Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation

Nonlinear Dynamics - In this article, we study some soliton-type analytical solutions of Schrödinger equation, with their numerical treatment by Galerkin finite element method. First of all,...     

Stability of the stationary solutions of the Allen–Cahn equation with non-constant stiffness

We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We assume the stiffness to be a positive function of the field and we discuss the stability of the stationary solutions proving both linear and local non-linear stability.     

Experimental analysis on membrane wrinkling under biaxial load – Comparison with bifurcation analysis

This paper presents a detailed experimental study of the formation and evolution of the wrinkle pattern that form in flat elastic and isotropic membranes under the action of in-plane tension. The experiments were carried out on a cruciform specimen stretched along two uncoupled axes using various loading paths. The wrinkled shapes of the membrane were digitized by using a full-field measurement based on the fringe analysis method.Over this experiment, several phenomena were observed: the mechanism of wrinkle division, the influence of the membrane thickness on the wrinkle pattern, and the reproducibility of a kinematic configuration of wrinkles.The main result is that non-unique wrinkle shapes have been observed over repeated experiments for nominally identical boundary conditions. The uncertainty of the experimental wrinkle shape has been explained using comparison with the results of a post-buckling finite element analysis.     

Lateral buckling of beams with shear deformations – A hyperelastic formulation

The equilibrium and buckling equations are derived for the lateral buckling of a prismatic straight beam. A consistent finite strain constitutive law is used, which is based on a hyperelastic model for an isotropic material. The kinematics of the cross-sectional deformations are based on a Timoshenko type beam displacement of the cross-sectional plane using Euler angles and two shear finite rotations coupled with warping taken normal to the displaced plane. Also derived are the second order approximations to the displacements, curvatures, twist and internal actions. The constitutive relationships for the internal actions reveal new coupling terms between the bending moments, torsion and bimoment, which are functions of the cross-sectional warping and shear deformations. New Wagner type nonlinear torsion terms are derived which are functions of the warping of the cross-sectional plane, and are coupled to the twisting and shear deformations of the cross-section. Solutions are determined for the lateral buckling of a prismatic monosymmetric beam under pure bending and the flexural–torsional buckling under axial compression. For the flexural–torsional buckling problem it is found that the Euler type column buckling formula is consistent with Haringx’s column buckling formula while the torsional buckling formula is different to conventional equations. The second variation of the total potential is also derived. The effects of shear deformations are explored by examining the non-dimensional lateral buckling equation for a simply supported beam.     

The microbuckling failure of Dyneema~? composites under compression

Two grades of Dyneema~?composite laminates with the commercial designations of HB26 and HB50 were cut into blocks with or without an edge crack and compressed in the longitudinal fiber direction. The cracked and uncracked specimens show similar compressive responses including failure pattern and failure load. The two grades of Dyneema~? composites exhibits different failure modes: a diffuse, sinusoidal buckling pattern for Dyneema~? HB50 due to its weak matrix constituent and a kink band for Dyneema~? HB26 due to its relatively stronger matrix constituent. An effective finite element model is used to simulate the collapse of Dyneema~? composites, and the sensitivity of laminate compressive responses to the overall effective shear modulus, interlaminar shear strength, thickness and imperfection angle are investigated. The change of failure mode from kink band to sinusoidal buckling pattern by decreasing the interlaminar shear strength is validated by the finite element analyses.     

Particle resolved direct numerical simulation of a liquid–solid fluidized bed: Comparison with experimental data

Particle-resolved direct numerical simulations of a 3-D liquid–solid fluidized bed experimentally investigated by Aguilar-Corona (2008) have been performed at different fluidization velocities (corresponding to a range of bed solid volume fraction between 0.1 and 0.4), using Implicit Tensorial Penalty Method. Particle Reynolds number and Stokes number are O(100) and O(10), respectively. In this paper, we compare the statistical quantities computed from numerical results with the experimental data obtained with 3-D trajectography and High Frequency PIV. Fluidization law predicted by the numerical simulations is in very good agreement with the experimental curve and the main features of trajectories and Lagrangian velocity signal of the particles are well reproduced by the simulations. The evolution of particle and flow velocity variances as a function of bed solid volume fraction is also well captured by the simulations. In particular, the numerical simulations predict the right level of anisotropy of the dispersed phase fluctuations and its independence of bed solid volume fraction. They also confirm the high value of the ratio between the fluid and the particle phase fluctuating kinetic energy. A quick analysis suggests that the fluid velocity fluctuations are mainly driven by fluid–particle wake interactions (pseudo-turbulence) whereas the particle velocity fluctuations derive essentially from the large scale flow motion (recirculation). Lagrangian autocorrelation function of particle fluctuating velocity exhibits large-scale oscillations, which are not observed in the corresponding experimental curves, a difference probably due to a statistical averaging effect. Evolution as a function of the bed solid volume fraction and the collision frequency based upon transverse component of particle kinetic energy correctly matches the experimental trend and is well fitted by a theoretical expression derived from Kinetic Theory of Granular Flows.     

