Thermal and tensile loading effects on size-dependent vibration response of traveling nanobeam by wavelet-based spectral element modeling

This paper presents a novel approach for size-dependent vibration response of a nanostructure under tensile and thermal loads, traveling in its axial direction at a constant velocity. The traveling nanostructure is modeled as an Euler–Bernoulli nanobeam based on modified couple stress theory. Wavelet-based spectral element model (WSEM) is performed for analyzing vibration of the system. Imposing WSEM reduces the governing partial differential equation of the system to a set of ordinary differential equations. The roles of nanobeam velocity, tensile and thermal loads on vibration and wave characteristics, and divergence/flutter instability are scrutinized by WSEM. The validity and accuracy of resulting responses are inspected by comparing with numeric values obtained from spectral element and finite element methods, and whenever possible, with those available in the literature.     

An experimental and analytical treatment of matrix cracking in cross-ply laminates

The development of damage in cross-ply Hercules AS4/3502 graphite/epoxy laminates has been investigated. Specific endeavors were to identify the mechanisms for initiation and growth of matrix cracks and to determine the effect of matrix cracking on the stiffness loss in cross-ply laminates. Two types of matrix cracks were identified. These include both straight and curved cracks. The experimental study of matrix crack damage revealed that the curved cracks formed after the straight cracks and followed a repeatable pattern of location and orientation relative to the straight cracks. Therefore, it was postulated that the growth mechanism for curved cracks is driven by the stress state resulting from the formation of the straight cracks. This phenomenon was analytically investigated by a finite-element model of straight cracks in a cross-ply laminate. The finite-element results provide supporting evidence for the postulated growth mechanism. The experimental study also revealed that the number of curved cracks increased with the number of consecutive 90-deg plies. Finally, experimental results show as much as 10-percent degradation in axial stiffness due to matrix cracking in cross-ply graphite/epoxy laminates.     

Quantitative characterization of the interfacial adhesion of Ni thin film on steel substrate: A compression-induced buckling delamination test

A compression-induced buckling delamination test is employed to quantitatively characterize the interfacial adhesion of Ni thin film on steel substrate. It is shown that buckles initiate from edge flaws and surface morphologies exhibit symmetric, half-penny shapes. Taking the elastoplasticity of film and substrate into account, a three-dimensional finite element model for an edge flaw with the finite size is established to simulate the evolution of energy release rates and phase angles in the process of interfacial buckling-driven delamination. The results show that delamination propagates along both the straight side and curved front. The mode II delamination plays a dominant role in the process with a straight side whilst the curved front experiences almost the pure mode I. Based on the results of finite element analysis, a numerical model is developed to evaluate the interfacial energy release rate, which is in the range of 250–315 J/m² with the corresponding phase angle from −41° to −66°. These results are in agreement with the available values determined by other testing methods, which confirms the effectiveness of the numerical model.     

Modeling low Reynolds number incompressible flows with curved boundaries using SPH

In this paper, a method that can be used to model low Reynolds number incompressible flows with curved boundaries using SPH was presented. In contrast to that usually used for the flows with flat and straight boundaries, the hydrostatic pressure gradient is treated as a variable body force in this method, and thus, it can be applied to simulate the flows with curved boundaries. Three numerical examples of low Reynolds number incompressible flows, including Poiseuille flow, flow in a section of blood vessel with a local expansion, and flow between inclined plates were calculated to test the method. The results obtained with the proposed method were in good agreement with the analytical solutions. It implies that the method presented in this paper can be successfully used to simulate low Reynolds number incompressible flows not only with flat and straight boundaries but also with curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Two-point correlation and integral scale of spatially advancing curved channel flow at friction-velocity-based Reynolds number 550

