Convergence of Equilibria of Thin Elastic Plates – The Von Kármán Case

We study the behaviour of thin elastic bodies of fixed cross-section and of height h, with h → 0. We show that critical points of the energy functional of nonlinear three-dimensional elasticity converge to critical points of the von Kármán functional, provided that their energy per unit height is bounded by Ch ⁴ (and that the stored energy density function satisfies a technical growth condition). This extends recent convergence results for absolute minimizers.     

The time-dependent von K��rm��n plate equation as a limit of 3d nonlinear elasticity

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation.     

Asymptotic modelling of a Signorini problem of generalized Marguerre-von Kármán shallow shells

Chacha and Bensayah [Asymptotic modeling of a Coulomb frictional Signorini problem for the von Kármán plates, C. R. Mécanique 336 (2008), pp. 846–850] have studied the asymptotic modelling of Coulomb frictional unilateral contact problem between an elastic nonlinear von Kármán plate and a rigid obstacle. The main result obtained is that the leading term of the asymptotic expansion is characterized by a two-dimensional Signorini problem but without friction. In this article, we extend this study to the case of a shallow shell under generalized Marguerre-von Kármán conditions.     

Conforming finite element methods for the von Kármán equations

In this paper, we consider canonical von Kármán equations that describe the bending of thin elastic plates defined on polygonal domains. A conforming finite element method is employed to approximate the displacement and Airy stress functions. Optimal order error estimates in energy, H ¹ and L ² norms are deduced. The results of numerical experiments confirm the theoretical results obtained.     

Rigorous Bounds for the Föppl—von Kármán Theory of Isotropically Compressed Plates

Summary. We study the F?ppl—von Kármán theory for isotropically compressed thin plates in a geometrically linear setting, which is commonly used to model weak buckling of thin films. We consider generic smooth domains with clamped boundary conditions, and obtain rigorous upper and lower bounds on the minimum energy linear in the plate thickness σ . This energy is much lower than previous estimates based on certain dimensional reductions of the problem, which had lead to energies of order 1+σ (scalar approximation) or σ ^2/3 (two-component approximation). Received August 7, 2000; accepted September 8, 2000 %%%Online publication November 15, 2000 Communicated by Robert V. Kohn     

Numerical analysis of the generalized von Kármán equations

The ‘generalized von Kármán equations’ 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. 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. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The von Kármán theory for incompressible elastic shells

We rigorously derive the von Kármán shell theory for incompressible materials, starting from the 3D nonlinear elasticity. In case of thin plates, the Euler-Lagrange equations of the limiting energy functional reduce to the incompressible version of the classical von Kármán equations, obtained formally in the limit of Poisson’s ratio ν → 1/2. More generally, the midsurface of the shell to which our analysis applies, is only assumed to have the following approximation property: ${\mathcal C^3}$ first order infinitesimal isometries are dense in the space of all W ^2,2 infinitesimal isometries. The class of surfaces with this property includes: subsets of ${\mathbb R^2}$ , convex surfaces, developable surfaces and rotationally invariant surfaces. Our analysis relies on the methods and extends the results of Conti and Dolzmann (Calc Var PDE 34:531–551, 2009, Lewicka et al. (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX:253–295, 2010, Friesecke et al. (Comm. Pure. Appl. Math. 55, no. 2, 1461–1506, 2002).     

Von Kármán equations in L p spaces

We show the existence of solution in L ^p spaces for a generalized form of the classical Von Kármán equations where the coefficients of nonlinear terms are variable. We use Campanato's near operators theory.     

Conservative numerical methods for the Full von Kármán plate equations

This article is concerned with the numerical solution of the full dynamical von Kármán plate equations for geometrically nonlinear (large‐amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von Kármán system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semidiscrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretizations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1948–1970, 2015     

Un théorème d'existence pour les équations de von Kármán généralisées

Using techniques from formal asymptotic analysis, the first two authors have recently identified “generalized von Kármán equations”, which constitute a two-dimensional model for a nonlinearly elastic plate where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, the remaining portion being free.In this Note, we establish an existence theorem for these equations. To this end, we notably adapt a compactness method due to J.-L. Lions.     

On the boundary value problem of the biharmonic operator on domains with angular corners

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.     

Axial Compression of a Thin Elastic Cylinder: Bounds on the Minimum Energy Scaling Law

We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression. We focus exclusively on the setting where the radius of the core is greater than or equal to the natural radius of the cylinder. We consider two cases: the “large mandrel” case, where the radius of the core exceeds that of the cylinder, and the “neutral mandrel” case, where the radii of the core and cylinder are the same. In the large mandrel case, our upper and lower bounds match in their scaling with respect to thickness, compression, and the magnitude of pre‐strain induced by the core. We construct three types of axisymmetric wrinkling patterns whose energy scales as the minimum in different parameter regimes, corresponding to the presence of many wrinkles, few wrinkles, or no wrinkles at all. In the neutral mandrel case, our upper and lower bounds match in a certain regime in which the compression is small as compared to the thickness; in this regime, the minimum energy scales as that of the unbuckled configuration. We achieve these results for both the von Kármán–Donnell model and a geometrically nonlinear model of elasticity. © 2017 Wiley Periodicals, Inc.     

