Magnetoelastic buckling of tokamak superconducting toroidal field coils

Based on the perturbed vector potential theory and principle of stationary potential energy, a finite element method is applied to finding the critical current of the magnetoelastic buckling of the toroidal field coils of the fusion reactor. Stress is laid on the effects of the tremendous inplane tension induced magnetically and the perturbation history of the current vectors.     

超导线圈在强磁场作用下的弹塑性变形分析

根据超导线圈受电磁力、热膨胀、弯曲应力作用的受力机制,首先理论研究了超导线圈在发生弹塑性变形时内部的应力、应变、位移,然后使用两段直线之间由过渡曲线相连接的简化模型作为超导线的应力-应变曲线的近似特性。利用有限元方法,数值模拟了电磁力作用下典型的Nb3Sn超导线材在不同边界条件下的应力、应变、位移分布情况。计算结果定量显示了超导线的弹塑性变形特性同电流密度的关系,初步预测了超导线圈发生塑性变形的区域和扩展以及支撑结构对超导磁体力学行为的影响。     

Flexible shell-theory and buckling of toroidal shells and tubes

Summary Flexible shells, designed to sustain elastic displacements, have a semimembrane type of deformation, described by simplified equations. In what follows, flexible shell equations are derived and used to analyse bending of curved tubes and buckling of tubes and toroidal shells under external pressure, as interdependent problems. Design formulas and curves are presented.

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Übersicht Flexible Schalen, die elastischen Verformungen ausgesetzt sind, genügen einem Semi-Membranspannungszustand. Vereinfachte Gleichungen für diesen Zustand werden hergeleitet. Damit werden die Probleme des Beulens von Torusschalen unter Außendruck, ein Spezialfall des allgemeineren Problems der Biegung und der Stabilität von Rohren, untersucht. Spannungen, Deformationen und kritischer Außendruck werden in Abhängigkeit von der Rohrkrümmung und von dem Abstand zwischen Querrippen oder Flanschen bestimmt.

     

HT—7U超导托克马克纵场线圈等效模量的数值分析

HT－7U托克马克装置的纵场线圈是由多种材料组成的具有周期性分布的大型复杂结构，线圈可视为由超导线、支撑结构和绝缘材料组成的复合材料，结构极其复杂。整个线圈工作在液氦温区。在设计阶段对其宏观等效力学性能进行数值分析计算是十分必要的，但要对整个线圈直接进行有限元分析或实验是极其困难的，主要利用均匀化方法对其进行等效处理，从而分析计算其宏观等效模量，为线圈的设计和评估提供参考依据。另外本文对均匀化方法的边界条件进行了一些改进，提出了更加合理的三维边界条件。     

Bending and buckling of thin strain gradient elastic beams

Bending of strain gradient elastic thin beams is studied adopting Bernoulli-Euler principle. Simple linear strain gradient elastic theory with surface energy is employed. The governing beam equations with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin beams. Those terms are missing from the existing strain gradient beam theories. Those terms increase highly the stiffness of the thin beam. The buckling problem of the thin beams is also discussed.     

Bending and buckling of a class of nonlinear fiber composite rods

An investigation of the mechanics of bending and buckling is carried out for a class of nonlinear fiber composite rods composed of embedded unidirectional fibers parallel to the rod axis. The specific class of composite considered is one in which the fibers interact with the matrix through a nonlinear Needleman-type cohesive zone [Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. ASME J. Appl. Mech. 54, 525-531; Needleman, A., 1992. Micromechanical modelling of interfacial decohesion. Ultramicroscopy 40, 203-214]. The primary decohesive mechanism active in bending and buckling of these composite rods is shear slip along the fiber-matrix interfaces allowing the use of a previously developed constitutive relation for antiplane shear response [Levy, A.J., 2000b. The fiber composite with nonlinear interface—part II: antiplane shear. ASME J. Appl. Mech. 67, 733-739]. The formulation requires the specification of a potential interface force-slip law that is assumed to permit interface failure in shear.Four cases of the bending and shearing of beams (concentrated or uniform load on a cantilever or a simply supported beam) are analyzed, each of which exhibits qualitatively distinct response. For certain values of interface parameters, the beam deflection or its gradient at a fixed location can change discontinuously with load. Furthermore, for interface parameter values within a certain range, singular surfaces will exist in uniformly loaded beams where there is a non-uniform distribution of shear stress along the beam length. These singular surfaces divide the beam into regions of maximal and minimal fiber slip and propagate with a rate that varies inversely as the square of the applied load. For other parameter values, singular surfaces will not exist and fiber slip will be diffuse.For the class of nonlinear composite considered, bifurcation and imperfection buckling of pinned-pinned columns is analyzed. For bifurcation buckling, a nonlinear eigenvalue problem is derived and the solution is obtained by Galerkin's method. It is demonstrated that critical loads are influenced by the initial slope, and hence the linear portion, of the interface force-slip relation but the post-buckling response, which in some sense resembles that of plastic buckling, is affected by the entire interface constitutive relation. Imperfection buckling is analyzed in a similar manner by assuming a slight initial curvature of the rod. Sensitivity of the response to imperfection magnitude is discussed as well.     

Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube

This paper studies the pure bending and bending-induced local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube (SWNT). The Airy stress-function method was employed to analyse the deformation of the matrix, and the cross-sectional change of the SWNT in bending was taken into account. A particular consideration was given to the effect of the SWNT’s radial flexibility on the strain/stress states and buckling. It was found that in thicker matrix layers the SWNT buckles locally at smaller bending angles and greater flattening ratios. This causes higher strains/stresses in the surrounding matrix and in turn degrades the strength of the nanocomposite structure.     

The simplified buckling criterion applied to plates with partial edge loading

The development is outlined of the first author's Simplified Plate Buckling Criterion which is based on nonlinear theory. For the case of uniaxial, symmetric, partial-edge loading, the work of the first author is compared with the numerical solutions of the second author and with the results of Yamaki by energy theory. The Simplified Criterion gives results with a minimum of computational effort well within practical accuracy requirements.

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Zusammenfassung Die Ableitung des auf nichtlineare Theorie begründeten vereinfachten Plattenbeulungskriterium des ersten Verfassers ist skizziert. Für den Fall der einachsig, symmetrisch, teilweis belagerten Platte werden die Ergebnisse des ersten Verfasser mit den rein numerischen Ergebnissen des zweiten Verfasser und mit dem auf Energietheorie begründeten Ergebnissen Yamakis verglichen. Mit sehr minimaler Rechnungsbemühung kann das vereinfachte Plattenbeulungskriterium Koeffiziente erzielen deren Exaktheit innerhalb der üblichen Sicherheitsgrenze liegt.

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Based in part on a paper presented to the 19th Midwestern Mechanics Conference, Ohio State University, Columbus, Ohio, Sept. 9–11, 1985.     

Magnetoelastic buckling of thin plates in a uniform transverse magnetic field

The buckling of thin soft ferromagnetic elastic plates in a uniform transverse magnetic field is discussed. Based on a linearized version of the general magnetoelastic equations, a plate equation is derived together with the pertinent boundary conditions. By searching a nontrivial solution of this homogeneous system, a buckling value can be found. As an example, the circular plate is considered. In this paper, Maxwell stresses are used, but it is shown that identical results can be obtained by using other models (especially Amperian current) for the magnetoelastic stresses.

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Zusammenfassung In dieser Arbeit wird das Knicken dünner ferromagnetischer, elastischer Platten in uniformen magnetischen Feldern behandelt. Die Ableitung einer Plattengleichung zusammen mit den bezüglichen Randesbedingungen wird auf die linearisierten Gleichungen für weiche ferromagnetische Medien begründet. Durch Lösung dieses Systems wird eine Knicklast ermittelt. Die Methode wird anhand des Beispieles der kreisförmigen Platte illustriert. In dieser Abhandlung werden Maxwell Spannungen verwendet, aber es wird gezeigt dasz andere Modelle (insonderheit Amperesche Stromstärke) für die magnetoelastischen Spannungen das gleiche Resultat bilden.

     

Numerical study on three-dimensional flow field of continuously rotating detonation in a toroidal chamber

Gaseous detonation propagating in a toroidal chamber was numerically studied for hydrogen/oxygen/nitrogen mixtures. The numerical method used is based on the three-dimensional Euler equations with detailed finiterate chemistry. The results show that the calculated streak picture is in qualitative agreement with the picture recorded by a high speed streak camera from published literature. The three-dimensional flow field induced by a continuously rotating detonation was visualized and distinctive features of the rotating detonations were clearly depicted. Owing to the unconfined character of detonation wavelet, a deficit of detonation parameters was observed. Due to the effects of wall geometries, the strength of the outside detonation front is stronger than that of the inside portion. The detonation thus propagates with a constant circular velocity. Numerical simulation also shows three-dimensional rotating detonation structures, which display specific feature of the detonationshock combined wave. Discrete burning gas pockets are formed due to instability of the discontinuity. It is believed that the present study could give an insight into the interesting properties of the continuously rotating detonation, and is thus beneficial to the design of continuous detonation propulsion systems.     

Influence of the hall effect on plasma compressibility in a strong toroidal magnetic field

The development of the tearing instability is studied in the presence of a high toroidal magnetic field and a high plasma conductivity. The variation of the plasma density is shown to be significant in this case. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 3–9, March–April, 1998.     