Two-point correlation and integral scale of curved channel turbulent flow were examined for onset and growth of large-scale vortex. Away from the two walls of the straight channel, the two-point correlation of streamwise velocity has large negative values around local minimum at y/δ = 1.6 and those of radial and spanwise components take small negative values at the same area. The large-scale motion in the straight channel is thus indicated to have conspicuous streamwise component and weak secondary motion. In the curved channel, two-point correlation of radial velocity decreased around the minimum away from the wall (y/δ = 1.6) and large negative values extended to the whole channel width. It is thus indicated that the large-scale vortex originated from the large-scale structures with conspicuous streamwise component in the outer-wall side, grew and expanded to the full channel width of the curved part. The integral scale integrating two-point correlation conditionally on the negative values greatly decreased to denote the onset and growth of the large-scale motion quantitatively. The local minima of two-point correlation for radial velocity were used as indicator of the large-scale vortex and near-wall small-scale vortex. The magnitude of it at the channel center area was as low as one thirds of those of near-wall area in the straight channel, and however increased and surpassed those of near-wall area in the curved channel. Therefore, large-scale vortices were weaker than the small-scale motion in the straight part of the channel, and they grew and surpassed the small-scale motion in strength in the curved part of the channel.     

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.     

Self-exciting vibrations and Hopf's bifurcation in non-linear stability analysis of rail vehicles in a curved track

The main objective of this article is to present the authors' view of and results on non-linear lateral stability of rail vehicles in a curved track. Three elements are exploited in order to secure this objective. Firstly, physical genesis of the problem is discussed, and its similarity to straight track analysis is emphasised. Results of the theories of self-exciting vibrations and bifurcation are the key elements here. Secondly, the method suitable for analysis in a curved track is presented. New necessary elements, extending the better established methods for straight track are clearly mentioned and described. The methodology of building original stability maps, being the basis for the analysis and valid for whole range of curve radii and straight track is represented. Thirdly, a sample of the analysis is shown in order to give the idea how the method can be utilised. The case study refers to the influence of wheel/rail profiles on the stability in circularly curved track and straight track as well. Two different pairs of wheel/rail profiles are used and the corresponding results compared. The main contributions of the article are: a discussion of the physical nature of phenomena related to the stability in a curved tracks, and the method (procedure) established for the reasons of the analysis. Another and more general contribution is our say in the hot polemics on the advisability of stability analysis in curves and the advantages of the non-linear critical speed over the linear one.     

Geometric nonlinear formulation for curved beams with varying curvature

Based on exact Green strain of spatial curved beam, the nonlinear strain-displacement relation for plane curved beam with varying curvature is derived. Instead of using the previous straight beam elements, curved beam elements are used to approximate the curved beam with varying curvature. Based on virtual work principle, rigid-flexible coupling dynamic equations are obtained. Physical experiments were carried out to capture the large overall motion and the strain of curved beam to verify the present rigid-flexible coupling formulation for curved beam based on curved beam element. Numerical results obtained from simulations were compared with those results from the physical experiments. In order to illustrate the effectiveness of the curved beam element methodology, the simulation results of present curved beam elements are compared with those obtained by previous straight beam elements. The dynamic behavior of a slider-crank mechanism with an initially curved elastic connecting rod is investigated. The advantage of employing generalized-α method is pointed out and the special nonlinear dynamic characteristics of the curved beam are concluded.     

The elastic theory of thin-walled open cross sections with local deformations

A set of governing equations for nonlinear theory of spatially curved elastic beams of thin-walled open cross section composed of straight rectangular elements is presented explicitly in the Lagrangian form. It is shown that local deformations, i.e. in-plane distortion of the cross section may easily be taken into account by the use of the analytical model proposed by Epstein and Murray. The essential feature which distinguishes the present work from Epstein and Murray's is the use of an auxiliary element when the axial curve of beams is not located on the cross section. This enables us to select arbitrarily the axial curve of rods. For the engineering theory of rods, the simplified governing equations for the nonlinear and linear theories with and without local deformations are derived from the rigorous nonlinear theory by employing the thinness assumption. It is also shown that the reduced linear theory without local deformations agrees with the Vlasov theory.     

The curved beam element and its convergence rate

The quasi-conforming element of the curved beam and shallow curved beam is given in this paper. Numerical examples illustrate that the quasi-conforming elements of the curved beam and shallow curved beam which is used to approximate the curved beam have better accuracy than the straight beam element. The curved beam element constructed by displacement method can not satisfy rigid body motion condition and the very fine grids have to be used in order to satisfy rigid body motion condition approximately.In this paper it is proved that the straight beam element and the quasi-conforming element of the curved beam and shallow curved beam, when element size is reduced infinitely, have convergence rate with the same order O(l²) and when regular elements are used l is the element length.The Project Supported by National Natural Science Foundation of China.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

Compact waves on planar elastic rods

Planar Kirchhoff elastic rods with non-linear constitutive relations are shown to admit traveling wave solutions with compact support. The existence of planar compact waves is a general property of all non-linearly elastic intrinsically straight rods, while intrinsically curved rods do not exhibit this type of behavior.     