Asymptotic stability for a non-autonomous full von Kármán beam with thermo-viscoelastic damping

In this paper, we study the asymptotic behavior for a one-dimensional non-autonomous full von Kármán beam with a thermo-viscoelastic damping in the internal feedback. By introducing a suitable energy and some Lyapunov functionals, under some restrictions on the non-autonomous functions and the relaxation function, we show the asymptotic behavior of the solution and establish a general decay result for the energy.     

A short note on the derivation of the elastic von Kármán shell theory

We derive the Γ-limit of scaled elastic energies h^?4E ^h (u ^h ) associated with deformations u ^h of a family of thin shells \({S^h} = \left\{ {z = x + t\vec n\left( x \right);x \in S, - g_1^h\left( x \right) < t < g_2^h\left( x \right)} \right\}\). The obtained von Kármán theory is valid for a general sequence of boundaries g ₁ ^h , g ₂ ^h converging to 0 in an appropriate manner as h vanishes. Our analysis relies on the techniques and extends the results in [10] and [11].     

Generic bifurcation with application to the von Kármán equations

We examine the possible types of generic bifurcation than can occur for a three-parameter family of mappings from a Banach space into itself. Specifically, the general form of the bifurcation equations arising from the von Kármán equations for the buckling of a rectangular plate is investigated. Chow, Hale, and Mallet-Paret (Applications of generic bifurcation. II, Arch. Rational Mech. Anal.67 (1976)) studied the bifurcation of solutions to these equations in a two-parameter setting. These parameters were related to the normal loading and to the compressive thrust applied at the ends of the plate. We introduce a third bifurcation parameter by considering the length of the plate as variable. The generic hypotheses of Chow et al. no longer apply in this three-parameter setting, but modifications and extensions of these hypotheses permit a characterization of the three-parameter bifurcation diagram. The bifurcation sheets of this diagram appear as a natural generalization of the finite collection of arcs comprising the two-parameter diagram. As an example of this theory, an actual three-parameter bifurcation diagram is constructed for a specific form of the von Kármán equations.     

Justification de la théorie non linéaire de Kirchhoff–Love,comme application d'une nouvelle méthode d'inversion singulièreJustification of the nonlinear Kirchhoff–Love theory,as the application of a new singular inverse method

In the framework of nonlinear elasticity, we consider a three-dimensional plate made of a St Venant–Kirchhoff isotropic and homogeneous material of thickness 2ε and periodic in the two other directions. By a change of scales, the problem can be mapped on a fixed open set, and seen as a nonlinear singular perturbation problem. We introduce a new singular inverse method. Applying this method, we prove that for fixed and small enough exterior forces, the three-dimensional displacement converges to the solution of the nonlinear Kirchhoff–Love theory of plate as the thickness 2ε tends to zero. The limit plate model contains in particular that of von Kármán. We also give a quantitative estimate of the convergence. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615–620.     

On the compatibility relation for the Föppl–von Kármán plate equations

By using a coordinate-free approach we propose a new derivation of the compatibility equation for the Föppl–von Kármán nonlinear plate theory.     

The Γ-limit of a finite-strain Cosserat model for asymptotically thin domains and a consequence for the Cosserat couple modulus μc

We study the behaviour of a geometrically exact 3D Cosserat continuum model for an asymptotically flat domain. Despite the inherent nonlinearity, the Γ-limit of a corresponding canonically rescaled problem on a domain with constant thickness can be explicitly calculated. This “membrane” limit exhibits no bending contributions scaling with h ³ (similar to classical approaches) but features a transverse shear resistance scaling with h for strictly positive Cosserat couple modulus μ_c > 0. This result is physically inacceptable for a zero-thickness “membrane” limit model. Therefore it is suggested that the physically consistent value of the Cosserat couple modulus μ_c is zero. In this case, however, the Γ-limit looses coercivity for the midsurface deformation in H ^1,2(ω , ℝ³). For numerical purposes then, a transverse shear resistance can be reintroduced, establishing coercivity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the multiplicity of solutions to Marguerre–von Kármán membrane equations

We build explicitly an infinite number of equilibrium solutions of unloaded Marguerre–von Kármán membrane shells. This construction is based upon the existence of three elementary solutions, together with the solution of a Monge–Ampère equation associated with a partition of the reference configuration of the shell. These solutions are characterized as stationary points of energy functionals depending on the partition.