Inextensible Membrane Research Articles

The morphology of internal elastic lamina corrugations in arteries under physiological conditions

Background: In elastic and resistance arteries, an elastin-rich membrane, the Internal Elastic Lamina (IEL), separates the tunica intima from the underlying tunica media. The IEL often appears wrinkled or corrugated in histological images. These corrugations are sometimes ascribed to vessel contraction ex vivo, and to fixation artifacts, and therefore regarded as not physiologically relevant. We examine whether the IEL remains corrugated even under physiological conditions.Methods: The diameters of carotid arteries of anesthetized pigs were measured by ultrasound. The arteries were then excised, inflated within a conical sleeve, fixed, and imaged by confocal microscopy. The conical sleeve allows fixing each artery across a wide range of diameters which bracket its ultrasound diameter. Thus the study was designed to quantify how corrugations change with diameter for a single artery, and test whether corrugations exist when the fixed artery matches the ultrasound diameter.Results: At diameters below the ultrasound diameter (i.e. when the artery was constricted as compared to ultrasound conditions), the IEL corrugations were found to decrease significantly with increasing diameter, but not fully flatten at the ultrasound diameter. The contour length of the IEL was found to be roughly 10% larger than the circumference of the artery measured by ultrasound. The physiological diameter is likely to be even smaller than the ultrasound diameter since ultrasound was conducted with the animal under general anesthesia which leads to vasodilation, suggesting a higher level of corrugation under physiological conditions. For arterial cross sections constricted below the ultrasound diameter, the IEL contour length decreased roughly with the square root of the diameter.Conclusion: The primary conclusions of this study are: a) the IEL is corrugated when the artery is constricted and flattens as the artery diameter increases, b) the IEL is corrugated under physiological conditions and has a contour length at least 10% more than the physiological arterial diameter, and c) the IEL despite being relatively stiffer than the surrounding arterial layers, does not behave like an inextensible membrane.

Development of governing partial differential equations of reinforcing thin films

In this paper, a novel interface model for elastic isotropic thin films perfectly bonded to the surrounding elastic media is introduced. Considering an average stress and displacement through the thin film thickness, the shear stress discontinuity across the interface is expressed in terms of spatial and time derivatives of displacement components. The derived interfacial equations are presented in a general form. As such, they are applicable to asymmetric elastodynamic problems of anisotropic solids with thin film interfaces. Employing the introduced interface model and Hankel transform technique, static Green's functions of an elastic isotropic half-space reinforced by a buried thin film are obtained. Results of some special cases, including axisymmetric loading, inextensible membrane, unreinforced half-space, and surface-coated half-space, are also presented. The robustness and effectiveness of the proposed interface model are shown by two solved numerical examples, and some elastic responses are plotted to show the impact of thin film stiffness. According to the numerical results, modeling elastic thin films as inextensible membranes can significantly alter the elastic responses.

A theoretical modelling of strengthening mechanism in graphene-metal nanolayered composites

Graphene-metal nanolayered composites (GMNCs) are a new generation of nano-structural composites characterized by a very high density of graphene reinforced interfaces (GRI) between metal nanolayers. Compared to traditional graphene flake reinforced composites, GMNCs have much higher strength, toughness and ductility due to the excellent ability of GRI on constraining dislocation motion and crack propagation. Despite numerous experimental and numerical studies on the mechanical behavior of GMNCs, the underlying strengthening mechanism is still not fully understood due to the absence of appropriate theoretical model. This paper proposes a continuum mechanics based theoretical model to explain the strengthening mechanism in GMNCs. In this model, the metal matrix and the GRI are simulated as homogenous elastic medium of infinite extend and inextensible thin membrane of zero thickness, respectively. Using the theoretical model, two boundary value problems namely (i) A circular prismatic dislocation loop approaching to the GRI and (ii) A mixed mode I/II penny-shaped crack near the GRI are formulated to reveal the two key strengthening mechanism: dislocation blocking and crack shielding, respectively. The two problems are solved analytically by the Generalized Kelvin's Solution (GKS) based method for 3D elasticity and Fredholm integral integration technique. Exact closed form solution for the Peach-Koehler (P-K) force on the dislocation loop is obtained. An efficient numerical scheme is developed to solve the Fredholm integral integration for the crack problem with very high accuracy. It is shown that our theoretical model can well capture and explain the strengthening mechanism observed in experiments. Moreover, the dominant role of Poisson's ratio on the strengthening efficiency is also revealed by our model. This finding implies the exciting possibility that the strength of GMNCs can be tailored by controlling the Poisson's ratio of the metal matrix. The present theoretical modeling can provide valuable insights into the mechanics-based design of GMNCs.