On the bifurcations of the Lamé solutions in plane-strain elasticity

We consider the in-plane bifurcations experienced by the Lamé solutions corresponding to an elastic annulus subjected to radial tension on the curved boundaries. Numerical investigations of the relevant incremental problem reveal two main bifurcation modes: a long-wave local deformation around the central hole of the domain, or a material wrinkling-type instability along the same boundary. Strictly speaking, the latter scenario is related to the violation of the Shapiro–Lopatinskij condition in an appropriate traction boundary-value problem. It is further shown that the main features of this material instability mode can be found by using a singular-perturbation strategy.     

An aeroelastic model for composite rotor blades with straight and swept tips. Part I: Aeroelastic stability in hover

An analytical model for predicting the aeroelastic behavior of composite rotor blades with straight and swept tips is presented. The blade is modeled by beam type finite elements along the elastic axis. A single finite element is used to model the swept tip. The non-linear equations of motion for the finite element model are derived using Hamilton's principle and based on a moderate deflection theory and accounts for: arbitrary cross-sectional shape, pretwist, generally anisotropic material behavior, transverse shears and out-of-plane warping. Numerical results illustrating the effects of tip sweep, anhedral and composite ply orientation on blade aeroelastic behavior are presented. It is shown that composite ply orientation has a substantial effect on blade stability. At low thrust conditions, certain ply orientations can cause instability in the lag mode. The flap-torsion coupling associated with tip sweep can also induce aeroelastic instability in the blade. This instability can be removed by appropriate ply orientation in the composite construction.     

Non-linear dynamics of curved beams. Part 2, numerical analysis and experiments

The aim of this work is to formulate a model for the study of the dynamics of curved beams undergoing large oscillations. In Part 1, the interest was oriented to the formulation of a consistent analytical model and to obtain the equations of motion in weak form. In Part 2, a case-study is considered and the response for various initial curved configurations, obtained by varying the initial curvature, is analyzed. Both the free and the forced problems are considered: the linear free dynamics are studied to detect how the initial configuration affects the modal properties and to enlighten the typical phenomena of frequency coalescence and avoidance; the forced dynamics are then studied for different internal resonance conditions to enlighten the phenomenon of the dynamic instability under a shear periodic tip follower force and to describe the various classes of post-critical motion. The results of experimental tests conducted on a slightly imperfect straight beam prototype are eventually discussed.     

Modelling and simulation of rigid bodies transportation by means of rotating flexible rollers

The paper deals with a modelling and simulation approach intended to the solution of a problem of rigid bodies transportation using rotating rollers. It is motivated by problematic phenomenon of ceramic tiles transportation inside long kilns during firing. The row of tiles, which is initially straight at the kiln beginning, is coming out curved at the kiln end or after the passage through a certain part of the kiln. The main aim of the work is the development of the modelling and simulation method for the verification of the problems arising from the tile movement through very long kilns and for the understanding of the problem causation. The multibody dynamics approach is chosen for the development of a comprehensive methodology for the numerical simulation of the tiles and rollers movement. Two approaches to the modelling of the system of tiles and rollers using the SIMPACK simulation tool are shown in the paper. The more suitable one is based on the Polygonal Contact Model developed for the fast and efficient contact analysis between complexly shaped rigid bodies. It can be concluded that the problems of the curved row of tiles are influenced by the flexibility of bent rollers and also occur for ideally cylindrical rollers. The presented simulation methodology can be generally used for the investigation of similar transportation systems.     

On the mechanism by which nose bluntness suppresses second-mode instability

A physical mechanism by which nose bluntness suppresses second-mode instability is proposed.Considered are 7 degree half-angle straight cones with nose bluntness radii of 0.15 mm, 3.556 mm,5 mm, 9.525 mm, 12.7 mm and 25.4 mm at tunnel conditions relevant to the AFOSR-Notre Dame Large Mach 6 Quiet Tunnel. It is shown that second-mode suppression is achieved via entropy layer modulation of the basic state density gradient. A weakening of the density gradient disrupts the acoustic resonance necessary to sustain second-mode growth. These results are consistent with the thermoacoustic interpretation which posits that second-mode instability can be modeled as thermoacoustic resonance of acoustic energy trapped within an acoustic impedance well.Furthermore, the generalized inflection point criterion of Lees and Lin is applied to develop a criterion for the existence of second-mode instability based on the strength of the basic state density gradient.     