Comparison between Kelvin–Helmholtz instability experiments on OMEGA and simulation results using the CRASH code

The Center for Radiative Shock Hydrodynamics (CRASH) at the University of Michigan has developed a Eulerian radiation-hydrodynamics code with dynamic adaptive mesh refinement, CRASH, which can model high-energy-density laser-driven experiments. One of these experiments, performed previously on the OMEGA laser facility, was designed to produce and observe the Kelvin–Helmholtz instability. The target design included low-density carbonized-resorcinol-formaldehyde (CRF) foam layered on top of polyamide–imide plastic, with a sinusoidal perturbation on the interface and with the assembled materials encased in beryllium. The results of a series of CRASH simulations of these Kelvin–Helmholtz instability experiments are presented. These simulation results show good agreement both quantitatively and qualitatively with the experimental data.     

Peeling experiments of ductile thin films along ceramic substrates – Critical assessment of analytical models

Two types of peeling experiments are performed in the present research. One is for the Al film/Al₂O₃ substrate system with an adhesive layer between the film and the substrate. The other one is for the Cu film/Al₂O₃ substrate system without adhesive layer between the film and the substrate, and the Cu films are electroplated onto the Al₂O₃ substrates. For the case with adhesive layer, two kinds of adhesives are selected, which are all the mixtures of epoxy and polyimide with mass ratios 1:1.5 and 1:1, respectively. The relationships between energy release rate, the film thickness and the adhesive layer thickness are measured during the steady-state peeling process. The effects of the adhesive layer on the energy release rate are analyzed. Using the experimental results, several analytical criteria for the steady-state peeling based on the bending model and on the two-dimensional finite element analysis model are critically assessed. Through assessment of analytical models, we find that the cohesive zone criterion based on the beam bend model is suitable for a weak interface strength case and it describes a macroscale fracture process zone case, while the two-dimensional finite element model is effective to both the strong interface and weak interface, and it describes a small-scale fracture process zone case.     

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function $(P,F)$ should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.     

Moiré analysis of thick composites with embedded optical fibers

Moiré interferometry is used with Fourier transform fringe analysis to investigate the strain fields in a region local to 125-m uncoated silica optical fibers embedded in a quasi-isotropic graphite/PEEK thick composite compression specimens. Analysis of several regions in several specimens showed no measurable strain concentrations resulting from the embedded optical fibers, even though the optical fibers clearly alter the local microarchitecture of the host material system.Paper was presented at the 1993 SEM Spring Conference on Experimental Mechanics held in Dearborn, MI on June 7–9.     

Infinitely many solutions of p-Laplacian equations with limit subcritical growth

We discussed a class of p-Laplacian boundary problems on a bounded smooth domain,the nonlinearity is odd symmetric and limit subcritical growing at infinite.A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified.We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical.     

Pulling out a peptide chain from β-sheet crystallite: Propagation of instability of H-bonds under shear force

Anti-parallel β-sheet crystallite as the main component of silk fibroin has attracted much attention due to its superior mechanical properties. In this study, we examine the processes of pulling a peptide chain from β-sheet crystallite using steered molecular dynamics simulations to investigate the rupture behavior of the crystallite. We show that the failure of β-sheet crystallite was accompanied by a propagation of instability of hydrogen-bonds(H-bonds) in the crystallite. In addition, we find that there is an optimum size of the crystallite at which the H-bonds can work cooperatively to achieve the highest shear strength. In addition, we find that the stiffness of loading device and the loading rates have significant effects on the rupture behavior of β-sheet crystallite.The stiff loading device facilitates the rebinding of the Hbond network in the stick-slip motion between the chains,while the soft one suppresses it. Moreover, the rupture force of β-sheet crystallites decreases with loading rate. Particularly, when the loading rate decreases to a critical value, the rupture force of the β-sheet crystallite becomes independent of the loading rates. This study provides atomistic details of rupture behaviors of β-sheet crystallite, and, therefore, sheds valuable light on the underlying mechanism of the superior mechanical properties of silk fibroin.     

Effective Young’s modulus of biopolymer composites with imperfect interface

The effective elastic behaviour of a biopolymer composite with random microstructure is studied. The effective elasticity modulus is computed as function of the protein fraction with the hypothesis of an imperfect interface. This interface is handled considering normal and tangential stiffness parameters. The methodology assumes the direct import of real microstructures of starch–protein (zein) composite, obtained using Confocal Laser Scanning Microscopy (CLSM), into a finite element model. A static linear analysis is conducted to determine the influence of interface rigidities, zein fraction and interface quantity on the effective properties. Predictions show that the effective modulus is nonlinearly correlated to the product of stiffness parameters for which the normal stiffness is the most influential parameter. The effective property dependence with respect to interface related variables and zein fraction shows a logarithmic influence of the stiffness parameters when increasing zein fraction.     

