Analytical solution for large deflections of a cantilever beam under nonconservative load based on homotopy analysis method

In this article, large deflection and rotation of a nonlinear beam subjected to a coplanar follower static loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). The present solution can be used in wide range of load and length for beams under large deformations. The results obtained from HAM are compared with those results obtained by fourth order Range Kutta method. Finally, the load‐displacement characteristics of a uniform cantilever under a follower force normal to the deformed beam axis are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:541–553, 2011     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations. Received: February 24, 2005     

Instability of solitary waves on Euler’s elastica

Stability of solitary waves in a thin inextensible and unshearable rod of infinite length is studied. Solitary-wave profile of the elastica of such a rod without torsion has the form of a planar loop and its speed depends on a tension in the rod. The linear instability of a solitary-wave profile subject to perturbations escaping from the plane of the loop is established for a certain range of solitary-wave speeds. It is done using the properties of the Evans function, an analytic function on the right complex half-plane, that has zeros if and only if there exist the unstable modes of the linearization around a solitary-wave solution. The result follows from comparison of the behaviour of the Evans function in some neighbourhood of the origin with its asymptotic at infinity. The explicit computation of the leading coefficient of the Taylor series of the Evans function near the origin is performed by means of the symbolic computer language. Received: April 6, 2004; revised: December 12, 2004     

不可压饱和多孔弹性梁的大挠度非线性数学模型

在孔隙流体仅存在沿梁轴线方向扩散的假定下,建立了微观不可压饱和多孔弹性梁大挠度问题的非线性数学模型.利用Galerkin截断法,研究了固定端不可渗透、自由端可渗透的饱和多孔弹性悬臂梁在自由端突加集中载荷作用下的非线性弯曲,得到了梁骨架的挠度、弯矩以及孔隙流体压力等效力偶等的时间响应和沿轴线的分布.比较了大挠度非线性和小挠度线性理论的结果,揭示了两者间的差异.研究发现大挠度理论的结果小于相应的小挠度理论结果,并且,大挠度理论的结果趋于其稳态值的时间小于相应的小挠度理论结果趋于其稳态值的时间.     

Failure analysis of highly predeformed beams used as flexure hinges

The trend to extend the working ranges of flexure hinges implies large deformations during operation. To conduct a failure analysis the total deformation is decomposed into desired deformation and deviations. In particular, a flexure hinge of leaf-spring type is examined. It is modeled by the theory of elastica. The resulting boundary value problem is solved numerically for the static case by Ritz's method. It is discretized into trial functions and their free coefficients are determined from the minimum of potential energy by optimization methods. The crucial point is that the elastic energy stored in the beam is formulated intrinsically, while the potential of external conservative loads is formulated in a space-fixed coordinate system. The well-known special case of buckling of a straight cantilever beam is used for verification. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On frequency–amplitude dependences for surface and internal standing waves

The analytic approach proposed by Sekerzh-Zenkovich [On the theory of standing waves of finite amplitude, Dokl. Akad. Nauk USSR 58 (1947) 551–554] is developed in the present study of standing waves. Generalizing the solution method, a set of standing wave problems are solved, namely, the infinite- and finite-depth surface standing waves and the infinite- and finite-depth internal standing waves. Two-dimensional wave motion of an irrotational incompressible fluid in a rectangular domain is considered to study weakly nonlinear surface and internal standing waves. The Lagrangian formulation of the problems is used and the fifth-order perturbation solutions are determined. Since most of the approximate analytic solutions to these problems were obtained using the Eulerian formulation, the comparison of the results, as an example the analytic frequency–amplitude dependences, obtained in Lagrangian variables with the corresponding ones known in Eulerian variables has been carried out in the paper. The analytic frequency–amplitude dependences are in complete agreement with previous results known in the literature. Computer algebra procedures were written for the construction of asymptotic solutions. The application of the model constructed in Lagrangian formulation to a set of different problems shows the ability to correctly reproduce and predict a wide range of situations with different characteristics and some advantages of Lagrangian particle models (for example, the bigger radius of convergence of an expansion parameter than in Eulerian variables, simplification of the boundary conditions, parametrization of a free boundary).     

Two-material optimal design for nonlinear elastica

The work considers an optimal design problem in the context of nonlinear elastica. More specifically, we deal with finding the best way of mixing fixed amounts of two different elastic materials, so as to minimize the tip deflection of a cantilever beam loaded on its free extreme under the assumption of large deflections. Applying an optimality criteria method to the relaxed problem, simulations give us numerical evidence that the original design problem admits classical solutions (i.e. there is no microstructure) and those are the same as the respective ones for the case of small deflections.     