Swinging and tumbling of multicomponent vesicles in flow

Biological membranes are host to proteins and molecules which may form domain-like structures resulting in spatially varying material properties. Vesicles with such heterogeneous membranes can exhibit intricate shapes at equilibrium and rich dynamics when placed into a flow. Under the assumption of small deformations and a two-dimensional system, we develop a reduced-order model to describe the fluid-structure interaction between a viscous background shear flow and an inextensible membrane with spatially varying bending stiffness and spontaneous curvature. Material property variations of a critical magnitude, relative to the flow rate and internal/external viscosity contrast, can set off a qualitative change in the vesicle dynamics. A membrane of nearly constant bending stiffness or spontaneous curvature undergoes a small amplitude swinging motion (which includes tangential tank-treading), while for large enough material variations the dynamics pass through a regime featuring tumbling and periodic phase-lagging of the membrane material, and ultimately for very large material variation to a rigid-body tumbling behaviour. Distinct differences are found for even and odd spatial modes of domain distribution. Full numerical simulations are used to probe the theoretical predictions, which appear valid even when studying substantially deformed membranes.

Hydrodynamics of a semipermeable inextensible membrane under flow and confinement

Lipid bilayer membranes have a native (albeit small) permeability for water molecules. Under an external load, provided that the bilayer structure stays intact and does not suffer from poration or rupture, a lipid membrane deforms and its water influx/efflux is often assumed negligible in the absence of osmolarity. In this work we use boundary integral simulations to investigate the effects of water permeability on the vesicle hydrodynamics due to a mechanical load, such as the viscous stress from an external flow deforming a vesicle membrane in free space or pushing it through a confinement. Incorporating the membrane permeability into the framework of Helfrich free energy for an inextensible, elastic membrane as a model for a semipermeable vesicle, we illustrate that, in the absence of an osmotic stress gradient, the semipermeable vesicle is affected by water influx/efflux over a sufficiently long time or under a strong confinement. Our simulations quantify the conditions for water permeation to be negligible in terms of the time scales, flow strength, and confinement. These results shed light on how microfluidic confinement can be utilized to estimate membrane permeability.

A contact problem for a poroelastic halfspace containing an embedded inextensible membrane

SummaryThe paper examines the axisymmetric problem of the indentation of a poroelastic halfspace that is reinforced with an inextensible permeable/impermeable membrane located at a finite depth by a rigid indenter. The constitutive behavior of the poroelastic halfspace is described by the three‐dimensional theory of poroelasticity proposed by M.A. Biot. The contact conditions between the indenter and the poroelastic halfspace are varied to accommodate both adhesive/frictionless contact and impermeable/permeable conditions. The formulation of the mixed boundary value problems uses the stress function approaches applicable to semi‐infinite domains. Successive applications of Laplace and Hankel integral transforms are used to reduce the mixed boundary value problems to sets of coupled Fredholm integral equations of the second kind. These integral equations are solved using numerical approaches, applicable both for the solution of the systems of coupled equations and for Laplace transform inversion, to examine the time‐dependent displacement of the rigid indenter. The analytical‐numerical estimates for the time‐dependent displacements of the rigid indenter are compared with results obtained using a finite element approach.

Folded-Tube Soft Pneumatic Actuators for Bending.

