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.     

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).     

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.     

Asymptotic behavior to a von Kármán equations of memory type with acoustic boundary conditions

We study the stability of solutions to a von Kármán plate model of memory type with acoustic boundary conditions. We establish the general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve on some earlier results-exponential or polynomial decay rates.     

A mixed variational principle for the Föppl–von Kármán equations

A mixed variational principle is proposed for deducing the Föppl–von Kármán equations governing the large deflections of thin elastic plates or shallow shells. Proper boundary conditions are found for the case of applied in-plane tractions and displacements, and simple mechanical interpretations are achieved. Numerical implementation is carried out, along with examples and comparisons with the classical formulation in terms of displacements.     

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.     

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 necessary and sufficient condition for bifurcation in the von Kármán equations

The paper is devoted to the study of bifurcation in the von Kármán equations with two parameters that describe the behaviour of a thin round elastic plate lying on an elastic base under the action of a compressing force. The problem appears in the mechanics of elastic constructions. We prove the necessary and sufficient condition for bifurcation at points of the set of trivial solutions. Our proof is based on reducing the von Kármán equations to an operator equation in Banach spaces with a nonlinear Fredholm map of index 0 and applying the Crandall-Rabinowitz theorem on simple bifurcation points or a finite-dimensional reduction and degree theory. RID="h1" ID="h1"This research was supported by grant BW of UG no. 5100-5-0153-1 and by grant KBN no. 5 P03A 020 20.     

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.     

Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerre-von Kármán shallow shell equations

By Galerkin finite element method, we show the global existence and uniqueness of weak solution to the nonlinear viscoelastic full Marguerre-von Karman shallow shell equations.     

Energy Decay for von Kármán-Gurtin-Pipkin System

This paper aims to prove the asymptotic behavior of the solution for the thermo-elastic von Karman system where the thermal conduction is given by Gurtin-Pipkins law. Existence and uniqueness of the solution are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional.Therefore, the stability result clarified that the solutions energy functional decays exponentially at infinite time.     

Group analysis of the von Kármán–Howarth equation. Part I. Submodels

We study the von Kármán–Howarth (KH) equation by group theoretical methods. This scalar partial differential equation involves two dependent variables (closure problem) and, it has been derived from the Navier–Stokes equations. The equivalence Lie algebra L has been found to be infinite-dimensional and, it is spanned by the four operators. The subalgebra of L is spanned by the three operators. Furthermore, ideal comprises one operator. Optimal systems of one-, two- and three-dimensional subalgebras have been obtained. Normalizers for the one- and two-dimensional subalgebras have been calculated. Finally we have obtained the submodels of the KH equation corresponding to optimal system of one- and two-dimensional subalgebras. This merely suggests alternative solutions to the closure problem of isotropic turbulence.     

On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate

We study a dynamic contact problem for a thermoelastic von Kármán plate vibrating against a rigid obstacle. The plate is subjected to a perpendicular force and to a heat source. The dynamics is described by a hyperbolic variational inequality for deflections. The parabolic equation for a thermal strain resultant contains the time derivative of the deflection. We formulate a weak solution of the system and verify its existence using the penalization method. A detailed analysis of the velocity, acceleration, and reaction force of the solution is given. The singular nature of the dynamic contact makes it necessary to treat the acceleration and contact force as time-dependent measures with nonzero singular parts in the zones of contact. Accordingly, the velocity field over the plate suffers (global) jumps at a countable number of times with natural physical interpretations of the signs of the jumps.     

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     

Application of Topological Degree to the Study of Bifurcation in von Kármán Equations

The aim of this paper is to illustrate the use of topological degree for the study of bifurcation in von Kármán equations with two real positive parameters and for a thin elastic disk lying on the elastic base under the action of a compressing force, which may be written in the form of an operator equation F(x, , ) = 0 in some real Banach spaces X and Y. The bifurcation problem that we study is a mathematical model for a certain physical phenomenon and it is very important in the mechanics of elastic constructions. We reduce the bifurcation problem in the solution set of equation F(x, , ) = 0 at a point (0, ₀, ₀) X × IR ₊ ² to the bifurcation problem in the solution set of a certain equation in IR ⁿ at a point (0, ₀, ₀) IR ⁿ × IR ₊ ², where n = dim Ker F _x (0, ₀, ₀) and F _x (0, ₀, ₀): X Y is a Fréchet derivative of F with respect to x at (0, ₀, ₀). To solve the bifurcation problem obtained as a result of reduction, we apply homotopy and degree theory.     

Instabilities and bifurcations of von Kármán similarity solutions in swirling viscoelastic flow

We derive an asymptotic model that describes the swirling flow of a viscoelastic fluid between a rotating cone and a stationary plate when the gap angle, , is small and inertia is neglected. The model, which uses the Phan-Thien Tanner (PTT) constitutive law, is valid in the limit a 0 and for Deborah number, De, order unity. We show that the model admits similarity solutions of von Kármán type. A solution corresponding to a viscometric flow is obtained. This base flow, which exhibits shear thinning if the PTT parameter 0, is linearly stable if the Deborah number De is less than a critical value De _c and unstable if De > De _c . The critical Deborah number is a decreasing function of the retardation parameter , and an increasing function of . The method of Lyapunov-Schmidt is used to determine the nature of bifurcation when De is close to De _c . Our analysis shows that there is a supercritical pitchfork bifurcation at De=De _c .     

Global stabilization of a dynamic von Kármán plate with nonlinear boundary feedback

We consider a fully nonlinear von Kármán system with, in addition to the nonlinearity which appears in the equation, nonlinear feedback controls acting through the boundary as moments and torques. Under the assumptions that the nonlinear controls are continuous, monotone, and satisfy appropriate growth conditions (however, no growth conditions are imposed at the origin), uniform decay rates for the solution are established. In this fully nonlinear case, we do not have, in general, smooth solutions even if the initial data are assumed to be very regular. However, rigorous derivation of the estimates needed to solve the stabilization problem requires a certain amount of regularity of the solutions which is not guaranteed. To deal with this problem, we introduce a regularization/approximation procedure which leads to an approximating problem for which partial differential equation calculus can be rigorously justified. Passage to the limit on the approximation reconstructs the estimates needed for the original nonlinear problem.The material of M. A. Horn is based upon work partially supported under a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship. I. Lasiecka was partially supported by National Science Foundation Grant NSF DMS-9204338.