Structure and rheology of fiber-laden membranes via integration of nematodynamics and membranodynamics

This paper presents a study of the structure and dynamics of rigid fiber-laden deformable curved fluid membranes based on an viscoelastic model that integrates the statics of anisotropic membranes, the planar nematodynamics of fibers and the dynamics of isotropic membranes. Fiber-laden membranes arise frequently in biological systems, such as the plant cell wall and in protein–lipid bilayers. Based on the membrane's force and torque balance equations and the fiber's balance of molecular fields, a viscoelastic anisotropic model that provides the governing equations for the membrane's velocity and curvature and the fiber structure (fiber orientation and order) is found. A Helmholtz free energy that incorporates the tension/bending/and torsion membrane elasticity, the Landau–de Gennes fiber ordering, and fiber order-membrane curvature interactions is used to derive elastic moments, torques, and stresses. The corresponding viscous stresses and moments include the Boussinesq–Scriven contributions as well as bending, torsion, and rotational dissipation. A spectral decomposition leads to the main viscoelastic material functions for anisotropic fluid membranes. Applications of the rheological model to cylindrical growth and cylindrical axial stretching show that competing curvo-phobic, curvo-philic interactions under extensional flow predict transitions between axial and azimuthal fiber arrangements, of interest to cellulose fiber orientation in plant morphogenesis.     

Torsional,flexural, and torsional-flexural buckling modes of a cylindrical shell under combined loading

Stability problems for cylindrical shells under various loading modes were considered in numerous papers. A detailed analysis of such problems can be found, e.g., in the monograph [1]. We refer to the solutions presented in this monograph as classical.For long cylindrical shells in axial compression, one of the buckling modes is the purely beam flexural mode similar to the classical buckling mode of a straight rod. It is well known that it can be studied by using the nonlinear or linearized equations of the membrane theory of shells. In [2], it was shown that, on the basis of such equations constructed starting from the noncontradictory version of geometrically nonlinear elasticity relations in the quadratic approximation [3], under the separate action of the axial compression, external pressure, and torsion, there are also previously unknown nonclassical buckling modes, most of which are shear ones.In the present paper, we show that the use of the above equations for cylindrical shells under compression and external pressure with simultaneous pure torsion or bending permits revealing the earlier unknown torsional, beam flexural, and beam torsional-flexural buckling modes, which are nonclassical, just as those found in [2]. The second of these buckling modes is realized when axially compressing forces are formed in the shell with simultaneous torsion, and the third of them is realized under compression combined with pure bending.It was found that, earlier than the classical buckling modes, the torsional buckling modes can be realized for relatively short shells with small shear rigidity in the tangent plane, while the second and third buckling modes can be realized for relatively long shells.     

Torsional rigidity of shells of revolution

In this paper, the general equations of equilibrium for axisymmetrical deformation including the torsional deformation of revolutional shells are derived. It is shown that the shearing stress distribution due to torsion is independent of other stress components including those of membrane stress and bending stress. In this paper, the torsional deformation is considered to be represented by membrane action only, and also by the combined action of bending membrane deformation. It is shown that the main contribution of torsional rigidity is that related to membrane action.     

FINITE ELEMENT MODELING AND ROBUST VIBRATION CONTROL OF TWO-HINGED PLATE USING BONDED PIEZOELECTRIC SENSORS AND ACTUATORS

Active vibration control for a kind of two-hinged plate is developed in this paper. A finite element model for the hinged plate integrated with distributed piezoelectric sensors and actuators is derived, including bending and torsional modes of vibration. In this model, the hinges are simplified as regular plate elements to facilitate operation. The state space representations for bending and torsional vibrations are obtained. Based on two low-order models of the bending and torsional motion, two H ∞ robust controllers are designed for suppressing the vibrations of the bending and torsional modes, respectively. The simulation results indicate the effectiveness and feasibility of the designed H ∞ controllers. The vibration magnitudes of the low-order modes can be reduced without affecting the high frequency modes.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. II: Combination resonance

In Part I of this work nonlinear coupling between torsional motion and both in-plane and out-of-plane flexural motion was examined for inextensional beams in the presence of a one-to-one internal resonance. Here the nonlinear response of the system considered in Part I is investigated for the case of an internal combination resonance involving modes associated with bending in two directions and torsion. The analysis presented is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities and account for torsional dynamics.     

开口厚壁截面短构件的约束扭转1）

为了分析开口厚壁截面短构件的约束扭转问题,采用统一分析梁模型与有限节线法,对T形和L形厚壁截面短构件约束扭转时横截面的翘曲和应力分布情况等问题进行了分析研究.算例计算结果表明：开口厚壁截面短构件存在与其横截面形心位置不一致的扭转（弯曲）中心,构件在不过扭转中心的外力作用下会产生弯扭耦合变形,其横截面将产生不均匀翘曲,横截面上的翘曲正应力和扭转剪应力均呈非线性分布.     

