Incompressible Membrane Research Articles

Indentation of circular hyperelastic membrane with hole by cylindrical indenter

We consider the problem of indentation of a thin circular elastic plate with a hole in the centre by a cylindrical indenter under large strains. We use the nonlinear theory of isotropic incompressible hyperelastic membranes and consider a quasi-static process of indentation. We use Coulomb’s law to describe the friction at a contact. The goals of this study are to determine and to compare the effects of friction, material parameters and pre-stretch of the membrane on the indentation process.

Interpretable data-driven modeling of hyperelastic membranes.

We proposed an approach for interpretable data-driven modeling of an isotropic incompressible hyperelastic membrane deformation. The approach is based on the response functions in terms of the Laplace stretch and the finite element method, where response functions are partial derivatives of a hyperelastic potential with respect to the chosen strain measure. The Laplace stretch as the strain measure allows us to recover directly the response functions from experimental data and construct automatically data-driven constitutive relations. All needed formulas were obtained explicitly. We tested our approach for membrane inflation with data-driven constitutive relations based on the perforated membrane extension tests.

Frame tension governs the thermal fluctuations of a fluid membrane: new evidence.

Two different tensions can be defined for a fluid membrane: the internal tension, γ, conjugated to the real membrane area in the Hamiltonian, and the frame tension, τ, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by γ. However, using rotational invariance arguments, several studies argued that the fluctuation spectrum must be governed by the frame tension τ instead. These studies disagree on the origin of the result obtained with the standard description yet: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the simplest case of a one-dimensional membrane, for which the arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by τ, and show that the origin of the issue with the standard description is a miscounting of membrane configurations. The origin itself of this miscounting depends on the thermodynamic ensemble in which calculations are made.

Numerical modelling of wrinkled hyperelastic membranes with topologically complex internal boundary conditions

Several soft biological tissues and artificial materials are characterised by a mechanical behaviour described by two-dimensional structural systems sustaining in-plane forces. Within the framework of finite strain elasticity, in this paper the formulation and finite element implementation of a hyperelastic incompressible membrane is presented. Focus is placed on the behaviour of membranes presenting holes and internal cuts. A new efficient algorithm is presented to describe topologically complex internal boundaries along which dislocation-like distributions are prescribed, so as to allow a one-to-one progressive joining of boundary material points. The classical Ogden’s model is modified into a relaxed version in order to accommodate the no-compression response of thin membranes due to wrinkling. Three applicative examples are presented to illustrate the potential of the method proposed.

Time-Dependent Response of Incompressible Membranes Based on Quasi-Linear Viscoelasticity Theory

In this work, large deformation of incompressible, hyperelastic membranes based on the quasi-linear viscoelasticity (QLV) theory is formulated. Time integration algorithm and the expression for consistent fourth-order tangent tensor are presented. The formulation covers isotropic as well as anisotropic polymeric and biological materials. To solve numerical examples, a nonlinear finite element formulation in the Lagrangian framework is developed. Finally, several examples are provided to investigate the applicability of the present formulation. It is found that the numerical results are in good agreement with the analytical and experimental results available in the literature.

Membranes from the Material Assuming Nonlinear Elastic Properties if Exposed to Aggressive Environment

The article considers the behavior of a membrane of an incompressible material under the influence of a two-sided diffusion fracture mechanism, which manifests itself in a change in the initial modulus of elasticity. The object of study is incompressible material membrane. The relationship between deformations and displacements was established in the framework of the nonlinear theory of elasticity in accordance with the Karman formalism. For the phenomenological model, the maximum tensile stresses arising in a nonlinear elastic membrane are obtained, depending on the level of aggressiveness of the medium and the intensity of the acting uniformly distributed load. The resulting solution reflects the operation of the membrane, taking into account the accumulation of defects affecting the stress state. Nonlinearity of the results is manifested to a greater extent as defects accumulate.

Mechanics of deformation of malaria-infected red blood cells

Plasmodium Falciparum (pf) Malaria is one of the life-threatening infections for human red blood cells (RBCs), which deteriorates the topology of the corresponding bi-layer membranes and causes a 10-fold increase in their respective shear modulus during the well-distinguished Ring, Trophozoite and Schizont stages of infection progression. Previous efforts to characterize the bulk shear stiffness of pf-iRBC membranes include both in-vitro stretching tests and few simulations that enabled partial description of the membrane elasticity while assuming a uniform shear modulus. Although these results provided good insights into the axial-deformation of pf-iRBCs, the computed transverse diameter did not show similar agreement with the experimental values. The aim of the present work is to build a computational model that simulates the stretching tests of healthy and infected RBCs to better understand the mechanics of disease progression and its influence on the elastic properties of RBC membranes. For this purpose, a new patching technique is developed to mimic the infection progression through the decomposition of the cell membrane into infected pair patches and quasi-normal membrane segments. The incompressible membranes are modeled using the non-linear hyper-elastic Skalak constitutive model implemented through a VUMAT subroutine within the framework of a 3-D ABAQUS/Explicit finite-element model. In the advanced Schizont stage, a spheroidal-like geometry with uniform shear modulus is assumed, whereas in the other two stages, sizable circular patches of 2.4 and 4 µm in diameters, respectively, with adaptive shear moduli are implemented to replicate the stiffer pair-patches. The Skalak model indicates an 8-fold increase in the bulk shear modulus of the Schizont-cell beside showing better agreement with the published experimental results. Interestingly, for all other intermediate stages of infection, the bulk shear modulus is found to increase linearly with the percent area-infection, whereas nearly-constant shear modulus in the range of 14 µm is obtained for the quasi-normal membrane segments.

