Inextensible Membranes Research Articles

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.

Modular Design of Multistable Pneumatic Structures from a Flat Pattern of Air Pouches

We introduce a novel family of structures that can be inflated from a plane to form a three-dimensional surface using articulated straight pouches with inextensible membranes. Specifically, we propose a modular design using a quadrangle surrounded by four slender pouches, whose inflation causes the surface to form a saddle shape. Then we show a fabrication method using heated nylon films with heat adhesives separated by paper mask patterns inside. We first show a geometric model to explain and predict the out-of-plane deformation and the bi-stability effect inherited in this structure. Through a geometric analysis and an elastic simulation of a pouch, we show that lowering the width-length ratio of pouches reduces the imperfection of shapes. We further propose a method for controlling the height of saddles using concentric patterns. We then extend structural modules into multiple-grid systems and identify the multistable behaviors of the overall structure.

Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

We propose a numerical framework tailored for simulating the dynamics of vesicles with inextensible membranes, which mimic red blood cells, immersed in a non‐Newtonian fluid environment. A penalty method is proposed to handle the inextensibility constraint by relaxation, allowing a simple computer implementation and an affordable computational load compared to the full mixed formulation. To handle the high‐order derivatives in the stress jump across the membrane, which arise due to the high geometric order of the Helfrich functional, we employ higher degree finite elements for spatial discretization. The time integration scheme relies on the double composition of the Crank–Nicolson scheme to achieve faster fourth‐order convergence behavior. Additionally, an adaptive time‐stepping strategy based on a third‐order temporal integration error estimation is implemented. We address the main features of the proposed method, which is benchmarked against existing numerical and experimental results. Furthermore, we investigate the influence of non‐Newtonian rheology on the system dynamics.

On equilibrium states of fluid membranes

The paper studies the equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Using a continuum description of the membrane's motions based on the surface Navier–Stokes equations with bending forces, the paper derives differential equations governing the mechanical equilibrium. The equilibrium conditions are found to be independent of lateral viscosity and relate tension, pressure, and tangential velocity of the fluid. These conditions suggest that either the lateral fluid motion ceases or non-decaying stationary flow of mass can only be supported by surfaces with Killing vector fields, such as axisymmetric shapes. A shape equation is derived that extends the classical Helfrich model with an area constraint to membranes of non-negligible mass. Furthermore, the paper suggests a simple numerical method to compute solutions of the shape equation. Numerical experiments conducted reveal a diverse family of equilibrium configurations. The stability of equilibrium states involving lateral flow of mass remains an unresolved question.

Frequency reconfigurable antenna actuated by inflated triangular structure: Design and analysis

This paper proposes a frequency reconfigurable triangular antenna actuated by an inflated triangular structure. The open path antenna is transformed from an open type to a closed structure by inflating. Inflatable structures are easy to manufacture by fusing 2 inextensible membranes together along a defined pattern of lines. However, the prediction of their deployed shape remains a challenge. To solve the pattern changed problem, guided by geometric analyses and local buckle characteristics, the inflated triangular structure has been designed and verified by experiment and simulation. In the process of transformation of the antenna, the resonant frequency of the antenna is changed because this frequency is determined by the conformational change. The resonant frequency changes from GHz to kHz when the design of initial structure sizes is from millimeter to meter. The measured peak gains, the frequency, and the radiation direction are also reconfigurable by the initial size. Finally, the reconfigurable resonator array is presented, which is coupled to electric fields to absorb all incident radiation. In this work, the changed pattern design by inflating is applied to the antenna design, and its frequency reconfigurability is achieved. Through the electricity performance analysis of the reconfigurable antenna, precise manufacturing will be possible and provide guidance for manufacturing frequency reconfigurable antennas.