Mechanical model for fiber-laden membranes

An integrated mechanical model for fiber-laden membranes is presented and representative predictions of relevance to cellulose ordering and orientation in the plant cell wall are presented. The model describes nematic liquid crystalline self-assembly of rigid fibers on an arbitrarily curved fluid membrane. The mechanics of the fluid membrane is described by the Helfrich bending-torsion model, the fiber self-assembly is described by the 2D Landau-de Gennes quadrupolar Q-tensor order parameter model, and the fiber-membrane interactions (inspired by an extension of the 2D Maier-Saupe model to curved surfaces) include competing curvo-philic (curvature-seeking) and curvo-phobic (curvature-avoiding) effects. Analysis of the free energy reveals three fiber orientation regimes: (a) along the major curvature, (b) along the minor curvature, (c) away from the principal curvatures, according to the competing curvo-philic and curvo-phobic interactions. The derived shape equation (normal stress balance) now includes curvature-nematic ordering contributions, with both bending and torsion renormalizations. Integration of the shape and nematic order equations gives a complete model whose solution describes the coupled membrane shape/fiber order state. Applications to cylindrical membranes, relevant to the plant cell wall, shows how growth decreases the fiber order parameter and moves the fibers’ director from the axial direction towards the azimuthal orientation, eventually leading to a state of stress predicted by pure membranes. The ubiquitous 54.7° cellulose fibril orientation with respect to the long axis in a cylindrical plant cell wall is shown to be predicted by the preset model when the ratio of curvo-phobic and curvo-philic interactions is in the range of the cylinder radius.     

混凝土结构扭转理论研究的进展

本文回顾了国内外混凝土结构扭转理论的历史发展,尤其是最近十几年来的进展。文中就有代表性的斜弯理论和空间桁架理论进行了述评。并按照结构构件受纯扭、弯扭及弯剪扭联合作用等加载方式,对于素混凝土、钢筋混凝土和预应力混凝土结构扭转的计算方法进行了归纳。最后对我国开展混凝土结构扭转理论的研究提出了建议。      

On the static and dynamic stability of thin beam conveying fluid

In this paper, numerical investigation of the statical and dynamical stability of aligned and misaligned viscoelastic cantilevered beam is performed with a terminal nozzle in the presence of gravity in two cases: (1) effect of fluid velocity on the flutter boundary of beam conveying fluid and (2) effect of gravity on the buckling boundary of beam conveying fluid. The beam is assumed to have a large width-to-thickness ratio, so the out-of-plane bending rigidity is far higher than the in-plane bending and torsional rigidities. Gravity vector is considered in the vertical direction. Thus, deflection of the beam because of the gravity effect couples the in-plane bending and torsional equations. The beam is modeled by Euler–Bernoulli beam theory, with the flow-induced inertia, Coriolis and centrifugal forces along the beam considered as a distributed load along the beam. Furthermore, the end nozzle is regarded as a lumped mass and modeled as a follower axial force. The extended Hamilton’s principle and the Galerkin method are utilized to derive the bending–torsional equations of motion. The coupled equations of motion are solved as eigenvalue problems. Also, several cases are examined to study the impact of gravity, beam inclination angle, mass ratio, nozzle aspect ratio, bending-to-bending rigidity ratio and bending-to-torsional rigidity ratio on flutter and buckling margin of the system.

     

Quasi-steady prediction of coupled bending–torsion flutter under rotating stall

A method is presented in this paper to predict cascade flutter under subsonic stalled flow condition in a quasi-steady manner. The ability to predict the occurrence of aeroelastic flutter is highly important from the compressor design point of view. In the present work, the well known Moore–Greitzer compression system model is used to evaluate the flow under rotating stall and the linearized aerodynamic theory of Whitehead is used to estimate the blade loading. The cascade stability is then predicted by solving the structural model, which is posed as a complex eigenvalue problem. The possibility of occurrence of flutter in both bending and torsional modes is considered and the latter is found to be the dominant one, under subsonic stalled flow, for a large range of frequency ratios examined. It is also shown that the design of compressor blades at frequency ratios close to unity may result in rapid initiation of torsional flutter in the presence of stalled flow. A frequency ratio of 0.9 is primarily emphasized for most part of the study as many interesting features are revealed and the results are physically interpreted. Roughly a pitchfork pattern of energy distribution appears to occur between bending mode and torsional mode which ensures that only one flutter mode is possible at any instant in time. A bifurcation from bending flutter to torsional flutter is shown to occur during which the frequency of the two vibrating modes appear to coalesce for a very short period of time.     

复合受力下焊接栓钉SRC柱抗扭承载力计算

为研究压、弯、剪、扭复合受力下焊接栓钉型钢混凝土(SRC)柱的抗扭承载力计算方法,基于滞回性能试验结果,分析了试验过程和破坏形态,利用了空间桁架理论与叠加原理,提出了复合受力下焊接栓钉SRC柱抗扭计算模型和强度统一方程.结果表明:所提出的强度统一方程和抗扭承载力计算方法的计算结果与试验结果均值误差均在3％内,吻合较好.     

