21 publications

A one-dimensional model for elasto-capillary necking

A one-dimensional model for elasto-capillary necking

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability

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