Helical buckling of a whirling conducting rod in a uniform magnetic field

We study the effect of a magnetic field on the behaviour of a conducting elastic rod subject to a novel set of boundary conditions that, in the case of a transversely isotropic rod, give rise to exact helical post-buckling solutions. The equations used are the geometrically exact Kirchhoff equations and both static (buckling) and dynamic (whirling) instability are considered. Critical loads are obtained explicitly and are given by a surprisingly simple formula. By solving the linearised equations about the (quasi-)stationary solutions we also find secondary instabilities described by (Hamiltonian-)Hopf bifurcations, the usual signature of incipient ‘breathing’ modes. The boundary conditions can also be used to generate and study helical solutions through traditional non-magnetic buckling due to compression, twist or whirl.     

环形激波聚焦流场特性的数值研究

针对环形激波聚焦过程产生的高温、高压特性,采用间断有限元方法模拟了环形激波在同轴圆柱形激波管内的聚焦流场特性。计算结果表明,采用间断有限元方法能够有效地捕捉激波聚焦过程形成的二次激波、涡环、三波交点和球面双马赫反射等主要流动特征。此外,通过改变环形管道内外半径对聚焦流场进行模拟发现,环形管道外径对中心轴线上聚焦峰值压力的大小和位置影响较小,环形管道内径对中心轴线上聚焦峰值压力的大小和位置影响较大。计算结果可以为工程应用提供一定的理论指导。     

Dynamic and buckling analysis of a thin elastic-plastic square plate in a uniform temperature field

The nonlinear models of the elastic and elasticlinear strain-hardening square plates with four immovably simply-supported edges are established by employing Hamilton‘s Variational Principle in a uniform temperature field. The unilateral equilibrium equations satisfied by the plastically buckled equilibria are also established. Dynamics and stability of the elastic and plastic plates are investigated analytically and the buckled equilibria are investigated by employing Galerkin-Ritz‘s method. The vibration frequencies, the first critical temperature differences of instability or buckling, the elastically buckled equilibria and the extremes depending on the final loading temperature difference of the plastically buckled equillibria of the plate are obtained. The results indicate that the critical buckling value of the plastic plate is lower than its critical instability value and the critical value of its buckled equilibria turning back to the trivial equilibrium are higher than the value. However, three critical values of the elastic plate are equal. The unidirectional snap-through may occur both at the stress-strain boundary of elasticity and plasticity and at the initial stage of unloading of the plastic plate.     

Nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under time-dependent torsional loads

The nonlinear analysis with an analytical approach on dynamic torsional buckling of stiffened functionally graded thin toroidal shell segments is investigated. The shell is reinforced by inside stiffeners and surrounded by elastic foundations in a thermal environment and under a time-dependent torsional load. The governing equations are derived based on the Donnell shell theory with the von K′arm′an geometrical nonlinearity,the Stein and McE lman assumption, the smeared stiffeners technique, and the Galerkin method. A deflection function with three terms is chosen. The thermal parameters of the uniform temperature rise and nonlinear temperature conduction law are found in an explicit form. A closed-form expression for determining the static critical torsional load is obtained. A critical dynamic torsional load is found by the fourth-order Runge-Kutta method and the Budiansky-Roth criterion. The effects of stiffeners, foundations, material,and dimensional parameters on dynamic responses of shells are considered.     

Bending of a circular plate on a nonuniform elastic foundation in partial contact with the plate

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 162–169, January–February, 1989.     

Vortex ring modelling of toroidal bubbles

During the collapse of a bubble near a surface, a high-speed liquid jet often forms and subsequently impacts upon the opposite bubble surface. The jet impact transforms the originally singly-connected bubble to a toroidal bubble, and generates circulation in the flow around it. A toroidal bubble simulation is presented by introducing a vortex ring seeded inside the bubble torus to account for the circulation. The velocity potential is then decomposed into the potential of the vortex ring and a remnant potential. Because the remnant potential is continuous and satisfies the Laplace equation, it can be modelled by the boundary-integral method, and this circumvents an explicit domain cut and associated numerical treatment. The method is applied to study the collapse of gas bubbles in the vicinity of a rigid wall. Good agreement is found with the results of Best (J. Fluid Mech. 251 79–107, 1993), obtained by a domain cut method. Examination of the pressure impulse on the wall during jet impact indicates that the high-speed liquid jet has a significant potential for causing damage to a surface. There appears to be an optimal initial distance where the liquid jet is most damaging.     

Analysis of dynamical buckling and post buckling for beams by finite segment method

IntroductionThe stability or dynamical buckling of structures is the parametrically exacted vibration ofstructures,and itis a nonlinear vibration problem[1].Dynamical stabilityof beams was studiedusing Galerkin variation method by Russian scholar,and B.B.…