充液弯管连续/离散模型振动的动刚度解法

由充液弯管三维振动模型切入，应用动刚度法构建了弯管及直管单元的振动求解方法，进而用于组装求解充液管系的振动，可同时适用于含弯管单元的连续模型或只含直管单元的离散模型；通过算例对比，证明动刚度法比传递矩阵法和有限元法在计算效率和精度上有所提升；与充液L型管道振动实验测得的加速度频响曲线对比，验证了本文对于管道组装的计算方法的有效性，此外还分析了连续模型和离散模型的区别及适用范围。     

Wake-boundary layer interaction subject to convex and concave curvatures and adverse pressure gradient

Measurements of mean velocity and turbulent quantities have been carried out when the wake of a symmetrical airfoil interacts with the boundary layer on the (i) walls of a straight duct/diffuser and (ii) convex and concave walls of a curved duct/diffuser. The effects of adverse pressure gradient and of curvatures on the interaction are studied separately and in combination. Six cases are considered, viz. with (i) neither pressure gradient nor curvature, (ii) adverse pressure gradient and no curvature, (iii) and (iv) convex curvature with zero and adverse pressure gradients, respectively, (v) and (vi) concave curvature with zero and adverse pressure gradients, respectively. For the flows with curvature, the curvature parameter δ/R is 0.023, and for the flows with adverse pressure gradient, the Clauser pressure gradient parameter β is 0.62. The individual influences of adverse pressure gradient and convex and concave curvatures on the boundary layer are similar to those observed by earlier investigations. It is further observed that the combined effect of concave/convex curvature and the adverse pressure gradient causes higher turbulence intensities than the sum of the individual effects. The effect of curvature is to make the wake asymmetric, and in combination with adverse pressure gradient the asymmetry increases. It is observed that the adverse pressure gradient causes faster wake–boundary-layer interaction. Comparing measurements in a straight duct, a curved duct, a curved diffuser and a straight diffuser, it is seen that the convex curvature reduces the boundary layer thickness. The asymmetry in wake development compensates for this effect and the wake–boundary-layer interaction on a convex surface is almost the same as that on a straight surface. In the case of interaction with the boundary layer on a concave surface, the curvature increases the boundary layer thickness and causes enhanced turbulence intensities. However, the asymmetry in wake is such that the extent of wake is lower towards the boundary layer side. As a result, the wake–boundary-layer interaction on concave surface is almost the same as on a straight surface. The interaction is faster in the presence of adverse pressure gradient. Received: 16 June 2000 / Accepted: 17 May 2001     

Wake dynamics of external flow past a curved circular cylinder with the free-stream aligned to the plane of curvature

The fundamental mechanism of vortex shedding past a curved cylinder has been investigated at a Reynolds number of 100 using three-dimensional spectral/hp computations. Two different configurations are presented herein: in both cases the main component of the geometry is a circular cylinder whose centreline is a quarter of a ring and the inflow direction is parallel to the plane of curvature. In the first set of simulations the cylinder is forced to transversely oscillate at a fixed amplitude, while the oscillation frequency has been varied around the Strouhal value. Both geometries exhibit in-phase vortex shedding, with the vortex cores bent according to the body's curvature, although the wake topology is markedly different. In particular, the configuration that was found to suppress the vortex shedding in absence of forced motion exhibits now a primary instability in the near wake. A second set of simulations has been performed imposing an oscillatory roll to the curved cylinder, which is forced to rotate transversely around the axis of its bottom section. This case shows entirely different wake features from the previous one: the vortex shedding appears to be out-of-phase along the body's span, with straight cores that tend to twist after being shed and manifest a secondary spanwise instability. Further, the damping effect stemming from the transverse planar motion of the part of the cylinder parallel to the flow is no longer present, leading to a positive energy transfer from the fluid to the structure.