Recycling augmented Lagrangian preconditioner in an incompressible fluid solver

AbstractThe paper discusses a reuse of matrix factorization as a building block in the Augmented Lagrangian (AL) and modified AL preconditioners for nonsymmetric saddle point linear algebraic systems. The strategy is applied to solve two‐dimensional incompressible fluid problems with efficiency rates independent of the Reynolds number. The solver is then tested to simulate motion of a surface fluid, an example of a two‐dimensional flow motivated by an interest in lateral fluidity of inextensible viscous membranes. Numerical examples include the Kelvin–Helmholtz instability problem posed on the sphere and on the torus. Some new eigenvalue estimates for the AL preconditioner are derived.

An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes

We present in this paper a fully implicit finite element method tailored for the numerical modeling of inextensible fluidic membranes in a surrounding Newtonian fluid. We consider a highly simplified version of the Canham-Helfrich model for phospholipid membranes, in which the bending force and spontaneous curvature are disregarded. The coupled problem is formulated in a fully Eulerian framework and the membrane motion is tracked using the level set method. The resulting nonlinear problem is solved by a Newton-Raphson strategy, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the proposed method. We show that stability is maintained for significantly larger time steps with respect to an explicit decoupling method.

Thrust efficiency of harmonically oscillating flexible flat plates

We consider the efficiency of thrust-producing inextensible membranes with variable bending rigidities. The present study is a numerical investigation of the thrust generation and flow-field characteristics of a two-dimensional flapping flexible membrane, fixed at its leading edge. To study the time-dependent response of the membranes, a fluid/structure solver that couples a compact finite-difference immersed boundary method flow solver with a thin-membrane structural solver was developed. Using a body-fitted grid, external forcing to the structure is calculated from the boundary fluid dynamics. A systematic series of runs of the fluid/structure solver was performed in order to obtain a clear picture of the thrust-producing characteristics of membranes with bending rigidities ranging betweenEI= 5 × 10−6andEI= 2 × 10−5and structural mass coefficients between ρsh= 0.01 and ρsh= 0.04, for a Reynolds number ofRe= 851.

Equilibrium shapes of inflated inextensible membranes

This paper proposes an effective method for directly determining the final equilibrium shapes of closed inextensible membranes subjected to internal pressures. With reference to new high-performance textile materials, we assume that the mechanical response of a fabric membrane can be accurately represented by regarding it as a two-state material. In the active state, the membrane is subject to tensile stresses and is virtually inextensible; vice versa, in the passive state it is unable to sustain any compressive stress, so it contracts freely. Equilibrium of the membrane in the final configuration is enforced by recourse to the minimum total potential energy principle. The Lagrange multipliers method is used to solve the minimum problem by accounting for the aforesaid nonlinear constitutive law. The set of governing equations is solved for the unknown coordinates of the equilibrium surface points. Closed form solutions are obtained for fully wrinkled cylindrical and axisymmetric membranes under homogeneous boundary conditions, while a simple iterative procedure is used to numerically solve cases of axisymmetric membranes under various inhomogeneous boundary conditions. The soundness of the proposed method is verified by comparing the results with solutions available in the literature.

Development of a Non-Pneumatic Wheel

Abstract The pneumatic tire has proven to be the dominant design for rolling transport since shortly after its invention by John Boyd Dunlop in 1888. This paper identifies four critical characteristics that led to the success of the pneumatic tire: low energy loss on rough surfaces, low vertical stiffness, low contact pressure, and low mass. A non-pneumatic structure is proposed that exhibits these four critical characteristics while breaking some of the most restrictive design constraints imposed by pneumatic mechanics. This structure consists of a circular beam composed of two inextensible membranes separated by a relatively low modulus elastic material. The beam is connected to a hub by thin elastic spokes. This structure forming an integrated tire∕wheel is called a Tweel™.

