Wrinkle-free solutions of circular membrane problems

Axisymmetric deformations of a plane circular membrane subjected to an axial surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering the small-finite-deflection theory of Föppl-Hencky as well as a simplified version of Reissner's nonlinear theory of thin shells of revolution, the determination of the principal stresses in the membrane is shown to reduce to the solution of a nonlinear second-order ODE. In the Föppl case, the basic equation becomes singular when the membrane edge is free of traction. Nevertheless, existence and uniqueness of nonnegative solutions for zero edge traction is proved in both Föppl and Reissner models. Thereby, a limit curve for the radial displacement at the boundary is induced in the Reissner case, which subdivides the actual parameter range into complementary domains of existence and nonexistence of tensile solutions. The limit curve is studied, analytically and numerically. Finally, a maximum principle is established in order to determine the more restricted subdomain of those parameters which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.     

Analytic and numerical solutions in the theory of elastic plates

ABSTRACT

Numerical approximations of the solution of a boundary value problem when an exact solution is not available can be constructed by means of a variety of methods. In this paper, we present a technique that is based on the integral representation of the solution of an elliptic problem and the properties of the associated layer potentials. The procedure is illustrated in application to the mathematical model of bending of plates with transverse shear deformation in a finite domain, in the presence of Dirichlet, Neumann, and Robin conditions prescribed on the boundary.     

Radially isospectral annular membranes

^** Email: h.gottlieb{at}griffith.edu.au By utilizing transformations of both the radial coordinate andthe radial wave function, densities of annular membranes whichare radially isospectral to any given radial density are produced.In particular, new families of annular membrane densities arefound which are radially isospectral to a uniform membrane.Some new generalizations for completely isospectral annularmembranes (including angle-dependent densities) are also given.     

Fundamental solutions in the linear theory of consolidation for elastic solids with double porosity

This paper concerns the theory of consolidation for elastic solids with double porosity, and the governing fully coupled linear quasi-static equations are considered. The system of these equations is based on the equilibrium equations for a solid, conservation of fluid mass, the effective stress concept, and Darcy’s law for material with double porosity. Two levels of spatial cases of consolidation theory for a solid with double porosity are considered: equations of steady vibrations and equations of equilibrium. The fundamental solutions of these equations are constructed by means of elementary functions. Finally, the basic properties of these solutions are established.     

Axisymmetric isospectral annular plates and membranes

A simpler ‘conditional’ form of the vibrating plateequation for which all the physical parameters may be spatiallyvarying is found when the relationship D(1 – v) = constantis satisfied between flexural rigidity and Poisson's ratio.A method of coordinate inversion is used to find a family ofinhomogeneous clamped annular plates (with axisymmetric spatiallyvarying density, thickness, Young's modulus, and Poisson's ratio)which have the same complete frequency spectrum as a standardhomogeneous clamped annular plate of the same inner and outerradii. In particular, these inhomogeneous plates have inversefourthpower radial dependence of density. General radial power-lawtransformations are also considered, leading to conclusionsfor the axisymmetric vibration spectra. Various results forcircular and annular membranes also emerge from the analysis.     

Asymptotically radial solutions in expanding annular domains

In this paper we consider the problem

$\left\{\begin{array}{ll}-\Delta u=u^{p}\quad {\rm in}\, \Omega_R,\\ u=0 \quad \quad \quad {\rm on}\, \partial\Omega_R,\quad\quad\quad (0.1)\end{array}\right.$

where p > 1 and Ω _R is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as \({R \longrightarrow \infty}\). The main goal of the paper is to prove, for large R, the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.

     

Incompressible nonlinearly elastic thin membranes

Nonlinearly elastic thin membrane models are derived for hyperelastic incompressible materials using Γ-convergence arguments. We obtain an integral representation of the limit two-dimensional energy owing to a result of singular functionals relaxation due to Ben Belgacem [ESAIM Control Optim. Calc. Var. 5 (2000) 71–85 (electronic)]. To cite this article: K. Trabelsi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Continuation of some solutions in potential theory and their applications to elastic contact and crack problems

The following mixed boundary value problem is considered: arbitrary tangential displacements are prescribed inside a circle, while the tangential and normal stresses outside the circle are zero. In this case, a direct and simple formula is derived for the tangential displacements outside the circle in terms of prescribed displacements inside, thus making the tangential displacement known all over the boundary. The original problem is no longer mixed, and the complete solution becomes readily available. Another solution for the case, when arbitrary tangential displacements are prescribed outside a circle, is derived in a similar manner. The reciprocal theorem is used to derive the continuation formulae for the tangential stresses inside and outside a circle. Application of these results to contact and crack problems is demonstrated.     

