Isotropic Elastic Membrane Research Articles

The Large Deflection Solution of Circular Elastic Membrane under Composite Loads

In this paper, the large deflection problem of axisymmetric isotropic elastic membranes subjected to the composite loads is studied. The Hencky transformation is extended and the general solution of large deflection of the membranes under the composite loads is obtained. The general solution is given in different forms in different values of loads. The obtained results of the numerical values of elastic characters can be applied in practice.

Variational modelling of nematic elastomer foundations

We compute the [Formula: see text]-limit of energy functionals describing mechanical systems composed of a thin nematic liquid crystal elastomer sustaining a homogeneous and isotropic elastic membrane. We work in the regime of infinitesimal displacements and model the orientation of the liquid crystal according to the order tensor theories of both Frank and De Gennes. We describe the asymptotic regime by analysing a family of functionals parametrised by the vanishing thickness of the membranes and the ratio of the elastic constants, establishing that, in the limit, the system is represented by a two-dimensional integral functional interpreted as a linear membrane on top of a nematic active foundation involving an effective De Gennes optic tensor which allows for low order states. The latter can suppress shear energy by formation of microstructure as well as act as a pre-strain transmitted by the foundation to the overlying film.

Contact probing of stretched membranes and adhesive interactions: graphene and other two-dimensional materials.

Contact probing is the preferable method for studying mechanical properties of thin two-dimensional (2D) materials. These studies are based on analysis of experimental force–displacement curves obtained by loading of a stretched membrane by a probe of an atomic force microscope or a nanoindenter. Both non-adhesive and adhesive contact interactions between such a probe and a 2D membrane are studied. As an example of the 2D materials, we consider a graphene crystal monolayer whose discrete structure is modelled as a 2D isotropic elastic membrane. Initially, for contact between a punch and the stretched circular membrane, we formulate and solve problems that are analogies to the Hertz-type and Boussinesq frictionless contact problems. A general statement for the slope of the force–displacement curve is formulated and proved. Then analogies to the JKR (Johnson, Kendall and Roberts) and the Boussinesq–Kendall contact problems in the presence of adhesive interactions are formulated. General nonlinear relations among the actual force, displacements and contact radius between a sticky membrane and an arbitrary axisymmetric indenter are derived. The dimensionless form of the equations for power-law shaped indenters has been analysed, and the explicit expressions are derived for the values of the pull-off force and corresponding critical contact radius.

Monotonicity and stability of optimal solutions of a minimization problem

This paper is concerned with a minimization problem modeling the minimum displacement of an isotropic elastic membrane subject to a vertical force such as a load distribution. In addition to proving existence and uniqueness of optimal solutions, we show that these solutions are monotone and stable, in a certain sense. The main mathematical tool used in the analysis is the tangent cones from convex analysis, which helps to derive the optimality condition. Our results are compatible with physical expectations.

Modeling the indentation and puncturing of inflated elastic membranes by rigid indenters

In this paper energy argument and maximum local deformation criteria are used to predict the puncturing of inflated elastic membranes by a rigid indenter. The direct membrane theory and a particular Varga type strain energy function are used to model the indentation and puncturing of an isotropic spherical elastic membrane enclosing an incompressible fluid by a rigid frictionless cylindrical indenter. The non-linear equilibrium equations are formulated and numerical procedure is developed to obtain solutions. A global mode of failure is established by considering the resultant strain energy in the punctured membrane. An additional local mode of failure is identified in which the membrane fails once local deformation reaches a critical value. It is observed that the indenter radius, initial membrane inflation and material properties determine the conditions under which puncturing occurs.

In-plane dynamics of membranes having constant curvature

We have studied the dynamics of in-plane displacement of a class of two-dimensional homogeneous and isotropic elastic membranes having constant curvature. The equation of motion of a general in-plane displacement field has been derived in covariant form using the variational formulation. The contribution of curvature on the dynamics of the field variable is clearly observed in the field equation. We observe that for certain membranes with constant curvature, the curvature term introduces a quadratic potential with the sign of the coupling constant being decided by the Ricci curvature of the membrane. Using the Helmholtz theorem , we reduce the field equation to two decoupled scalar equations which exhibit an interesting structure. We consider some examples of membranes with positive, and negative curvatures and comment on the nature of solutions expected. We analyze the deformation of the membranes in terms of the dilatation, shear and vorticity of the displacement field. ► For membranes with periodic boundary conditions curvature act as a quadratic potential proportional to its Ricci scalar. ► Dilatoric and deviatoric wave equations are decoupled using Helmholz theorem. ► Eigenspectrum and modes of the spherical membrane are determined exactly. ► Deformation potentials of hyperbolic membrane are found exactly in terms of conical functions. ► Wave modes of hyperbolic membrane are found to be dispersive with certain cut-off frequencies.

