On the Bending and Stretching Elasticity of Biopolymer Filaments

Elastic filaments play an important role in the behaviour of cells and biological tissues. In this paper a two-dimensional nonlinear elastic framework, incorporating both bending and stretching, for the behaviour of biopolymer filaments treated as one-dimensional continua is developed. Explicit formulas for the extension-force relationship are obtained which include dependence on the initial end-to-end distance of the filament, unlike some existing models in the literature of, for example, the worm-like chain. The approach adopted allows treatment of both flexible and semi-flexible filaments and has the flexibility to accommodate different degrees of approximation. A key ingredient in the application of the model is inclusion of a body force term in the equilibrium equation. This is essential for finding non-trivial solutions of the governing equations and boundary conditions for filaments under tension. This highlights certain inconsistencies in the mechanics evident in the biophysics literature. Since the behaviour of individual filaments has a strong influence on the behaviour of networks of filaments the theory developed here can serve as a basis for analyzing the elasticity of networks such as actin and other filamentous biopolymer networks.     

On Vortex Tube Stretching and Instabilities in an Inviscid Fluid

We study instabilities that are present in two models that retain some of the dynamics of vortex tube stretching in the motion of a fluid in 3 dimensions. Both models are governed by a 2-dimensional PDE and are hence more tractable than the full 3-dimensional Euler equations. The first model is the so called surface quasi-geostrophic equation. The second model is a class of 3-dimensional flows that are invariant with respect to one spatial coordinate. Both models appear in the context of a rapidly rotating fluid. Instabilities due to an effect analogous to vortex tube stretching are detected: these instabilities are in the linearised equations in the first model and in the nonlinear equations in the second model. Such instabilities are absent, or weaker, in strictly 2-dimensional fluid motion.     

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at , tends as to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value. (Accepted April 22, 1996)     

On the Fiber Stretch in Shearing Deformations of Fibrous Soft Materials

The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a single family of oriented parallel fibers is examined. The analysis is purely kinematical and the results are valid for both compressible and incompressible materials. It is shown that for a given amount of shear, for all fiber orientation angles in the range \(0 < \theta < \pi /4\), the fiber stretch increases with increasing \(\theta\) whereas in the range \(\pi /4 < \theta < \pi /2\), this is no longer the case and there is a particular fiber orientation for which the fiber stretch is a maximum. For a particular amount of shear corresponding to a special angle of shear (a “magic” angle of \(35.26^{\circ}\)), the fiber-orientation angle at which the fiber stretch is a maximum is its geometric complement namely a magic angle of \(54.74^{\circ}\). The results are also valid for torsion of a circular cylinder reinforced with a single family of helically wound fibers.     

Stretching polymer chains

This paper is based on a presentation given at the de Gennes discussion meeting held at Chamonix in February 2009. The paper gives a personal review of the way developments relating to the stretching of polymer chains has taken place over the last 40 years. de Gennes was very influential in relation to chain dynamics concepts at the University of Bristol, where many pioneering concepts relating to chain stretching were developed by the late Sir Charles Frank and Andrew Keller. The paper reviews basic concepts on extensional rheology, droplet deformation, chain extension from extensional flow and the achievement of high chain extension for high-modulus polyethylene. The paper also reviews recent developments concerning the influence of chain stretching on polymer melt processing.     

镁合金MB2在高应力三轴度下的拉伸破坏行为研究

通过光滑试件及不同曲率半径缺口圆柱试件的拉伸试验,实现对镁合金MB2的单向及多向应力状态加载。结合数值模拟分析,研究了不同试件在拉伸加载过程中应力状态的变化。以应力三轴度为参数,给出了镁合金MB2等效破坏应变的变化规律,在应力三轴度-等效破坏应变空间建立了镁合金MB2的失效破坏准则。利用扫描电镜对试件断口形貌进行观察,分析了导致材料宏观延性变化的微观损伤机理,对不同应力状态下镁合金MB2的失效破坏行为做出了合理解释。     

偏振相移干涉栅投影技术及其应用

本文提出偏振相移干涉栅投影技术并成功地用于三维物面形状和大变形物体的位移测量，得到较为满意的实验结果。     

Stretching skin: The physiological limit and beyond

The goal of this paper is to establish a novel computational model for skin to characterize its constitutive behavior when stretched within and beyond its physiological limits. Within the physiological regime, skin displays a reversible, highly non-linear, stretch locking, and anisotropic behavior. We model these characteristics using a transversely isotropic chain network model composed of eight wormlike chains. Beyond the physiological limit, skin undergoes an irreversible area growth triggered through mechanical stretch. We model skin growth as a transversely isotropic process characterized through a single internal variable, the scalar-valued growth multiplier. To discretize the evolution of growth in time, we apply an unconditionally stable, implicit Euler backward scheme. To discretize it in space, we utilize the finite element method. For maximum algorithmic efficiency and optimal convergence, we suggest an inner Newton iteration to locally update the growth multiplier at each integration point. This iteration is embedded within an outer Newton iteration to globally update the deformation at each finite element node. To illustrate the characteristic features of skin growth, we first compare the two simple model problems of displacement- and force-driven growth. Then, we model the process of stretch-induced skin growth during tissue expansion. In particular, we compare the spatio-temporal evolution of stress, strain, and area gain for four commonly available tissue expander geometries. We believe that the proposed model has the potential to open new avenues in reconstructive surgery and rationalize critical process parameters in tissue expansion, such as expander geometry, expander size, expander placement, and inflation timing.     

