Non-uniform inflation of a cylindrical elastic membrane and direct determination of the strain energy function

The theory of finite elastic deformations of an isotropic body, in which a completely general strainenergy function is used, is applied to the problem of a small twist superposed upon a finite extension of a cylinder which has a constant cross-section. The law which relates the force necessary to produce the large extension, with the torsional modulus for the small torsion superposed on that extension, is given by a simple general formula. When the material is incompressible the corresponding law is independent of the particular form of the strain-energy function which applies to the material. When the cylinder is not a circular cylinder or a circular cylindrical tube the twisting couple vanishes for a certain value of the extension ratio, this value being independent of the particular form of the strain-energy function when the material is incompressible. The problems of a small twist superposed upon a hydrostatic pressure, or upon a combined hydrostatic pressure and tension, are also solved. Attention is then confined to isotropic incompressible rubber-like materials using a strain-energy function suggested by Mooney, and the second-order effects which accompany the torsion of cylinders of constant cross-sections are examined. The problem is reduced to the determination of two functions of a complex variable which are regular in the cross-section of the cylinders and which satisfy a suitable boundary condition on the boundary of the cross-section. The solution is expressed as an integral equation and applications are made to cylinders with various cross-sections. This theory is then generalized to include the second-order effects in torsion superposed upon a finite extension of the cylinders. Complex variables are used throughout this part of the paper, and the problem is reduced to the determination of four canonical functions of a complex variable, these functions being the solutions of certain integral equations. An explicit solution is given for an elliptical cylinder but without using the integral equations.

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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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Strain energy function for biological tissues

Journal of Biomechanics ◽

10.1016/0021-9290(70)90038-2 ◽

1970 ◽

Vol 3 (6) ◽

pp. 547-550 ◽

Cited By ~ 35

Author(s):

Perng-Fei Gou

Keyword(s):

Energy Function ◽

Strain Energy ◽

Strain Energy Function ◽

Biological Tissues

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Bifurcation of rotating circular cylindrical elastic membranes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100056772 ◽

1980 ◽

Vol 87 (2) ◽

pp. 357-376 ◽

Cited By ~ 3

Author(s):

D. M. Haughton ◽

R. W. Ogden

Keyword(s):

Axial Force ◽

Strain Energy ◽

Elastic Strain Energy ◽

Strain Energy Function ◽

Angular Speed ◽

Elastic Membrane ◽

Axial Extension ◽

End Conditions ◽

Elastic Membranes ◽

Realistic Choice

SummaryBifurcation from a finitely deformed circular cylindrical configuration of a rotating circular cylindrical elastic membrane is examined. It is found (for a physically realistic choice of elastic strain-energy function) that the angular speed attains a maximum followed by a minimum relative to the increasing radius of the cylinder for either a fixed axial extension or fixed axial force.At fixed axial extension (a) a prismatic mode of bifurcation (in which the cross-section of the cylinder becomes uniformly non-circular) may occur at a maximum of the angular speed provided the end conditions on the cylinder allow this; (b) axisyim-metric modes may occur before, at or after the angular speed maximum depending on the length of the cylinder and the magnitude of the axial extension; (c) an asymmetric or ‘wobble’ mode is always possible before either (a) or (b) as the angular speed increases from zero for any length of cylinder or axial extension. Moreover, ‘wobble’ occurs at lower angular speeds for longer cylinders.At fixed axial force the results are similar to (a), (b) and (c) except that an axisym-metric mode necessarily occurs between the turning points of the angular speed.

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