Towards linear modal analysis for an L-shaped beam: Equations of motion

We consider an L-shaped beam structure and derive all the equations of motion considering also the rotary inertia terms. We show that the equations are decoupled in two motions, namely the in-plane bending and out-of-plane bending with torsion. In neglecting the rotary inertia terms the torsional equation for the secondary beam is fully decoupled from the other equations for out-of-plane motion. A numerical modal analysis was undertaken for two models of the L-shaped beam, considering two different orientations of the secondary beam, and it was shown that the mode shapes can be grouped into these two motions: in-plane bending and out-of-plane motion. We compared the theoretical natural frequencies of the secondary beam in torsion with finite element results which showed some disagreement, and also it was shown that the torsional mode shapes of the secondary beam are coupled with the other out-of-plane motions. These findings confirm that it is necessary to take rotary inertia terms into account for out-of-plane bending. This work is essential in order to perform accurate linear modal analysis on the L-shaped beam structure.     

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.     

Approximate equations for torsional waves in a nonlinear elastic rod with weak viscoelasticity

Approximate equations are derived for nonlinear torsional waves propagating along a thin circular viscoelastic rod. Ignoring the thermal effect, ‘nearly elastic’ compressible viscoelastic solids are considered in which a weak dependence of stresses on a history of strain is assumed. With the assumption that the rod is subjected to a finite angle of torsion, but that the rod is thin, the displacement is sought in a power series of the radial coordinate. The effects of geometrical and material nonlinearity give rise to the normal stress effect, which introduces deformations in the cross sectional and longitudinal dimensions of rod. Taking account of both the effect of nonlinearity and that of viscoelasticity, one dimensional approximate equations are obtained for the angle of torsion coupled with the longitudinal deformation.     

Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability

Traditional torsional resonators, often obtaining the viscoelastic moduli of complex fluids only at one or several given discrete frequencies, lack the continuously varying frequency capability. This is an obvious disadvantage of the traditional torsional resonator technique. This paper presents an improved strategy, based on our previous discrete-frequency-measuring method (Wang et al., J Rheol 52:999–1011, 2008), to overcome such restriction and thus accomplish the continuously varying frequency capability of the traditional torsional resonator for measuring the viscoelastic properties of complex fluids. The feasibility of this strategy is demonstrated with the Newtonian fluids (several water–glycerol solutions) of viscosities varying from 10 to 1,400 cp by using our homemade torsion resonator apparatus in the 10 ~ 2,500 rad/s frequency range (continuous frequencies). Some results for typical viscoelastic polymers (two polyethylene oxide (PEO) aqueous solutions) are also given. Additionally, a comparison of the PEO results is made with the common rheometer technique. It is demonstrated that this improved strategy could enable the traditional torsional resonators, with one oscillating resonance mode, to work as the microrheological technique and the common rheometer technique in the continuous frequency range.     

Mechanics of fluid-activated,clustered satellite bellows

A bellows, or a closed thin-walled elastic tube with corrugated walls, undergoes longitudinal extension when subjected to internal fluid pressure. Investigated herein is the mechanical behavior of several pressurized bellows in clusters, which are designed to bend and twist as well as to extend and compress longitudinally. Bellows in clusters can be employed as robotic limbs, such as manipulator arms and legs for walking machines. For limb bending, analysis shows that there is an optimal geometry for satellite bellows, or a set of identical bellows clustered longitudinally about a central core. For limb torsion, the bellows are clustered in a cylindrical helix whose angle is chosen to produce the desired load–displacement relationships, for instance the highest rotation for a given torque. For both bending and torsional limbs, experimental results are included that exhibit the predicted mechanical behavior.     

Free vibrations of a thin elastica by normal modes

In this work we investigate the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system. The system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes. Using a variety of perturbation techniques in conjunction with the computer algebra system MACSYMA, we obtain approximate expressions for a diversity of periodic motions, including nonlinear normal modes, elliptic orbits and non-local modes. The latter motions, which involve both bending and torsional motions in a 2:1 ratio, correspond to behavior previously observed in experiments by Cusumano.     

A formulation of the centre of twist and shear for nonhomogeneous beam

A formula will be derived for the torsional stiffness of conical membrane shell made from linear elastic homogeneous isotropic material. The torsional stiffness of the truncated conical membrane shell is defined by the following manner: it is that value which is obtained as the square of the applied torque divided by two times strain energy in the state of pure torsion.     

Dynamic analysis for axially moving viscoelastic panels

In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-flow end for clamped–clamped and clamped-simply supported panels. The numerical results suggest that the moving viscoelastic panel undergoes divergence instability for low values of viscosity. They also show that the critical panel velocity increases when viscosity is increased and that the viscoelastic panel does not experience instability with a sufficiently high viscosity coefficient. For the cases with low viscosity, the modes and velocities corresponding to divergence instability are found numerically. We also report that the value of bending rigidity (bending stiffness) affects the distance between the divergence velocity and the flutter velocity: the higher the bending rigidity, the larger the distance.     

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.The torsional storage and loss moduli are determined under quasistatic conditions as functions of frequency and axial static pre-strain. The time-strain separability is found to be a resonable approximation in a relatively limited range of static prestrain magnitudes and frequencies considered for the natural gum rubbers investigated. The experimental methodology is discussed in some detail.