This article presents a model and experimental results for a new class of bending soft pneumatic actuators (SPAs) made from a single section of thin-walled flexible tubing. The actuator is formed by creasing the tubing into fin-like rectangular folds and inserting the folds through regularly spaced slits in a high-strength fabric. When pressurized, the folds inflate and press against one another to create a bending moment. The actuator bends with a constant-length arc formed by the fabric. The walls of the tubing behave like an inextensible membrane. This membrane reinforcement allows the actuator to create large motions with limited material strain. Accordingly, the actuator can actuate over large bending angles without generating large elastic restoration forces in its structure. Spacing the slits for the folds closer together increases the bending moment for the same angle and pressure. Three folded-tube SPAs with different fold-spacings were experimentally characterized for this work. The strongest of the three was able to produce >190 N of tension and an estimated 10 Nm of bending moment with only 60 kPa. Even after bending 160°, the bending moment was still ∼5.7 Nm at the same pressure. The model presented in this work describes the moment per unit pressure of the actuators over the measured range of motion. For the 40 and 60 kPa tests, the model differed from the data with an average absolute error of <13% for all three actuators.

The Simulation of an Automotive Air Spring Suspension Using a Pseudo-Dynamic Procedure

This paper describes a numerical solution to characterize the deformation of a bellows-type air spring for automotive suspensions. In a first step, the shell structure is modeled as a practically inextensible membrane that has virtually no bending stiffness; the structure has only a pneumatic-elastic deformation due to the compressibility of the pressurized air. In a second step, a finite element modeling of the device using a commercial code is carried out in order to validate the first model. Complementing this work, an experimental procedure based on a pseudo-dynamic technique was implemented to simulate the behavior of the pneumatic suspension bellows subjected to dynamic loads. The method consists of a combined numeric/experimental procedure simulating a suddenly applied load.

A composite material with inextensible-membrane-type interface

We consider a model of a composite material with “inextensible membrane type” interface conditions. An analytic solution of a stationary heat conduction problem in an unbounded doubly periodic two-dimensional composite whose matrix and inclusions consist of isotropic temperature-dependent materials is given. In the case where the conductive properties of the inclusions are proportional to those of the matrix, the problem is transformed into a fully linear boundary value problem for doubly periodic analytic functions. The solution makes it possible to calculate the average properties over the unit cell and discuss the effective conductivity of the composite. We present numerical examples to indicate some peculiarities of the solution.

The steady motion of a closely fitting vesicle in a tube

A singular perturbation theory is developed for the steady, inertialess motion of a lipid-bilayer vesicle flowing through a narrow tube. The vesicle is treated as a sac of fluid enclosed by an inextensible membrane that admits a bending stiffness. Matched asymptotic expansions are developed in terms of a clearance parameter $\unicode[STIX]{x1D716}\ll 1$ in order to calculate the flow field and vesicle shape. Mild restrictions are applied to the ratio of viscosities $\unicode[STIX]{x1D705}$ and the ratio of bending stresses to viscous stresses $\unicode[STIX]{x1D6FD}$; in particular, the theory holds for $\unicode[STIX]{x1D705}=o(\unicode[STIX]{x1D716}^{-1/2})$ and $\unicode[STIX]{x1D6FD}=O(\unicode[STIX]{x1D716}^{-1})$. The ratio of the vesicle length to the tube radius $\ell$ is included as a parameter and asymptotic solutions in the limit of negligible bending stiffness are developed for long, cylindrical vesicles and short, spherical vesicles. The main result of the theory is a prediction for the vesicle speed and extra pressure drop due to the presence of the vesicle in the tube. The effects of confinement, vesicle length, and membrane bending elasticity are examined. The theoretical predictions show good agreement with experimental measurements reported for vesicles and red blood cells in highly confined channel flow. Previously reported models for red blood cells (Secomb et al., J. Fluid Mech., vol. 163, 1986, pp. 405–423; Halpern &amp; Secomb, J. Fluid Mech., vol. 203, 1989, pp. 381–400) are clarified and extended in light of the new theory.

Equilibrium configuration of a bounded inextensible membrane subject to solar radiation pressure

The equilibrium shape of a thin inextensible membrane subject to solar radiation pressure under given boundary constraints is studied. The membrane is assumed to be insusceptible to elastic deformation and to have negligible bending resistance, and its steady-state shape is therefore described by a developable surface (i.e., a surface of zero Gaussian curvature), resulting from an equilibrium between radiation pressure and membrane tension forces. A quantitative understanding of the mechanics of such membranes is essential in characterizing the dynamics of solar sail spacecraft that use sail wing tip displacement as an attitude control mode. The analysis in this paper develops a theoretical foundation for the billowed wing shape. Under reasonable simplifying assumptions, the key result is that solar radiation pressure and a given wing tip displacement yield a billowed solar sail wing with the shape of a generalized cylinder (i.e., a developable ruled surface, whose rulings are all parallel, rather than a general developable with variable ruling directions). The base curve geometry for the solar sail is also determined as the solution to a boundary value problem. The results presented herein allow the shape of the billowed membrane to be computed to any desired precision, for any given tip displacement.