Magnetoelastic deformation of a circular membrane: Wrinkling and limit point instabilities

We study the inflation of a weakly magnetizable isotropic incompressible circular membrane in the presence of magnetic field generated by a magnetic dipole. Following the approach in recent papers by Reddy and Saxena (Int. J. Sol. Struct. 136, 203–219, 2018; Int. J. Non-Lin. Mech. 95, 248–263, 2017) we start with a variational formulation, solving the resulting governing equations to determine the equilibria and checking the second variation condition for stability. Conjecture of possibility of multiple equilibria under a single coupled load, and attaining elastic and magnetic limit points made in the above two papers is confirmed in the present work for a circular membrane. Another main focus of this work is on the determination of wrinkling instability in the membrane due to magnetoelastic stresses. Wrinkles along one or both in-plane directions of membrane appear in a majority of loading scenarios due to compressive Maxwell stresses. Our computations demonstrate that wrinkles arise in the central region when dipole and inflation are in the same direction and in the annular region close to the edges when the dipole and inflation are in opposite directions.

Instabilities in the axisymmetric magnetoelastic deformation of a cylindrical membrane

We study the inflation of a weakly magnetizable isotropic incompressible cylindrical membrane and the effects of an external magnetic field generated by a current carrying wire placed along the axis of cylinder. A variational formulation based on magnetization is used and the computational results obtained by using four elastic constitutive models (neo-Hookean, Mooney–Rivlin, Ogden, and Arruda–Boyce) are studied and compared. Cylinders of various aspect ratios are studied in each case. Our study shows that the external magnetic field alters the elastic limit point, does not lead to equilibrium solutions below certain value of internal pressure, and can give rise to multiple equilibrium states for a given value of pressure. Presence of magnetic limit point, a phenomenon recently reported in the literature is reconfirmed. Magnetic limit point is a state where a further strengthening of the applied magnetic field at a given pressure does not yield any static equilibrium state. In this case it is detected when the cylindrical membrane deflates into the volume enclosed by itself. We also observe a quadratic relation between the defined magnetic energy parameter and the internal pressure at the magnetic limit point. Relaxed form of the strain energy density is used to account for wrinkling in this case of inward inflation. A finite difference method coupled with an arc-length technique is used for the computations and the stability of the solution is determined from the second variation.

Remarks on the space of volume preserving embeddings

Let (N,g) be a Riemannian manifold. Given a compact, connected and oriented submanifold M of N, we define the space of volume preserving embeddings Embμ(M,N) as the set of smooth embeddings f:M↪N such that f⁎μf=μ, where μf (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient metric g (the orientation on f(M) being induced by f).In this article, we use the Nash–Moser inverse function Theorem to show that the set of volume preserving embeddings in Embμ(M,N) whose mean curvature is nowhere vanishing forms a tame Fréchet manifold, and determine explicitly the Euler–Lagrange equations of a natural class of Lagrangians.As an application, we generalize the Euler equations of an incompressible fluid to the case of an “incompressible membrane” of arbitrary dimension moving in N.

Simultaneous production of high-quality water and electrical power from aqueous feedstock’s and waste heat by high-pressure membrane distillation

A new membrane distillation (MD) concept (MemPower) has been developed for the simultaneous production of high-quality water from various aqueous feedstocks with cogeneration of mechanical power (electricity). Driven by low-grade heat (waste, solar, geothermal, etc.) a pressurized distillate can be produced by operating TNO’s Memstill® process at high hydraulic pressures. These pressures are theoretically limited by the liquid entry pressure (LEP) of the membrane. The proof of principle has been shown and is based on the transport of water vapor against a hydraulic pressure gradient. Various commercially available membranes have been evaluated in order to obtain high yields in water flux and power densities. Power densities have been measured which are sufficient to drive the pumps in MD. This allows standalone Memstill® units without electricity consumption to be possible, which are fully driven by waste heat. The application of new incompressible hydrophobic membranes, combining a high permeance with a high LEP, will allow for much higher power densities.