A micro–macro method for predicting the shear strength of brittle rock under compressive loading

Microcracks have great significance for shear strength of brittle rock in compression. A major challenge of this area is to establish the correlation of microcracks and macroscopic shear strength. A new micro–macro method is presented to predict the shear strength of brittle rock in compression. This method incorporates the microcrack model suggested by Ashby, Mohr–Coulomb failure criterion and a crack-strain relation. This crack–strain relation is presented to link the crack growth and axial strain by combining the micro and macro definitions from rock damage. The shear strength and stress–strain relationship of Jinping marble are theoretically investigated in detail. The rationality of this suggested method is verified by using the experimental results founded on Jinping marble. Effects of the initial microcrack size, friction coefficient and confining pressure on internal friction angle, cohesion, and shear strength are also discussed.     

Multilamellar vesicles (“onions”) under shear quench: pathway of discontinuous size growth

The size growth of multilamellar vesicles (MLV; “onions”) of a non-ionic surfactant system composed of 40 wt.% C₁₀E₃ in D₂O was investigated by shear quench experiments, i.e. by reducing the shear rate. Structural changes were monitored by means of viscosity, rheo-small angle neutron scattering, rheo-small angle light scattering and optical microscopy. Discontinuous growth of vesicle size was observed in addition to the continuous process reported previously. In the discontinuous size growth, first lamellar domains are formed, which afterwards transform into larger MLVs. The lamellar domains coexist with the original small vesicles leading to a bimodal size distribution. The transition of the lamellar domains into new MLVs follows the same pathway as was observed for the MLV formation from aligned lamellar phases.     

Mechanical behavior of hexagonal honeycombs under low-velocity impact – theory and simulations

Based on the cells’ collapse mechanisms of the hexagonal honeycombs revealed from the numerical simulations under the low-velocity impact, an analytical model is established to deduce the crushing strength of the honeycomb and the stress at the supporting end both as functions of impact velocity, cell size, cell-wall angle, and the mechanical properties of the base material. The results show that the honeycomb’s crushing strength increases with the impact velocity, while the supporting stress decreases with the increase of the impact velocity. Combining with the dynamic predictions under the high-velocity impact in our previous work (Hu and Yu, 2010), the crushing strength of the honeycombs can be analytically predicted over wide range of crushing velocities. The analytical expression of the critical velocity is also obtained, which offers the boundary for the application of the functions of the honeycomb’s crushing strength under the low-velocity and the high-velocity impacts. All of the analytical predictions are in good agreement with the numerical simulation results.     

A nonlinear numerical simulation of a lab centrifuge with internal damping

This paper is about numerical simulations of dissipation processes in rotor shaft joints of rotor systems. Based on measurement results a nonlinear simulation model of a lab centrifuge is stated. The effects of internal damping in combination with nonlinear stiffness and friction in the rotor shaft joint of the lab centrifuge are worked out. It is shown that the nonlinearities cause the amplitudes to rest limited once increased amplitudes due to internal damping appear. One focus is the derivation of suitable force laws describing the mechanisms of the components within the connection.     

Numerical simulation and experimental verification of silicone oil flow over magnetic fluid under applied magnetic field

Two-layer flow of magnetic fluid and non-magnetic silicone oil was simulated numerically. The continuity equation, momentum equations, kinematic equation, and magnetic potential equation were solved in two-dimensional Cartesian coordinate. PLIC （piecewise linear integration calculation） VOF （volume of fluid） scheme was employed to track the free interface. Surface tension was treated via a continuous surface force （CSF） model that ensures robustness and accuracy. The influences of applied magnetic field, inlet velocity profile, initial surface disturbance of interface and surface tension were analyzed. The computed interface shapes at different conditions were compared with experimental observation.     

Overall longitudinal shear elastic modulus of a 1–3 composite with anisotropic constituents

The overall properties of a binary elastic periodic fiber-reinforced composite, with transversely isotropic constituents in an anti-plane-strain deformation state, are studied here for a cell periodicity of square type. This analysis considers four different orientations of the axis of transverse isotropy of constituents with respect to the direction of fibers. Each case is characterized by very simple closed-form expressions for the effective coefficients, which were obtained using the asymptotic homogenization method. Local problems defined on a periodic square unit cell are solved using Weierstrassian and Natanzon’s functions and perturbation theory relative to small anisotropy. In the isotropic limit, comparison with rigorous bounds and some well-known mixing rules are made. Also, comparisons with finite element calculations show that the derived closed-form formulae provide excellent results even for large anisotropy.