Dispersion Spectrum of an Anisotropic Waveguide with Sector-Shaped Cross Section and Fixed Boundary

The dispersion equations are derived analytically for normal elastic waves in a longitudinally anisotropic cylindrical waveguide with circular cross-section and a sector-shaped channel of arbitrary angular measure. The boundary surfaces of the channel are covered with flexible, inextensible membranes. The cylindrical portion of the lateral surface is rigidly fixed. A numerical analysis of the dispersion equations reveals spectral branches of traveling normal waves with different types of symmetry of cross-sectional displacements and different circumferential wave numbers. Some features of the spectrum due to variation in the angular measure are established

Equilibrium States of Inextensible Membranes under Gravity

We discuss the possible shapes and stress distributions of an inextensible membrane acted on by a gravitational force. It is shown that there are smooth shapes which cannot be an equilibrium solution under gravity for any stress distribution. In addition there are infinitely many shapes which are equilibrium solutions under gravity and for which the stress distribution is unique. Finally, there are infinitely many solution surfaces – each consisting of straight line generators parallel to the direction of gravity. These solutions are possible under infinitely many different stress distributions. In several cases we will solve the equations explicitly to find actual shapes and stress distributions.

Theory for Conical Membrane Wings of High Aspect Ratio

The inviscid aerodynamics of high-aspect-ratio wings having conical surfaces that are inextensible membranes capable of carrying compressive stresses and supported along two straight generators are investigated. For a given shape, the aerodynamics of the wing are found using lifting-line theory and thin-airfoil theory. The observation that the line of action of the membrane tension along each supporting generator is the same gives rise to a global constraint between the membrane shape, the pitching moment, and the root bending moment. This constraint enables an approximate solution to be found for the equilibrium membrane shape without needing to consider the full equations of equilibrium for the membrane. The functional form of the relationships between the wing shape and its force and moment coefficients is found to be analogous to those for the corresponding two-dimensional problem. The wing shapes considered in detail are the wing shape needed for elliptic loading, a triangular planform and an optimum wing at off-design conditions

The behavior of closed inextensible membranes in linear and quadratic shear flows

The dynamics of a spheroidal vesicle, bounded by an inextensible membrane, is analyzed as a function of the enclosed fluid viscosity, and of the membrane mechanical properties. The two situations in which a bending rigidity and a shear elasticity are the potential energy source in the membrane have been compared, and the stability properties of quasi-spherical vesicle shapes discussed. The transition from tank-treading to flipping motion in an external shear flow, has been studied as a function of the vesicle internal viscosity and of the strength of the shear quadratic component. The transverse lift strength has been calculated in both cases of a tank-treading and of a flipping motion regime.

Ultrastructure and development of the extensible abdominal intersegmental membranes of the female migratory locust

The cuticle of the retracted extensible membrane of the female locust is 300 μm thick and below its highly folded epicuticle there is a zone (5 μm thick) of helicoidally oriented laminae of microfibrils (lamellae), an elastomer layer (180 μm thick) of microfibrils with no preferred orientation and a subcuticular zone (1 μm thick). The epidermal cell layer has an extensive system of junctional specializations and pore canals traverse the cuticle to the helicoidally oriented lamellae. In the newly ecdysed adult the elastomer layer is absent and the helicoidally oriented lamellae are incomplete. Essentially the membrane consists of an elastomer layer contained between two stable layers cross-linked by pore canals, one to the other. When in the plasticized state the membrane combines low stiffness with high extensibility and during extension the elastomer layer flows. Recovery is effected by muscles and when the two stable layers have returned to their unstretched states the fluid elastomer is again evenly distributed. There is an increase in the water content and in the volume of the cuticle when it is fully extended. The ultrastructure of the extensible membrane is compared with those of the inextensible membranes from male and female locusts.

Finite deformation and stability behaviour of spherical inflatables subjected to axi-symmetric hydrostatic loading

This paper deals with the large deflection and stability behaviour of inextensible spherical air-supported membranes subjected to an accumulating ponding fluid. It is assumed that ponding fluid is available to fill an initial axi-symmetric deformation around the apex and that due to such accumulation deflections increase until under certain conditions collapse of the structure occurs.The problem is reduced to a set of differential equations which are solved numerically to determine the parameters defining the deflected wrinkled shape of the membrane. A simple expression for use in design of such structures is obtained and the results presented in non-dimensional graphic form.

ANALYSIS OF FOLDABILITY IN EXPANDABLE STRUCTURES

Presentation of outlines for a general theory of large, deformations including folding of arbitrary inextensible membranes