Finite Inflation of a Bonded Toroid

A general approach to the numerical solutions for axially symmetric membrane problem is presented. The formulation of the problem leads to a system of first-order nonlinear differential equations. These equations are formulated such that the numerical integration can be carried out for any form of strain-energy function. Solutions to these equations are feasible for various boundary conditions. In this paper, these equations are applied to the problem of a bonded toroid under inflation. A bonded toroid, which is in the shape of a tubeless tire, has its two circular edges rigidly bonded to a rim. The Runge-Kutta method is employed to solve the system of differential equations, in which Mooney’s form of strain-energy function is adopted.

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Generalized shear deformations for isotropic incompressible hyperelastic materials

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000000151x ◽

1977 ◽

Vol 20 (2) ◽

pp. 129-141 ◽

Cited By ~ 4

Author(s):

James M. Hill

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Shear Deformations ◽

Partial Differential ◽

Hyperelastic Materials ◽

Restricted Form ◽

Consistent Solution

AbstractFor isotropic incompressible hyperelastic materials the single function characterizing generalized shear deformations or as they are sometimes called anti-plane strain deformations must satisfy two distinct partial differential equations. Knowles [5] has recently given a necessary and sufficient condition for the strain–energy function of the material which if satisfied ensures that the two equations have consistent solutions. It is shown here for the general material not satisfying Knowles' criterion that the only possible consistent solution of the two partial differential equations are those which are already known to exist for all strain–energy functions. More general types of generalized shear deformations for such meterials are shown to exist only for special or restricted form ot the strain-energy function. In derving these results we also obtain an alternative derivation of Knowles' criterion.

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General Deformations of Neo-Hookean Membranes

Journal of Applied Mechanics ◽

10.1115/1.3422977 ◽

1973 ◽

Vol 40 (1) ◽

pp. 7-12 ◽

Cited By ~ 12

Author(s):

W. H. Yang ◽

C. H. Lu

Keyword(s):

Differential Equations ◽

Strain Energy ◽

Numerical Solutions ◽

Nonlinear Partial Differential Equations ◽

Strain Energy Function ◽

Finite Deformations ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Special Cases ◽

Convergent Algorithm

A set of three nonlinear partial-differential equations is derived for general finite deformations of a thin membrane. The material that composes the membrane is assumed to be hyperelastic. Its mechanical property is represented by the neo-Hookean strain-energy function. The equations reduce to special cases known in the literature. A fast convergent algorithm is developed. The numerical solutions to the finite-difference approximation of the differential equations are computed iteratively with a trivial initial iterant. As an example, the problem of inflating a rectangular membrane with fixed edges by a uniform pressure applied on one side is presented. The solutions and their convergence are displayed and discussed.

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A note on the finite plane-strain equations for isotropic incompressible materials

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100030280 ◽

1955 ◽

Vol 51 (2) ◽

pp. 363-367 ◽

Cited By ~ 9

Author(s):

J. E. Adkins

Keyword(s):

Differential Equations ◽

Plane Strain ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Theory Of Elasticity ◽

Finite Plane ◽

Incompressible Materials ◽

Stress Components ◽

Strain Invariants

For elastic deformations beyond the range of the classical infinitesimal theory of elasticity, the governing differential equations are non-linear in form, and orthodox methods of solution are not usually applicable. Simplifying features appear, however, when a restriction is imposed either upon the form of the deformation, or upon the form of strain-energy function employed to define the elastic properties of the material. Thus in the problems of torsion and flexure considered by Rivlin (4, 5, 6) it is possible to avoid introducing partial differential equations into the analysis, while in the theory of finite plane strain developed by Adkins, Green and Shield (1) the reduction in the number of dependent and independent variables involved introduces some measure of simplicity. Some further simplification is achieved when the strain-energy function can be considered as a linear function of the strain invariants as postulated by Mooney(2) for incompressible materials. In the present paper the plane-strain equations for a Mooney material are reduced to symmetrical forms which do not involve the stress components, and some special solutions of these equations are derived.

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Joined dissimilar orthotropic elastic cylindrical membranes under internal pressure and longitudinal tension

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000008912 ◽

1993 ◽

Vol 34 (3) ◽

pp. 296-317 ◽

Cited By ~ 7

Author(s):

V. G. Hart ◽

Jingyu Shi

Keyword(s):

