Buckling of a non-Euclidean annular plate

The effect of free edges of a monoatomic graphene sheet leads to excess edge energy due to the reconstruction of dangling bonds. Molecular static calculations show, that individual carbon atoms near the edge are displaced out of plane for relaxed nanoribbons [1]. In this work we are considering the effect of excess edge energy for almost circular graphene patches. To tackle this problem in the framework of continuum mechanics we are modelling the edge effect with a non-Euclidean plate model. A linear stability analysis of the flat configuration leads to the stability boundary in the parameter plane. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modelling of twin helical spring under tensile loading

This paper presents a 3D geometric modelling of a twin helical spring and its finite element analysis to study the spring mechanical behaviour under tensile axial loading. The spiralled shape graphic design is achieved through the use of Computer Aided Design (CAD) tools, of which a finite element model is generated. Thus, a 3D 18-dof pentaedric elements are employed to discretise the complex “wired-shape” of the spring, allowing the analysis of the mechanical response of the twin spiralled helical spring under an axial load. The study provides a clear match between the evolution of the theoretical and the numerical tensile and compression normal stresses, being of sinusoidal behaviour. The overall equivalent stress isovalues increases radially from 0° to 180°, being maximal on the internal radial zone at the section 180°. On the other hand, the minimum stress level is located in the centre of the filament cross section.     

Buckling for an elastoplastic plate with an incremental constitutive relation

We study the buckling of an elastoplastic plate loaded on its edges by a planar thrust(t)F(x ₁ ,x ₂ ) depending on time. We consider an incremental constitutive relation which permits the distinction between loading and unloading. This could be obtained by taking a Prandtl-Reuss relation, but, for technical reasons, we restrict ourselves to an approximation of this relation. We give a necessary condition for the existence of a nontrivial solution (together with the zero one) after a timet ₀ ; an important feature is that this condition takes into account the whole story of the loading and cannot be expressed in terms of its value at a single time. We prove the existence of bifurcated branches, for some sufficient condition involving the slope of at timet ₀ , and describe their local behavior; we point out the fact that buckling can actually occur when the loading is decreasing.Part of this work was done while the authors were visiting the I.M.A. in Minneapolis, Minnesota, USA.     

Vibrations of an annular plate under the influence of a force rotating at a constant velocity

An exact solution is obtained for a dynamic problem concerning the bending of circular and annular plates under the influence of a force rotating at a constant angular velocity. It is shown that this problem is related to the problem of the bending of a plate under the influence of a harmonic load. Velocities leading to resonance are found and their first values are calculated for circular hinged and rigidly fastened plates.Translated from Dinamicheskie Sistemy, No. 4, pp. 62–65, 1985.     

Stress distribution in an annular retarded-elastic plate with cylindrical anisotropy

Formulas for the stresses and displacements in an annular plate with cylindrical anisotropy are obtained with allowance for the time factor. Different cases of loading of the inside and outside edges of the plate are examined.Mekhanika Polimerov, Vol. 2, No. 3, pp. 403–412, 1966     

Temperature stresses in an annular anisotropic plate with temperature-dependent heat transfer coefficient

The nonsteady temperature stresses are determined for an annular plate, an infinite plate with a circular opening, and a circular plate, whose coefficients of heat transfer from the lateral surfaces are functions of temperature.Physico-Mechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 5, pp. 949–950, September–October, 1971.     

The thermally stressed state of a round plate heated by an annular heat source

Using Hankel and Laplace transforms we solve an axisymmetric dynamic problem of thermoelasticity for a round plate under the action of an annular heat source. We carry out a numerical study of the temperature field and stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 58–60.     

Buckling of clamped circular arches subjected to a point load

Summary This note presents the critical values of the downward point load applied at the crown of clamped circular arches with inextensible centroidal line. The calculations are based on the Euler theory of the initially curved elastica undergoing large deflections. In contrast to the two-highed circular arches the buckling is not always of the sidesway type; for most practical dimensions of the clamped arch the instability is characterized by snap-through.

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Zusammenfassung Mit Hilfe der nichtlinearen Eulerschen Theorie wird die Stabilität des beidseitig eingespannten und durch eine mittige Einzelkraft belasteten Kreisbogenträgers untersucht. Man nimmt dabei an, der Bogen sei genügend schlandk und die Dehnung der Stabachse vernachlässigbar klein. Im Gegensatz zum Zweigelenkbogen zeigt sich die Instabilität des steilen, eingespannten Bogenträgers eher in einem Durchschlagen als im seitlichen Ausweichen.