Stretching strongly confined semiflexible polymer chain

By the so-called wormlike chain （WLC） model in polymer physics envision- ing an isotropic rod that is continuously flexible, the force-extension relations of semi- flexible polymer chains strongly constrained by various confinements are theoretically investigated, including a slab-like confinement where the polymer chains are sandwiched between two parallel impenetrable walls, and a capped nanochannel confinement with a circular or rectangular cross-section where the chains are bounded in three directions. The Brownian dynamics （BD） simulations based on the generalized bead-rod （GBR） model are performed to verify the theoretical predictions.     

Brinkman Flow Past a Stretching Sheet

The problem of steady viscous flow of an incompressible fluid over a flat deformable sheet in a porous medium, when the sheet is stretched in its own plane is revisited. An exact solution is recovered for the two-dimensional case and a totally analytic approximate solution is developed for the axisymmetric case. Stretching rate of two-dimensional case is assumed as double the stretching rate of axisymmetric case. The analytical expressions of residual errors, horizontal, vertical velocity distributions, stream lines, vorticity lines, pressure distributions have been obtained and plotted. The values of skin friction, entrainment velocity, boundary layer thickness, momentum thickness and energy thickness have been tabulated. For the first time, two-dimensional and axisymmetric cases are compared by means of a unified scale.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

绝热剪切研究进展

从实验技术、分析手段和数值模拟技术等三方面对近年来绝热剪切研究相关的工作成果进行了总结和评述,指出了现有工作的不足,即:尚不明了绝热剪切带内微观组织演化规律及尚未建立相应的力学模型;绝热剪切带的静水压相关性仍有待进一步探索;描述绝热剪切的自组织行为的模型需进一步完善;绝热剪切数值模拟精度有待提高等。最后,结合国内外最新的研究动态对绝热剪切研究的发展趋势提出了自己的观点。     

Homotopy Classes of Self-Maps of Annuli, Generalised Twists and Spin Degree

Let X be a [generalised] annulus and consider the space of continuous self-maps of X, that is,

白光散斑剪切照相的统计分析

本文运用统计光学的理论，详细研究了白光散斑剪切照相的统计规律，导出了白光散斑剪切照相法测量位移导数场的条件，指出了物面粗糙度，光源相干性及光路对称性等对散斑条纹的影响。     

Stretching and polarizing a dielectric gel immersed in a solvent

This paper studies a gel formed by a network of cross-linked polymers and a species of mobile molecules. The gel is taken to be a dielectric, in which both the polymers and the mobile molecules are non-ionic. We formulate a theory of the gel in contact with a solvent made of the mobile molecules, and subject to electromechanical loads. A free-energy function is constructed for an ideal dielectric gel, including contributions from stretching the network, mixing the polymers and the small molecules, and polarizing the gel. We show that the free-energy function is non-convex, leading to instabilities. We also show that mechanical constraint markedly affects the behavior of the gel.     

Spin of an almost-axial,almost-symmetric,torque-free gyrostat near an osculation condition

Summary Some spacecraft can be modeled as torque-free spinning gyrostats, and in practice these often are designed as nominally symmetric about a principal axis with an internal angular momentum aligned with the same axis. As actually constructed, they are almost-symmetric, almost-aligned. A general solution for the polhodes is known as a computational algorithm, though not as a literal expression. The general solution does not simplify in the slightly off-nominal case. This paper develops an accurate literal expression for polhodes sufficiently close to an osculation points in the latter case. Errors in the endcap solutions are of the order of I % for the end-cap in the several test problems examined.

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Übersicht Als Modelle für Raumschiffe werden meist Gyrostaten verwendet, die im allgemeinen bezüglich einer Hauptachse symmetrisch mit einem inneren Drallvektor längs dieser Achse entworfen werden. Reale Konstruktionen sind jedoch nur fast-symmetrisch und fast-ausgerichtet. Die dann entstehenden Polkurven können zwar in bekannter Weise ausgerechnet werden, jedoch sind bisher keine expliziten Formeln dafür bekannt. Die allgemeinen Lösungen werden für geringe Abweichungen vom symmetrischen, ausgerichteten Fall nicht einfacher. Deshalb wird hier ein Ausdruck für die Polkurven in der Umgebung stationärer Lösungen für schwach abweichende Fälle abgeleitet. Die Fehler dieser Lösungen liegen in den durchgerechneten Fällen bei etwa I %.

     

Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials

The classical problem of simple shear in nonlinear elasticity has played an important role as a basic pilot problem involving a homogeneous deformation that is rich enough to illustrate several key features of the nonlinear theory, most notably the presence of normal stress effects. Here our focus is on certain ambiguities in the formulation of simple shear arising from the determination of the arbitrary hydrostatic pressure term in the normal stresses for the case of an incompressible isotropic hyperelastic material. A new formulation in terms of the principal stretches is given. An alternative approach to the determination of the hydrostatic pressure is proposed here: it will be required that the stress distribution for a perfectly incompressible material be the same as that for a slightly compressible counterpart. The form of slight compressibility adopted here is that usually assumed in the finite element simulation of rubbers. For the particular case of a neo-Hookean material, the different stress distributions are compared and contrasted.     

Analysis of Stretching of Elastoplastic Samples and Necking with Edge Effects

Large deformations of elastoplastic cylindrical rods and shells of different thicknesses under tension are studied. The effect of sample geometry and the dependence of stresses on deformations under uniaxial tension on edge effects and necking are estimated. The applicability of the Considere criterion for determining the instance of stability loss of plastic tensile strain is analyzed. The computational experiments confirm that the role of similarity in the processes of nonuniform tensile strain of samples can be played by the ratio of the tangent of the slope of the true strain diagram to the stress intensity.