Internal Pressure ◽

Energy Function ◽

Strain Energy ◽

Numerical Solutions ◽

Circumferential Stress ◽

Strain Energy Function ◽

The Body ◽

Surgical Implants ◽

Elastic Membranes ◽

Preliminary Study

AbstractFollowing work in an earlier paper, the theory of finite deformation of elastic membranes is applied to the problem of two initially-circular semi-infinite cylindrical membranes of the same radius but of different material, joined longitudinally at a cross-section. The body is inflated by constant interior pressure and is also extended longitudinally. The exact solution found for an arbitrary material is now specialised to the orthotropic case, and the results are interpreted for forms of the strain-energy function introduced by Vaishnav and by How and Clarke in connection with the study of arteries. Also considered in this context is the similar problem where two semi-infinite cylindrical membranes of the same material are separated by a cuff of different material. Numerical solutions are obtained for various pressures and longitudinal extensions. It is shown that discontinuities in the circumferential stress at the joint can be reduced by suitable choice of certain coefficients in the expression defining the strain-energy function. The results obtained here thus solve the problem of static internal pressure loading in extended dissimilar thin orthotropic tubes, and may also be useful in the preliminary study of surgical implants in arteries.

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Mechanical power and hyperelasticity

10.1093/oso/9780198567783.003.0003 ◽

2017 ◽

Author(s):

David J. Steigmann

Keyword(s):

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Mechanical Power ◽

Elastic Materials ◽

Cyclic Motion

This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.

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Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

Journal of Elasticity ◽

10.1007/s10659-021-09823-x ◽

2021 ◽

Author(s):

Afshin Anssari-Benam ◽

Andrea Bucchi ◽

Giuseppe Saccomandi

Keyword(s):

Experimental Data ◽

Energy Function ◽

Limit Point ◽

Strain Energy ◽

Cylindrical Shells ◽

Strain Energy Function ◽

Model Parameters ◽

Wide Range ◽

Cylindrical Tubes ◽

Gent Model

AbstractThe application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

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Mechanical characterization of a woven multi-layered hyperelastic composite laminate under uniaxial loading

Journal of Composite Materials ◽

10.1177/00219983211011528 ◽

2021 ◽

pp. 002199832110115

Author(s):

Shaikbepari Mohmmed Khajamoinuddin ◽

Aritra Chatterjee ◽

MR Bhat ◽

Dineshkumar Harursampath ◽

Namrata Gundiah

Keyword(s):

Composite Materials ◽

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Mechanical Tests ◽

Stress Strain ◽

Aerospace Applications ◽

Envelope Material ◽

The Individual ◽

High Dielectric

We characterize the material properties of a woven, multi-layered, hyperelastic composite that is useful as an envelope material for high-altitude stratospheric airships and in the design of other large structures. The composite was fabricated by sandwiching a polyaramid Nomex® core, with good tensile strength, between polyimide Kapton® films with high dielectric constant, and cured with epoxy using a vacuum bagging technique. Uniaxial mechanical tests were used to stretch the individual materials and the composite to failure in the longitudinal and transverse directions respectively. The experimental data for Kapton® were fit to a five-parameter Yeoh form of nonlinear, hyperelastic and isotropic constitutive model. Image analysis of the Nomex® sheets, obtained using scanning electron microscopy, demonstrate two families of symmetrically oriented fibers at 69.3°± 7.4° and 129°± 5.3°. Stress-strain results for Nomex® were fit to a nonlinear and orthotropic Holzapfel-Gasser-Ogden (HGO) hyperelastic model with two fiber families. We used a linear decomposition of the strain energy function for the composite, based on the individual strain energy functions for Kapton® and Nomex®, obtained using experimental results. A rule of mixtures approach, using volume fractions of individual constituents present in the composite during specimen fabrication, was used to formulate the strain energy function for the composite. Model results for the composite were in good agreement with experimental stress-strain data. Constitutive properties for woven composite materials, combining nonlinear elastic properties within a composite materials framework, are required in the design of laminated pretensioned structures for civil engineering and in aerospace applications.

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Finite Deflections of a Nonlinearly Elastic Bar

Journal of Applied Mechanics ◽

10.1115/1.3408488 ◽

1970 ◽

Vol 37 (1) ◽

pp. 48-52 ◽

Cited By ~ 40

Author(s):

J. T. Oden ◽

S. B. Childs

Keyword(s):

Differential Equations ◽

Axial Force ◽

Classical Theory ◽

Numerical Solutions ◽

Nonlinear Differential Equations ◽

Finite Deflections ◽

Hyperbolic Tangent ◽

Elastic Bar ◽

Elastoplastic Materials ◽

Nonlinearly Elastic

The problem of finite deflections of a nonlinearly elastic bar is investigated as an extension of the classical theory of the elastica to include material nonlinearities. A moment-curvature relation in the form of a hyperbolic tangent law is introduced to simulate that of a class of elastoplastic materials. The problem of finite deflections of a clamped-end bar subjected to an axial force is given special attention, and numerical solutions to the resulting system of nonlinear differential equations are obtained. Tables of results for various values of the parameters defining the material are provided and solutions are compared with those of the classical theory of the elastica.

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