Block toeplitz operators with rational symbol analytic at infinity

This paper concerns the problem of explicit inversion of a block Toeplitz operator with rational and analytic at infinity symbol. The necessary and sufficient conditions for the invertibility and explicit formulas for the inverse are given in terms of the realization of the symbol.     

纯弯曲矩形截面梁Ⅰ型单边裂纹端部的应力应变场及裂纹失稳扩展临界应力的计算

本文应用文［１］的分析方法，研究了纯弯曲矩形载面梁Ⅰ型单边裂纹端部的应力应变场，给出了裂纹尖端的应力应变分量和计算裂纹端部弹性变形区和变形强化区宽度的公式以及计算裂纹失稳扩展临界应力的方程组。最后用计算实例对裂纹失稳扩展临界应力方程组进行了验证，最大误差不超过０．１８％．     

Singularities at the tip of a crack on the interface of piezoelectric bodies

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezoelectric media under various boundary conditions on the crack surfaces. The singularity exponents form the spectrum of a certain polynomial pencil, and although explicit formulas are not available, this spectrum is described completely though. The mathematical results apply to problems in fracture mechanics. In this way the Griffith formulas are obtained for increments of energy functionals due to the growth of the crack, and the notion of energy release matrix is introduced. Normalization conditions for bases of singular solutions are proposed to adapt them to energy, stress, and deformation fracture criteria. Connections between these bases are determined, and additional properties of the deformation basis related to the notion of electric surface enthalpy are established. Bibliography: 44 titles. Dedicated to Vsevolod Alekseevich Solonnikov Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 241–271.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

Quantized Weyl algebras at roots of unity

We classify the centers of the quantized Weyl algebras that are polynomial identity algebras and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.     

On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space

In this paper, we obtain accurate analytic free vibration solutions of rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected.     

Busy periods of discrete-time queues using the Lagrange implicit function theorem

In this paper, we study the busy-period distribution for discrete-time queues assuming a Bernoulli arrival process, with arbitrary service-time and batch-arrival distributions. We derive explicit analytic formulas for these distributions using the Lagrange Implicit Function Theorem applied to probability generating functions. The convenient coefficient operator notation used in these formulas leads to a computationally efficient method for obtaining the distributions in their entirety from these analytic formulas.     

Determination of the maximum stresses in a cantilever composed of material with deformation anisotropy

The results of a theoretical investigation of the state of stress of a cantilever with allowance for deformation anisotropy are presented. Expressions for determining the bending stresses in a beam with a profile having a specific radius of curvature at the support are obtained on the basis of the hypothesis of broken-line (cylindrical) sections. The expressions obtained are extended to the case of a beam made of material with viscoelastic properties. A numerical example is given.Institute of Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel'. Translated from Mekhanika Polimerov, No. 3, pp. 453–458, May–June, 1976.     

Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads

This paper considers a functionally graded cantilever beam with different modulus in tension and compression. The beam is subjected to bending loads, including pure bending, shear force at the free end and uniform pressure on the upper lateral, respectively. Its modulus values in tension and compression both change with the thickness coordinate as arbitrary functions, which could bring the beam a broader range of applications in engineering. The problem is treated as a plane stress case and described by Airy stress function. By using semi-inverse method, the elastic solutions for the beam are obtained, which can be easily degenerated into the ones for homogeneous beams. An example is finally presented to show the effect of nonhomogeneous materials with different modulus on the elastic field in a cantilever beam.     

Bending of a nonlinear orthotropic cantilever beam

The method of elastic solutions is employed to investigate the plane problem of the deformation of a cantilever beam of orthotropic glass-reinforced plastic under a concentrated load with allowance for the non-linear properties of the material. The first approximation of the stress function is given and the stress distribution over the cross section is calculated for a specific GRP.Mekhanika Polimerov, Vol. 2, No. 5, pp. 773–778, 1966     

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.     

Inverse spectral problems for difference operators with rational scattering matrix function

In this paper we obtain explicit formulas for the coefficients of a second order difference block operator if its spectral or its scattering functions are rational matrix functions analytic and invertible on the unit circle. The solutions are given in terms of realizations of the spectral or scattering function.     

压电梁的多项式解(Ⅱ)--典型问题解析解

从正交各向异性压电介质平面问题，对于材料3个特征根互不相等情况下，以3个拟调和位移函数表达位移、电势、应力和电位移的通解出发，利用调和多项式的显式表达式，结合试凑法，给出了平面压电梁的若干典型问题的解析解，包括悬臂压电梁自由端作用横向集中力和点电荷，悬臂雎电梁表面作用线性电势和均布载荷，以及两端简支压电梁作用均布载荷等的解析解．