Elastic characterization of the gerbil pars flaccida from in situ inflation experiments

In hearing science, finite element modelling is used commonly to study the mechanical behaviour of the middle ear. Correct quantitative elasticity parameters are an important input in these models. However, up till now, no large deformation elastic characterization of the pars flaccida, a small part of the tympanic membrane, has been carried out. In this paper, an elastic characterization of the gerbil pars flaccida is presented. The gerbil is used frequently as animal model in middle ear mechanics research. Characterization was done via inverse analysis of in situ static pressure inflation experiments. As a first approach, the pars flaccida was modelled as a linear homogeneous isotropic elastic membrane, which resulted in an average Young's modulus of <E> = (41.0 ± 0.4) kPa. It was found that linear elastic modelling cannot describe inflation stagnation at high pressures. Therefore, in a second approach, the Veronda-Westmann hyperelastic model was introduced. This was able to describe curve stagnation, the mean parameters that were found are <C1> = (3.1 ± 0.4) kPa and <C2> = (2.5 ± 0.2). Finally, in situ strain was considered in the finite element models which resulted in a better description of the behaviour for small pressures. Incorporating this, the optimal Veronda-Westmann parameters are <CεR1> = (2.6 ± 0.6) kPa, <CεR2> = (1.4 ± 0.2) kPa for a radial in situ strain of <εR> = (12 ± 2)%. In conclusion, this paper shows that a linear elastic material is not appropriate to describe pars flaccida's behaviour in the quasi-static pressure regime, that the currently used membrane stiffness estimates do not hold for large deformations and that incorporating an in situ strain in the models is necessary for a good description for small static pressures.

Computational modeling of reconstructive surgery: The effects of the natural tension on skin wrinkling

A computational model is presented for the simulation of procedures of reconstructive surgery characterized by the excision of a cutaneous defect and the closure and suture of the wound edges. The skin is modeled as a plane membrane with zero flexural stiffness. The membrane undergoes large deformations and is characterized by a Fung type constitutive response in biaxial tension. Skin wrinkling, which is a typical outcome of the surgery in the form of extrusion of the wound edges and dog-ears, is considered through a modification of the elastic potential as originally proposed by Pipkin's Relaxed Energy Density theory [A.C. Pipkin, The relaxed energy density for isotropic elastic membranes, IMA J. Appl. Math. 36 (1986) 85–99; A.C. Pipkin, Relaxed energy density for large deformations of membranes, IMA J. Appl. Math. 52 (1994) 197–308]. The post-buckling analysis of a stretched annular membrane performed by Geminard et al. [Wrinkle formations in axi-symmetrically stretched membranes, Eur. Phys. J. E 15 (2004) 117–126] is used to validate the model under conditions similar to those of the surgery and to discuss the influence of a pre-existing tension in the membrane on the extension of the wrinkled regions. The model is applied to simulate different surgical procedures and investigate the effects of the natural state of the skin and the shape and size of the excisions. The results explain and validate current practice.

Mechanical Response of Single Plant Cells to Cell Poking: A Numerical Simulation Model

AbstractCell poking is an experimental technique that is widely used to study the mechanical properties of plant cells. A full understanding of the mechanical responses of plant cells to poking force is helpful for experimental work. The aim of this study was to numerically investigate the stress distribution of the cell wall, cell turgor, and deformation of plant cells in response to applied poking force. Furthermore, the locations damaged during poking were analyzed. The model simulates cell poking, with the cell treated as a spherical, homogeneous, isotropic elastic membrane, filled with incompressible, highly viscous liquid. Equilibrium equations for the contact region and the non‐contact regions were determined by using membrane theory. The boundary conditions and continuity conditions for the solution of the problem were found. The force‐deformation curve, turgor pressure and tension of the cell wall under cell poking conditions were obtained. The tension of the cell wall circumference was larger than that of the meridian. In general, maximal stress occurred at the equator around. When cell deformation increased to a certain level, the tension at the poker tip exceeded that of the equator. Breakage of the cell wall may start from the equator or the poker tip, depending on the deformation. A nonlinear model is suitable for estimating turgor, stress, and stiffness, and numerical simulation is a powerful method for determining plant cell mechanical properties.(Managing editor: Wei Wang)

