Three-dimensional Nonlinear Theory Research Articles

Wrinkling of a film/substrate bilayer with periodic material properties: An assessment of the Winkler foundation model

The Winkler foundation model is often used to analyze the wrinkling of a film/substrate bilayer under compression, and it can be rigorously justified when both the film and substrate are homogeneous and the film is much stiffer than the substrate. We assess the validity of this model when the substrate is still homogeneous but the film has periodic material properties in the direction parallel to the interface. More precisely, we assume that each unit cell is piecewise homogeneous, and each piece can be described by the Euler–Bernoulli beam theory. We provide analytical results for the critical compression when the substrate is viewed as a Winkler foundation with stiffness modeled either approximately (as in some previous studies) or exactly (using the Floquet theory). The analytical results are then compared with those from Abaqus simulations based on the three-dimensional nonlinear elasticity theory in order to assess the validity of the Euler–Bernoulli beam theory and the Winkler foundation model in the current context.

Dynamic Response Analysis of a Bulk Carrier by Nonlinear Hydroelastic Method

With increasing demands for huge ship dimensions and the wide use of high-strength steel, the influence of slamming and elastic structure on structural strength cannot be ignored. Therefore, in this paper, a three-dimensional (3D) nonlinear hydroelastic theory is introduced, in which the nonlinear hydrostatic restoring force caused by instantaneous wetted surface as well as slamming force are taken into consideration, and the bending moments with/without slamming effects are calculated, respectively. Numerical simulations of the dynamic response of a flexible hull at different speeds are carried out using the finite element analysis software MSC/PATRAN. By comparison with the results of classical beam theory, the accuracy of the dynamic analysis method is studied. Finally, the dynamic response method is compared with the quasi-static method and classical beam theory. By analyzing and quantifying the influence of forward speed and nonlinear factors on structural responses, the reasonable applicable conditions for different methods are discussed, which can be used as reference in the structure design of bulk carriers.

Branching of twins in shape memory alloys revisited

We study the branching of twins appearing in shape memory alloys at the interface between austenite and martensite. In the framework of three-dimensional non-linear elasticity theory, we propose an explicit, low-energy construction of the branched microstructure, generally applicable to any shape memory material without restrictions on the symmetry class of martensite or on the geometric parameters of the interface. We show that the suggested construction follows the expected energy scaling law, i.e., that (for the surface energy of the twins being sufficiently small) the branching leads to energy reduction. Furthermore, the construction can be modified to capture different features of experimentally observed microstructures without violating this scaling law. By using a numerical procedure, we demonstrate that the proposed construction is able to predict realistically the twin width in a Cu-Al-Ni single crystal and to estimate an upper bound to the number of the branching generations.

Buckling prognosis for thin elastic shallow shells

This article addresses several problems that are related to the elastic stability of thin shells and that are due to inconsistencies between the experimental data and predictions based on the equations of the shallow-shell theory. The above contradictions are solved within the three-dimensional nonlinear theory of elasticity. In particular, the dynamic approach enables a prediction of the moment of bifurcation for very thin shells under momentless stress through the analysis of asymptotically built two-dimensional equations. Appearance and development of the dents and patterns in initially ideal shells that precede the buckling are also explained in this article. The dents represent solitonic waves that are detectible by acoustic devices. Therefore, it is possible to reduce the risk of failure of thin shells by using acoustic devices to monitor their conditions. The article covers two types of experiments with thin shells when the loads are close to buckling. These experiments enable to assess the safety buckling factor under technical operation of shells.

УСТОЙЧИВОСТЬ ЦИЛИНДРА ИЗ МАТЕРИАЛА МУРНАГАНА ПРИ РАСТЯЖЕНИИ, СЖАТИИ И РАЗДУВАНИИ

The problem of equilibrium and stability of a hollow cylinder subjected to simultaneous uniaxial tension/compression and inflation is considered within the framework of the three-dimensional nonlinear theory of elasticity. To describe the mechanical properties of the material of the cylinder five-constant Murnaghan model is used. By the semi-inverse method the three-dimensional problem is reduced to the study of a nonlinear boundary value problem for an ordinary second-order differential equation. For most sets of material parameters known from the literature, the presence of a falling section in the stretching/inflation diagram, indicating the possible existence of instability zones even in the area of tensile stresses, has been found numerically. The stability analysis was carried out using a bifurcation approach based on linearization of the equilibrium equations in the neighborhood of the constructed solution by means of the method of imposing a small strain on a finite one. The value of a particular deformation characteristic, for which non-trivial solutions of a homogeneous boundary-value problem exist for the equations of neutral equilibrium obtained in the linearization process, was identified with the critical value of the loading parameter, i.e. value at which the system loses stability. As a rule, the coefficient of stretching/shortening of the cylinder and the coefficient of increase/decrease of its internal or external radius were chosen as such parameters. On the plane of the above-mentioned deformation characteristics the areas of stability under tension and compression, as well as under compression by external force and inflation by internal pressure, are constructed. The forms of possible of stability loss depending on the type of stress state are constructed, and the effect on the stability of material and geometric parameters is studied.