On the stability of finitely deformed elastic membranes

Zusammenfassung Die Arbeit befaßt sich mit einigen Stabilitätsproblemen bei elastischen Membranen unter endlichen Deformationen und konservativer Belastung. Im Teil I wird die Änderung der Verzögerungsenergie diskutiert, hervorgerufen durch eine kleine der endlichen Anfangsverformung übergelagerten Deformation. Im Hinblick auf spätere numerische Anwendungen folgt die Herleitung von empirischen Formänderungsfunktionen aus experimentellen Resultaten von Treloar für vulkanisierte Gummimembranen. Für ein Blatt unter gleichmäßigem Zug wird die Stabilität bei fehlender Belastung oder bei eingespanntem Rand untersucht.     

On the stability of finitely deformed elastic membranes

Summary The stability of a uniformly inflated and extended circular thin tube of elastic material is examined. Stability criteria are obtained for fixed ends and for a tube with a fixed end and an end with a rigid closure. The criteria apply to tubes under constant internal pressure, inflated by a gas, or filled with an incompressible fluid. The stability of a spherical membrane inflated by a gas is also investigated. Numerical results are presented for an empirical strain-energy function for rubber-like materials.

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Zusammenfassung Der Teil II bringt die Untersuchung eines aufgeblasenen dünnen Schlauches mit Kreisquerschnitt aus elastischem Material, für den sich bei festen Enden und im Fall mit einem festen Ende und einem starren Verschluß Stabilitätskriterien ergeben. Diese Kriterien lassen sich auf Rohre, die mit einem Gas oder mit einer inkompressiblen Flüssigkeit gefüllt sind, anwenden. Ferner wird die Stabilität einer mit Gas aufgeblasenen sphärischen Membran betrachtet. Abschließend werden numerische Resultate für eine empirische Formänderungsfunktion für gummiähnliche Materialien angegeben.

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For Part I see Z. ang. Math. Phys.22, 1016 (1971).     

Forced nonlinear oscillations of elastic membranes

An initial-boundary-value problem is considered for the spatially two-dimensional damped Boussinesq equation with a forcing term. The problem in question is argued to serve as a model for describing small nonlinear oscillations of a circular membrane under the influence of acoustic pressure. Eigenfunction expansion method is used for constructing the global-in-time solution of the problem in question. Existence and uniqueness follow from the construction. Long-time asymptotics is computed on the basis of the obtained series representation. Numerical simulations are conducted. The algorithm shows excellent convergence properties.     

Nonperturbative solutions in the electroweak theory

We apply the Bogoliubov compensation principle to the gauge electroweak interaction. We demonstrate a spontaneous appearance of an anomalous gauge-invariant effective three-boson interaction. The nontrivial solution of the compensation equations uniquely determines the form factor of the anomalous interaction and the parameters of the theory including the value of the gauge electroweak coupling in satisfactory agreement with its experimental value.     

Fundamental solutions in the theory of microfluids

This paper is concerned with the theory of non-heat-conducting microfluids. We consider a specialized microfluid continuum theory in which the micromotions consist of the intrinsic rotations and stretch. First, we establish a representation of Galerkin type for the solutions to the field equations. Then, we derive the fundamental solutions for the differential system governing the motion in the case of steady vibrations.      

Transmission problems in the theory of elastic hemitropic materials

The purpose of this article is to investigate mixed transmission-boundary value problems for the differential equations of the theory of hemitropic (chiral) elastic materials. We consider the differential equations corresponding to the time harmonic dependent case, the so called pseudo-oscillation equations. Applying the potential method and the theory of pseudodifferential equations we prove uniqueness and existence theorems of solutions to the Dirichlet, Neumann and mixed transmission-boundary value problems for piecewise homogeneous hemitropic composite bodies and analyze their regularity properties. We investigate also interface crack problems and establish almost best regularity results.     

Non-linear bending analysis of multi-layer orthotropic annular/circular graphene sheets embedded in elastic matrix in thermal environment based on non-local elasticity theory

In this paper, the large deflection of multi-layer orthotropic annular/circular graphene sheets is investigated based on the non-local elasticity theory. The plate is considered to be in thermal environment. The equilibrium equations are derived in terms of generalized displacements and rotations considering the FSDT non-local elasticity theory and the van der Waals interaction between the layers. In order to solve the governing equations, the differential quadrature method (DQM) which is an accurate numerical method and a new semi-analytical polynomial method (SAPM) are applied. By applying DQM or SAPM, the ODE's would be converted to non-linear algebraic equations. In continue, the Newton–Raphson iterative scheme is applied to solve the obtained non-linear algebraic equations. The results of DQM and SAPM are compared. Although, the SAPM's formulation is considerably simpler than DQM, however, the results of two methods are so close to each other. The results are validated with the other available researches. The effect of small scale parameter, temperature effects on non-local results, the value of van der Waals interaction between the layers for bi-layer and triple layers graphene sheet, different values of elastic foundation matrix and load for various small scale parameters, the comparison between local and non-local deflections and linear to non-linear analysis are investigated.