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).     

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.     

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 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.     

Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system

The existence of solutions is proved for a full system of dynamic von Kármán equations expressing vibrations of geometrically nonlinear viscoelastic plate, the viscosity of which has the character of a short memory. The system models the behaviour of a bridge. The in-plane acceleration terms are taken into account. The boundary contact conditions for plane displacements and possibly the contact with the rigid support are considered.     

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.     

Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system

The existence of solutions is proved for a full system of dynamic von Kármán equations expressing vibrations of geometrically nonlinear viscoelastic plate, the viscosity of which has the character of a short memory. The system models the behaviour of a bridge. The in-plane acceleration terms are taken into account. The boundary contact conditions for plane displacements and possibly the contact with the rigid support are considered.     

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.     

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.     

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.     

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.     

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.     

关于一阶剪切板的Kármán型精化理论

依据Mindlin_Reissner理论 ,着重研究一阶剪切板的K rm n型精化理论 ,并推导出仅以挠度和应力函数为未知变量的广义K rm n型大挠度方程 ,适用于复合材料上复合构造剪切板的非线性分析·　在当前板的精化理论中 ,消去的两个转角以隐含形式作用于板的整体变形·　这一工程理论适用于各种计及横向剪切复合材料板、正交各向异性中厚板和夹层板等的线性和非线性分析·　容易发现 ,针对具体的工程应用 ,由该方程可直接获得相应的退化形式 ,并且与文献所载的一致·     

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.     

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.     

Buckling of a von Kármán Plate Adhesively Connected to a Rigid Support Allowing for Delamination: Existence and Multiplicity Results

The present paper deals with an eigenvalue problem for a hemivariational inequality, arising in the study of a mechanical problem: the buckling of a von Kármán plate adhesively connected to a rigid support with delamination effects. For this eigenvalue problem an existence result is obtained by applying a critical point method suitable for nonconvex nonsmooth functions. Further, a result concerning the multiplicity of solutions is proved. The mechanical interpretation of these results is briefly discussed.     

Asymptotic justification of dynamical equations for generalized Marguerre–von Kármán anisotropic shallow shells

In this paper, using technics from asymptotic analysis, we show that the three-dimensional dynamical model for a non-linearly elastic shallow shell made of anisotropic material, with boundary conditions of generalized Marguerre–von Kármán’s type, reduces to two-dimensional dynamical model.