A three-dimensional nonlinear beam–wave interaction theory for common traveling wave tubes

A three-dimensional (3-D) nonlinear theory model of beam–wave interaction for common traveling wave tubes (TWTs) is described. The equation governing the amplitude of electromagnetic wave is derived analogously to Poynting’s theorem. The electron dynamics are treated using the 3-D Lorentz force equations. RF space charge fields are obtained from solutions of the Helmholtz equations. In the model, the RF field profiles for cold structure are represented by the digitized RF field calculated by a finite-element software, HFSS. Because the digitized RF field can be obtained in the same way, the 3-D simulation code can be used to simulate common TWTs, such as helix TWTs, coupled-cavity TWTs, and even the folded waveguide TWTs. Results from the 3-D code are compared with those from 1-D code, experiment and the existing analytical theories for three types of slow wave structures.

Cabling in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

Experimental and Numerical Analysis of Hull Girder Vibrations and Bow Impact of a Large Ship Sailing in Waves

It is of great importance to evaluate the hull structural vibrations response of large ships in extreme seas. Studies of hydroelastic response of an ultra large ship have been conducted with comparative verification between experimental and numerical methods in order to estimate the wave loads response considering hull vibration and water impact. A segmented self-propelling model with steel backbone system was elaborately designed and the experiments were performed in a tank. Time domain numerical simulations of the ship were carried out by using three-dimensional nonlinear hydroelasticity theory. The results from the computational analyses have been correlated with those from model tests.

A well-posed finite-strain model for thin elastic sheets with bending stiffness

An accurate, well-posed two-dimensional model incorporating stretching and bending effects, suitable for analyzing the wrinkling pattern in stretched sheets, is derived from three-dimensional nonlinear elasticity theory.

A three-dimensional time-dependent theory for helix traveling wave tubes in beam-wave interaction

This paper presents a three-dimensional time-dependent nonlinear theory of helix traveling wave tubes for beam-wave interaction. The radio frequency electromagnetic fields are represented as the superposition of azimuthally symmetric waves in a vacuum sheath helix. Coupling impedance is introduced to the electromagnetic field equations' stimulating sources, which makes the theory easier and more flexible to realize. The space charge fields are calculated by electron beam space-charge waves expressed as the superposition solutions of Helmholtz equations. The focusing forces due to either a solenoidal field or a periodic permanent magnetic field is also included. The dynamical equations of electrons are Lorentz equations associating with electromagnetic fields, focusing fields and space-charge fields. The numerically simulated results of a tube are presented.

On the influence of the gravity force on the stability of an inhomogeneous layer under biaxial extension-compression

In the present paper, in the framework of the three-dimensional nonlinear theory of elasticity, we study the stability of a heavy layer under biaxial extension-compression. The elastic properties of the layer are assumed to be inhomogeneous along thickness and are described by a semilinear material model. We study the stability by using the bifurcation approach. By solving the linearized equilibrium equations, we obtain the critical curves and the stability domain in the plane of the loading parameters, for which we take the material elongation ratios along the coordinate axes lying in the layer plane. We analyze the influence of the layer thickness, specific weight, and material parameters on buckling. In particular, we find that, when studying stability, it is expedient to take the gravity force into account only if the layer rigidity decreases with increasing depth.

Verification of the earthquake resistance of the arch-gravity dam of the Sayano-Shushenskaya HPP using the body of mathematics of the three-dimensional nonlinear wave theory of earthquake resistance

Verification of the earthquake resistance of the arch-gravity dam of the Sayano-Shushenskaya HPP using the body of mathematics of the three-dimensional nonlinear wave theory of earthquake resistance

Free electron laser with curved parallel plate waveguide

A free electron laser (FEL) amplifier with a curved parallel plate waveguide and a planar wiggler is presented. A set of operating equations for this FEL amplifier is derived by using the three-dimensional nonlinear theory. The characteristics including the evolution of power, efficiency and bandwidth of the FEL operating at the frequency of 94 GHz are numerically analyzed. The effects of electron energy spread and wiggler taper on saturation efficiency are also studied.