The mechanism of shape instability for a vesicle in extensional flow

AbstractWhen a flexible vesicle is placed in an extensional flow (planar or uniaxial), it undergoes two unique sets of shape transitions that to the best of the authors’ knowledge have not been observed for droplets. At intermediate reduced volumes (i.e. intermediate particle aspect ratio) and high extension rates, the vesicle stretches into an asymmetric dumbbell separated by a long, cylindrical thread. At low reduced volumes (i.e. high particle aspect ratio), the vesicle extends symmetrically without bound, in a manner similar to the breakup of liquid droplets. During this ‘burst’ phase, ‘pearling’ occasionally occurs, where the vesicle develops a series of periodic beads in its central neck. In this paper, we describe the physical mechanisms behind these seemingly unrelated instabilities by solving the Stokes flow equations around a single, fluid-filled particle whose interfacial dynamics is governed by a Helfrich energy (i.e. the membranes are inextensible with bending resistance). By examining the linear stability of the steady-state shapes, we determine that vesicles are destabilized by curvature changes on its interface, similar to the Rayleigh–Plateau phenomenon. This result suggests that the vesicle’s initial geometry plays a large role in its shape transitions under tension. The stability criteria calculated by our simulations and scaling analyses agree well with available experiments. We hope that this work will lend insight into the stretching dynamics of other types of biological particles with nearly incompressible membranes, such as cells.

Amoeboid Swimming: A Generic Self-Propulsion of Cells in Fluids by Means of Membrane Deformations

Microorganisms, such as bacteria, algae, or spermatozoa, are able to propel themselves forward thanks to flagella or cilia activity. By contrast, other organisms employ pronounced changes of the membrane shape to achieve propulsion, a prototypical example being the Eutreptiella gymnastica. Cells of the immune system as well as dictyostelium amoebas, traditionally believed to crawl on a substratum, can also swim in a similar way. We develop a model for these organisms: the swimmer is mimicked by a closed incompressible membrane with force density distribution (with zero total force and torque). It is shown that fast propulsion can be achieved with adequate shape adaptations. This swimming is found to consist of an entangled pusher-puller state. The autopropulsion distance over one cycle is a universal linear function of a simple geometrical dimensionless quantity A/V(2/3) (V and A are the cell volume and its membrane area). This study captures the peculiar motion of Eutreptiella gymnastica with simple force distribution.

Predicting the influence of operating conditions on DCMD flux and thermal efficiency for incompressible and compressible membrane systems

Membrane distillation (MD) is a hybrid of thermal distillation and membrane separation. As a thermal distillation process, MD is an energy intensive technique. Therefore, it is important to select optimum operation parameters to maximise both the flux and energy efficiency, even though low grade or waste heat maybe employed in the process. In this paper, a model originally developed for Direct Contact Membrane Distillation (DCMD) using a compressible membrane is employed to predict the thermal efficiency and flux for both compressible and incompressible membrane systems. The predicted thermal efficiency was also compared with experimental results, and the errors were mostly within the range of experimental variation (±5%). As a result of membrane compressibility, the thermal conductivity and vapour permeability of the membrane were altered, causing a change in flux and energy efficiency. If a compressible membrane is used, it was found from the predicted results, that increasing stream velocity will reduce the thermal efficiency and even reduce flux once the increased pressure associated with the higher flows is in the range to compress the membrane. Increasing temperature either on feed side or on permeate side will increase thermal efficiency. In scaling up the process using a compressible membrane, it is necessary to reduce the pressure drop across the module in the DCMD process to maximise energy efficiency and flux. Since the operating temperature is normally not adjustable when low grade waste heat is employed, optimising the flowrates and pressure drops are the key factors for optimising flux and thermal efficiency.

On skin expansion

This article discusses skin expansion without considering cellular growth of the skin. An in vivo analysis was carried out that involved expansion at three different sites on one patient, allowing for the observation of the relaxation process. Those measurements were used to characterize the human skin of the thorax during the surgical process of skin expansion. A comparison between the in vivo results and the numerical finite elements model of the expansion was used to identify the material elastic parameters of the skin of the thorax of that patient. Delfino's constitutive equation was chosen to model the in vivo results. The skin is considered to be an isotropic, homogeneous, hyperelastic, and incompressible membrane. When the skin is extended, such as with expanders, the collagen fibers are also extended and cause stiffening in the skin, which results in increasing resistance to expansion or further stretching. We observed this phenomenon as an increase in the parameters as subsequent expansions continued. The number and shape of the skin expanders used in expansions were also studied, both mathematically and experimentally. The choice of the site where the expansion should be performed is discussed to enlighten problems that can lead to frustrated skin expansions. These results are very encouraging and provide insight into our understanding of the behavior of stretched skin by expansion. To our knowledge, this study has provided results that considerably improve our understanding of the behavior of human skin under expansion.