Axisymmetric response of a bi-material full-space reinforced by an interfacial thin film

Analytical treatment of a linear elastic isotropic bi-material full-space reinforced by an interfacial thin film under axisymmetric normal loading is addressed. The thin film is modeled as an extensible membrane perfectly bonded to the half-spaces. By virtue of Love’s potential function and Hankel integral transform, elastic fields of the system are explicitly written in the form of semi-infinite line integrals. The analytical results are verified by the special cases corresponding to the surface stiffened half-space and classical bi-material problem. The limiting cases of reinforced homogeneous full-space and inextensible membrane are presented and discussed. The proposed formulation is also applicable for studying reinforced auxetic materials with negative Poisson’s ratio. The surface/interface effect on the elastic responses of two perfectly bonded half-spaces is also simulated by assigning equivalent surface elastic constant to the membrane stiffness. Effects of thin film stiffness, material properties, loading depth, and surface/interface effect are studied by some numerical examples.

Amoeboid swimming in a channel.

Several micro-organisms, such as bacteria, algae, or spermatozoa, use flagellar or ciliary activity to swim in a fluid, while many other micro-organisms instead use ample shape deformation, described as amoeboid, to propel themselves either by crawling on a substrate or swimming. Many eukaryotic cells were believed to require an underlying substratum to migrate (crawl) by using membrane deformation (like blebbing or generation of lamellipodia) but there is now increasing evidence that a large variety of cells (including those of the immune system) can migrate without the assistance of focal adhesion, allowing them to swim as efficiently as they can crawl. This paper details the analysis of amoeboid swimming in a confined fluid by modeling the swimmer as an inextensible membrane deploying local active forces (with zero total force and torque). The swimmer displays a rich behavior: it may settle into a straight trajectory in the channel or navigate from one wall to the other depending on its confinement. The nature of the swimmer is also found to be affected by confinement: the swimmer can behave, on average over one swimming cycle, as a pusher at low confinement, and becomes a puller at higher confinement, or vice versa. The swimmer's nature is thus not an intrinsic property. The scaling of the swimmer velocity V with the force amplitude A is analyzed in detail showing that at small enough A, V∼A(2)/η(2) (where η is the viscosity of the ambient fluid), whereas at large enough A, V is independent of the force and is determined solely by the stroke cycle frequency and the swimmer size. This finding starkly contrasts with models where motion is based on ciliary and flagellar activity, where V∼A/η. To conclude, two definitions of efficiency as put forward in the literature are analyzed with distinct outcomes. We find that one type of efficiency has an optimum as a function of confinement while the other does not. Future perspectives are outlined.

Derivation of design charts based on the two-dimensional structural analysis of geotextile tubes

Analytical solutions for modeling geotextile tubes during the filling process and approximation method to determine the densified tube shape are reviewed. The geotextile tube filling analysis is based on Plaut & Suherman\'s two-dimensional solution for geotextile tubes having a weightless and frictionless inextensible membrane resting on a rigid horizontal foundation subjected to internal and external hydrostatic pressures. The approximation for the densified tube shape developed by Leshchinsky et al. was adopted. A modified method for approximating the densified tube shape based on an areal-strain deformation analysis is introduced. Design diagrams useful for approximating geotextile tube measurements in the design process are provided.

An inextensible membrane at the interface of a transversely isotropic bi-material full-space

A theoretical formulation is presented for the determination of the dynamic interaction of a horizontally loaded inextensible circular membrane embedded at the interface of a transversely isotropic bi-material full-space, using cylindrical co-ordinate system and applying Hankel integral transforms in the radial direction and Fourier series, the problem will be changed to a system of four separate integral equations, which, in turn, are reduced to a pair of Fredholm equations of the second kind that are amenable to numerical treatments. The impedance functions have been evaluated in dynamic case, which can be directly used in soil–structure-interaction and engineering problems. It is shown that the present solutions are analytical and numerically in exact agreement with different degenerate cases. To validate the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy. The numerical scheme for solving Fredholm equation is also presented.

