Edge-buckling in stretched thin films under in-plane bending

The wrinkling instabilities of a stretched rectangular thin film subjected to in-plane bending are investigated within the framework of the linearised Donnell-von Kármán bifurcation equation for thin plates. One of our principal objectives is to assess the role played by the finite bending stiffness of the film on the linear wrinkling mechanism. To this end, we employ a non-homogeneous linear pre-bifurcation solution and cast the corresponding eigenvalue problem as a singularly-perturbed differential equation with variable coefficients. Numerical simulations of this problem reveal the existence of two different regimes for the behaviour of the lowest eigenvalue. Based on this observation, a WKB analysis is carried out in order to capture the dependence of the critical wrinkling load on the wavelength of the localised oscillatory buckling pattern and the stiffness of the elastic film. The validity of the analytical results is corroborated by independent numerical computations of the eigenvalues using the method of compound matrices.     

The bending of a circular membrane on a linearly deformed foundation

The axisymmetric problem of the bending of a circular transversely-loaded membrane (i.e., a thin plate having no flexural stiffness), which lies without friction on a linearly deformed foundation, where there is contact over the whole area of the membrane, is considered. The problem is reduced to the combined investigation of a differential equation for the bending of the membrane and an integral equation of the first kind with an irregular kernel in the unknown contact pressure. The method of special orthonormalized polynomials and the regular asymptotic “large λ” method are used to solve the problem.     

Boundary layer analysis for the eigenvalues of hanging rod

The motion of a naturally straight inextensible flexible elastic hanging rod is formulated and then linearized about the straight solution. To solve this equation by separation of variables, an eigenvalue problem is derived. When the stiffness of the rod is small, the eigenvalue equation is a singular perturbation problem. This paper is devoted to solving this eigenvalue problem by boundary layer analysis when the stiffness is suitably small, especially on the analytic approximate solutions of the first several eigenvalues and eigenfunctions. The first three eigenvalues are also compared with the numerical results computed by a finite difference method. The excellent agreement shows the efficiency of the boundary layer analysis.     

A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite deflection analysis of elastic plate by the boundary element method

An integral equation formulation for finite deflection analysis of thin elastic plates is presented, based on general nonlinear differential equations which are equivalent to the von Kármán equations and by virtue of generalized Green identities. Boundary element discretization is applied and a relaxation iterative approach is employed to solve the nonlinear plate bending problems. A number of numerical examples are given; the results of computation are compared with the analytical solutions and good agreement is observed. It appears that the approach developed in this paper is effective.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

Fehlerabschätzungen für Eigenwertnäherungen nach der Ersatzkernmethode bei Integralgleichungen

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by replacing the original kernel by an approximate kernel. This procedure results in a linear algebraic eigenvalue problem. In this paper we investigate the order of convergence of this method for simple eigenvalues and corresponding eigenfunctions. Our results confirm some numerically observed superconvergence phenomena.

     

On the convergence of the Nyström method for the integral equation eigenvalue problem

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and a framework for its error analysis was introduced by Noble [15]. In this paper the convergence of the method is considered when the integral operator is a compact operator from a Banach spaceX intoX.     

Duality Theory in Nonlinear Buckling Analysis for von Kármán Equations

The nonlinear eigenvalue problem in buckling analysis is studied for von Kármán plates. By using the general duality theory developed by Gao-Strang [1, 2] it is proved that the stability criterion for the bifurcated state depends on a reduced complementary gap function. The duality theory is established for nonlinear bifurcation problems. This theory shows that the nonlinear eigenvalue problem is eventually equivalent to a coupled quadratic dual optimization problem. A series of equivalent variational principles are constructed and a lower bound theorem for the first eigenvalue of the buckling factor is proved.     

Asymptotic Behavior of a Structure Made by a Plate and a Straight Rod

This paper is devoted to describing the asymptotic behavior of a structure made by a thin plate and a thin perpendicular rod in the framework of nonlinear elasticity. The authors scale the applied forces in such a way that the level of the total elastic energy leads to the Von-K′arm′an’s equations (or the linear model for smaller forces) in the plate and to a one-dimensional rod-model at the limit. The junction conditions include in particular the continuity of the bending in the plate and the stretching in the rod at the junction.     