Application of variational principles to the axial extension of a circular cylindical nonlinearly elastic membrane

In this paper stationary potential-energy and complementary-energy principles are formulated for boundary-value problems for compressible or incompressible nonlinearly elastic membranes, and full justification for adoption of the complementary principle is provided. The stationary principles are then extended to extremum principles, which provide upper and lower bounds on the energy functional associated with the solution of a given problem. The principles are then illustrated by their application to the nonlinear problem of the axially symmetric static deformation of an isotropic elastic membrane. In its undeformed natural configuration the membrane has the form of a circular cylindrical surface. The cylinder is subject to a prescribed (tensile) axial force with the ends of the cylinder constrained so that their radii remain constant. The alternative boundary condition in which the axial displacement of the ends is prescribed instead of the axial force is also considered. The extremum principles are applied first without restriction on the form of strain-energy function in order to obtain primitive bounds on the energy of Voigt and Reuss type commonly used in composite-material mechanics. Then, for particular forms of strain-energy function, specific bounds are obtained by selecting suitable trial deformation and stress fields and the bounds are optimized using a numerical procedure (which is readily adapted for other forms of strain-energy function). It is found that these bounds are very close and hence give a good estimate of the actual energy. The associated deformed geometry of the membrane is described together with the resulting principal stresses.

Finite Element Analysis of Wrinkling Membranes

A new iterative scheme is proposed for finite element analysis of wrinkling or tension structures. The scheme is based upon the observation that there exists an invariant relationship, due to the uniaxial tensile stress state of wrinkling, between some of the strain components referred to the local frame aligned with wrinkling in a region where wrinkling occurs. This enables us to update the stress state and the internal forces correctly taking into account the existence of wrinkling. The finite element implementation of the scheme is straightforward and simple, and only minor modifications of the existing total Lagrangian finite element codes for membranes are needed. The validity of the scheme is demonstrated via numerical examples for the torsion of a membrane and the quasi-static inflation of an automotive airbag, both made of isotropic or anisotropic elastic membranes. The examples suggest that the present iterative scheme has a good convergence characteristic even for a large loading step.

Axisymmetric elastic membranes subjected to fluid loading

We consider axisymmetric isotropic elastic membranes that are loaded by an incompressible liquid. The resulting deformation which contains the liquid is assumed to be axisymmetric. We consider undeformed plane disks of varying thickness and also hemispherical cups. We show how the resulting deformation can be controlled by the initial thickness variation of the disk. In all cases there is a critical loading where the membrane undergoes a very rapid change of shape, from being almost concave to having an hourglass shape. For a hemispherical cup, the deformation produces a wrinkled region which diminishes with increased loading but briefly reappears as the membrane traverses the unstable region. The onset of instability appears to be governed by the equibiaxial deformation at the apex of the deformed container. Instability is initiated when the deformation reaches the corresponding value of the first bifurcation point for a spherical membrane.

A B‐spline finite element for the dynamic analysis of sandwich shells of revolution

Presents a new B‐spline finite element for the dynamic analysis of unsymmetrical sandwich shells of revolution. The formulation takes account of the membrane and bending effects in isotropic or orthotropic elastic facings, and membrane, bending and transverse shearing effects in an isotropic or othotropic elastic core. Both geometry and local displacements are interpolated by a set of B‐spline functions. The main aspects added by the sandwich structure of the element are the transverse shearing and membrane‐bending coupling effects in the core. These are well represented by a set of new variables which are the mean end relative in‐plane displacements of the facing middle surfaces. Together with the transverse displacement, these variables constitute the degrees of freedom (dofs) of this new B‐spline sandwich element. The finite elements are grouped into super‐elements with C1 continuity to obtain the whole finite element model. For each super‐element a total of five dofs per node is then obtained except for its end nodes where the derivatives of these dofs with respect to the meridional co‐ordinate are added. This choice reduces to a minimum the total number of dofs in comparison to existing sandwich elements. Evaluates the efficiency and accuracy of the proposed element through several benchmark examples. Compares the results with the analytical and numerical solutions found in the literature. A very satisfactory behaviour of the element was observed in all test cases.