Well-Posedness of Hydrodynamics on the Moving Elastic Surface

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.

A Parametric Study and Simulations in Quantifying Human Skin Hyperelastic Parameters

Skin deformation behaviour is complex, that exhibits similarly to an anisotropic, non-homogenous and viscoelastic material. Nevertheless, the understanding of skin characteristics and behaviour is important in many applications. This study attempts to investigate and simulate skin deformation under tensile loading using the integration of experimental-computational approaches. An experimental procedure has been conducted to measure skin deformation using motion capture system. Based on the information and data obtained from the measurement, finite element models of skin are developed using ABAQUS to mimic the experimental procedure. In this study, skin is modeled as an isotropic, incompressible and hyperelastic membrane that underwent large deformation. A systematic parametric study is constructed to perform finite element simulations using various mesh size and element types. The results obtained from the simulations are compared to the experimental and the best match curve constitutes skin material parameters. The final parameter shows that the best Ogden's values were at 10Pa and 110 for Ogden's coefficient and exponent respectively. It can be concluded that the integration of experiment and numerical approaches has successfully quantified the skin properties.

Rheology of a dilute suspension of liquid-filled elastic capsules

Rheology of a dilute suspension of liquid-filled elastic capsules in linear shear flow is studied by three-dimensional numerical simulations using a front-tracking method. This study is motivated by a recent discovery that a suspension of viscous vesicles exhibits a shear viscosity minimum when the vesicles undergo an unsteady vacillating-breathing dynamics at the threshold of a transition between the tank-treading and tumbling motions. Here we consider capsules of spherical resting shape for which only a steady tank-treading motion is observed. A comprehensive analysis of the suspension rheology is presented over a broad range of viscosity ratio (ratio of internal-to-external fluid viscosity), shear rate (or, capillary number), and capsule surface-area dilatation. We find a result that the capsule suspension exhibits a shear viscosity minimum at moderate values of the viscosity ratio, and high capillary numbers, even when the capsules are in a steady tank-treading motion. It is further observed that the shear viscosity minimum exists for capsules with area-dilating membranes but not for those with nearly incompressible membranes. Nontrivial results are also observed for the normal stress differences which are shown to decrease with increasing capillary number at high viscosity ratios. Such nontrivial results neither can be predicted by the small-deformation theory nor can be explained by the capsule geometry alone. Physical mechanisms underlying these results are studied by decomposing the particle stress tensor into a contribution due to the elastic stresses in the capsule membrane and a contribution due to the viscosity differences between the internal and suspending fluids. It is shown that the elastic contribution is shear-thinning, but the viscous contribution is shear thickening. The coupling between the capsule geometry and the elastic and viscous contributions is analyzed to explain the observed trends in the bulk rheology.

Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space

The dynamics of a membrane is a coupled system of a moving elastic surface and an incompressible membrane fluid. The difficulties in analyzingsuch a system include the nonlinearity of the curved space (geometric nonlinearity), the nonlinearity of the fluid dynamics (fluid nonlinearity), and the coupling to the surface incompressibility. In the two-dimensional case, the fluid vanishes and the system reduces to a coupling of a wave equation and an elliptic equation. Here we prove the local existence and uniqueness of the solution to the system by constructing a suitable discrete scheme and proving the compactness of the discrete solutions. The risk of blowing up due to the geometric nonlinearity is overcome by the bending elasticity.

Inflation of a circular elastomeric membrane into a horizontally semi-infinite liquid reservoir of finite vertical depth: Quasi-static deformation model

Similar in both form and function to a classical circular membrane inflation experiment, cell-elastomer composite diaphragm inflation (CDI) experiments have recently been introduced as a methodology for exploring the mechanobiological properties of cell–cell anchoring junctions and cytoskeletal protein networks within a living epithelial sheet. Despite outward similarities, CDI experiments are unique from classical membrane inflation in that the loading associated with the inflation process, though axisymmetric, is inherently non-uniform. In order to better account for the effects of non-uniform loading in a CDI experiment, we introduce a model for the quasi-static inflation of a prestretched clamped circular isotropic incompressible hyperelastic membrane into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth. A set of coupled nonlinear first-order differential equations is shown to govern the inflation process. Assuming a Mooney–Rivlin (MR) constitutive model, the governing equations are formulated into a boundary value problem, and a procedure is developed for finding numerical solutions that quantify discrete membrane inflation states in terms of deformations, displacements, curvatures, stress resultants, and an inflation pressure distribution. Through iteration of the discrete state solution procedure, we further demonstrate how to generate a polynomial interpolating function that approximates the continuous volume–pressure response of a MR membrane. Select numerical examples are used to simulate the inflation response of a polydimethylsiloxane (PDMS) elastomer membrane used in an actual CDI experiment.