Wrinkling Behavior of Highly Stretched Rectangular Elastic Films via Parametric Global Bifurcation

We consider the wrinkling of highly stretched, thin rectangular sheets—a problem that has attracted the attention of several investigators in recent years, nearly all of which employ the classical Föppl–von Kármán (F–K) theory of plates. We first propose a rational model that correctly accounts for large mid-plane strain. We then carefully perform a numerical bifurcation/continuation analysis, identifying stable solutions (local energy minimizers). Our results in comparison to those from the F–K theory (also obtained herewith) show: (i) For a given fine thickness, only a certain range of aspect ratios admit stable wrinkling; for a fixed length (in the highly stretched direction), wrinkling does not occur if the width is too large or too small. In contrast, the F–K model erroneously predicts wrinkling in those very same regimes for sufficiently large applied macroscopic strain. (ii) When stable wrinkling emerges as the applied macroscopic strain is steadily increased, the amplitude first increases, reaches a maximum, decreases, and then returns to zero again. In contrast, the F–K model predicts an ever-increasing wrinkling amplitude as the macroscopic strain is increased. We identify (i) and (ii) as global isola-center bifurcations—in terms of both the macroscopic-strain parameter and an aspect-ratio parameter. (iii) When stable wrinkling occurs, for fixed parameters, the transverse pattern admits an entire orbit of neutrally stable (equally likely) possibilities: These include reflection symmetric solutions about the mid-plane, anti-symmetric solutions about the mid-line (a rotation by π radians about the mid-line leaves the wrinkled shape unchanged) and a continuously evolving family of shapes “in-between”, say, parametrized by an arbitrary phase angle, each profile of which is neither reflection symmetric nor anti-symmetric.     

Nonlinear bending and buckling for strain gradient elastic beams

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions are derived through a variational method. New terms are considered, already introduced for linear cases, indicating the importance of the cross-section area, in addition to moment of inertia in bending of thin beams. Those terms strongly increase the stiffness of the thin beam. The non-linear theory is applied to buckling problems of thin beams, especially in the study of the postbuckling behaviour.     

On the eigenvalue problem of a class of linear partial difference equations

In this paper, the eigenvalue problem of a class of linear partial difference equations is studied. The results concern the existence of eigenvalues, their character (real, positive), as well as the behavior of its eigenfunctions (positivity, oscillation). Moreover a theorem is given concerning the existence of a unique solution of an associated non-homogeneous partial difference equation. The results generalize previously known results for ordinary linear difference equations. The method used is a functional-analytic one, which transforms the eigenvalue problem for the difference equation into the equivalent problem of the eigenvalues of an operator defined on an abstract separable Hilbert space.     

A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Analysis of load transfer in thermo–elastic membranes

The flexural stiffness can be negligible in the analysis of extremely thin walled shell structures. Modeling these structures as membranes simplifies the theoretical formulation and reduces the computational effort. However, in case of compressive in–plane loads, the prediction of the load transition behavior by means of the membrane theory may be incorrect, if the wrinkling phenomenon is not taken into account. Therefore wrinkling algorithms were established in the past. Thermal strains influence the occurrence of wrinkling and the state of membrane forces. In order to analyze thermo-mechanical effects in conjunction with membrane wrinkling, the Roddeman wrinkling theory was modified. For small strains the incorporation of thermal effects into the wrinkling algorithm is straight forward. A method for large strains was developed and elaborated for thermoelastic rubber–like materials. The wrinkling algorithm is easy to implement into existing FE-programs. Results of numerical analysis are presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

超声速蒙皮颤振微分积分方程的特征值问题

根据二维线化理论讨论超声速薄钣的动力稳定性,导致一类新颖的数学物理问题:非自共轭Volterra型四阶微分积分方程的复特征值问题.求得这一气动弹性系统的严格解.与其它近似分析对比,本法的临界曲线与实验数据符合良好,在低超声速范围不存在发散问题.此外,在数学物理实质方面,发现:(1)颤振频谱与固有频谱有互为间隔现象;(2)临界Mach数有简并现象.指出本法可以推广应用于三维机翼模型和燃气轮中叶栅的超声速颤振问题.