Point loads on a hemispherical elastic membrane

We analyze axisymmetric equilibrium deformations of a pressurized, hemispherical, isotropic elastic membrane subjected to a concentrated force at its pole. Wrinkling of the membrane is taken into account by using a relaxed energy derived from Ogden's three-term strain energy function. The special features of the theory in wrinkled regions of the membrane are used to reduce the analysis to quadratures. Existence of solutions requires that the uniaxial stress-stretch relation satisfy a certain growth condition for large stretches.

Biaxial Testing of Membrane Biomaterials: Testing Equipment and Procedures

A testing facility for measuring the biaxial mechanical properties of highly deformable membranes is described. Forces are applied, via strain-gauge force transducers, to four points on each side of an initially square 12 to 25 mm membrane sample to produce biaxial extensions of up to 80 percent of undeformed length. Strain is estimated from the displacement of markers bounding a 1 to 2 mm central square. The accuracy of stress and strain field measurements has been assessed by finite element analysis of a biaxially-loaded isotropic elastic membrane. Major advantages of the present system over those previously described in the literature are that 1) sample mounting procedures are simplified, 2) there is provision for independent adjustment of stress field uniformity and measurement of the applied point forces and 3) faster strain rates can be imposed on the relatively small samples tested.

Normal configurations and the nonlinear elastostatic problems of bending, torsion, expansion, and eversion for compressible bodies

First, we decompose a (deformed) equilibrium configuration into a disjoint union of a one-parameter family of surfaces, called lamina surfaces. We require that all of these lamina surfaces be two-dimensional universal solutions for homogeneous isotropic elastic membranes. Then the two tangential equations of equilibrium are satisfied without any body force, universally for all isotropic elastic (membrane) materials

Tension-field theory

A general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations. The principal contribution is a partial differential equation describing a geometrical property of tension trajectories. This is one of a system of two equations which describes the state of stress independently of the deformation. This system is strongly elliptic at any stable solution, whereas the deformation is described by a system of parabolic type. Controllable solutions, i. e. those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone, are obtained. The general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples. Existing small strain theories are shown to correspond to a singular limit of the general theory, at which the underlying system changes from elliptic to parabolic type.

Axisymmetric tension fields

Finite deformations with wrinkling for isotropic elastic membranes of arbitrary initial shape are discussed. The shape of the deformed membrane in a wrinkled region is governed by a parabolic equation that does not involve material properties, and the stress field is governed by an elliptic system that involves the scalar force-stretch function. For axisymmetric deformations of surfaces of revolution, solution of these equations involves only integration and function inversion.

Free and constrained inflation of elastic membranes in relation to thermoforming — non-axisymmetric problems

This paper deals with the inflation of elastic membranes of general shapes in the context of thermoforming of heated polymeric sheets against relatively cold moulding surfaces. A previous paper (1) has dealt with axisymmetric problems. As before, both theoretical and experimental investigations are reported. The theoretical part consists of a self-contained finite element formulation for large deformation, free and constrained inflation of isotropic elastic membranes. The computer program developed on the basis of presented formulation is applied to free and constrained inflation of plane elliptical membranes of aspect ratios 2 and 4. The material model adopted is that of a single constant neo-Hookean elastic material. The constrained analyses are carried out for inflation against: (a) cylindrical walls (90 degree elliptical cones), (b) 60 degree elliptical cones, and (c) horizontal plates. Separate analyses are performed by assuming the contact to be either slipless or frictionless. The theoretical results are compared with the experimental ones for free inflation and constrained inflation cases (a) and (c).

On the stability of asymmetric deformations of a symmetrically-tensioned elastic sheet

Bifurcation from a symmetrically to an unsymmetrically deformed homogeneous configuration of a plane incompressible isotropic elastic membrane subject to equal in-plane tensions is examined. Critical values of the tension at which bifurcation occurs are obtained for a general form of strain-energy function, and the nature of the deformation in the neighbourhood of the critical point is studied in detail. The stability of the global unsymmetric bifurcation branch is assessed. and secondary branching into either a homogeneous or a non-homogeneous mode of deformation is found to be possible under certain conditions.