Nonlinear study on Raman regime free electron laser with elliptical waveguide

A free electron laser (FEL) amplifier with an elliptical waveguide and a planar wiggler is proposed. A set of coupled operating equations for this FEL operating in the Raman regime is derived using three-dimensional nonlinear theory. The numerical simulation code is developed, and the characteristics of this FEL amplifier are analysed numerically. Our numerical simulations are conducted to model a 34.6 GHz amplifier with an electron beam with an energy of 3.1 MeV. The results indicate that the elliptical waveguide is more suitable for a FEL with a planar wiggler system than a rectangular waveguide.

Free electron laser amplifier with elliptic-groove waveguide

Based on an analysis of the wave propagation characteristics of an elliptic-groove waveguide, the use of this waveguide as a free electron laser (FEL) amplifier high frequency structure is proposed. A set of operating equations for this FEL amplifier is derived by using three-dimensional nonlinear theory. A numerical simulation code is developed, and the characteristics including the evolution of power, efficiency and bandwidth of the FEL amplifier are analysed numerically. The effects of electron energy spread and wiggler taper on the saturation efficiency are also considered.

On worm-like chain models within the three-dimensional continuum mechanics framework

In this paper, we review critically some basic models derived from the statistics of long-chain molecules and then discuss the status of such models within the three-dimensional nonlinear theory of elasticity. We draw attention to some deficiencies of certain worm-like chain (WLC) models when viewed within the three-dimensional continuum mechanics framework. Modifications of the WLC models motivated by such considerations and avoiding these deficiencies are then discussed and shown to correspond well with data generated by the exact WLC model.

Review of hydroelasticity theories for global response of marine structures

Existing hydroelastic theories are reviewed. The theories are classified into different types: two-dimensional linear theory, two-dimensional nonlinear theory, three-dimensional linear theory and three-dimensional nonlinear theory. Applications to analysis of very large floating structures (VLFS) are reviewed and discussed in details. Special emphasis is placed on papers from China and Japan (in native languages) as these papers are not generally publicly known in the rest of the world.

Three-dimensional nonlinear analysis of creep in concrete filled steel tube columns

This paper proposes a based on 3D-VLE (three-dimensional nonlinear viscoelastic theory) three-parameters viscoe- lastic model for studying the time-dependent behaviour of concrete filled steel tube (CFT) columns. The method of 3D-VLE was developed to analyze the effects of concrete creep behavior on CFT structures. After the evaluation of the parameters in the proposed creep model, experimental measurements of two prestressed reinforced concrete beams were used to investigate the creep phenomenon of three CFT columns under long-term axial and eccentric load was investigated. The experimentally obtained time-dependent creep behaviour accorded well with the curves obtained from the proposed method. Many factors (such as ratio of long-term load to strength, slenderness ratio, steel ratio, and eccentricity ratio) were considered to obtain the regularity of influence of concrete creep on CFT structures. The analytical results can be consulted in the engineering practice and design.

MODELING HETEROGENEOUS MARTENSITIC WIRES

We give a direct derivation of a theory of martensitic heterogeneous wires in the zero thickness and homogenization limit via a convergence result. We start from three-dimensional nonlinear hyperelasticity theory augmented by a term of interfacial energy of the van der Waals type. The derivation involves no a priori choice of asymptotic expansion or Ansatz. It yields a wire theory with two Cosserat vector fields. A formal derivation is given of higher theories for homogeneous wires, which yields one corrector for the deformation of the central line and two correctors for the Cosserat vector fields. Finally, we present a few numerical results.

Three-dimensional nonlinear theory of travelling wave tubes and simulation

A three-dimensional (3D) nonlinear theory of travelling wave tubes (TWTs) is developed, which includes a fundamental radio frequency (RF) and harmonics. When the instantaneous bandwidth exceeds an octave, the harmonic is generated and the mutual coupling between the harmonic and the fundamental RF can be observed in TWTs due to nonlinear interaction between the electron beam and the RF. At low frequencies the harmonic has an obvious effect. Based upon Tien's disc model, a plastic 3D super-particle model is proposed to improve the nonlinear analysis of TWTs. Numerical results employing a periodic magnetic focusing field are presented.