Isogeometric boundary integral analysis of drops and inextensible membranes in isoviscous flow

The Boundary Integral Method (BIM) is applied to investigate the dynamics of a single drop and an inextensible membrane in isoviscous shear flow. The novelty of this work resides in the application of Isogeometric Analysis (IGA) to define the interface of the deformable objects. The employed B-spline basis functions facilitate the direct evaluation of surface normal vectors and curvatures, as required by the BIM. Collocation and L2-projection methods are implemented to approximate the velocity of the B-spline control points. In particular, a comparison between these two methods for the case of the drop is reported and shows that the collocation method provides faster and more stable results. The collocation method is also applied for the determination of the surface tension in an inextensible membrane. A series of simulations is conducted to verify the isogeometric approach, and various computational aspects are studied.

A necklace model for vesicles simulations in 2D

SUMMARYThe aim of this paper is to propose a new numerical model to simulate 2D vesicles interacting with a Newtonian fluid. The inextensible membrane is modeled by a chain of circular rigid particles, which are maintained in cohesion by using two different types of forces. First, a spring force is imposed between neighboring particles in the chain. Second, in order to model the bending of the membrane, each triplet of successive particles is submitted to an angular force.Numerical simulations of vesicles in shear flow have been run using FEM and the FreeFem++ software. Exploring different ratios of inner and outer viscosities, we recover the well‐known ‘tank‐treading’ and ‘tumbling’ motions predicted by a theory and experiments. Moreover, for the first time, 2D simulations of the ‘vacillating‐breathing’ regime without special ingredient such as thermal fluctuations are recovered. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Long-wave dynamics of an inextensible planar membrane in an electric field

AbstractIn this paper the dynamics of an inextensible capacitive elastic membrane under an electric field is investigated in the long-wave (lubrication) leaky dielectric framework, where a sixth-order nonlinear differential equation with an integral constraint is derived. Steady equilibrium profiles for a non-conducting membrane in a direct current (DC) field are found to depend only on the membrane excess area and the volume under the membrane. Linear stability analysis on a tensionless flat membrane in a DC field gives the growth rate in terms of membrane conductance and electric properties in the bulk. Numerical simulations of a capacitive conducting membrane under an alternating current (AC) field elucidate how variation of the membrane tension correlates with the nonlinear membrane dynamics. Different membrane dynamics, such as undulation and flip-flop, are found at different electric field strength and membrane area. In particular a travelling wave on the membrane is found as a response to a periodic AC field in the perpendicular direction.

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.

Membrane tension homeostasis of epithelial cells through surface area regulation in response to osmotic stress

Osmotic stress poses one of the most fundamental challenges to living cells. Particularly, the largely inextensible plasma membrane of eukaryotic cells easily ruptures under in-plane tension calling for sophisticated strategies to readily respond to osmotic stress. We describe how epithelial cells react and adapt mechanically to the exposure to hypotonic and hypertonic solutions in the context of a confluent monolayer. Site-specific indentation experiments in conjunction with tether pulling on individual cells have been carried out with an atomic force microscope to reveal spatio-temporal changes in membrane tension and surface area. We found that cells compensate for an increase in lateral tension due to hypoosmotic stress by sacrificing excess of membrane area stored in protrusions and invaginations such as microvilli and caveolae. At mild hypotonic conditions lateral tension increases partly compensated by surface are regulation, i.e. the cell sacrifices some of its membrane reservoirs. A loss of membrane–actin contacts occurs upon exposure to stronger hypotonic solutions giving rise to a drop in lateral tension. Tension release recovers on longer time scales by an increasing endocytosis, which efficiently removes excess membrane from the apical side to restore the initial pre-stress. Hypertonic solutions lead to shrinkage of cells and collapse of the apical membrane onto the cortex. Exposure to distilled water leads to stiffening of cells due to removal of excess surface area and tension increase due to elevated osmotic pressure across the plasma membrane.