     

Electroelastic state of an inhomogeneous piezoceramic layer under symmetric loading

A procedure for solving three-dimensional mixed symmetric problems of electroelasticity theory for a piezoceramic layer with an inclusion, which is weakened with a through hole, is proposed. The boundary-value problem is reduced to a system of 12k (k = 1, 2, …) integrodifferential equations. Expressions for stresses characterizing the stress state of the inhomogeneous layer are found. Calculation results for characteristic stresses are presented.     

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

We consider the axisymmetric deformation of an initially spherical,porous vesicle with incompressible membrane having finite resistanceto in-plane shearing, as the vesicle is compressed between parallelplates. We adopt a thin-shell balance-of-forces formulationin which the mechanical properties of the membrane are describedby a single dimensionless parameter, C, which is the ratio ofthe membrane's resistance to shearing to its resistance to bending.This results in a novel free-boundary problem which we solvenumerically to obtain vesicle shapes as a function of plateseparation, h. For small deformations, the vesicle contactseach plate over a small circular area. At a critical value ofplate separation, h_TC, there is a transcritical bifurcationfrom which a new branch of solutions emerges, representing buckledvesicles which contact each plate along a circular curve. Forthe values of C investigated, we find that the transcriticalbifurcation is subcritical and that there is a further saddle-nodebifurcation (fold) along the branch of buckled solutions ath = h_SN (where h_SN > h_TC). The resulting bifurcation structureis commensurate with a hysteresis loop in which a sudden transitionfrom an unbuckled solution to a buckled one occurs as h is decreasedthrough h_TC and a further sudden transition, this time froma buckled solution to an unbuckled one, occurs as h is increasedthrough h_SN. We find that h_SN and h_TC increase with C, thatis, vesicles that resist shear are more prone to buckling.     

Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under uniaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported stepped and stiffened rectangular plate with uniformly distributed compressive forces in one direction is discussed. The plate is divided into 400 rectangular meshes. The partial derivatives are approximated using central difference formula. Three hundred and sixty-one equations are formed and using MATLAB the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped plates and the results are presented in tables and graphs for ready use by designers. Two separate tables are presented in comparison with the exact buckling factor for stepped rectangular plate by Xiang, Wei and Wang. The results give a close agreement.     

The stressed state of a thermosensitive annular plate of variable thickness

We find an exact solution of the problem of displacements and stresses in a polar-orthotropic annular plate of variable thickness with physico-mechanical properties of the material depending on temperature. The solutions are obtained as generalized and degenerate hypergeometric functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 42–45.     

Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under Biaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported (a) stepped and stiffened rectangular thin plate (b) linear and non-linear variation of thickness (c) uniformly distributed compressive forces in both directions (d) uniformly distributed compressive force in y direction and non-uniform distribution of compressive force in x-direction is discussed. The thin plate is divided into 900 rectangular meshes. The partial derivatives are approximated using central difference formula. Eight hundred and forty one equations are formed and using the program developed and the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped and non prismatic plates and the results are presented in tables and graphs for ready use by designers. Buckling factors for some cases are presented in the form of three separate tables and compared with the values obtained by Xiang, Wei and Wang. The results are in close agreement.     

An accurate method to predict the stress concentration in composite laminates with a circular hole under tensile loading

The paper presents a theoretical-numerical hybrid method for determining the stresses distribution in composite laminates containing a circular hole and subjected to uniaxial tensile loading. The method is based upon an appropriate corrective function allowing a simple and rapid evaluation of stress distributions in a generic plate of finite width with a hole based on the theoretical stresses distribution in an infinite plate with the same hole geometry and material. In order to verify the accuracy of the method proposed, various numerical and experimental tests have been performed by considering different laminate lay-ups; in particular, the experimental results have shown that a combined use of the method proposed and the well-know point-stress criterion leads to reliable strength predictions for GFRP or CFRP laminates with a circular hole. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 531–570, July–August, 2007.     

弹性杆的动态屈曲模态

本文提出了屈曲相关初缺陷的概念,采用最优模态分析方法,由Bernolli-Euler梁方程出发给出了弹性杆在齐次边条件下的动态屈曲模态、由此对两端固支弹性杆的动态屈曲模态进行了讨论.